Applied Mathematics Research
http://hdl.handle.net/10023/96
20160828T16:47:11Z

Tracking the evolution of cancer cell populations through the mathematical lens of phenotypestructured equations
http://hdl.handle.net/10023/9363
Background: A thorough understanding of the ecological and evolutionary mechanisms that drive the phenotypic evolution of neoplastic cells is a timely and key challenge for the cancer research community. In this respect, mathematical modelling can complement experimental cancer research by offering alternative means of understanding the results of in vitro and in vivo experiments, and by allowing for a quick and easy exploration of a variety of biological scenarios through in silico studies. Results: To elucidate the roles of phenotypic plasticity and selection pressures in tumour relapse, we present here a phenotypestructured model of evolutionary dynamics in a cancer cell population which is exposed to the action of a cytotoxic drug. The analytical tractability of our model allows us to investigate how the phenotype distribution, the level of phenotypic heterogeneity, and the size of the cell population are shaped by the strength of natural selection, the rate of random epimutations, the intensity of the competition for limited resources between cells, and the drug dose in use. Conclusions: Our analytical results clarify the conditions for the successful adaptation of cancer cells faced with environmental changes. Furthermore, the results of our analyses demonstrate that the same cell population exposed to different concentrations of the same cytotoxic drug can take different evolutionary trajectories, which culminate in the selection of phenotypic variants characterised by different levels of drug tolerance. This suggests that the response of cancer cells to cytotoxic agents is more complex than a simple binary outcome, i.e., extinction of sensitive cells and selection of highly resistant cells. Also, our mathematical results formalise the idea that the use of cytotoxic agents at high doses can act as a doubleedged sword by promoting the outgrowth of drug resistant cellular clones. Overall, our theoretical work offers a formal basis for the development of anticancer therapeutic protocols that go beyond the ‘maximumtolerateddose paradigm’, as they may be more effective than traditional protocols at keeping the size of cancer cell populations under control while avoiding the expansion of drug tolerant clones.
This work was supported in part by the French National Research Agency through the “ANR blanche” project Kibord [ANR13BS010004].
20160823T00:00:00Z
Lorenzi, Tommaso
Chisholm, Rebecca H.
Clairambault, Jean
Background: A thorough understanding of the ecological and evolutionary mechanisms that drive the phenotypic evolution of neoplastic cells is a timely and key challenge for the cancer research community. In this respect, mathematical modelling can complement experimental cancer research by offering alternative means of understanding the results of in vitro and in vivo experiments, and by allowing for a quick and easy exploration of a variety of biological scenarios through in silico studies. Results: To elucidate the roles of phenotypic plasticity and selection pressures in tumour relapse, we present here a phenotypestructured model of evolutionary dynamics in a cancer cell population which is exposed to the action of a cytotoxic drug. The analytical tractability of our model allows us to investigate how the phenotype distribution, the level of phenotypic heterogeneity, and the size of the cell population are shaped by the strength of natural selection, the rate of random epimutations, the intensity of the competition for limited resources between cells, and the drug dose in use. Conclusions: Our analytical results clarify the conditions for the successful adaptation of cancer cells faced with environmental changes. Furthermore, the results of our analyses demonstrate that the same cell population exposed to different concentrations of the same cytotoxic drug can take different evolutionary trajectories, which culminate in the selection of phenotypic variants characterised by different levels of drug tolerance. This suggests that the response of cancer cells to cytotoxic agents is more complex than a simple binary outcome, i.e., extinction of sensitive cells and selection of highly resistant cells. Also, our mathematical results formalise the idea that the use of cytotoxic agents at high doses can act as a doubleedged sword by promoting the outgrowth of drug resistant cellular clones. Overall, our theoretical work offers a formal basis for the development of anticancer therapeutic protocols that go beyond the ‘maximumtolerateddose paradigm’, as they may be more effective than traditional protocols at keeping the size of cancer cell populations under control while avoiding the expansion of drug tolerant clones.

Evolution of magnetic helicity during eruptive flares and coronal mass ejections
http://hdl.handle.net/10023/9320
During eruptive solar flares and coronal mass ejections, a nonpotential magnetic arcade with much excess magnetic energy goes unstable and reconnects. It produces a twisted erupting flux rope and leaves behind a sheared arcade of hot coronal loops. We suggest that: the twist of the erupting flux rope can be determined from conservation of magnetic flux and magnetic helicity and equipartition of magnetic helicity. It depends on the geometry of the initial preeruptive structure. Two cases are considered, in the first of which a flux rope is not present initially but is created during the eruption by the reconnection. In the second case, a flux rope is present under the arcade in the preeruptive state,and the e.ect of the eruption and reconnection is to add an amount of magnetic helicity that depends on the fluxes of the rope and arcade and the geometry.
Funding: UK STFC, High Altitude Observatory and Montana State University.
20160801T00:00:00Z
Priest, Eric Ronald
Longcope, D W
Janvier, M
During eruptive solar flares and coronal mass ejections, a nonpotential magnetic arcade with much excess magnetic energy goes unstable and reconnects. It produces a twisted erupting flux rope and leaves behind a sheared arcade of hot coronal loops. We suggest that: the twist of the erupting flux rope can be determined from conservation of magnetic flux and magnetic helicity and equipartition of magnetic helicity. It depends on the geometry of the initial preeruptive structure. Two cases are considered, in the first of which a flux rope is not present initially but is created during the eruption by the reconnection. In the second case, a flux rope is present under the arcade in the preeruptive state,and the e.ect of the eruption and reconnection is to add an amount of magnetic helicity that depends on the fluxes of the rope and arcade and the geometry.

Properties of the prominence magnetic field and plasma distributions as obtained from 3D wholeprominence fine structure modeling
http://hdl.handle.net/10023/9203
Aims. We analyze distributions of the magnetic field strength and prominence plasma (temperature, pressure, plasma beta, and mass) using the 3D wholeprominence fine structure model. Methods. The model combines a 3D magnetic field configuration of an entire prominence, obtained from nonlinear forcefree field simulations, with a detailed semiempirically derived description of the prominence plasma. The plasma is located in magnetic dips in hydrostatic equilibrium and is distributed along multiple fine structures within the 3D magnetic model. Results. We show that in the modeled prominence, the variations of the magnetic field strength and its orientation are insignificant on scales comparable to the smallest dimensions of the observed prominence fine structures. We also show the ability of the 3D wholeprominence fine structure model to reveal the distribution of the prominence plasma, with respect to its temperature within the prominence volume. This provides new insights into the composition of the prominencecorona transition region. We further demonstrate that the values of the plasma beta are small throughout the majority of the modeled prominence when realistic photospheric magnetic flux distributions and prominence plasma parameters are assumed. While this is generally true, we also find that in the region with the deepest magnetic dips, the plasma beta may increase towards unity. Finally, we show that the mass of the modeled prominence plasma is in good agreement with the mass of observed noneruptive prominences.
20160801T00:00:00Z
Gunar, Stanislav
Mackay, Duncan Hendry
Aims. We analyze distributions of the magnetic field strength and prominence plasma (temperature, pressure, plasma beta, and mass) using the 3D wholeprominence fine structure model. Methods. The model combines a 3D magnetic field configuration of an entire prominence, obtained from nonlinear forcefree field simulations, with a detailed semiempirically derived description of the prominence plasma. The plasma is located in magnetic dips in hydrostatic equilibrium and is distributed along multiple fine structures within the 3D magnetic model. Results. We show that in the modeled prominence, the variations of the magnetic field strength and its orientation are insignificant on scales comparable to the smallest dimensions of the observed prominence fine structures. We also show the ability of the 3D wholeprominence fine structure model to reveal the distribution of the prominence plasma, with respect to its temperature within the prominence volume. This provides new insights into the composition of the prominencecorona transition region. We further demonstrate that the values of the plasma beta are small throughout the majority of the modeled prominence when realistic photospheric magnetic flux distributions and prominence plasma parameters are assumed. While this is generally true, we also find that in the region with the deepest magnetic dips, the plasma beta may increase towards unity. Finally, we show that the mass of the modeled prominence plasma is in good agreement with the mass of observed noneruptive prominences.

Motives and tensions in the release of open educational resources : the UKOER program
http://hdl.handle.net/10023/9166
Open educational resources (OER) have been promoted as a path to universal education, supporting economic development and intercultural dialogue. However, to realise these benefits requires greater understanding of the factors that influence both OER supply and use. This paper examines an aspect of the supply side of the OER lifecycle – the motives prompting release – and the resultant tensions in the release process. It draws evidence from a major program of OER release projects (UKOER) funded by the UK government. The paper sets the UKOER program within the global context of OER initiatives. It uses grounded theory to identify five candidate motive types. Then, by mapping the actions evident in the UKOER program against an organisational framework derived from an activity system, it examines tensions or contradictions encountered by the projects, revealing unstated motives. The findings will be of interest to funders, institutions and educators releasing OER as they reveal potential limitations and barriers to realising the benefits of OER
It gives us pleasure to acknowledge the support of the UK Joint Information Systems Committee and Higher Education Academy, who funded the UKOER projects upon which this paper is based.
20160101T00:00:00Z
Falconer, Isobel Jessie
Littlejohn, Allison
McGill, Lou
Beetham, Helen
Open educational resources (OER) have been promoted as a path to universal education, supporting economic development and intercultural dialogue. However, to realise these benefits requires greater understanding of the factors that influence both OER supply and use. This paper examines an aspect of the supply side of the OER lifecycle – the motives prompting release – and the resultant tensions in the release process. It draws evidence from a major program of OER release projects (UKOER) funded by the UK government. The paper sets the UKOER program within the global context of OER initiatives. It uses grounded theory to identify five candidate motive types. Then, by mapping the actions evident in the UKOER program against an organisational framework derived from an activity system, it examines tensions or contradictions encountered by the projects, revealing unstated motives. The findings will be of interest to funders, institutions and educators releasing OER as they reveal potential limitations and barriers to realising the benefits of OER

Impact of an L5 magnetograph on nonpotential solar global magnetic field modeling
http://hdl.handle.net/10023/9154
We present the first theoretical study to consider what improvement could be obtained in global nonpotential modeling of the solar corona if magnetograph data were available from the L5 Lagrange point, in addition to from the direction of Earth. To consider this, we first carry out a "reference Sun'' simulation over two solar cycles. An important property of this simulation is that random bipole emergences are allowed across the entire solar surface at any given time (such as can occur on the Sun). Next we construct two "limited data'' simulations, where bipoles are only included when they could be seen from (i) an Earthbased magnetograph and (ii) either Earth or L5 based magnetographs. The improvement in reproducing the reference Sun simulation when an L5 view is available is quantified through considering global quantities in the limited data simulations. These include surface and polar flux, total magnetic energy, volume electric current, open flux and the number of flux ropes. Results show that when an L5 observational viewpoint is included, the accuracy of the global quantities in the limited data simulations can increase by 2640%. This clearly shows that a magnetograph at the L5 point could significantly increase the accuracy of global nonpotential modeling and with this the accuracy of future space weather forecasts.
20160712T00:00:00Z
Mackay, Duncan Hendry
Yeates, Anthony Robinson
Bocquet, FrancoisXavier
We present the first theoretical study to consider what improvement could be obtained in global nonpotential modeling of the solar corona if magnetograph data were available from the L5 Lagrange point, in addition to from the direction of Earth. To consider this, we first carry out a "reference Sun'' simulation over two solar cycles. An important property of this simulation is that random bipole emergences are allowed across the entire solar surface at any given time (such as can occur on the Sun). Next we construct two "limited data'' simulations, where bipoles are only included when they could be seen from (i) an Earthbased magnetograph and (ii) either Earth or L5 based magnetographs. The improvement in reproducing the reference Sun simulation when an L5 view is available is quantified through considering global quantities in the limited data simulations. These include surface and polar flux, total magnetic energy, volume electric current, open flux and the number of flux ropes. Results show that when an L5 observational viewpoint is included, the accuracy of the global quantities in the limited data simulations can increase by 2640%. This clearly shows that a magnetograph at the L5 point could significantly increase the accuracy of global nonpotential modeling and with this the accuracy of future space weather forecasts.

Explosive fragmentation of liquids in spherical geometry
http://hdl.handle.net/10023/9116
Rapid acceleration of a spherical shell of liquid following detonation of a high explosive causes the liquid to form fine jets that are similar in appearance to the particle jets that are formed during explosive dispersal of a packed layer of solid particles. Of particular interest is determining the dependence of the scale of the jetlike structures on the physical parameters of the system, including the fluid properties (e.g., density, viscosity, surface tension) and the ratio of the mass of the liquid to that of the explosive. The present paper presents computational results from a multimaterial hydrocode describing the dynamics of the explosive dispersal process. The computations are used to track the overall features of the dispersal of the liquid layer, including the wave dynamics, and motion of the spall and accretion layers. The results are compared with experimental results of spherical charges surrounded by a variety of different fluids, including water, glycerol, ethanol, and vegetable oil, which together encompass a significant range of fluid properties. The results show that the number of jet structures is not sensitive to the fluid properties, but primarily dependent on the mass ratio. Above a certain mass ratio of liquid fill to explosive burster (F/B), the number of jets is approximately constant and consistent with an empirical model based on the maximum thickness of the accretion layer. For small values of F/B, the number of liquid jets is reduced, in contrast with explosive powder dispersal, where small F/B yields a larger number of particle jets. A hypothetical explanation of these features based on nucleation of cavitation is explored numerically.
20160708T00:00:00Z
Milne, Alexander Mitchell
Longbottom, Aaron William
Frost, David
Loiseau, Jason
Goroshin, Samuel
Petel, Oren
Rapid acceleration of a spherical shell of liquid following detonation of a high explosive causes the liquid to form fine jets that are similar in appearance to the particle jets that are formed during explosive dispersal of a packed layer of solid particles. Of particular interest is determining the dependence of the scale of the jetlike structures on the physical parameters of the system, including the fluid properties (e.g., density, viscosity, surface tension) and the ratio of the mass of the liquid to that of the explosive. The present paper presents computational results from a multimaterial hydrocode describing the dynamics of the explosive dispersal process. The computations are used to track the overall features of the dispersal of the liquid layer, including the wave dynamics, and motion of the spall and accretion layers. The results are compared with experimental results of spherical charges surrounded by a variety of different fluids, including water, glycerol, ethanol, and vegetable oil, which together encompass a significant range of fluid properties. The results show that the number of jet structures is not sensitive to the fluid properties, but primarily dependent on the mass ratio. Above a certain mass ratio of liquid fill to explosive burster (F/B), the number of jets is approximately constant and consistent with an empirical model based on the maximum thickness of the accretion layer. For small values of F/B, the number of liquid jets is reduced, in contrast with explosive powder dispersal, where small F/B yields a larger number of particle jets. A hypothetical explanation of these features based on nucleation of cavitation is explored numerically.

Impact of flux distribution on elementary heating events
http://hdl.handle.net/10023/9109
Context. The complex magnetic field on the solar surface has been shown to contain a range of sizes and distributions of magnetic flux structures. The dynamic evolution of this magnetic carpet by photospheric flows provides a continual source of free magnetic energy into the solar atmosphere, that can subsequently be released by magnetic reconnection. Aims. We investigate how the distribution and number of magnetic flux sources impact the energy release and locations of heating through magnetic reconnection driven by slow footpoint motions. Methods. 3D MHD simulations using Lare3D are carried out, where fluxtubes are formed between positive and negative sources placed symmetrically on the lower and upper boundaries of the domain, respectively. The fluxtubes are subjected to rotational driving velocities on the boundaries and are forced to interact and reconnect. Results. Initially, simple flux distributions with two and four sources are compared. In both cases, central current concentrations are formed between the fluxtubes and Ohmic heating occurs. The reconnection and subsequent energy release is delayed in the four source case and is shown to produce more locations of heating, but with smaller magnitudes. Increasing the values of background field between the fluxtubes is shown to delay the onset of reconnection and increases the efficiency of heating in both the two and four source cases. The two fluxtube cases are always more energetic than the corresponding four fluxtube case, however the addition of the background field makes this disparity less significant. A final experiment with a larger number of smaller flux sources is considered and the field evolution and energetics are shown to be remarkably similar to the two source case, indicating the importance of the size and separation of the flux sources relative to the spatial scales of the velocity driver.
This work used the COSMA Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk. This equipment was funded by a BIS National Einfrastructure capital grant ST/K00042X/1, STFC capital grant ST/K00087X/1, DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National EInfrastructure. I.D.M was funded by the Science and Technology Facilities Council (UK). The research leading to these results has also received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation programme (grant agreement No. 647214). J.O was funded by the Science and Technology Facilities Council (UK) by Doctoral Grant [ST/K502327/1].
20160623T00:00:00Z
O'Hara, Jennifer Patricia
De Moortel, Ineke
Context. The complex magnetic field on the solar surface has been shown to contain a range of sizes and distributions of magnetic flux structures. The dynamic evolution of this magnetic carpet by photospheric flows provides a continual source of free magnetic energy into the solar atmosphere, that can subsequently be released by magnetic reconnection. Aims. We investigate how the distribution and number of magnetic flux sources impact the energy release and locations of heating through magnetic reconnection driven by slow footpoint motions. Methods. 3D MHD simulations using Lare3D are carried out, where fluxtubes are formed between positive and negative sources placed symmetrically on the lower and upper boundaries of the domain, respectively. The fluxtubes are subjected to rotational driving velocities on the boundaries and are forced to interact and reconnect. Results. Initially, simple flux distributions with two and four sources are compared. In both cases, central current concentrations are formed between the fluxtubes and Ohmic heating occurs. The reconnection and subsequent energy release is delayed in the four source case and is shown to produce more locations of heating, but with smaller magnitudes. Increasing the values of background field between the fluxtubes is shown to delay the onset of reconnection and increases the efficiency of heating in both the two and four source cases. The two fluxtube cases are always more energetic than the corresponding four fluxtube case, however the addition of the background field makes this disparity less significant. A final experiment with a larger number of smaller flux sources is considered and the field evolution and energetics are shown to be remarkably similar to the two source case, indicating the importance of the size and separation of the flux sources relative to the spatial scales of the velocity driver.

Null point distribution in global coronal potential field extrapolations
http://hdl.handle.net/10023/9063
Magnetic null points are points in space where the magnetic field is zero. Thus, they can be important sites for magnetic reconnection by virtue of the fact that they are weak points in the magnetic field and also because they are associated with topological structures, such as separators, which lie on the boundary between four topologically distinct flux domains and therefore are also locations where reconnection occurs. The number and distribution of nulls in a magnetic field acts as a measure of the complexity of the field. In this article, the numbers and distributions of null points in global potential field extrapolations from highresolution synoptic magnetograms are examined. Extrapolations from magnetograms obtained with the Michelson Doppler Imager (MDI) are studied in depth and compared with those from highresolution SOlar Longtime Investigations of the Sun (SOLIS) and Heliospheric Magnetic Imager (HMI). The falloff in the density of null points with height is found to follow a power law with a slope that differs depending on whether the data are from solar maximum or solar minimum. The distribution of null points with latitude also varies with the cycle as null points form predominantly over quietSun regions and avoid activeregion fields. The exception to this rule are the null points that form high in the solar atmosphere, and these null points tend to form over large areas of strong flux in active regions. From case studies of data acquired with the MDI, SOLIS, and HMI, it is found that the distribution of null points is very similar between data sets, except, of course, that there are far fewer nulls observed in the SOLIS data than in the cases from MDI and HMI due to its lower resolution.
SJE would like to thank the Isle of Man Government for support during her PhD and also for the financial support of the STFC.
20150718T00:00:00Z
Edwards, S.J.
Parnell, C.E.
Magnetic null points are points in space where the magnetic field is zero. Thus, they can be important sites for magnetic reconnection by virtue of the fact that they are weak points in the magnetic field and also because they are associated with topological structures, such as separators, which lie on the boundary between four topologically distinct flux domains and therefore are also locations where reconnection occurs. The number and distribution of nulls in a magnetic field acts as a measure of the complexity of the field. In this article, the numbers and distributions of null points in global potential field extrapolations from highresolution synoptic magnetograms are examined. Extrapolations from magnetograms obtained with the Michelson Doppler Imager (MDI) are studied in depth and compared with those from highresolution SOlar Longtime Investigations of the Sun (SOLIS) and Heliospheric Magnetic Imager (HMI). The falloff in the density of null points with height is found to follow a power law with a slope that differs depending on whether the data are from solar maximum or solar minimum. The distribution of null points with latitude also varies with the cycle as null points form predominantly over quietSun regions and avoid activeregion fields. The exception to this rule are the null points that form high in the solar atmosphere, and these null points tend to form over large areas of strong flux in active regions. From case studies of data acquired with the MDI, SOLIS, and HMI, it is found that the distribution of null points is very similar between data sets, except, of course, that there are far fewer nulls observed in the SOLIS data than in the cases from MDI and HMI due to its lower resolution.

The dependence of coronal loop heating on the characteristics of slow photospheric motions
http://hdl.handle.net/10023/9044
The Parker hypothesis assumes that heating of coronal loops occurs due to reconnection, induced when photospheric motions braid field lines to the point of current sheet formation. In this contribution we address the question of how the nature of photospheric motions affects the heating of braided coronal loops. We design a series of boundary drivers and quantify their properties in terms of complexity and helicity injection. We examine a series of longduration full resistive MHD simulations in which a simulated coronal loop, consisting of initially uniform field lines, is subject to these photospheric flows. Braiding of the loop is continually driven until differences in behavior induced by the drivers can be characterized. It is shown that heating is crucially dependent on the nature of the photospheric driver—coherent motions typically lead to fewer large energy release events, while more complex motions result in more frequent but less energetic heating events.
20160606T00:00:00Z
Ritchie, M. L.
WilmotSmith, A. L.
Hornig, G.
The Parker hypothesis assumes that heating of coronal loops occurs due to reconnection, induced when photospheric motions braid field lines to the point of current sheet formation. In this contribution we address the question of how the nature of photospheric motions affects the heating of braided coronal loops. We design a series of boundary drivers and quantify their properties in terms of complexity and helicity injection. We examine a series of longduration full resistive MHD simulations in which a simulated coronal loop, consisting of initially uniform field lines, is subject to these photospheric flows. Braiding of the loop is continually driven until differences in behavior induced by the drivers can be characterized. It is shown that heating is crucially dependent on the nature of the photospheric driver—coherent motions typically lead to fewer large energy release events, while more complex motions result in more frequent but less energetic heating events.

A new technique for the photospheric driving of nonpotential solar coronal magnetic field simulations
http://hdl.handle.net/10023/9043
In this paper, we develop a new technique for driving global nonpotential simulations of the Sun's coronal magnetic field solely from sequences of radial magnetic maps of the solar photosphere. A primary challenge to driving such global simulations is that the required horizontal electric field cannot be uniquely determined from such maps. We show that an "inductive" electric field solution similar to that used by previous authors successfully reproduces specific features of the coronal field evolution in both single and multiple bipole simulations. For these cases, the true solution is known because the electric field was generated from a surface fluxtransport model. The match for these cases is further improved by including the noninductive electric field contribution from surface differential rotation. Then, using this reconstruction method for the electric field, we show that a coronal nonpotential simulation can be successfully driven from a sequence of ADAPT maps of the photospheric radial field, without including additional physical observations which are not routinely available.
20160523T00:00:00Z
Weinzierl, Marion
Yeates, Anthony
Mackay, Duncan Hendry
Henney, Carl
Arge, C. Nick
In this paper, we develop a new technique for driving global nonpotential simulations of the Sun's coronal magnetic field solely from sequences of radial magnetic maps of the solar photosphere. A primary challenge to driving such global simulations is that the required horizontal electric field cannot be uniquely determined from such maps. We show that an "inductive" electric field solution similar to that used by previous authors successfully reproduces specific features of the coronal field evolution in both single and multiple bipole simulations. For these cases, the true solution is known because the electric field was generated from a surface fluxtransport model. The match for these cases is further improved by including the noninductive electric field contribution from surface differential rotation. Then, using this reconstruction method for the electric field, we show that a coronal nonpotential simulation can be successfully driven from a sequence of ADAPT maps of the photospheric radial field, without including additional physical observations which are not routinely available.

Solar cycle variation of magnetic flux ropes in a quasistatic coronal evolution model
http://hdl.handle.net/10023/9037
The structure of electric current and magnetic helicity in the solar corona is closely linked to solar activity over the 11year cycle, yet is poorly understood. As an alternative to traditional currentfree "potential field" extrapolations, we investigate a model for the global coronal magnetic field which is nonpotential and timedependent, following the buildup and transport of magnetic helicity due to flux emergence and largescale photospheric motions. This helicity concentrates into twisted magnetic flux ropes, which may lose equilibrium and be ejected. Here, we consider how the magnetic structure predicted by this modelin particular the flux ropesvaries over the solar activity cycle, based on photospheric input data from six periods of cycle 23. The number of flux ropes doubles from minimum to maximum, following the total length of photospheric polarity inversion lines. However, the number of flux rope ejections increases by a factor of eight, following the emergence rate of active regions. This is broadly consistent with the observed cycle modulation of coronal mass ejections, although the actual rate of ejections in the simulation is about a fifth of the rate of observed events. The model predicts that, even at minimum, differential rotation will produce sheared, nonpotential, magnetic structure at all latitudes.
20100501T00:00:00Z
Yeates, A. R.
Constable, J. A.
Martens, P. C. H.
The structure of electric current and magnetic helicity in the solar corona is closely linked to solar activity over the 11year cycle, yet is poorly understood. As an alternative to traditional currentfree "potential field" extrapolations, we investigate a model for the global coronal magnetic field which is nonpotential and timedependent, following the buildup and transport of magnetic helicity due to flux emergence and largescale photospheric motions. This helicity concentrates into twisted magnetic flux ropes, which may lose equilibrium and be ejected. Here, we consider how the magnetic structure predicted by this modelin particular the flux ropesvaries over the solar activity cycle, based on photospheric input data from six periods of cycle 23. The number of flux ropes doubles from minimum to maximum, following the total length of photospheric polarity inversion lines. However, the number of flux rope ejections increases by a factor of eight, following the emergence rate of active regions. This is broadly consistent with the observed cycle modulation of coronal mass ejections, although the actual rate of ejections in the simulation is about a fifth of the rate of observed events. The model predicts that, even at minimum, differential rotation will produce sheared, nonpotential, magnetic structure at all latitudes.

Coronal density structure and its role in wave damping in loops
http://hdl.handle.net/10023/9020
It has long been established that gradients in the Alfvén speed, and in particular the plasma density, are an essential part of the damping of waves in the magnetically closed solar corona by mechanisms such as resonant absorption or phase mixing. While models of wave damping often assume a fixed density gradient, in this paper the selfconsistency of such calculations is assessed by examining the temporal evolution of the coronal density. It is shown conceptually that for some coronal structures, density gradients can evolve in a way that the wave damping processes are inhibited. For the case of phase mixing we argue that: (a) wave heating cannot sustain the assumed density structure and (b) inclusion of feedback of the heating on the density gradient can lead to a highly structured density, although on long timescales. In addition, transport coefficients well in excess of classical are required to maintain the observed coronal density. Hence, the heating of closed coronal structures by global oscillations may face problems arising from the assumption of a fixed density gradient and the rapid damping of oscillations may have to be accompanied by a separate (nonwave based) heating mechanism to sustain the required density structuring.
This project has received funding from the Science and Technology Facilities Council (UK) and the European Research Council (ERC) under the European Unionʼs Horizon 2020 research and innovation program (grant agreement No 647214). The research leading to these results has also received funding from the European Commission Seventh Framework Programme (FP7/20072013) under the grant agreement SOLSPANET (project No. 269299, www.solspanet.eu/about).
20160519T00:00:00Z
Cargill, Peter
De Moortel, Ineke
Kiddie, Greg
It has long been established that gradients in the Alfvén speed, and in particular the plasma density, are an essential part of the damping of waves in the magnetically closed solar corona by mechanisms such as resonant absorption or phase mixing. While models of wave damping often assume a fixed density gradient, in this paper the selfconsistency of such calculations is assessed by examining the temporal evolution of the coronal density. It is shown conceptually that for some coronal structures, density gradients can evolve in a way that the wave damping processes are inhibited. For the case of phase mixing we argue that: (a) wave heating cannot sustain the assumed density structure and (b) inclusion of feedback of the heating on the density gradient can lead to a highly structured density, although on long timescales. In addition, transport coefficients well in excess of classical are required to maintain the observed coronal density. Hence, the heating of closed coronal structures by global oscillations may face problems arising from the assumption of a fixed density gradient and the rapid damping of oscillations may have to be accompanied by a separate (nonwave based) heating mechanism to sustain the required density structuring.

From onedimensional fields to Vlasov equilibria : Theory and application of Hermite Polynomials
http://hdl.handle.net/10023/8992
We consider the theory and application of a solution method for the inverse problem in collisionless equilibria, namely that of calculating a VlasovMaxwell equilibrium for a given macroscopic (fluid) equilibrium. Using Jeans' Theorem, the equilibria are expressed as functions of the constants of motion, in the form of a Maxwellian multiplied by an unknown function of the canonical momenta. In this case it is possible to reduce the inverse problem to inverting Weierstrass transforms, which we achieve by using expansions over Hermite Polynomials. Sufficient conditions are found which guarantee the convergence,boundedness and nonnegativity of the candidate solution, when satisfied. These conditions are obtained by elementary means, and it is clear how to put them into practice. Illustrative examples of the use of this method with both forcefree and non forcefree macroscopic equilibria are presented, including the full verification of a recently derived distribution function for the ForceFree Harris Sheet (Allanson et al. (2015)). In the effort to model equilibria with lower values of the plasma beta, solutions for the same macroscopic equilibrium in a new gauge are calculated, with numerical results presented for βpl = 0:05.
20160601T00:00:00Z
Allanson, Oliver Douglas
Neukirch, Thomas
Troscheit, Sascha
Wilson, Fiona
We consider the theory and application of a solution method for the inverse problem in collisionless equilibria, namely that of calculating a VlasovMaxwell equilibrium for a given macroscopic (fluid) equilibrium. Using Jeans' Theorem, the equilibria are expressed as functions of the constants of motion, in the form of a Maxwellian multiplied by an unknown function of the canonical momenta. In this case it is possible to reduce the inverse problem to inverting Weierstrass transforms, which we achieve by using expansions over Hermite Polynomials. Sufficient conditions are found which guarantee the convergence,boundedness and nonnegativity of the candidate solution, when satisfied. These conditions are obtained by elementary means, and it is clear how to put them into practice. Illustrative examples of the use of this method with both forcefree and non forcefree macroscopic equilibria are presented, including the full verification of a recently derived distribution function for the ForceFree Harris Sheet (Allanson et al. (2015)). In the effort to model equilibria with lower values of the plasma beta, solutions for the same macroscopic equilibrium in a new gauge are calculated, with numerical results presented for βpl = 0:05.

Emergence of nontwisted magnetic fields in the Sun : jets and atmospheric response
http://hdl.handle.net/10023/8990
Aims. We study the emergence of a nontwisted flux tube from the solar interior into the solar atmosphere. We investigate whether the length of the buoyant part of the flux tube (i.e. λ) affects the emergence of the field and the dynamics of the evolving magnetic flux system. Methods. We perform threedimensional (3D), timedependent, resistive, compressible magnetohydrodynamic (MHD) simulations using the Lare3D code. Results. We find that there are considerable differences in the dynamics of the emergence of a magnetic flux tube when λ is varied. In the solar interior, for larger values of λ, the rising magnetic field emerges faster and expands more due to its lower magnetic tension. As a result, its field strength decreases and its emergence above the photosphere occurs later than in the smaller λ case. However, in both cases, the emerging field at the photosphere becomes unstable in two places, forming two magnetic bipoles that interact dynamically during the evolution of the system. Most of the dynamic phenomena occur at the current layer, which is formed at the interface between the interacting bipoles. We find the formation and ejection of plasmoids, the onset of successive jets from the interface, and the impulsive heating of the plasma in the solar atmosphere. We discuss the triggering mechanism of the jets and the atmospheric response to the emergence of magnetic flux in the two cases.
The authors acknowledge support by the EU (IEF272549 grant) and the Royal Society. The present research has been cofinanced by the European Union (European Social FundESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) – Research Funding Program: Thales. Investing in knowledge society through the European Social Fund. This research has also been carried out in the frame of the research program of the RCAAM of the Academy of Athens and has been cofinanced by the Program “IKY Scholarships” of the Greek national funds through the Operational Program Education and Lifelong Learning of the NSRF through the European Social Fund of ESPA 20072013. Finally, the work reported in this article was additionally supported by the SOLARNET project, funded by the European Commisions FP7 Capacities Program, under the Grant Agreement 312495. The simulations were performed on the STFC and SRIF funded UKMHD cluster, at the University of St Andrews.
20151201T00:00:00Z
Syntelis, P.
Archontis, V.
Gontikakis, C.
Tsinganos, K.
Aims. We study the emergence of a nontwisted flux tube from the solar interior into the solar atmosphere. We investigate whether the length of the buoyant part of the flux tube (i.e. λ) affects the emergence of the field and the dynamics of the evolving magnetic flux system. Methods. We perform threedimensional (3D), timedependent, resistive, compressible magnetohydrodynamic (MHD) simulations using the Lare3D code. Results. We find that there are considerable differences in the dynamics of the emergence of a magnetic flux tube when λ is varied. In the solar interior, for larger values of λ, the rising magnetic field emerges faster and expands more due to its lower magnetic tension. As a result, its field strength decreases and its emergence above the photosphere occurs later than in the smaller λ case. However, in both cases, the emerging field at the photosphere becomes unstable in two places, forming two magnetic bipoles that interact dynamically during the evolution of the system. Most of the dynamic phenomena occur at the current layer, which is formed at the interface between the interacting bipoles. We find the formation and ejection of plasmoids, the onset of successive jets from the interface, and the impulsive heating of the plasma in the solar atmosphere. We discuss the triggering mechanism of the jets and the atmospheric response to the emergence of magnetic flux in the two cases.

Spontaneous reconnection at a separator current layer : 2. Nature of the waves and flows
http://hdl.handle.net/10023/8960
Sudden destabilisations of the magnetic field, such as those caused by spontaneous reconnection, will produce waves and/or flows. Here, we investigate the nature of the plasma motions resulting from spontaneous reconnection at a 3D separator. In order to clearly see the perturbations generated by the reconnection, we start from a magnetohydrostatic equilibrium containing two oppositelysigned null points joined by a generic separator along which lies a twisted current layer. The nature of the magnetic reconnection initiated in this equilibrium as a result of an anomalous resistivity is discussed in detail in \cite{Stevenson15_jgra}. The resulting sudden loss of force balance inevitably generates waves that propagate away from the diffusion region carrying the dissipated current. In their wake a twisting stagnationflow, in planes perpendicular to the separator, feeds flux back into the original diffusion site (the separator) in order to try to regain equilibrium. This flow drives a phase of slow weak impulsivebursty reconnection that follows on after the initial fastreconnection phase.
JEHS would like to thank STFC for financial support during her Ph.D and continued support after on the St Andrews SMTG’s STFC consortium grant. CEP also acknowledges support from this same grant.
20151210T00:00:00Z
E. H. Stevenson, Julie
E. Parnell, Clare
Sudden destabilisations of the magnetic field, such as those caused by spontaneous reconnection, will produce waves and/or flows. Here, we investigate the nature of the plasma motions resulting from spontaneous reconnection at a 3D separator. In order to clearly see the perturbations generated by the reconnection, we start from a magnetohydrostatic equilibrium containing two oppositelysigned null points joined by a generic separator along which lies a twisted current layer. The nature of the magnetic reconnection initiated in this equilibrium as a result of an anomalous resistivity is discussed in detail in \cite{Stevenson15_jgra}. The resulting sudden loss of force balance inevitably generates waves that propagate away from the diffusion region carrying the dissipated current. In their wake a twisting stagnationflow, in planes perpendicular to the separator, feeds flux back into the original diffusion site (the separator) in order to try to regain equilibrium. This flow drives a phase of slow weak impulsivebursty reconnection that follows on after the initial fastreconnection phase.

Spontaneous reconnection at a separator current layer : I. Nature of the reconnection
http://hdl.handle.net/10023/8959
Magnetic separators, which lie on the boundary between four topologicallydistinct flux domains, are prime locations in threedimensional magnetic fields for reconnection, especially in the magnetosphere between the planetary and interplanetary magnetic field and also in the solar atmosphere. Little is known about the details of separator reconnection and so the aim of this paper, which is the first of two, is to study the properties of magnetic reconnection at a single separator. Threedimensional, resistive magnetohydrodynamic numerical experiments are run to study separator reconnection starting from a magnetohydrostatic equilibrium which contains a twisted current layer along a single separator linking a pair of oppositepolarity null points. The resulting reconnection occurs in two phases. The first is short involving rapidreconnection in which the current at the separator is reduced by a factor of around 2.3. Most ($75\%$) of the magnetic energy is converted during this phase, via Ohmic dissipation, directly into internal energy, with just $0.1\%$ going into kinetic energy. During this phase the reconnection occurs along most of the separator away from its ends (the nulls), but in an asymmetric manner which changes both spatially and temporally over time. The second phase is much longer and involves slow impulsivebursty reconnection. Again Ohmic heating dominates over viscous damping. Here, the reconnection occurs in small localised bursts at random anywhere along the separator.
20160127T00:00:00Z
E. H. Stevenson, Julie
E. Parnell, Clare
Magnetic separators, which lie on the boundary between four topologicallydistinct flux domains, are prime locations in threedimensional magnetic fields for reconnection, especially in the magnetosphere between the planetary and interplanetary magnetic field and also in the solar atmosphere. Little is known about the details of separator reconnection and so the aim of this paper, which is the first of two, is to study the properties of magnetic reconnection at a single separator. Threedimensional, resistive magnetohydrodynamic numerical experiments are run to study separator reconnection starting from a magnetohydrostatic equilibrium which contains a twisted current layer along a single separator linking a pair of oppositepolarity null points. The resulting reconnection occurs in two phases. The first is short involving rapidreconnection in which the current at the separator is reduced by a factor of around 2.3. Most ($75\%$) of the magnetic energy is converted during this phase, via Ohmic dissipation, directly into internal energy, with just $0.1\%$ going into kinetic energy. During this phase the reconnection occurs along most of the separator away from its ends (the nulls), but in an asymmetric manner which changes both spatially and temporally over time. The second phase is much longer and involves slow impulsivebursty reconnection. Again Ohmic heating dominates over viscous damping. Here, the reconnection occurs in small localised bursts at random anywhere along the separator.

SSALMON  the Solar Simulations for the Atacama Large Millimeter Observatory Network
http://hdl.handle.net/10023/8874
The Solar Simulations for the Atacama Large Millimeter Observatory Network (SSALMON) was initiated in 2014 in connection with two ALMA development studies. The Atacama Large Millimeter/submillimeter Array (ALMA) is a powerful new tool, which can also observe the Sun at high spatial, temporal, and spectral resolution. The international SSALMONetwork aims at coordinating the further development of solar observing modes for ALMA and at promoting scientific opportunities for solar physics with particular focus on numerical simulations, which can provide important constraints for the observing modes and can aid the interpretation of future observations. The radiation detected by ALMA originates mostly in the solar chromosphere – a complex and dynamic layer between the photosphere and corona, which plays an important role in the transport of energy and matter and the heating of the outer layers of the solar atmosphere. Potential targets include active regions, prominences, quiet Sun regions, flares. Here, we give a brief overview over the network and potential science cases for future solar observations with ALMA.
20150101T00:00:00Z
Wedemeyer, S.
Bastian, T.
Brajša, R.
Barta, M.
Hudson, H.
Fleishman, G.
Loukitcheva, M.
Fleck, B.
Kontar, E.
De Pontieu, B.
Tiwari, S.
Kato, Y.
Soler, R.
Yagoubov, P.
Black, J.H.
Antolin, P.
Gunár, S.
Labrosse, N.
Benz, A.O.
Nindos, A.
Steffen, M.
Scullion, E.
Doyle, J.G.
Zaqarashvili, T.
Hanslmeier, A.
Nakariakov, V.M.
Heinzel, P.
Ayres, T.
Karlicky, M.
The Solar Simulations for the Atacama Large Millimeter Observatory Network (SSALMON) was initiated in 2014 in connection with two ALMA development studies. The Atacama Large Millimeter/submillimeter Array (ALMA) is a powerful new tool, which can also observe the Sun at high spatial, temporal, and spectral resolution. The international SSALMONetwork aims at coordinating the further development of solar observing modes for ALMA and at promoting scientific opportunities for solar physics with particular focus on numerical simulations, which can provide important constraints for the observing modes and can aid the interpretation of future observations. The radiation detected by ALMA originates mostly in the solar chromosphere – a complex and dynamic layer between the photosphere and corona, which plays an important role in the transport of energy and matter and the heating of the outer layers of the solar atmosphere. Potential targets include active regions, prominences, quiet Sun regions, flares. Here, we give a brief overview over the network and potential science cases for future solar observations with ALMA.

A computational framework for particle and whole cell tracking applied to a real biological dataset
http://hdl.handle.net/10023/8783
Cell tracking is becoming increasingly important in cell biology as it provides a valuable tool for analysing experimental data and hence furthering our understanding of dynamic cellular phenomena. The advent of highthroughput, highresolution microscopy and imaging techniques means that a wealth of large data is routinely generated in many laboratories. Due to the sheer magnitude of the data involved manual tracking is often cumbersome and the development of computer algorithms for automated cell tracking is thus highly desirable. In this work, we describe two approaches for automated cell tracking. Firstly, we consider particle tracking. We propose a few segmentation techniques for the detection of cells migrating in a nonuniform background, centroids of the segmented cells are then calculated and linked from frame to frame via a nearestneighbour approach. Secondly, we consider the problem of whole cell tracking in which one wishes to reconstruct in time whole cell morphologies. Our approach is based on fitting a mathematical model to the experimental imaging data with the goal being that the physics encoded in the model is rejected in the reconstructed data. The resulting mathematical problem involves the optimal control of a phasefield formulation of a geometric evolution law. Efficient approximation of this challenging optimal control problem is achieved via advanced numerical methods for the solution of semilinear parabolic partial differential equations (PDEs) coupled with parallelisation and adaptive resolution techniques. Along with a detailed description of our algorithms, a number of simulation results are reported on. We focus on illustrating the effectivity of our approaches by applying the algorithms to the tracking of migrating cells in a dataset which reflects many of the challenges typically encountered in microscopy data.
FY, CV, VS and AM acknowledge support from the Leverhulme Trust Research Project Grant (RPG2014149). The work of CV, VS and AM was partially supported by the Engineering and Physical Sciences Research Council, UK grant (EP/J016780/1). This work (AM, ZG, EH, RZ) has also received funding from the European Union's Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant agreement No 642866. The work of CV is partially supported by an EPSRC Impact Accelerator Account award.
20160524T00:00:00Z
Yang, Feng Wei
Venkataraman, Chandrasekhar
Styles, Vanessa
Kuttenberger, Verena
Horn, Elias
von Guttenberg, Zeno
Madzvamuse, Anotida
Cell tracking is becoming increasingly important in cell biology as it provides a valuable tool for analysing experimental data and hence furthering our understanding of dynamic cellular phenomena. The advent of highthroughput, highresolution microscopy and imaging techniques means that a wealth of large data is routinely generated in many laboratories. Due to the sheer magnitude of the data involved manual tracking is often cumbersome and the development of computer algorithms for automated cell tracking is thus highly desirable. In this work, we describe two approaches for automated cell tracking. Firstly, we consider particle tracking. We propose a few segmentation techniques for the detection of cells migrating in a nonuniform background, centroids of the segmented cells are then calculated and linked from frame to frame via a nearestneighbour approach. Secondly, we consider the problem of whole cell tracking in which one wishes to reconstruct in time whole cell morphologies. Our approach is based on fitting a mathematical model to the experimental imaging data with the goal being that the physics encoded in the model is rejected in the reconstructed data. The resulting mathematical problem involves the optimal control of a phasefield formulation of a geometric evolution law. Efficient approximation of this challenging optimal control problem is achieved via advanced numerical methods for the solution of semilinear parabolic partial differential equations (PDEs) coupled with parallelisation and adaptive resolution techniques. Along with a detailed description of our algorithms, a number of simulation results are reported on. We focus on illustrating the effectivity of our approaches by applying the algorithms to the tracking of migrating cells in a dataset which reflects many of the challenges typically encountered in microscopy data.

Existence, stability and formation of baroclinic triples in quasigeostrophic flows
http://hdl.handle.net/10023/8770
Hetons are baroclinic vortices able to transport tracers or species, and which have been observed at sea. This paper studies the offset collision of two identical hetons, often resulting in the formation of a baroclinic tripole, in a continuously stratified quasigeostrophic model. This process is of interest since it (temporarily or definitely) stops the transport of tracers contained in the hetons. Firstly, the structure, stationarity and nonlinear stability of baroclinic tripoles composed of an upper core and of two lower (symmetric) satellites are studied analytically for point vortices and numerically for finitearea vortices. The condition for stationarity of the point vortices is obtained and it is proven that the baroclinic point tripoles are neutral. Finitevolume stationary tripoles exist with marginal states having very elongated (figure8) upper core. In the case of vertically distant upper and lower cores, these latter can nearly joint near the center of the plane. These steady states are compared with their twolayer counterparts. Then, the nonlinear evolution of the steady states shows when they are often neutral (showing an oscillatory evolution); when they are unstable, they can either split into two hetons (by breaking of the upper core) or form a single heton (by merger of the lower satellites). These evolutions reflect the linearly unstable modes which can grow on the vorticity poles. Very tall tripoles can break up vertically due to the vertical shear mutually induced by the poles. Finally, the formation of such baroclinic tripoles from the offset collision of two identical hetons is investigated numerically. This formation occurs for hetons offset by less than the internal separation between their poles. The velocity shear during the interaction can lead to substantial filamentation by the upper core, thus forming small, upper satellites, vertically aligned with the lower ones. Finally, in the case of close and flat poles, this shear (or the baroclinic instability of the tripole) can be strong enough that the formed baroclinic tripole is shortlived and that hetons eventually emerge from the collision and drift away.
20151201T00:00:00Z
Reinaud, Jean Noel
Carton, Xavier
Hetons are baroclinic vortices able to transport tracers or species, and which have been observed at sea. This paper studies the offset collision of two identical hetons, often resulting in the formation of a baroclinic tripole, in a continuously stratified quasigeostrophic model. This process is of interest since it (temporarily or definitely) stops the transport of tracers contained in the hetons. Firstly, the structure, stationarity and nonlinear stability of baroclinic tripoles composed of an upper core and of two lower (symmetric) satellites are studied analytically for point vortices and numerically for finitearea vortices. The condition for stationarity of the point vortices is obtained and it is proven that the baroclinic point tripoles are neutral. Finitevolume stationary tripoles exist with marginal states having very elongated (figure8) upper core. In the case of vertically distant upper and lower cores, these latter can nearly joint near the center of the plane. These steady states are compared with their twolayer counterparts. Then, the nonlinear evolution of the steady states shows when they are often neutral (showing an oscillatory evolution); when they are unstable, they can either split into two hetons (by breaking of the upper core) or form a single heton (by merger of the lower satellites). These evolutions reflect the linearly unstable modes which can grow on the vorticity poles. Very tall tripoles can break up vertically due to the vertical shear mutually induced by the poles. Finally, the formation of such baroclinic tripoles from the offset collision of two identical hetons is investigated numerically. This formation occurs for hetons offset by less than the internal separation between their poles. The velocity shear during the interaction can lead to substantial filamentation by the upper core, thus forming small, upper satellites, vertically aligned with the lower ones. Finally, in the case of close and flat poles, this shear (or the baroclinic instability of the tripole) can be strong enough that the formed baroclinic tripole is shortlived and that hetons eventually emerge from the collision and drift away.

On the stability of continuously stratified quasigeostrophic hetons
http://hdl.handle.net/10023/8709
In this paper we examine the stability of quasigeostrophic hetons in a stably, continuously stratified fluid. To this purpose we first determinate numerically equilibrium states. Equilibrium hetons consist of two vortices of equal and opposite strength lying at different depths that are steadily translating without deforming. The situation is studied through a parameter space comprising the vertical offset between the vortices, their horizontal separation distance and their aspect ratio. The study first shows that the equilibrium vortices are not only strongly deformed in the vertical but that their instability modes are also varying within the height of the structures. The main purpose of the present contribution is to study families of equilibria which stem from the case of two vertically aligned cylindrical vortices. It is however shown that other branches of solutions exist with different properties. The paper concludes that hetons may be sensitive to baroclinic instabilities provided the separation distance between the poles of the hetons is moderate both in the horizontal and in the vertical directions. The hetons become stable and efficient ways to transport properties as fas as the poles are distant from one another. The critical separation distance in a nontrivial function of the radiustoheight aspect ratio of the poles.
20150430T00:00:00Z
Reinaud, Jean Noel
In this paper we examine the stability of quasigeostrophic hetons in a stably, continuously stratified fluid. To this purpose we first determinate numerically equilibrium states. Equilibrium hetons consist of two vortices of equal and opposite strength lying at different depths that are steadily translating without deforming. The situation is studied through a parameter space comprising the vertical offset between the vortices, their horizontal separation distance and their aspect ratio. The study first shows that the equilibrium vortices are not only strongly deformed in the vertical but that their instability modes are also varying within the height of the structures. The main purpose of the present contribution is to study families of equilibria which stem from the case of two vertically aligned cylindrical vortices. It is however shown that other branches of solutions exist with different properties. The paper concludes that hetons may be sensitive to baroclinic instabilities provided the separation distance between the poles of the hetons is moderate both in the horizontal and in the vertical directions. The hetons become stable and efficient ways to transport properties as fas as the poles are distant from one another. The critical separation distance in a nontrivial function of the radiustoheight aspect ratio of the poles.

Solar prominences embedded in flux ropes : morphological features and dynamics from 3D MHD simulations
http://hdl.handle.net/10023/8702
The temporal evolution of a solar prominence inserted in a threedimensional magnetic flux rope is investigated numerically. Using the model of Titov & Démoulin (1999) under the regime of weak twist, the cold and dense prominence counteracts gravity by modifying the initially forcefree magnetic configuration. In some cases a quasistationary situation is achieved after the relaxation phase, characterized by the excitation of standing vertical oscillations. These oscillations show a strong attenuation with time produced by the mechanism of continuum damping due to the in homogeneous transition between the prominence and solar corona. The characteristic period of the vertical oscillations does not depend strongly on the twist of the flux rope. Nonlinearity is the responsible for triggering the KelvinHelmholtz instability associated to the vertical oscillations and that eventually produces horizontal structures. Contrary to other configurations in which the longitudinal axis of the prominence is permeated by a perpendicular magnetic field, like in unsheared arcades, the orientation of the prominence along the flux rope axis prevents the development of RayleighTaylor instabilities and therefore the appearance of vertical structuring along this axis.
J.T. and R.S. acknowledge support from MINECO and UIB through a Ramón y Cajal grant. The authors acknowledge support by the Spanish MINECO and FEDER funds through project AYA201454485P. M.L. acknowledges the support by the Spanish Ministry of Economy and Competitiveness through projects AYA201124808, AYA201018029, and AYA201455078P. This work contributes to the deliverables identified in FP7 European Research Council grant agreement 277829, “Magnetic Connectivity through the Solar Partially Ionized Atmosphere” (PI: E. Khomenko). M.L., J.T., and J.L.B. also acknowledge support from the International Space Science Institute (ISSI) to the Team 314 on “LargeAmplitude Oscillation in prominences” led by M. Luna.
20160330T00:00:00Z
Terradas, J.
Soler, R.
Luna, M.
Oliver, R.
Ballester, J. L.
Wright, Andrew Nicholas
The temporal evolution of a solar prominence inserted in a threedimensional magnetic flux rope is investigated numerically. Using the model of Titov & Démoulin (1999) under the regime of weak twist, the cold and dense prominence counteracts gravity by modifying the initially forcefree magnetic configuration. In some cases a quasistationary situation is achieved after the relaxation phase, characterized by the excitation of standing vertical oscillations. These oscillations show a strong attenuation with time produced by the mechanism of continuum damping due to the in homogeneous transition between the prominence and solar corona. The characteristic period of the vertical oscillations does not depend strongly on the twist of the flux rope. Nonlinearity is the responsible for triggering the KelvinHelmholtz instability associated to the vertical oscillations and that eventually produces horizontal structures. Contrary to other configurations in which the longitudinal axis of the prominence is permeated by a perpendicular magnetic field, like in unsheared arcades, the orientation of the prominence along the flux rope axis prevents the development of RayleighTaylor instabilities and therefore the appearance of vertical structuring along this axis.

Copulae on products of compact Riemannian manifolds
http://hdl.handle.net/10023/8672
Abstract One standard way of considering a probability distribution on the unit n cube, [0 , 1]n , due to Sklar (1959), is to decompose it into its marginal distributions and a copula, i.e. a probability distribution on [0 , 1]n with uniform marginals. The definition of copula was extended by Jones et al. (2014) to probability distributions on products of circles. This paper defines a copula as a probability distribution on a product of compact Riemannian manifolds that has uniform marginals. Basic properties of such copulae are established. Two fairly general constructions of copulae on products of compact homogeneous manifolds are given; one is based on convolution in the isometry group, the other using equivariant functions from compact Riemannian manifolds to their spaces of square integrable functions. Examples illustrate the use of copulae to analyse bivariate spherical data and bivariate rotational data.
20150901T00:00:00Z
Jupp, P.E.
Abstract One standard way of considering a probability distribution on the unit n cube, [0 , 1]n , due to Sklar (1959), is to decompose it into its marginal distributions and a copula, i.e. a probability distribution on [0 , 1]n with uniform marginals. The definition of copula was extended by Jones et al. (2014) to probability distributions on products of circles. This paper defines a copula as a probability distribution on a product of compact Riemannian manifolds that has uniform marginals. Basic properties of such copulae are established. Two fairly general constructions of copulae on products of compact homogeneous manifolds are given; one is based on convolution in the isometry group, the other using equivariant functions from compact Riemannian manifolds to their spaces of square integrable functions. Examples illustrate the use of copulae to analyse bivariate spherical data and bivariate rotational data.

The effect of interstitial pressure on therapeutic agent transport : coupling with the tumor blood and lymphatic vascular systems
http://hdl.handle.net/10023/8648
Vascularized tumor growth is characterized by both abnormal interstitial fluid flow and the associated interstitial fluid pressure (IFP). Here, we study the effect that these conditions have on the transport of therapeutic agents during chemotherapy. We apply our recently developed vascular tumor growth model which couples a continuous growth component with a discrete angiogenesis model to show that hypertensive IFP is a physical barrier that may hinder vascular extravasation of agents through transvascular fluid flux convection, which drives the agents away from the tumor. This result is consistent with previous work using simpler models without blood flow or lymphatic drainage. We consider the vascular/interstitial/lymphatic fluid dynamics to show that tumors with larger lymphatic resistance increase the agent concentration more rapidly while also experiencing faster washout. In contrast, tumors with smaller lymphatic resistance accumulate less agents but are able to retain them for a longer time. The agent availability (areaunderthe curve, or AUC) increases for less permeable agents as lymphatic resistance increases, and correspondingly decreases for more permeable agents. We also investigate the effect of vascular pathologies on agent transport. We show that elevated vascular hydraulic conductivity contributes to the highest AUC when the agent is less permeable, but to lower AUC when the agent is more permeable. We find that elevated interstitial hydraulic conductivity contributes to low AUC in general regardless of the transvascular agent transport capability. We also couple the agent transport with the tumor dynamics to simulate chemotherapy with the same vascularized tumor under different vascular pathologies. We show that tumors with an elevated interstitial hydraulic conductivity alone require the strongest dosage to shrink. We further show that tumors with elevated vascular hydraulic conductivity are more hypoxic during therapy and that the response slows down as the tumor shrinks due to the heterogeneity and low concentration of agents in the tumor interior compared with the cases where other pathological effects may combine to flatten the IFP and thus reduce the heterogeneity. We conclude that dual normalizations of the micronevironment ? both the vasculature and the interstitium ? are needed to maximize the effects of chemotherapy, while normalization of only one of these may be insufficient to overcome the physical resistance and may thus lead to suboptimal outcomes.
20140821T00:00:00Z
Wu, Min
Frieboes, Hermann B.
Chaplain, Mark A. J.
McDougall, Steven R.
Cristini, Vittorio
Lowengrub, John S.
Vascularized tumor growth is characterized by both abnormal interstitial fluid flow and the associated interstitial fluid pressure (IFP). Here, we study the effect that these conditions have on the transport of therapeutic agents during chemotherapy. We apply our recently developed vascular tumor growth model which couples a continuous growth component with a discrete angiogenesis model to show that hypertensive IFP is a physical barrier that may hinder vascular extravasation of agents through transvascular fluid flux convection, which drives the agents away from the tumor. This result is consistent with previous work using simpler models without blood flow or lymphatic drainage. We consider the vascular/interstitial/lymphatic fluid dynamics to show that tumors with larger lymphatic resistance increase the agent concentration more rapidly while also experiencing faster washout. In contrast, tumors with smaller lymphatic resistance accumulate less agents but are able to retain them for a longer time. The agent availability (areaunderthe curve, or AUC) increases for less permeable agents as lymphatic resistance increases, and correspondingly decreases for more permeable agents. We also investigate the effect of vascular pathologies on agent transport. We show that elevated vascular hydraulic conductivity contributes to the highest AUC when the agent is less permeable, but to lower AUC when the agent is more permeable. We find that elevated interstitial hydraulic conductivity contributes to low AUC in general regardless of the transvascular agent transport capability. We also couple the agent transport with the tumor dynamics to simulate chemotherapy with the same vascularized tumor under different vascular pathologies. We show that tumors with an elevated interstitial hydraulic conductivity alone require the strongest dosage to shrink. We further show that tumors with elevated vascular hydraulic conductivity are more hypoxic during therapy and that the response slows down as the tumor shrinks due to the heterogeneity and low concentration of agents in the tumor interior compared with the cases where other pathological effects may combine to flatten the IFP and thus reduce the heterogeneity. We conclude that dual normalizations of the micronevironment ? both the vasculature and the interstitium ? are needed to maximize the effects of chemotherapy, while normalization of only one of these may be insufficient to overcome the physical resistance and may thus lead to suboptimal outcomes.

Recent advances in coronal heating
http://hdl.handle.net/10023/8643
The solar corona, the tenuous outer atmosphere of the Sun, is orders of magnitude hotter than the solar surface. This 'coronal heating problem' requires the identification of a heat source to balance losses due to thermal conduction, radiation and (in some locations) convection. The review papers in this Theo Murphy meeting issue present an overview of recent observational findings, large and smallscale numerical modelling of physical processes occurring in the solar atmosphere and other aspects which may affect our understanding of the proposed heating mechanisms. At the same time, they also set out the directions and challenges which must be tackled by future research. In this brief introduction, we summarize some of the issues and themes which reoccur throughout this issue.
20150528T00:00:00Z
De Moortel, I.
Browning, P.
The solar corona, the tenuous outer atmosphere of the Sun, is orders of magnitude hotter than the solar surface. This 'coronal heating problem' requires the identification of a heat source to balance losses due to thermal conduction, radiation and (in some locations) convection. The review papers in this Theo Murphy meeting issue present an overview of recent observational findings, large and smallscale numerical modelling of physical processes occurring in the solar atmosphere and other aspects which may affect our understanding of the proposed heating mechanisms. At the same time, they also set out the directions and challenges which must be tackled by future research. In this brief introduction, we summarize some of the issues and themes which reoccur throughout this issue.

Is magnetic topology important for heating the solar atmosphere?
http://hdl.handle.net/10023/8642
Magnetic fields permeate the entire solar atmosphere weaving an extremely complex pattern on both local and global scales. In order to understand the nature of this tangled web of magnetic fields, its magnetic skeleton, which forms the boundaries between topologically distinct flux domains, may be determined. The magnetic skeleton consists of null points, separatrix surfaces, spines and separators. The skeleton is often used to clearly visualize key elements of the magnetic configuration, but parts of the skeleton are also locations where currents and waves may collect and dissipate. In this review, the nature of the magnetic skeleton on both global and local scales, over solar cycle time scales, is explained. The behaviour of wave pulses in the vicinity of both nulls and separators is discussed and so too is the formation of current layers and reconnection at the same features. Each of these processes leads to heating of the solar atmosphere, but collectively do they provide enough heat, spread over a wide enough area, to explain the energy losses throughout the solar atmosphere? Here, we consider this question for the three different solar regions: active regions, openfield regions and the quiet Sun. We find that the heating of active regions and openfield regions is highly unlikely to be due to reconnection or wave dissipation at topological features, but it is possible that these may play a role in the heating of the quiet Sun. In active regions, the absence of a complex topology may play an important role in allowing large energies to build up and then, subsequently, be explosively released in the form of a solar flare. Additionally, knowledge of the intricate boundaries of openfield regions (which the magnetic skeleton provides) could be very important in determining the main acceleration mechanism(s) of the solar wind.
CEP and JT acknowledge the support of STFC through the St Andrew’s SMTG consolidated grant. JEHS is supported by STFC as a PhD student. SJE is supported STFC through the Durham University Impact Acceleration Account.
20150521T00:00:00Z
E. Parnell, C.
E. H. Stevenson, J.
Threlfall, J.
J. Edwards, S.
Magnetic fields permeate the entire solar atmosphere weaving an extremely complex pattern on both local and global scales. In order to understand the nature of this tangled web of magnetic fields, its magnetic skeleton, which forms the boundaries between topologically distinct flux domains, may be determined. The magnetic skeleton consists of null points, separatrix surfaces, spines and separators. The skeleton is often used to clearly visualize key elements of the magnetic configuration, but parts of the skeleton are also locations where currents and waves may collect and dissipate. In this review, the nature of the magnetic skeleton on both global and local scales, over solar cycle time scales, is explained. The behaviour of wave pulses in the vicinity of both nulls and separators is discussed and so too is the formation of current layers and reconnection at the same features. Each of these processes leads to heating of the solar atmosphere, but collectively do they provide enough heat, spread over a wide enough area, to explain the energy losses throughout the solar atmosphere? Here, we consider this question for the three different solar regions: active regions, openfield regions and the quiet Sun. We find that the heating of active regions and openfield regions is highly unlikely to be due to reconnection or wave dissipation at topological features, but it is possible that these may play a role in the heating of the quiet Sun. In active regions, the absence of a complex topology may play an important role in allowing large energies to build up and then, subsequently, be explosively released in the form of a solar flare. Additionally, knowledge of the intricate boundaries of openfield regions (which the magnetic skeleton provides) could be very important in determining the main acceleration mechanism(s) of the solar wind.

Depletion of nonlinearity in the pressure force driving NavierStokes flows : nonlinear depletion in NS flows
http://hdl.handle.net/10023/8620
The dynamics of the velocity norms uLq for q ≥ 3, in NavierStokes flows is studied. The pressure term that drives this dynamics has a high degree of nonlinear depletion, which owes its origin to a genuine negative correlation between u and ∇u, among other things. Under viscous effects, such depletion may give rise to mild growth of uLq. We explore the possibility of nonsingular growth of uLq.
20150417T00:00:00Z
Tran, Chuong Van
Yu, Xinwei
The dynamics of the velocity norms uLq for q ≥ 3, in NavierStokes flows is studied. The pressure term that drives this dynamics has a high degree of nonlinear depletion, which owes its origin to a genuine negative correlation between u and ∇u, among other things. Under viscous effects, such depletion may give rise to mild growth of uLq. We explore the possibility of nonsingular growth of uLq.

Memory versus effector immune responses in oncolytic virotherapies
http://hdl.handle.net/10023/8604
The main priority when designing cancer immunotherapies has been to seek viable biological mechanisms that lead to permanent cancer eradication or cancer control. Understanding the delicate balance between the role of effector and memory cells on eliminating cancer cells remains an elusive problem in immunology. Here we make an initial investigation into this problem with the help of a mathematical model for oncolytic virotherapy; although the model can in fact be made general enough to be applied also to other immunological problems. Our results show that longterm cancer control is associated with a large number of persistent effector cells (irrespective of the initial peak in effector cell numbers). However, this large number of persistent effector cells is sustained by a relatively large number of memory cells. Moreover, we show that cancer control from a dormant state cannot be predicted by the size of the memory population.
R.E. acknowledges support from an Engineering and Physical Sciences Research Council (UK) First Grant number EP/K033689/1
20150721T00:00:00Z
Macnamara, Cicely Krystyna
Eftimie, Raluca
The main priority when designing cancer immunotherapies has been to seek viable biological mechanisms that lead to permanent cancer eradication or cancer control. Understanding the delicate balance between the role of effector and memory cells on eliminating cancer cells remains an elusive problem in immunology. Here we make an initial investigation into this problem with the help of a mathematical model for oncolytic virotherapy; although the model can in fact be made general enough to be applied also to other immunological problems. Our results show that longterm cancer control is associated with a large number of persistent effector cells (irrespective of the initial peak in effector cell numbers). However, this large number of persistent effector cells is sustained by a relatively large number of memory cells. Moreover, we show that cancer control from a dormant state cannot be predicted by the size of the memory population.

On the latetime behaviour of a bounded, inviscid twodimensional flow
http://hdl.handle.net/10023/8603
Using complementary numerical approaches at high resolution, we study the latetime behaviour of an inviscid incompressible twodimensional flow on the surface of a sphere. Starting from a random initial vorticity field comprised of a small set of intermediatewavenumber spherical harmonics, we find that, contrary to the predictions of equilibrium statistical mechanics, the flow does not evolve into a largescale steady state. Instead, significant unsteadiness persists, characterised by a population of persistent smallscale vortices interacting with a largescale oscillating quadrupolar vorticity field. Moreover, the vorticity develops a stepped, staircase distribution, consisting of nearly homogeneous regions separated by sharp gradients. The persistence of unsteadiness is explained by a simple pointvortex model characterising the interactions between the four main vortices which emerge.
We thank the Kavli Institute for Theoretical Physics for supporting our participation in the 2014 Program “WaveFlow Interaction in Geophysics, Climate, Astrophysics, and Plasmas” where this work was initiated. The KITP is supported in part by the NSF Grant No. NSF PHY1125915. This work was also supported in part by the NSF under grant Nos. DMR1306806 and CCF1048701 (JBM and WQ).
20151101T00:00:00Z
Dritschel, David Gerard
Qi, Wanming
Marston, J.B.
Using complementary numerical approaches at high resolution, we study the latetime behaviour of an inviscid incompressible twodimensional flow on the surface of a sphere. Starting from a random initial vorticity field comprised of a small set of intermediatewavenumber spherical harmonics, we find that, contrary to the predictions of equilibrium statistical mechanics, the flow does not evolve into a largescale steady state. Instead, significant unsteadiness persists, characterised by a population of persistent smallscale vortices interacting with a largescale oscillating quadrupolar vorticity field. Moreover, the vorticity develops a stepped, staircase distribution, consisting of nearly homogeneous regions separated by sharp gradients. The persistence of unsteadiness is explained by a simple pointvortex model characterising the interactions between the four main vortices which emerge.

Whole cell tracking through the optimal control of geometric evolution laws
http://hdl.handle.net/10023/8582
Cell tracking algorithms which automate and systematise the analysis of time lapse image data sets of cells are an indispensable tool in the modelling and understanding of cellular phenomena. In this study we present a theoretical framework and an algorithm for whole cell tracking. Within this work we consider that "tracking" is equivalent to a dynamic reconstruction of the whole cell data (morphologies) from static image data sets. The novelty of our work is that the tracking algorithm is driven by a model for the motion of the cell. This model may be regarded as a simplification of a recently developed physically meaningful model for cell motility. The resulting problem is the optimal control of a geometric evolution law and we discuss the formulation and numerical approximation of the optimal control problem. The overall goal of this work is to design a framework for cell tracking within which the recovered data reflects the physics of the forward model. A number of numerical simulations are presented that illustrate the applicability of our approach.
This work (A.M., V.S. and C.V.) is supported by the Engineering and Physical Sciences Research Council, UK grant (EP/J016780/1) and the Leverhulme Trust Research Project Grant (RPG2014149). K.B. was partially supported by the Embirikion Foundation Grant (20112014) – Greece.
20150915T00:00:00Z
Blazakis, Konstantinos N.
Madzvamuse, Anotida
ReyesAldasoro, Constantino Carlos
Styles, Vanessa
Venkataraman, Chandrasekhar
Cell tracking algorithms which automate and systematise the analysis of time lapse image data sets of cells are an indispensable tool in the modelling and understanding of cellular phenomena. In this study we present a theoretical framework and an algorithm for whole cell tracking. Within this work we consider that "tracking" is equivalent to a dynamic reconstruction of the whole cell data (morphologies) from static image data sets. The novelty of our work is that the tracking algorithm is driven by a model for the motion of the cell. This model may be regarded as a simplification of a recently developed physically meaningful model for cell motility. The resulting problem is the optimal control of a geometric evolution law and we discuss the formulation and numerical approximation of the optimal control problem. The overall goal of this work is to design a framework for cell tracking within which the recovered data reflects the physics of the forward model. A number of numerical simulations are presented that illustrate the applicability of our approach.

Solar coronal electron heating by shortwavelength dispersive shear Alfvén waves
http://hdl.handle.net/10023/8581
The electron heating of the solar coronal plasma has remained one of the most important problems in solar physics. An explanation of the electron heating rests on the identification of the energy source and appropriate physical mechanisms via which the energy can be channelled to the electrons. Our objective here is to present an estimate for the electron heating rate in the presence of finite amplitude shortwavelength (in comparison with the ion gyroradius) dispersive shear Alfven (SWDSA) waves that propagate obliquely to the ambient magnetic field direction in the solar corona. Specifically, it is demonstrated that SWDSA waves can significantly contribute to the solar coronal electron heating via collisionless heating involving SWDSA waveelectron interactions.
This work was partially supported by the STFC through the Centre for Fundamental Physics (CfFP) at Rutherford Appleton Laboratory, Chilton, Didcot, UK. BE acknowledges support by the Engineering and Physical Sciences Research Council (EPSRC), UK, Grant no EP/M009386/1.
20150915T00:00:00Z
Bingham, R.
Shukla, P. K.
Eliasson, B.
Cairns, A.
Cairns, R Alan
The electron heating of the solar coronal plasma has remained one of the most important problems in solar physics. An explanation of the electron heating rests on the identification of the energy source and appropriate physical mechanisms via which the energy can be channelled to the electrons. Our objective here is to present an estimate for the electron heating rate in the presence of finite amplitude shortwavelength (in comparison with the ion gyroradius) dispersive shear Alfven (SWDSA) waves that propagate obliquely to the ambient magnetic field direction in the solar corona. Specifically, it is demonstrated that SWDSA waves can significantly contribute to the solar coronal electron heating via collisionless heating involving SWDSA waveelectron interactions.

The role of dimerisation and nuclear transport in the Hes1 gene regulatory network
http://hdl.handle.net/10023/8458
Hes1 is a member of the family of basic helixloophelix transcription factors and the Hes1 gene regulatory network (GRN) may be described as the canonical example of transcriptional control in eukaryotic cells, since it involves only the Hes1 protein and its own mRNA. Recently, the Hes1 protein has been established as an excellent target for an anticancer drug treatment, with the design of a small molecule Hes1 dimerisation inhibitor representing a promising if challenging approach to therapy. In this paper, we extend a previous spatial stochastic model of the Hes1 GRN to include nuclear transport and dimerisation of Hes1 monomers. Initially, we assume that dimerisation occurs only in the cytoplasm, with only dimers being imported into the nucleus. Stochastic simulations of this novel model using the URDME software show that oscillatory dynamics in agreement with experimental studies are retained. Furthermore, we find that our model is robust to changes in the nuclear transport and dimerisation parameters. However, since the precise dynamics of the nuclear import of Hes1 and the localisation of the dimerisation reaction are not known, we consider a second modelling scenario in which we allow for both Hes1 monomers and dimers to be imported into the nucleus, and we allow dimerisation of Hes1 to occur everywhere in the cell. Once again, computational solutions of this second model produce oscillatory dynamics in agreement with experimental studies. We also explore sensitivity of the numerical solutions to nuclear transport and dimerisation parameters. Finally, we compare and contrast the two different modelling scenarios using numerical experiments that simulate dimer disruption, and suggest a biological experiment that could distinguish which model more faithfully captures the Hes1 GRN.
20140401T00:00:00Z
Sturrock, Marc
Hellander, Andreas
Aldakheel, Sahar
Petzold, Linda
Chaplain, Mark A. J.
Hes1 is a member of the family of basic helixloophelix transcription factors and the Hes1 gene regulatory network (GRN) may be described as the canonical example of transcriptional control in eukaryotic cells, since it involves only the Hes1 protein and its own mRNA. Recently, the Hes1 protein has been established as an excellent target for an anticancer drug treatment, with the design of a small molecule Hes1 dimerisation inhibitor representing a promising if challenging approach to therapy. In this paper, we extend a previous spatial stochastic model of the Hes1 GRN to include nuclear transport and dimerisation of Hes1 monomers. Initially, we assume that dimerisation occurs only in the cytoplasm, with only dimers being imported into the nucleus. Stochastic simulations of this novel model using the URDME software show that oscillatory dynamics in agreement with experimental studies are retained. Furthermore, we find that our model is robust to changes in the nuclear transport and dimerisation parameters. However, since the precise dynamics of the nuclear import of Hes1 and the localisation of the dimerisation reaction are not known, we consider a second modelling scenario in which we allow for both Hes1 monomers and dimers to be imported into the nucleus, and we allow dimerisation of Hes1 to occur everywhere in the cell. Once again, computational solutions of this second model produce oscillatory dynamics in agreement with experimental studies. We also explore sensitivity of the numerical solutions to nuclear transport and dimerisation parameters. Finally, we compare and contrast the two different modelling scenarios using numerical experiments that simulate dimer disruption, and suggest a biological experiment that could distinguish which model more faithfully captures the Hes1 GRN.

Effects of thermal conduction and compressive viscosity on the period ratio of the slow mode
http://hdl.handle.net/10023/8423
Aims: Increasing observational evidence of wave modes brings us to a closer understanding of the solar corona. Coronal seismology allows us to combine wave observations and theory to determine otherwise unknown parameters. The period ratio, P1/2P2, between the period P1 of the fundamental mode and the period P2 of its first overtone, is one such tool of coronal seismology and its departure from unity provides information about the structure of the corona. Methods: We consider analytically the effects of thermal conduction and compressive viscosity on the period ratio for a longitudinally propagating sound wave. Results: For coronal values of thermal conduction the effect on the period ratio is negligible. For compressive viscosity the effect on the period ratio may become important for some short hot loops. Conclusions: Damping typically has a small effect on the period ratio, suggesting that longitudinal structuring remains the most significant effect.
C.K.M. acknowledges financial support from the CarnegieTrust. Discussions with Dr. I. De Moortel and Prof. A. W. Hood are gratefully acknowledged
20100601T00:00:00Z
Macnamara, Cicely Krystyna
Roberts, Bernard
Aims: Increasing observational evidence of wave modes brings us to a closer understanding of the solar corona. Coronal seismology allows us to combine wave observations and theory to determine otherwise unknown parameters. The period ratio, P1/2P2, between the period P1 of the fundamental mode and the period P2 of its first overtone, is one such tool of coronal seismology and its departure from unity provides information about the structure of the corona. Methods: We consider analytically the effects of thermal conduction and compressive viscosity on the period ratio for a longitudinally propagating sound wave. Results: For coronal values of thermal conduction the effect on the period ratio is negligible. For compressive viscosity the effect on the period ratio may become important for some short hot loops. Conclusions: Damping typically has a small effect on the period ratio, suggesting that longitudinal structuring remains the most significant effect.

Particleincell simulations of collisionless magnetic reconnection with a nonuniform guide field
http://hdl.handle.net/10023/8386
Results are presented of a first study of collisionless magnetic reconnection starting from a recently found exact nonlinear forcefree VlasovMaxwell equilibrium. The initial state has a Harris sheet magnetic field profile in one direction and a nonuniform guide field in a second direction, resulting in a spatially constant magnetic field strength as well as a constant initial plasma density and plasma pressure. It is found that the reconnection process initially resembles guide field reconnection, but that a gradual transition to antiparallel reconnection happens as the system evolves. The time evolution of a number of plasma parameters is investigated, and the results are compared with simulations starting from a Harris sheet equilibrium and a Harris sheet plus constant guide field equilibrium.
20160302T00:00:00Z
Wilson, Fiona
Neukirch, Thomas
Hesse, Michael
Harrison, Michael G.
Stark, Craig R.
Results are presented of a first study of collisionless magnetic reconnection starting from a recently found exact nonlinear forcefree VlasovMaxwell equilibrium. The initial state has a Harris sheet magnetic field profile in one direction and a nonuniform guide field in a second direction, resulting in a spatially constant magnetic field strength as well as a constant initial plasma density and plasma pressure. It is found that the reconnection process initially resembles guide field reconnection, but that a gradual transition to antiparallel reconnection happens as the system evolves. The time evolution of a number of plasma parameters is investigated, and the results are compared with simulations starting from a Harris sheet equilibrium and a Harris sheet plus constant guide field equilibrium.

Apparent crossfield superslow propagation of magnetohydrodynamic waves in solar plasmas
http://hdl.handle.net/10023/8377
In this paper we show that the phase mixing of continuum Alfvén waves and/or continuum slow waves in magnetic structures of the solar atmosphere as, e.g., coronal arcades, can create the illusion of wave propagation across the magnetic eld. This phenomenon could be erroneously interpreted as fast mag netosonic waves. The crossfield propagation due to phase mixing of continuum waves is apparent because there is no real propagation of energy across the magnetic surfaces. We investigate the continuous Alfvén and slow spectra in 2D Cartesian equilibrium models with a purely poloidal magnetic field. We show that apparent superslow propagation across the magnetic surfaces in solar coronal structures is a consequence of the existence of continuum Alfvén waves and continuum slow waves that naturally live on those structures and phase mix as time evolves. The apparent crossfield phase velocity is related to the spatial variation of the local Alfvén/slow frequency across the magnetic surfaces and is slower than the Alfvén/sound velocities for typical coronal conditions. Understanding the nature of the apparent crossfield propagation is important for the correct analysis of numerical simulations and the correct interpretation of observations.
TK was supported by the Program for Leading Graduate School, MEXT, Japan. This work was supported by JSPS KAKENHI Grant Number 15H03640. RS acknowledges support from MINECO through project AYA201454485P and from FEDER funds. RS also acknowledges support from MINECO through a ‘Juan de la Cierva’ grant, from MECD through project CEF110012, and from the ‘Vicerectorat d’Investigació Postgrau’ of the UIB. JT acknowledges support from the Spanish Ministerio de Educación y Ciencia through a Ramón y Cajal grant. JT acknowledges support from MINECO through project AYA201454485P and from FEDER funds. MG was supported by IAP P7/08 CHARM (Belspo) and the GOA2015014 (KU Leuven). TVD was supported by an Odysseus grant of the FWO Vlaanderen, the IAP P7/08 CHARM (Belspo) and the GOA2015014 (KU Leuven)
20151015T00:00:00Z
Kaneko, T
Goossens, Marcel
Soler, Roberto
Terradas, Jaume
Van Doorsselaere, Tom
Yokoyama, T
Wright, Andrew Nicholas
In this paper we show that the phase mixing of continuum Alfvén waves and/or continuum slow waves in magnetic structures of the solar atmosphere as, e.g., coronal arcades, can create the illusion of wave propagation across the magnetic eld. This phenomenon could be erroneously interpreted as fast mag netosonic waves. The crossfield propagation due to phase mixing of continuum waves is apparent because there is no real propagation of energy across the magnetic surfaces. We investigate the continuous Alfvén and slow spectra in 2D Cartesian equilibrium models with a purely poloidal magnetic field. We show that apparent superslow propagation across the magnetic surfaces in solar coronal structures is a consequence of the existence of continuum Alfvén waves and continuum slow waves that naturally live on those structures and phase mix as time evolves. The apparent crossfield phase velocity is related to the spatial variation of the local Alfvén/slow frequency across the magnetic surfaces and is slower than the Alfvén/sound velocities for typical coronal conditions. Understanding the nature of the apparent crossfield propagation is important for the correct analysis of numerical simulations and the correct interpretation of observations.

Stability analysis and simulations of coupled bulksurface reactiondiffusion systems
http://hdl.handle.net/10023/8349
In this article, we formulate new models for coupled systems of bulksurface reactiondiffusion equations on stationary volumes. The bulk reactiondiffusion equations are coupled to the surface reactiondiffusion equations through linear Robintype boundary conditions. We then state and prove the necessary conditions for diffusiondriven instability for the coupled system. Owing to the nature of the coupling between bulk and surface dynamics, we are able to decouple the stability analysis of the bulk and surface dynamics. Under a suitable choice of model parameter values, the bulk reactiondiffusion system can induce patterning on the surface independent of whether the surface reactiondiffusion system produces or not, patterning. On the other hand, the surface reactiondiffusion system cannot generate patterns everywhere in the bulk in the absence of patterning from the bulk reactiondiffusion system. For this case, patterns can be induced only in regions close to the surface membrane. Various numerical experiments are presented to support our theoretical findings. Our most revealing numerical result is that, Robintype boundary conditions seem to introduce a boundary layer coupling the bulk and surface dynamics.
20150308T00:00:00Z
Madzvamuse, Anotida
Chung, Andy H. W.
Venkataraman, Chandrasekhar
In this article, we formulate new models for coupled systems of bulksurface reactiondiffusion equations on stationary volumes. The bulk reactiondiffusion equations are coupled to the surface reactiondiffusion equations through linear Robintype boundary conditions. We then state and prove the necessary conditions for diffusiondriven instability for the coupled system. Owing to the nature of the coupling between bulk and surface dynamics, we are able to decouple the stability analysis of the bulk and surface dynamics. Under a suitable choice of model parameter values, the bulk reactiondiffusion system can induce patterning on the surface independent of whether the surface reactiondiffusion system produces or not, patterning. On the other hand, the surface reactiondiffusion system cannot generate patterns everywhere in the bulk in the absence of patterning from the bulk reactiondiffusion system. For this case, patterns can be induced only in regions close to the surface membrane. Various numerical experiments are presented to support our theoretical findings. Our most revealing numerical result is that, Robintype boundary conditions seem to introduce a boundary layer coupling the bulk and surface dynamics.

Stellar coronal response to differential rotation and flux emergence
http://hdl.handle.net/10023/8298
We perform a numerical parameter study to determine what effect varying differential rotation and flux emergence has on a star's nonpotential coronal magnetic field. In particular we consider the effects on the star's surface magnetic flux, open magnetic flux, mean azimuthal field strength, coronal free magnetic energy, coronal heating and flux rope eruptions. To do this, we apply a magnetic flux transport model to describe the photospheric evolution, and couple this to the nonpotential coronal evolution using a magnetofrictional technique. A flux emergence model is applied to add new magnetic flux on to the photosphere and into the corona. The parameters of this flux emergence model are derived from the solar flux emergence profile, however the rate of emergence can be increased to represent higher flux emergence rates than the Sun's. Overall we find that flux emergence has a greater effect on the nonpotential coronal properties compared to differential rotation, with all the aforementioned properties increasing with increasing flux emergence rate. Although differential rotation has a lesser effect on the overall coronal properties compared to flux emergence, varying differential rotation does alter the coronal structure. As the differential rotation rate increases, the corona becomes more open, and more nonpotential.
GPSG would like to thank the STFC for financial support. DHM would like to thank the STFC and the Leverhulme Trust for financial support. Simulations were carried out on a STFC/SRIF funded UKMHD cluster at St Andrews.
20160114T00:00:00Z
Gibb, Gordon Peter Samuel
Mackay, Duncan Hendry
Jardine, Moira Mary
Yeates, A. R.
We perform a numerical parameter study to determine what effect varying differential rotation and flux emergence has on a star's nonpotential coronal magnetic field. In particular we consider the effects on the star's surface magnetic flux, open magnetic flux, mean azimuthal field strength, coronal free magnetic energy, coronal heating and flux rope eruptions. To do this, we apply a magnetic flux transport model to describe the photospheric evolution, and couple this to the nonpotential coronal evolution using a magnetofrictional technique. A flux emergence model is applied to add new magnetic flux on to the photosphere and into the corona. The parameters of this flux emergence model are derived from the solar flux emergence profile, however the rate of emergence can be increased to represent higher flux emergence rates than the Sun's. Overall we find that flux emergence has a greater effect on the nonpotential coronal properties compared to differential rotation, with all the aforementioned properties increasing with increasing flux emergence rate. Although differential rotation has a lesser effect on the overall coronal properties compared to flux emergence, varying differential rotation does alter the coronal structure. As the differential rotation rate increases, the corona becomes more open, and more nonpotential.

Head on collisions between two quasigeostrophic hetons in a continuously stratified fluid
http://hdl.handle.net/10023/8219
We examine the interactions between two threedimensional quasigeostrophic hetons. The hetons are initially translating towards one another. We address the effect of the vertical distance between the two poles (vortices) constituting each heton, on the interaction. We also examine the influence of the horizontal separation between the poles within each heton. In this investigation, the two hetons are facing each other. Two configurations are possible depending on the respective location of the likesigned poles of the hetons. When they lie at the same depth, we refer to the configuration as symmetric; the antisymmetric configuration corresponds to oppositesigned poles at the same depth. The first step in the investigation uses point vortices to represent the poles of the hetons. This approach allows to rapidly browse the parameter space and to estimate the possible heton trajectories. For a symmetric pair, hetons either reverse their trajectory or recombine and escape perpendicularly depending of their horizontal and vertical offsets. On the other hand, antisymmetric hetons recombine and escape perpendicularly as samedepth dipoles. In a second part, we focus on finite core hetons (with finite volume poles). These hetons can deform and may be sensitive to horizontal shear induced deformations, or to baroclinic instability. These destabilisations depend on the vertical and horizontal offsets between the various poles, as well as on their widthtoheight aspect ratios. They can modify the volume of the poles via vortex merger, breaking and/or shearing out; they compete with the advective evolution observed for singular (point) vortices. Importantly, hetons can break down or reconfigure before they can drift away as expected from a point vortex approach. Thus a large variety of behaviours is observed in the parameter space. Finally, we briefly illustrate the behaviour of tall hetons which can be unstable to an azimuthal mode l=1 when many vertical modes of deformation are present on the heton.
Date of Acceptance : 21/07/2015
20150901T00:00:00Z
Reinaud, Jean Noel
Carton, Xavier
We examine the interactions between two threedimensional quasigeostrophic hetons. The hetons are initially translating towards one another. We address the effect of the vertical distance between the two poles (vortices) constituting each heton, on the interaction. We also examine the influence of the horizontal separation between the poles within each heton. In this investigation, the two hetons are facing each other. Two configurations are possible depending on the respective location of the likesigned poles of the hetons. When they lie at the same depth, we refer to the configuration as symmetric; the antisymmetric configuration corresponds to oppositesigned poles at the same depth. The first step in the investigation uses point vortices to represent the poles of the hetons. This approach allows to rapidly browse the parameter space and to estimate the possible heton trajectories. For a symmetric pair, hetons either reverse their trajectory or recombine and escape perpendicularly depending of their horizontal and vertical offsets. On the other hand, antisymmetric hetons recombine and escape perpendicularly as samedepth dipoles. In a second part, we focus on finite core hetons (with finite volume poles). These hetons can deform and may be sensitive to horizontal shear induced deformations, or to baroclinic instability. These destabilisations depend on the vertical and horizontal offsets between the various poles, as well as on their widthtoheight aspect ratios. They can modify the volume of the poles via vortex merger, breaking and/or shearing out; they compete with the advective evolution observed for singular (point) vortices. Importantly, hetons can break down or reconfigure before they can drift away as expected from a point vortex approach. Thus a large variety of behaviours is observed in the parameter space. Finally, we briefly illustrate the behaviour of tall hetons which can be unstable to an azimuthal mode l=1 when many vertical modes of deformation are present on the heton.

Particle dynamics in a nonflaring solar active region model
http://hdl.handle.net/10023/8203
The aim of this work is to investigate and characterise particle behaviour in a (observationallydriven) 3D MHD model of the solar atmosphere above a slowly evolving, nonflaring active region. We use a relativistic guidingcentre particle code to investigate particle acceleration in a single snapshot of the 3D MHD simulation. Despite the lack of flarelike behaviour in the active region, direct acceleration of electrons and protons to nonthermal energies (≲ 42 MeV) was found, yielding spectra with highenergy tails which conform to a power law. Examples of particle dynamics, including particle trapping caused by local electric rather than magnetic field effects, are observed and discussed, together with implications for future experiments which simulate nonflaring active region heating and reconnection.
20160301T00:00:00Z
Threlfall, J.
A. Bourdin, Ph.
Neukirch, T.
E. Parnell, C.
The aim of this work is to investigate and characterise particle behaviour in a (observationallydriven) 3D MHD model of the solar atmosphere above a slowly evolving, nonflaring active region. We use a relativistic guidingcentre particle code to investigate particle acceleration in a single snapshot of the 3D MHD simulation. Despite the lack of flarelike behaviour in the active region, direct acceleration of electrons and protons to nonthermal energies (≲ 42 MeV) was found, yielding spectra with highenergy tails which conform to a power law. Examples of particle dynamics, including particle trapping caused by local electric rather than magnetic field effects, are observed and discussed, together with implications for future experiments which simulate nonflaring active region heating and reconnection.

Magnetohydrostatic modelling of stellar coronae
http://hdl.handle.net/10023/8067
We introduce to the stellar physics community a method of modelling stellar coronae that can be considered to be an extension of the potential field. In this approach, the magnetic field is coupled to the background atmosphere. The model is magnetohydrostatic and is a balance between the Lorentz force, the pressure gradient and gravity. Analytical solutions are possible and we consider a particular class of equilibria in this paper. The model contains two free parameters and the effects of these on both the geometry and topology of the coronal magnetic field are investigated. A demonstration of the approach is given using a magnetogram derived from Zeeman–Doppler imaging of the 0.75 M⊙ Mdwarf star GJ 182.
20160211T00:00:00Z
MacTaggart, David
Gregory, Scott
Neukirch, Thomas
Donati, JeanFrancois
We introduce to the stellar physics community a method of modelling stellar coronae that can be considered to be an extension of the potential field. In this approach, the magnetic field is coupled to the background atmosphere. The model is magnetohydrostatic and is a balance between the Lorentz force, the pressure gradient and gravity. Analytical solutions are possible and we consider a particular class of equilibria in this paper. The model contains two free parameters and the effects of these on both the geometry and topology of the coronal magnetic field are investigated. A demonstration of the approach is given using a magnetogram derived from Zeeman–Doppler imaging of the 0.75 M⊙ Mdwarf star GJ 182.

Particle acceleration at reconnecting separator current layers
http://hdl.handle.net/10023/8001
The aim of this work is to investigate and characterise particle behaviour in a 3D MHD model of a reconnecting magnetic separator. We use a relativistic guidingcentre testparticle code to investigate electron and proton acceleration in snapshots from 3D MHD separator reconnection experiments, and compare the results with findings from an analytical separator reconnection model studied in a previous investigation. The behaviour (and acceleration) of large distributions of particles are examined in detail for both analytical and numerical separator reconnection models. Differences in acceleration sites are recovered and discussed, together with the dependence of final particle energy ranges upon the dimensions of the models and the stage of the (timedependent) MHD reconnection event. We discuss the implications of these results for observed magnetic separators in the solar corona.
20160101T00:00:00Z
Threlfall, J.
E. H. Stevenson, J.
E. Parnell, C.
Neukirch, T.
The aim of this work is to investigate and characterise particle behaviour in a 3D MHD model of a reconnecting magnetic separator. We use a relativistic guidingcentre testparticle code to investigate electron and proton acceleration in snapshots from 3D MHD separator reconnection experiments, and compare the results with findings from an analytical separator reconnection model studied in a previous investigation. The behaviour (and acceleration) of large distributions of particles are examined in detail for both analytical and numerical separator reconnection models. Differences in acceleration sites are recovered and discussed, together with the dependence of final particle energy ranges upon the dimensions of the models and the stage of the (timedependent) MHD reconnection event. We discuss the implications of these results for observed magnetic separators in the solar corona.

A model for selection of eyespots on butterfly wings
http://hdl.handle.net/10023/7904
Unsolved Problem The development of eyespots on the wing surface of butterflies of the family Nympalidae is one of the most studied examples of biological pattern formation. However, little is known about the mechanism that determines the number and precise locations of eyespots on the wing. Eyespots develop around signaling centers, called foci, that are located equidistant from wing veins along the midline of a wing cell (an area bounded by veins). A fundamental question that remains unsolved is, why a certain wing cell develops an eyespot, while other wing cells do not. Key Idea and Model We illustrate that the key to understanding focus point selection may be in the venation system of the wing disc. Our main hypothesis is that changes in morphogen concentration along the proximal boundary veins of wing cells govern focus point selection. Based on previous studies, we focus on a spatially twodimensional reactiondiffusion system model posed in the interior of each wing cell that describes the formation of focus points. Using finite element based numerical simulations, we demonstrate that variation in the proximal boundary condition is sufficient to robustly select whether an eyespot focus point forms in otherwise identical wing cells. We also illustrate that this behavior is robust to small perturbations in the parameters and geometry and moderate levels of noise. Hence, we suggest that an anteriorposterior pattern of morphogen concentration along the proximal vein may be the main determinant of the distribution of focus points on the wing surface. In order to complete our model, we propose a two stage reactiondiffusion system model, in which an onedimensional surface reactiondiffusion system, posed on the proximal vein, generates the morphogen concentrations that act as nonhomogeneous Dirichlet (i.e., fixed) boundary conditions for the twodimensional reactiondiffusion model posed in the wing cells. The twostage model appears capable of generating focus point distributions observed in nature. Result We therefore conclude that changes in the proximal boundary conditions are sufficient to explain the empirically observed distribution of eyespot focus points on the entire wing surface. The model predicts, subject to experimental verification, that the source strength of the activator at the proximal boundary should be lower in wing cells in which focus points form than in those that lack focus points. The model suggests that the number and locations of eyespot foci on the wing disc could be largely controlled by two kinds of gradients along two different directions, that is, the first one is the gradient in spatially varying parameters such as the reaction rate along the anteriorposterior direction on the proximal boundary of the wing cells, and the second one is the gradient in source values of the activator along the veins in the proximaldistal direction of the wing cell.
The authors acknowledge financial support from the EPSRC grant EP/J016780/1. AM and CV acknowledge financial support from the Leverhulme Trust Research Project Grant (RPG2014149). This research was started while CV was visiting Japan as a 2013 Japanese Society for the Promotion of Science (JSPS) Summer Fellow (http://www.jsps.go.jp/). This research was finalized whilst TS, CV and AM were participants in the Isaac Newton Institute Program, Coupling Geometric PDEs with Physics for Cell Morphology, Motility and Pattern Formation. This work (AM) has received funding from the European Union Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant agreement No 642866. AM was partially supported by a grant from the Simons Foundation.
20151104T00:00:00Z
Sekimura, Toshio
Venkataraman, Chandrasekhar
Madzvamuse, Anotida
Unsolved Problem The development of eyespots on the wing surface of butterflies of the family Nympalidae is one of the most studied examples of biological pattern formation. However, little is known about the mechanism that determines the number and precise locations of eyespots on the wing. Eyespots develop around signaling centers, called foci, that are located equidistant from wing veins along the midline of a wing cell (an area bounded by veins). A fundamental question that remains unsolved is, why a certain wing cell develops an eyespot, while other wing cells do not. Key Idea and Model We illustrate that the key to understanding focus point selection may be in the venation system of the wing disc. Our main hypothesis is that changes in morphogen concentration along the proximal boundary veins of wing cells govern focus point selection. Based on previous studies, we focus on a spatially twodimensional reactiondiffusion system model posed in the interior of each wing cell that describes the formation of focus points. Using finite element based numerical simulations, we demonstrate that variation in the proximal boundary condition is sufficient to robustly select whether an eyespot focus point forms in otherwise identical wing cells. We also illustrate that this behavior is robust to small perturbations in the parameters and geometry and moderate levels of noise. Hence, we suggest that an anteriorposterior pattern of morphogen concentration along the proximal vein may be the main determinant of the distribution of focus points on the wing surface. In order to complete our model, we propose a two stage reactiondiffusion system model, in which an onedimensional surface reactiondiffusion system, posed on the proximal vein, generates the morphogen concentrations that act as nonhomogeneous Dirichlet (i.e., fixed) boundary conditions for the twodimensional reactiondiffusion model posed in the wing cells. The twostage model appears capable of generating focus point distributions observed in nature. Result We therefore conclude that changes in the proximal boundary conditions are sufficient to explain the empirically observed distribution of eyespot focus points on the entire wing surface. The model predicts, subject to experimental verification, that the source strength of the activator at the proximal boundary should be lower in wing cells in which focus points form than in those that lack focus points. The model suggests that the number and locations of eyespot foci on the wing disc could be largely controlled by two kinds of gradients along two different directions, that is, the first one is the gradient in spatially varying parameters such as the reaction rate along the anteriorposterior direction on the proximal boundary of the wing cells, and the second one is the gradient in source values of the activator along the veins in the proximaldistal direction of the wing cell.

Magnetostatic modelling of the mixed plasma Beta solar atmosphere based on SUNRISE/IMaX data
http://hdl.handle.net/10023/7887
Our aim is to model the 3D magnetic field structure of the upper solar atmosphere, including regions of nonnegligible plasma beta. We use highresolution photospheric magnetic field measurements from SUNRISE/IMaX as boundary condition for a magnetostatic magnetic field model. The high resolution of IMaX allows us to resolve the interface region between photosphere and corona, but modelling this region is challenging for the following reasons. While the coronal magnetic field is thought to be forcefree (the Lorentzforce vanishes), this is not the case in the mixed plasma β environment in the photosphere and lower chromosphere. In our model, pressure gradients and gravity forces are taken selfconsistently into account and compensate the nonvanishing Lorentzforce. Above a certain height (about 2 Mm) the nonmagnetic forces become very weak and consequently the magnetic field becomes almost forcefree. Here we apply a linear approach, where the electric current density consists of a superposition of a fieldline parallel current and a current perpendicular to the Sun’s gravity field. We illustrate the prospects and limitations of this approach and give an outlook for an extension towards a nonlinear model.
TN acknowledges support by the U.K.’s Science and Technology Facilities Council and would like to thank the MPS for its hospitality during a visit in December 2014.
20151201T00:00:00Z
Wiegelmann, Thomas
Neukirch, Thomas
Nickeler, Dieter
Solanki, Sami
Martinez Pillet, Valentin
Borrero, Juan Manule
Our aim is to model the 3D magnetic field structure of the upper solar atmosphere, including regions of nonnegligible plasma beta. We use highresolution photospheric magnetic field measurements from SUNRISE/IMaX as boundary condition for a magnetostatic magnetic field model. The high resolution of IMaX allows us to resolve the interface region between photosphere and corona, but modelling this region is challenging for the following reasons. While the coronal magnetic field is thought to be forcefree (the Lorentzforce vanishes), this is not the case in the mixed plasma β environment in the photosphere and lower chromosphere. In our model, pressure gradients and gravity forces are taken selfconsistently into account and compensate the nonvanishing Lorentzforce. Above a certain height (about 2 Mm) the nonmagnetic forces become very weak and consequently the magnetic field becomes almost forcefree. Here we apply a linear approach, where the electric current density consists of a superposition of a fieldline parallel current and a current perpendicular to the Sun’s gravity field. We illustrate the prospects and limitations of this approach and give an outlook for an extension towards a nonlinear model.

The appearance, motion, and disappearance of threedimensional magnetic null points
http://hdl.handle.net/10023/7868
While theoretical models and simulations of magnetic reconnection often assume symmetry such that the magnetic null point when present is colocated with a flow stagnation point, the introduction of asymmetry typically leads to nonideal flows across the null point. To understand this behavior, we present exact expressions for the motion of threedimensional linear null points. The most general expression shows that linear null points move in the direction along which the magnetic field and its time derivative are antiparallel. Null point motion in resistive magnetohydrodynamics results from advection by the bulk plasma flow and resistive diffusion of the magnetic field, which allows nonideal flows across topological boundaries. Null point motion is described intrinsically by parameters evaluated locally; however, global dynamics help set the local conditions at the null point. During a bifurcation of a degenerate null point into a nullnull pair or the reverse, the instantaneous velocity of separation or convergence of the nullnull pair will typically be infinite along the null space of the Jacobian matrix of the magnetic field, but with finite components in the directions orthogonal to the null space. Not all bifurcating nullnull pairs are connected by a separator. Furthermore, except under special circumstances, there will not exist a straight line separator connecting a bifurcating nullnull pair. The motion of separators cannot be described using solely local parameters because the identification of a particular field line as a separator may change as a result of nonideal behavior elsewhere along the field line.
N.A.M. acknowledges support from NASA grants NNX11AB61G, NNX12AB25G, and NNX15AF43G; NASA contract NNM07AB07C; and NSF SHINE grants AGS1156076 and AGS1358342 to SAO. C.E.P. acknowledges support from the St Andrews 2013 STFC Consolidated grant.
20151030T00:00:00Z
A. Murphy, Nicholas
Parnell, Clare Elizabeth
Haynes, Andrew Lewis
While theoretical models and simulations of magnetic reconnection often assume symmetry such that the magnetic null point when present is colocated with a flow stagnation point, the introduction of asymmetry typically leads to nonideal flows across the null point. To understand this behavior, we present exact expressions for the motion of threedimensional linear null points. The most general expression shows that linear null points move in the direction along which the magnetic field and its time derivative are antiparallel. Null point motion in resistive magnetohydrodynamics results from advection by the bulk plasma flow and resistive diffusion of the magnetic field, which allows nonideal flows across topological boundaries. Null point motion is described intrinsically by parameters evaluated locally; however, global dynamics help set the local conditions at the null point. During a bifurcation of a degenerate null point into a nullnull pair or the reverse, the instantaneous velocity of separation or convergence of the nullnull pair will typically be infinite along the null space of the Jacobian matrix of the magnetic field, but with finite components in the directions orthogonal to the null space. Not all bifurcating nullnull pairs are connected by a separator. Furthermore, except under special circumstances, there will not exist a straight line separator connecting a bifurcating nullnull pair. The motion of separators cannot be described using solely local parameters because the identification of a particular field line as a separator may change as a result of nonideal behavior elsewhere along the field line.

Nearthreshold electron injection in the laserplasma wakefield accelerator leading to femtosecond bunches
http://hdl.handle.net/10023/7750
The laserplasma wakefield accelerator is a compact source of high brightness, ultrashort duration electron bunches. Selfinjection occurs when electrons from the background plasma gain sufficient momentum at the back of the bubbleshaped accelerating structure to experience sustained acceleration. The shortest duration and highest brightness electron bunches result from selfinjection close to the threshold for injection. Here we show that in this case injection is due to the localized charge density buildup in the sheath crossing region at the rear of the bubble, which has the effect of increasing the accelerating potential to above a critical value. Bunch duration is determined by the dwell time above this critical value, which explains why single or multiple ultrashort electron bunches with little dark current are formed in the first bubble. We confirm experimentally, using coherent optical transition radiation measurements, that single or multiple bunches with femtosecond duration and peak currents of several kiloAmpere, and femtosecond intervals between bunches, emerge from the accelerator.
We gratefully acknowledge the support of the UK EPSRC (grant no. EP/J018171/1), the EU FP7 programmes: the Extreme Light Infrastructure (ELI) project, the LaserlabEurope (no. 284464), and the EUCARD2 project (no. 312453).
20150917T00:00:00Z
Islam, M.R.
Brunetti, E.
Shanks, R.P.
Ersfeld, B.
Issac, R.C.
Cipiccia, S.
Anania, M.P.
Welsh, G.H.
Wiggins, S.M.
Noble, A.
Cairns, R Alan
Raj, G.
Jaroszynski, D.A.
The laserplasma wakefield accelerator is a compact source of high brightness, ultrashort duration electron bunches. Selfinjection occurs when electrons from the background plasma gain sufficient momentum at the back of the bubbleshaped accelerating structure to experience sustained acceleration. The shortest duration and highest brightness electron bunches result from selfinjection close to the threshold for injection. Here we show that in this case injection is due to the localized charge density buildup in the sheath crossing region at the rear of the bubble, which has the effect of increasing the accelerating potential to above a critical value. Bunch duration is determined by the dwell time above this critical value, which explains why single or multiple ultrashort electron bunches with little dark current are formed in the first bubble. We confirm experimentally, using coherent optical transition radiation measurements, that single or multiple bunches with femtosecond duration and peak currents of several kiloAmpere, and femtosecond intervals between bunches, emerge from the accelerator.

Future capabilities of CME polarimetric 3D reconstructions with the METIS instrument : a numerical test
http://hdl.handle.net/10023/7748
Context. Understanding the 3D structure of coronal mass ejections (CMEs) is crucial for understanding the nature and origin of solar eruptions. However, owing to the optical thinness of the solar corona we can only observe the line of sight integrated emission. As a consequence the resulting projection effects hide the true 3D structure of CMEs. To derive information on the 3D structure of CMEs from whitelight (total and polarized brightness) images, the polarization ratio technique is widely used. The soontobelaunched METIS coronagraph on board Solar Orbiter will use this technique to produce new polarimetric images. Aims. This work considers the application of the polarization ratio technique to synthetic CME observations from METIS. In particular we determine the accuracy at which the position of the centre of mass, direction and speed of propagation, and the column density of the CME can be determined along the line of sight. Methods. We perform a 3D MHD simulation of a flux rope ejection where a CME is produced. From the simulation we (i) synthesize the corresponding METIS whitelight (total and polarized brightness) images and (ii) apply the polarization ratio technique to these synthesized images and compare the results with the known density distribution from the MHD simulation. In addition, we use recent results that consider how the position of a single blob of plasma is measured depending on its projected position in the plane of the sky. From this we can interpret the results of the polarization ratio technique and give an estimation of the error associated with derived parameters. Results. We find that the polarization ratio technique reproduces with high accuracy the position of the centre of mass along the line of sight. However, some errors are inherently associated with this determination. The polarization ratio technique also allows information to be derived on the real 3D direction of propagation of the CME. The determination of this is of fundamental importance for future space weather forecasting. In addition, we find that the column density derived from whitelight images is accurate and we propose an improved technique where the combined use of the polarization ratio technique and whitelight images minimizes the error in the estimation of column densities. Moreover, by applying the comparison to a set of snapshots of the simulation we can also assess the errors related to the trajectory and the expansion of the CME. Conclusions. Our method allows us to thoroughly test the performance of the polarization ratio technique and allows a determination of the errors associated with it, which means that it can be used to quantify the results from the analysis of the forthcoming METIS observations in white light (total and polarized brightness). Finally, we describe a satellite observing configuration relative to the Earth that can allow the technique to be efficiently used for space weather predictions.
D.H.M. would like to thank STFC and the Leverhulme Trust for their financial support. P.P. would like to thank STFC and the Leverhulme Trust. The computational work for this paper was carried out on the joint STFC and SFC (SRIF) funded cluster at the University of St Andrews (Scotland, UK).
20151001T00:00:00Z
Pagano, Paolo
Bemporad, A
Mackay, Duncan Hendry
Context. Understanding the 3D structure of coronal mass ejections (CMEs) is crucial for understanding the nature and origin of solar eruptions. However, owing to the optical thinness of the solar corona we can only observe the line of sight integrated emission. As a consequence the resulting projection effects hide the true 3D structure of CMEs. To derive information on the 3D structure of CMEs from whitelight (total and polarized brightness) images, the polarization ratio technique is widely used. The soontobelaunched METIS coronagraph on board Solar Orbiter will use this technique to produce new polarimetric images. Aims. This work considers the application of the polarization ratio technique to synthetic CME observations from METIS. In particular we determine the accuracy at which the position of the centre of mass, direction and speed of propagation, and the column density of the CME can be determined along the line of sight. Methods. We perform a 3D MHD simulation of a flux rope ejection where a CME is produced. From the simulation we (i) synthesize the corresponding METIS whitelight (total and polarized brightness) images and (ii) apply the polarization ratio technique to these synthesized images and compare the results with the known density distribution from the MHD simulation. In addition, we use recent results that consider how the position of a single blob of plasma is measured depending on its projected position in the plane of the sky. From this we can interpret the results of the polarization ratio technique and give an estimation of the error associated with derived parameters. Results. We find that the polarization ratio technique reproduces with high accuracy the position of the centre of mass along the line of sight. However, some errors are inherently associated with this determination. The polarization ratio technique also allows information to be derived on the real 3D direction of propagation of the CME. The determination of this is of fundamental importance for future space weather forecasting. In addition, we find that the column density derived from whitelight images is accurate and we propose an improved technique where the combined use of the polarization ratio technique and whitelight images minimizes the error in the estimation of column densities. Moreover, by applying the comparison to a set of snapshots of the simulation we can also assess the errors related to the trajectory and the expansion of the CME. Conclusions. Our method allows us to thoroughly test the performance of the polarization ratio technique and allows a determination of the errors associated with it, which means that it can be used to quantify the results from the analysis of the forthcoming METIS observations in white light (total and polarized brightness). Finally, we describe a satellite observing configuration relative to the Earth that can allow the technique to be efficiently used for space weather predictions.

Corotating interaction regions as seen by the STEREO Heliospheric Imagers 2007 – 2010
http://hdl.handle.net/10023/7746
NASA’s Solar Terrestrial Relations Observatory (STEREO) mission has coincided with a pronounced solar minimum. This allowed for easier detection of corotating interaction regions (CIRs). CIRs are formed by the interaction between fast and slow solarwind streams ejected from source regions on the solar surface that rotate with the Sun. Highdensity plasma blobs that have become entrained at the stream interface can be tracked out to large elongations in data from the Heliospheric Imager (HI) instruments onboard STEREO. These blobs act as tracers of the CIR itself such that their HI signatures can be used to estimate CIR source location and radial speed. We estimate the kinematic properties of solarwind transients associated with 40 CIRs detected by the HI instrument onboard the STEREOA spacecraft between 2007 and 2010. We identify insitu signatures of these transients at L1 using the Advanced Composition Explorer (ACE) and compare the insitu parameters with the HI results. We note that solarwind transients associated with CIRs appear to travel at or close to the slow solarwind speed preceding the event as measured in situ. We also highlight limitations in the commonly used analysis techniques of solarwind transients by considering variability in the solar wind.
T.M. Conlon and A.O. Williams were supported by an STFC, UK studentship and S.E. Milan was supported by STFC grant ST/K001000/1. Date of Acceptance: 08/08/2015
20150801T00:00:00Z
Conlon, Thomas Michael
Milan, S.E.
Davies, J.A.
Williams, A.O.
NASA’s Solar Terrestrial Relations Observatory (STEREO) mission has coincided with a pronounced solar minimum. This allowed for easier detection of corotating interaction regions (CIRs). CIRs are formed by the interaction between fast and slow solarwind streams ejected from source regions on the solar surface that rotate with the Sun. Highdensity plasma blobs that have become entrained at the stream interface can be tracked out to large elongations in data from the Heliospheric Imager (HI) instruments onboard STEREO. These blobs act as tracers of the CIR itself such that their HI signatures can be used to estimate CIR source location and radial speed. We estimate the kinematic properties of solarwind transients associated with 40 CIRs detected by the HI instrument onboard the STEREOA spacecraft between 2007 and 2010. We identify insitu signatures of these transients at L1 using the Advanced Composition Explorer (ACE) and compare the insitu parameters with the HI results. We note that solarwind transients associated with CIRs appear to travel at or close to the slow solarwind speed preceding the event as measured in situ. We also highlight limitations in the commonly used analysis techniques of solarwind transients by considering variability in the solar wind.

Observational signatures of waves and flows in the solar corona
http://hdl.handle.net/10023/7722
Propagating perturbations have been observed in extended coronal loop structures for a number of years, but the interpretation in terms of slow (propagating) magnetoacoustic waves and/or as quasiperiodic upflows remains unresolved. We used forwardmodelling to construct observational signatures associated with a simple slow magnetoacoustic wave or periodic flow model. Observational signatures were computed for the 171 Å Fe ix and the 193 Å Fe xii spectral lines. Although there are many differences between the flow and wave models, we did not find any clear, robust observational characteristics that can be used in isolation (i.e. that do not rely on a comparison between the models). For the waves model, a relatively rapid change of the average line widths as a function of (shallow) lineofsight angles was found, whereas the ratio of the line width amplitudes to the Doppler velocity amplitudes is relatively high for the flow model. The most robust observational signature found is that the ratio of the mean to the amplitudes of the Doppler velocity is always higher than one for the flow model. This ratio is substantially higher for flows than for waves, and for the flows model used in the study is exactly the same in the 171 Å Fe ix and the 193 Å Fe xii spectral lines. However, these potential observational signatures need to be treated cautiously because they are likely to be modeldependent.
DM acknowledges support of a Royal Society University Research Fellowship and a KU Leuven Research Council senior research fellowship (SF/12/008). The research leading to these results has also received funding from the European Commission Seventh Framework Programme (FP7/20072013) under the grant agreement SOLSPANET (project No. 269299, www.solspanet.eu/solspanet ). TVD has been sponsored by an Odysseus grant of the FWO Vlaanderen. The research was performed in the context of the IAP P7/08 CHARM (Belspo) and the GOA2015014 (KU Leuven). TVD acknowledges the funding from the FP7 ERG grant with number 276808.
20150201T00:00:00Z
De Moortel, I.
Antolin, P.
Van Doorsselaere, T.
Propagating perturbations have been observed in extended coronal loop structures for a number of years, but the interpretation in terms of slow (propagating) magnetoacoustic waves and/or as quasiperiodic upflows remains unresolved. We used forwardmodelling to construct observational signatures associated with a simple slow magnetoacoustic wave or periodic flow model. Observational signatures were computed for the 171 Å Fe ix and the 193 Å Fe xii spectral lines. Although there are many differences between the flow and wave models, we did not find any clear, robust observational characteristics that can be used in isolation (i.e. that do not rely on a comparison between the models). For the waves model, a relatively rapid change of the average line widths as a function of (shallow) lineofsight angles was found, whereas the ratio of the line width amplitudes to the Doppler velocity amplitudes is relatively high for the flow model. The most robust observational signature found is that the ratio of the mean to the amplitudes of the Doppler velocity is always higher than one for the flow model. This ratio is substantially higher for flows than for waves, and for the flows model used in the study is exactly the same in the 171 Å Fe ix and the 193 Å Fe xii spectral lines. However, these potential observational signatures need to be treated cautiously because they are likely to be modeldependent.

Multiscale modelling of cancer progression and treatment control : the role of intracellular heterogeneities in chemotherapy treatment
http://hdl.handle.net/10023/7714
Cancer is a complex, multiscale process involving interactions at intracellular, intercellular and tissue scales that are in turn susceptible to microenvironmental changes. Each individual cancer cell within a cancer cell mass is unique, with its own internal cellular pathways and biochemical interactions. These interactions contribute to the functional changes at the cellular and tissue scale, creating a heterogenous cancer cell population. Anticancer drugs are effective in controlling cancer growth by inflicting damage to various target molecules and thereby triggering multiple cellular and intracellular pathways, leading to cell death or cellcycle arrest. One of the major impediments in the chemotherapy treatment of cancer is drug resistance driven by multiple mechanisms, including multidrug and cellcycle mediated resistance to chemotherapy drugs. In this article, we discuss two hybrid multiscale modelling approaches, incorporating multiple interactions involved in the subcellular, cellular and microenvironmental levels to study the effects of cellcycle, phasespecific chemotherapy on the growth and progression of cancer cells.
20150601T00:00:00Z
Chaplain, Mark Andrew Joseph
Powathil, Gibin
Cancer is a complex, multiscale process involving interactions at intracellular, intercellular and tissue scales that are in turn susceptible to microenvironmental changes. Each individual cancer cell within a cancer cell mass is unique, with its own internal cellular pathways and biochemical interactions. These interactions contribute to the functional changes at the cellular and tissue scale, creating a heterogenous cancer cell population. Anticancer drugs are effective in controlling cancer growth by inflicting damage to various target molecules and thereby triggering multiple cellular and intracellular pathways, leading to cell death or cellcycle arrest. One of the major impediments in the chemotherapy treatment of cancer is drug resistance driven by multiple mechanisms, including multidrug and cellcycle mediated resistance to chemotherapy drugs. In this article, we discuss two hybrid multiscale modelling approaches, incorporating multiple interactions involved in the subcellular, cellular and microenvironmental levels to study the effects of cellcycle, phasespecific chemotherapy on the growth and progression of cancer cells.

Systems oncology : towards patientspecific treatment regimes informed by multiscale mathematical modelling
http://hdl.handle.net/10023/7713
The multiscale complexity of cancer as a disease necessitates a corresponding multiscale modelling approach to produce truly predictive mathematical models capable of improving existing treatment protocols. To capture all the dynamics of solid tumour growth and its progression, mathematical modellers need to couple biological processes occurring at various spatial and temporal scales (from genes to tissues). Because effectiveness of cancer therapy is considerably affected by intracellular and extracellular heterogeneities as well as by the dynamical changes in the tissue microenvironment, any model attempt to optimise existing protocols must consider these factors ultimately leading to improved multimodal treatment regimes. By improving existing and building new mathematical models of cancer, modellers can play important role in preventing the use of potentially suboptimal treatment combinations. In this paper, we analyse a multiscale computational mathematical model for cancer growth and spread, incorporating the multiple effects of radiation therapy and chemotherapy in the patient survival probability and implement the model using two different cell based modelling techniques. We show that the insights provided by such multiscale modelling approaches can ultimately help in designing optimal patientspecific multimodality treatment protocols that may increase patients quality of life.
20150201T00:00:00Z
Powathil, Gibin G.
Swat, Maciej
Chaplain, Mark A. J.
The multiscale complexity of cancer as a disease necessitates a corresponding multiscale modelling approach to produce truly predictive mathematical models capable of improving existing treatment protocols. To capture all the dynamics of solid tumour growth and its progression, mathematical modellers need to couple biological processes occurring at various spatial and temporal scales (from genes to tissues). Because effectiveness of cancer therapy is considerably affected by intracellular and extracellular heterogeneities as well as by the dynamical changes in the tissue microenvironment, any model attempt to optimise existing protocols must consider these factors ultimately leading to improved multimodal treatment regimes. By improving existing and building new mathematical models of cancer, modellers can play important role in preventing the use of potentially suboptimal treatment combinations. In this paper, we analyse a multiscale computational mathematical model for cancer growth and spread, incorporating the multiple effects of radiation therapy and chemotherapy in the patient survival probability and implement the model using two different cell based modelling techniques. We show that the insights provided by such multiscale modelling approaches can ultimately help in designing optimal patientspecific multimodality treatment protocols that may increase patients quality of life.

Mathematical modelling of cancer invasion : implications of cell adhesion variability for tumour infiltrative growth patterns
http://hdl.handle.net/10023/7712
Cancer invasion, recognised as one of the hallmarks of cancer, is a complex, multiscale phenomenon involving many interrelated genetic, biochemical, cellular and tissue processes at different spatial and temporal scales. Central to invasion is the ability of cancer cells to alter and degrade an extracellular matrix. Combined with abnormal excessive proliferation and migration which is enabled and enhanced by altered cellcell and cellmatrix adhesion, the cancerous mass can invade the neighbouring tissue. Along with tumourinduced angiogenesis, invasion is a key component of metastatic spread, ultimately leading to the formation of secondary tumours in other parts of the host body. In this paper we explore the spatiotemporal dynamics of a model of cancer invasion, where cellcell and cellmatrix adhesion is accounted for through nonlocal interaction terms in a system of partial integrodifferential equations. The change of adhesion properties during cancer growth and development is investigated here through timedependent adhesion characteristics within the cell population as well as those between the cells and the components of the extracellular matrix. Our computational simulation results demonstrate a range of heterogeneous dynamics which are qualitatively similar to the invasive growth patterns observed in a number of different types of cancer, such as tumour infiltrative growth patterns (INF).
20141121T00:00:00Z
Domschke, Pia
Trucu, Dumitru
Gerisch, Alf
Chaplain, Mark A. J.
Cancer invasion, recognised as one of the hallmarks of cancer, is a complex, multiscale phenomenon involving many interrelated genetic, biochemical, cellular and tissue processes at different spatial and temporal scales. Central to invasion is the ability of cancer cells to alter and degrade an extracellular matrix. Combined with abnormal excessive proliferation and migration which is enabled and enhanced by altered cellcell and cellmatrix adhesion, the cancerous mass can invade the neighbouring tissue. Along with tumourinduced angiogenesis, invasion is a key component of metastatic spread, ultimately leading to the formation of secondary tumours in other parts of the host body. In this paper we explore the spatiotemporal dynamics of a model of cancer invasion, where cellcell and cellmatrix adhesion is accounted for through nonlocal interaction terms in a system of partial integrodifferential equations. The change of adhesion properties during cancer growth and development is investigated here through timedependent adhesion characteristics within the cell population as well as those between the cells and the components of the extracellular matrix. Our computational simulation results demonstrate a range of heterogeneous dynamics which are qualitatively similar to the invasive growth patterns observed in a number of different types of cancer, such as tumour infiltrative growth patterns (INF).

Stochastic modelling of chromosomal segregation : errors can introduce correction
http://hdl.handle.net/10023/7711
Cell division is a complex process requiring the cell to have many internal checks so that division may proceed and be completed correctly. Failure to divide correctly can have serious consequences, including progression to cancer. During mitosis, chromosomal segregation is one such process that is crucial for successful progression. Accurate segregation of chromosomes during mitosis requires regulation of the interactions between chromosomes and spindle microtubules. If left uncorrected, chromosome attachment errors can cause chromosome segregation defects which have serious effects on cell fates. In early prometaphase, where kinetochores are exposed to multiple microtubules originating from the two poles, there are frequent errors in kinetochoremicrotubule attachment. Erroneous attachments are classified into two categories, syntelic and merotelic. In this paper, we consider a stochastic model for a possible function of syntelic and merotelic kinetochores, and we provide theoretical evidence that merotely can contribute to lessening the stochastic noise in the time for completion of the mitotic process in eukaryotic cells.
20140701T00:00:00Z
Matzavinos, Anastasios
Roitershtein, Alexander
Shtylla, Blerta
Voller, Zachary
Liu, Sijia
Chaplain, Mark A.J.
Cell division is a complex process requiring the cell to have many internal checks so that division may proceed and be completed correctly. Failure to divide correctly can have serious consequences, including progression to cancer. During mitosis, chromosomal segregation is one such process that is crucial for successful progression. Accurate segregation of chromosomes during mitosis requires regulation of the interactions between chromosomes and spindle microtubules. If left uncorrected, chromosome attachment errors can cause chromosome segregation defects which have serious effects on cell fates. In early prometaphase, where kinetochores are exposed to multiple microtubules originating from the two poles, there are frequent errors in kinetochoremicrotubule attachment. Erroneous attachments are classified into two categories, syntelic and merotelic. In this paper, we consider a stochastic model for a possible function of syntelic and merotelic kinetochores, and we provide theoretical evidence that merotely can contribute to lessening the stochastic noise in the time for completion of the mitotic process in eukaryotic cells.

Mathematical modeling of tumor growth and treatment
http://hdl.handle.net/10023/7710
Using mathematical models to simulate dynamic biological processes has a long history. Over the past couple of decades or so, quantitative approaches have also made their way into cancer research. An increasing number of mathematical, physical, computational and engineering techniques have been applied to various aspects of tumor growth, with the ultimate goal of understanding the response of the cancer population to clinical intervention. Socalled in silico trials that predict patientspecific response to various dose schedules or treatment combinations and sequencing are on the way to becoming an invaluable tool to optimize patient care. Herein we describe fundamentals of mathematical modeling of tumor growth and tumorhost interactions, and summarize some of the seminal and most prominent approaches.
20140101T00:00:00Z
Enderling, Heiko
Chaplain, Mark A. J.
Using mathematical models to simulate dynamic biological processes has a long history. Over the past couple of decades or so, quantitative approaches have also made their way into cancer research. An increasing number of mathematical, physical, computational and engineering techniques have been applied to various aspects of tumor growth, with the ultimate goal of understanding the response of the cancer population to clinical intervention. Socalled in silico trials that predict patientspecific response to various dose schedules or treatment combinations and sequencing are on the way to becoming an invaluable tool to optimize patient care. Herein we describe fundamentals of mathematical modeling of tumor growth and tumorhost interactions, and summarize some of the seminal and most prominent approaches.

Mean field analysis of a spatial stochastic model of a gene regulatory network
http://hdl.handle.net/10023/7709
A gene regulatory network may be defined as a collection of DNA segments which interact with each other indirectly through their RNA and protein products. Such a network is said to contain a negative feedback loop if its products inhibit gene transcription, and a positive feedback loop if a gene product promotes its own production. Negative feedback loops can create oscillations in mRNA and protein levels while positive feedback loops are primarily responsible for signal amplification. It is often the case in real biological systems that both negative and positive feedback loops operate in parameter regimes that result in low copy numbers of gene products. In this paper we investigate the spatiotemporal dynamics of a single feedback loop in a eukaryotic cell. We first develop a simplified spatial stochastic model of a canonical feedback system (either positive or negative). Using a Gillespie's algorithm, we compute sample trajectories and analyse their corresponding statistics. We then derive a system of equations that describe the spatiotemporal evolution of the stochastic means. Subsequently, we examine the spatially homogeneous case and compare the results of numerical simulations with the spatially explicit case. Finally, using a combination of steadystate analysis and data clustering techniques, we explore model behaviour across a subregion of the parameter space that is difficult to access experimentally and compare the parameter landscape of our spatiotemporal and spatiallyhomogeneous models.
20151001T00:00:00Z
Sturrock, M.
Murray, P. J.
Matzavinos, A.
Chaplain, M. A. J.
A gene regulatory network may be defined as a collection of DNA segments which interact with each other indirectly through their RNA and protein products. Such a network is said to contain a negative feedback loop if its products inhibit gene transcription, and a positive feedback loop if a gene product promotes its own production. Negative feedback loops can create oscillations in mRNA and protein levels while positive feedback loops are primarily responsible for signal amplification. It is often the case in real biological systems that both negative and positive feedback loops operate in parameter regimes that result in low copy numbers of gene products. In this paper we investigate the spatiotemporal dynamics of a single feedback loop in a eukaryotic cell. We first develop a simplified spatial stochastic model of a canonical feedback system (either positive or negative). Using a Gillespie's algorithm, we compute sample trajectories and analyse their corresponding statistics. We then derive a system of equations that describe the spatiotemporal evolution of the stochastic means. Subsequently, we examine the spatially homogeneous case and compare the results of numerical simulations with the spatially explicit case. Finally, using a combination of steadystate analysis and data clustering techniques, we explore model behaviour across a subregion of the parameter space that is difficult to access experimentally and compare the parameter landscape of our spatiotemporal and spatiallyhomogeneous models.

An exact collisionless equilibrium for the ForceFree Harris Sheet with low plasma beta
http://hdl.handle.net/10023/7691
We present a first discussion and analysis of the physical properties of a new exact collisionless equilibrium for a onedimensional nonlinear forcefree magnetic field, namely, the forcefree Harris sheet. The solution allows any value of the plasma beta, and crucially below unity, which previous nonlinear forcefree collisionless equilibria could not. The distribution function involves infinite series of Hermite polynomials in the canonical momenta, of which the important mathematical properties of convergence and nonnegativity have recently been proven. Plots of the distribution function are presented for the plasma beta modestly below unity, and we compare the shape of the distribution function in two of the velocity directions to a Maxwellian distribution.
Funding: STFC Consolidated Grant [ST/K000950/1] (OA, TN & FW) and a Doctoral Training Grant [ST/K502327/1] (OA). EPSRC Doctoral Training Grant [EP/K503162/1] (ST).
20151001T00:00:00Z
Allanson, Oliver Douglas
Neukirch, Thomas
Wilson, Fiona
Troscheit, Sascha
We present a first discussion and analysis of the physical properties of a new exact collisionless equilibrium for a onedimensional nonlinear forcefree magnetic field, namely, the forcefree Harris sheet. The solution allows any value of the plasma beta, and crucially below unity, which previous nonlinear forcefree collisionless equilibria could not. The distribution function involves infinite series of Hermite polynomials in the canonical momenta, of which the important mathematical properties of convergence and nonnegativity have recently been proven. Plots of the distribution function are presented for the plasma beta modestly below unity, and we compare the shape of the distribution function in two of the velocity directions to a Maxwellian distribution.

3D wholeprominence fine structure modeling. II. Prominence evolution
http://hdl.handle.net/10023/7683
We use the new threedimensional (3D) wholeprominence fine structure model to study the evolution of prominences and their fine structures in response to changes in the underlying photospheric magnetic flux distribution. The applied model combines a detailed 3D prominence magnetic field configuration with a realistic description of the prominence plasma distributed along multiple fine structures. In addition, we utilize an approximate Hα visualization technique to study the evolution of the visible cool prominence plasma both in emission (prominence) and absorption (filament). We show that the initial magnetic field configuration of the modeled prominence is significantly disturbed by the changing position of a single polarity of a magnetic bipole as the bipole is advected toward the main body of the filament. This leads to the creation of a barb, which becomes the dominant feature visible in the synthetic Hα images of both the prominence and filament views. The evolution of the bipole also creates conditions that lead to the disappearance and reappearance of large portions of the main body. We also show that an archlike region containing a dark void (a bubble) can be naturally produced in the synthetic prominence Hα images. While not visible in terms of the magnetic field lines, it is due to a lack of Hα emission from lowpressure, lowdensity plasma located in shallow magnetic dips lying along the lines of sight intersecting the dark void. In addition, a quasivertical smallscale feature consisting of short and deep dips, piled one above the other, is produced.
20151020T00:00:00Z
Gunar, Stanislav
Mackay, Duncan Hendry
We use the new threedimensional (3D) wholeprominence fine structure model to study the evolution of prominences and their fine structures in response to changes in the underlying photospheric magnetic flux distribution. The applied model combines a detailed 3D prominence magnetic field configuration with a realistic description of the prominence plasma distributed along multiple fine structures. In addition, we utilize an approximate Hα visualization technique to study the evolution of the visible cool prominence plasma both in emission (prominence) and absorption (filament). We show that the initial magnetic field configuration of the modeled prominence is significantly disturbed by the changing position of a single polarity of a magnetic bipole as the bipole is advected toward the main body of the filament. This leads to the creation of a barb, which becomes the dominant feature visible in the synthetic Hα images of both the prominence and filament views. The evolution of the bipole also creates conditions that lead to the disappearance and reappearance of large portions of the main body. We also show that an archlike region containing a dark void (a bubble) can be naturally produced in the synthetic prominence Hα images. While not visible in terms of the magnetic field lines, it is due to a lack of Hα emission from lowpressure, lowdensity plasma located in shallow magnetic dips lying along the lines of sight intersecting the dark void. In addition, a quasivertical smallscale feature consisting of short and deep dips, piled one above the other, is produced.

Formation and largescale patterns of filament channels and filaments
http://hdl.handle.net/10023/7673
The properties and largescale patterns of filament channels and filaments are considered. Initially, the global formation locations of filament channels and filaments are discussed, along with their hemispheric pattern. Next, observations of the formation of filament channels and filaments are described where two opposing views are considered. Finally, the wide range of models that have been constructed to consider the formation of filament channels and filaments over long timescales are described, along with the origin of the hemispheric pattern of filaments.
2015ASSL..415..355M
20150101T00:00:00Z
Mackay, Duncan Hendry
The properties and largescale patterns of filament channels and filaments are considered. Initially, the global formation locations of filament channels and filaments are discussed, along with their hemispheric pattern. Next, observations of the formation of filament channels and filaments are described where two opposing views are considered. Finally, the wide range of models that have been constructed to consider the formation of filament channels and filaments over long timescales are described, along with the origin of the hemispheric pattern of filaments.

Particle energisation in a collapsing magnetic trap model : the relativistic regime
http://hdl.handle.net/10023/7603
Context. In solar flares, a large number of charged particles is accelerated to high energies. By which physical processes this is achieved is one of the main open problems in solar physics. It has been suggested that during a flare, regions of the rapidly relaxing magnetic field can form a collapsing magnetic trap (CMT) and that this trap may contribute to particle energisation. Aims. In this Research Note we focus on a particular analytical CMT model based on kinematic magnetohydrodynamics. Previous investigations of particle acceleration for this CMT model focused on the nonrelativistic energy regime. It is the specific aim of this Research Note to extend the previous work to relativistic particle energies. Methods. Particle orbits were calculated numerically using the relativistic guiding centre equations. We also calculated particle orbits using the nonrelativistic guiding centre equations for comparison. Results. For mildly relativistic energies the relativistic and nonrelativistic particle orbits mainly agree well, but clear deviations are seen for higher energies. In particular, the final particle energies obtained from the relativistic calculations are systematically lower than the energies reached from the corresponding nonrelativistic calculations, and the mirror points of the relativistic orbits are systematically higher than for the corresponding nonrelativistic orbits. Conclusions. While the overall behaviour of particle orbits in CMTs does not differ qualitatively when using the relativistic guiding centre equations, there are a few systematic quantitative differences between relativistic and nonrelativistic particle dynamics.
The authors acknowledge financial support by the UK’s Science and Technology Facilities Council through a Doctoral Training Grant (SEO) and Consolidated Grant ST/K000950/1 (SEO and TN).
20140701T00:00:00Z
Eradat Oskoui, S.
Neukirch, T.
Context. In solar flares, a large number of charged particles is accelerated to high energies. By which physical processes this is achieved is one of the main open problems in solar physics. It has been suggested that during a flare, regions of the rapidly relaxing magnetic field can form a collapsing magnetic trap (CMT) and that this trap may contribute to particle energisation. Aims. In this Research Note we focus on a particular analytical CMT model based on kinematic magnetohydrodynamics. Previous investigations of particle acceleration for this CMT model focused on the nonrelativistic energy regime. It is the specific aim of this Research Note to extend the previous work to relativistic particle energies. Methods. Particle orbits were calculated numerically using the relativistic guiding centre equations. We also calculated particle orbits using the nonrelativistic guiding centre equations for comparison. Results. For mildly relativistic energies the relativistic and nonrelativistic particle orbits mainly agree well, but clear deviations are seen for higher energies. In particular, the final particle energies obtained from the relativistic calculations are systematically lower than the energies reached from the corresponding nonrelativistic calculations, and the mirror points of the relativistic orbits are systematically higher than for the corresponding nonrelativistic orbits. Conclusions. While the overall behaviour of particle orbits in CMTs does not differ qualitatively when using the relativistic guiding centre equations, there are a few systematic quantitative differences between relativistic and nonrelativistic particle dynamics.

Multiscale modelling of the dynamics of cell colonies : insights into celladhesion forces and cancer invasion from in silico simulations
http://hdl.handle.net/10023/7571
Studying the biophysical interactions between cells is crucial to understanding how normal tissue develops, how it is structured and also when malfunctions occur. Traditional experiments try to infer events at the tissue level after observing the behaviour of and interactions between individual cells. This approach assumes that cells behave in the same biophysical manner in isolated experiments as they do within colonies and tissues. In this paper, we develop a multiscale multicompartment mathematical model that accounts for the principal biophysical interactions and adhesion pathways not only at a cell?cell level but also at the level of cell colonies (in contrast to the traditional approach). Our results suggest that adhesion/separation forces between cells may be lower in cell colonies than traditional isolated singlecell experiments infer. As a consequence, isolated singlecell experiments may be insufficient to deduce important biological processes such as singlecell invasion after detachment from a solid tumour. The simulations further show that kinetic rates and cell biophysical characteristics such as pressurerelated cellcycle arrest have a major influence on cell colony patterns and can allow for the development of protrusive cellular structures as seen in invasive cancer cell lines independent of expression levels of proinvasion molecules.
20150201T00:00:00Z
Schluter, Daniela K.
RamisConde, Ignacio
Chaplain, Mark A. J.
Studying the biophysical interactions between cells is crucial to understanding how normal tissue develops, how it is structured and also when malfunctions occur. Traditional experiments try to infer events at the tissue level after observing the behaviour of and interactions between individual cells. This approach assumes that cells behave in the same biophysical manner in isolated experiments as they do within colonies and tissues. In this paper, we develop a multiscale multicompartment mathematical model that accounts for the principal biophysical interactions and adhesion pathways not only at a cell?cell level but also at the level of cell colonies (in contrast to the traditional approach). Our results suggest that adhesion/separation forces between cells may be lower in cell colonies than traditional isolated singlecell experiments infer. As a consequence, isolated singlecell experiments may be insufficient to deduce important biological processes such as singlecell invasion after detachment from a solid tumour. The simulations further show that kinetic rates and cell biophysical characteristics such as pressurerelated cellcycle arrest have a major influence on cell colony patterns and can allow for the development of protrusive cellular structures as seen in invasive cancer cell lines independent of expression levels of proinvasion molecules.

Hopf bifurcation in a gene regulatory network model : molecular movement causes oscillations
http://hdl.handle.net/10023/7564
Gene regulatory networks, i.e. DNA segments in a cell which interact with each other indirectly through their RNA and protein products, lie at the heart of many important intracellular signal transduction processes. In this paper, we analyze a mathematical model of a canonical gene regulatory network consisting of a single negative feedback loop between a protein and its mRNA (e.g. the Hes1 transcription factor system). The model consists of two partial differential equations describing the spatiotemporal inter actions between the protein and its mRNA in a 1dimensional domain. Such intracellular negative feedback systems are known to exhibit oscillatory behavior and this is the case for our model, shown initially via computational simulations. In order to investigate this behavior more deeply, we undertake a linearized stability analysis of the steady states of the model. Our results show that the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. This shows that the spatial movement of the mRNA and protein molecules alone is sufficient to cause the oscillations. Our result has implications for transcription factors such as p53, NFκB and heat shock proteins which are involved in regulating important cellular processes such as inflammation, meiosis, apoptosis and the heat shock response, and are linked to diseases such as arthritis and cancer.
M.A.J.C. and M.S. gratefully acknowledge the support of the ERC Advanced Investigator Grant 227619, “M5CGS — From Mutations to Metastases: Multiscale Mathematical Modelling of Cancer Growth and Spread”. M.S. would also like to thank the support from the Mathematical Biosciences Institute at the Ohio State University and NSF Grant DMS0931642.
20150615T00:00:00Z
Chaplain, Mark
Ptashnyk, Mariya
Sturrock, Marc
Gene regulatory networks, i.e. DNA segments in a cell which interact with each other indirectly through their RNA and protein products, lie at the heart of many important intracellular signal transduction processes. In this paper, we analyze a mathematical model of a canonical gene regulatory network consisting of a single negative feedback loop between a protein and its mRNA (e.g. the Hes1 transcription factor system). The model consists of two partial differential equations describing the spatiotemporal inter actions between the protein and its mRNA in a 1dimensional domain. Such intracellular negative feedback systems are known to exhibit oscillatory behavior and this is the case for our model, shown initially via computational simulations. In order to investigate this behavior more deeply, we undertake a linearized stability analysis of the steady states of the model. Our results show that the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. This shows that the spatial movement of the mRNA and protein molecules alone is sufficient to cause the oscillations. Our result has implications for transcription factors such as p53, NFκB and heat shock proteins which are involved in regulating important cellular processes such as inflammation, meiosis, apoptosis and the heat shock response, and are linked to diseases such as arthritis and cancer.

Magnetospheric signatures of ionospheric density cavities observed by Cluster
http://hdl.handle.net/10023/7509
We present Cluster measurements of large amplitude electric fields corre lated with intense downward fieldaligned currents, observed during a nightside crossing of the auroral zone. The data are reproduced by a simple model of magnetosphereionosphere coupling which, under different conditions, can also produce a divergent electric field signature in the downward current region, or correlation between the electric and perturbed magnetic fields. We conclude that strong electric field associated with intense downward fieldaligned current, such as this observation, is a signature of ionospheric plasma depletion caused by the downward current. It is also shown that the electric field in the downward current region correlates with downward current density if a background field is present, e.g. due to magnetospheric convection.
AJBR ackowledges support from STFC under consolidated grant ST/K000993/1.
20150301T00:00:00Z
Russell, Alexander John Barkway
Karlsson, Tomas
Wright, Andrew Nicholas
We present Cluster measurements of large amplitude electric fields corre lated with intense downward fieldaligned currents, observed during a nightside crossing of the auroral zone. The data are reproduced by a simple model of magnetosphereionosphere coupling which, under different conditions, can also produce a divergent electric field signature in the downward current region, or correlation between the electric and perturbed magnetic fields. We conclude that strong electric field associated with intense downward fieldaligned current, such as this observation, is a signature of ionospheric plasma depletion caused by the downward current. It is also shown that the electric field in the downward current region correlates with downward current density if a background field is present, e.g. due to magnetospheric convection.

Strategies of eradicating glioma cells : a multiscale mathematical model with miR451AMPKmTOR control
http://hdl.handle.net/10023/7503
The cellular dispersion and therapeutic control of glioblastoma, the most aggressive type of primary brain cancer, depends critically on the migration patterns after surgery and intracellular responses of the individual cancer cells in response to external biochemical and biomechanical cues in the microenvironment. Recent studies have shown that a particular microRNA, miR451, regulates downstream molecules including AMPK and mTOR to determine the balance between rapid proliferation and invasion in response to metabolic stress in the harsh tumor microenvironment. Surgical removal of main tumor is inevitably followed by recurrence of the tumor due to inaccessibility of dispersed tumor cells in normal brain tissue. In order to address this multiscale nature of glioblastoma proliferation and invasion and its response to conventional treatment, we propose a hybrid model of glioblastoma that analyses spatiotemporal dynamics at the cellular level, linking individual tumor cells with the macroscopic behaviour of cell organization and the microenvironment, and with the intracellular dynamics of miR451AMPKmTOR signaling within a tumour cell. The model identifies a key mechanism underlying the molecular switches between proliferative phase and migratory phase in response to metabolic stress and biophysical interaction between cells in response to fluctuating glucose levels in the presence of blood vessels (BVs). The model predicts that cell migration, therefore efficacy of the treatment, not only depends on oxygen and glucose availability but also on the relative balance between random motility and strength of chemoattractants. Effective control of growing cells near BV sites in addition to relocalization of invisible migratory cells back to the resection site was suggested as a way of eradicating these migratory cells.
20150128T00:00:00Z
Kim, Yangjin
Powathil, Gibin
Kang, Hyunji
Trucu, Dumitru
Kim, Hyeongi
Lawler, Sean
Chaplain, Mark
The cellular dispersion and therapeutic control of glioblastoma, the most aggressive type of primary brain cancer, depends critically on the migration patterns after surgery and intracellular responses of the individual cancer cells in response to external biochemical and biomechanical cues in the microenvironment. Recent studies have shown that a particular microRNA, miR451, regulates downstream molecules including AMPK and mTOR to determine the balance between rapid proliferation and invasion in response to metabolic stress in the harsh tumor microenvironment. Surgical removal of main tumor is inevitably followed by recurrence of the tumor due to inaccessibility of dispersed tumor cells in normal brain tissue. In order to address this multiscale nature of glioblastoma proliferation and invasion and its response to conventional treatment, we propose a hybrid model of glioblastoma that analyses spatiotemporal dynamics at the cellular level, linking individual tumor cells with the macroscopic behaviour of cell organization and the microenvironment, and with the intracellular dynamics of miR451AMPKmTOR signaling within a tumour cell. The model identifies a key mechanism underlying the molecular switches between proliferative phase and migratory phase in response to metabolic stress and biophysical interaction between cells in response to fluctuating glucose levels in the presence of blood vessels (BVs). The model predicts that cell migration, therefore efficacy of the treatment, not only depends on oxygen and glucose availability but also on the relative balance between random motility and strength of chemoattractants. Effective control of growing cells near BV sites in addition to relocalization of invisible migratory cells back to the resection site was suggested as a way of eradicating these migratory cells.

Sunspot rotation. I : A consequence of flux emergence
http://hdl.handle.net/10023/7497
Context. Solar eruptions and high flare activity often accompany the rapid rotation of sunspots. The study of sunspot rotation and the mechanisms driving this motion are therefore key to our understanding of how the solar atmosphere attains the conditions necessary for large energy release. Aims. We aim to demonstrate and investigate the rotation of sunspots in a 3D numerical experiment of the emergence of a magnetic flux tube as it rises through the solar interior and emerges into the atmosphere. Furthermore, we seek to show that the subphotospheric twist stored in the interior is injected into the solar atmosphere by means of a definitive rotation of the sunspots. Methods. A numerical experiment is performed to solve the 3D resistive magnetohydrodynamic (MHD) equations using a LagrangianRemap code. We track the emergence of a toroidal flux tube as it rises through the solar interior and emerges into the atmosphere investigating various quantities related to both the magnetic field and plasma. Results. Through detailed analysis of the numerical experiment, we find clear evidence that the photospheric footprints or sunspots of the flux tube undergo a rotation. Significant vertical vortical motions are found to develop within the two polarity sources after the field emerges. These rotational motions are found to leave the interior portion of the field untwisted and twist up the atmospheric portion of the field. This is shown by our analysis of the relative magnetic helicity as a significant portion of the interior helicity is transported to the atmosphere. In addition, there is a substantial transport of magnetic energy to the atmosphere. Rotation angles are also calculated by tracing selected fieldlines; the fieldlines threading through the sunspot are found to rotate through angles of up to 353º over the course of the experiment. We explain the rotation by an unbalanced torque produced by the magnetic tension force, rather than an apparent effect.
ZS acknowledges the financial support of the Carnegie Trust for Scotland and CMM the support of the Royal Society of Edinburgh. This work used the DIRAC 1, UKMHD Consortium machine at the University of St Andrews and the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National Einfrastructure capital grant ST/K00042X/1, STFC capital grant ST/H008519/1, and STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National EInfrastructure.
20151012T00:00:00Z
Sturrock, Zoe
Hood, Alan William
Archontis, Vasilis
McNeill, Craig
Context. Solar eruptions and high flare activity often accompany the rapid rotation of sunspots. The study of sunspot rotation and the mechanisms driving this motion are therefore key to our understanding of how the solar atmosphere attains the conditions necessary for large energy release. Aims. We aim to demonstrate and investigate the rotation of sunspots in a 3D numerical experiment of the emergence of a magnetic flux tube as it rises through the solar interior and emerges into the atmosphere. Furthermore, we seek to show that the subphotospheric twist stored in the interior is injected into the solar atmosphere by means of a definitive rotation of the sunspots. Methods. A numerical experiment is performed to solve the 3D resistive magnetohydrodynamic (MHD) equations using a LagrangianRemap code. We track the emergence of a toroidal flux tube as it rises through the solar interior and emerges into the atmosphere investigating various quantities related to both the magnetic field and plasma. Results. Through detailed analysis of the numerical experiment, we find clear evidence that the photospheric footprints or sunspots of the flux tube undergo a rotation. Significant vertical vortical motions are found to develop within the two polarity sources after the field emerges. These rotational motions are found to leave the interior portion of the field untwisted and twist up the atmospheric portion of the field. This is shown by our analysis of the relative magnetic helicity as a significant portion of the interior helicity is transported to the atmosphere. In addition, there is a substantial transport of magnetic energy to the atmosphere. Rotation angles are also calculated by tracing selected fieldlines; the fieldlines threading through the sunspot are found to rotate through angles of up to 353º over the course of the experiment. We explain the rotation by an unbalanced torque produced by the magnetic tension force, rather than an apparent effect.

Evolution of field line helicity during magnetic reconnection
http://hdl.handle.net/10023/7485
We investigate the evolution of field line helicity for magnetic fields that connect two boundaries without null points, with emphasis on localized finiteB magnetic reconnection. Total ( relative) magnetic helicity is already recognized as an important topological constraint on magnetohydrodynamic processes. Field line helicity offers further advantages because it preserves all topological information and can distinguish between different magnetic fields with the same total helicity. Magnetic reconnection changes field connectivity and field line helicity reflects these changes; the goal of this paper is to characterize that evolution. We start by deriving the evolution equation for field line helicity and examining its terms, also obtaining a simplified form for cases where dynamics are localized within the domain. The main result, which we support using kinematic examples, is that during localized reconnection in a complex magnetic field, the evolution of field line helicity is dominated by a worklike term that is evaluated at the field line endpoints, namely, the scalar product of the generalized field line velocity and the vector potential. Furthermore, the flux integral of this term over certain areas is very small compared to the integral of the unsigned quantity, which indicates that changes of field line helicity happen in a wellorganized pairwise manner. It follows that reconnection is very efficient at redistributing helicity in complex magnetic fields despite having little effect on the total helicity.
This work was supported by the Science and Technology Facilities Council (UK) through consortium Grant Nos. ST/K000993/1 and ST/K001043 to the University of Dundee and Durham University.
20150301T00:00:00Z
Russell, A. J. B.
Yeates, A. R.
Hornig, G.
WilmotSmith, A. L.
We investigate the evolution of field line helicity for magnetic fields that connect two boundaries without null points, with emphasis on localized finiteB magnetic reconnection. Total ( relative) magnetic helicity is already recognized as an important topological constraint on magnetohydrodynamic processes. Field line helicity offers further advantages because it preserves all topological information and can distinguish between different magnetic fields with the same total helicity. Magnetic reconnection changes field connectivity and field line helicity reflects these changes; the goal of this paper is to characterize that evolution. We start by deriving the evolution equation for field line helicity and examining its terms, also obtaining a simplified form for cases where dynamics are localized within the domain. The main result, which we support using kinematic examples, is that during localized reconnection in a complex magnetic field, the evolution of field line helicity is dominated by a worklike term that is evaluated at the field line endpoints, namely, the scalar product of the generalized field line velocity and the vector potential. Furthermore, the flux integral of this term over certain areas is very small compared to the integral of the unsigned quantity, which indicates that changes of field line helicity happen in a wellorganized pairwise manner. It follows that reconnection is very efficient at redistributing helicity in complex magnetic fields despite having little effect on the total helicity.

On the theory of translationally invariant magnetohydrodynamic equilibria with anisotropic pressure and magnetic shear
http://hdl.handle.net/10023/7484
We present an improved formalism for translationally invariant magnetohydrodynamic equilibria with anisotropic pressure and currents with a field aligned component. The derivation of a GradShafranov type equation is given along with a constraint which links the shear field to the parallel pressure. The difficulties of the formalism are discussed and various methods of circumventing these difficulties are given. A simple example is then used to highlight the methods and difficulties involved.
Funding: STFC Doctoral Training Grant ST/K502327/1 (Jonathan Hodgson) and STFC Consolidated Grant ST/K000950/1 (Thomas Neukirch)
20150902T00:00:00Z
Hodgson, Jonathan David Brockie
Neukirch, Thomas
We present an improved formalism for translationally invariant magnetohydrodynamic equilibria with anisotropic pressure and currents with a field aligned component. The derivation of a GradShafranov type equation is given along with a constraint which links the shear field to the parallel pressure. The difficulties of the formalism are discussed and various methods of circumventing these difficulties are given. A simple example is then used to highlight the methods and difficulties involved.

Coronal heating in multiple magnetic threads
http://hdl.handle.net/10023/7259
Context. Heating the solar corona to several million degrees requires the conversion of magnetic energy into thermal energy. In this paper, we investigate whether an unstable magnetic thread within a coronal loop can destabilise a neighbouring magnetic thread. Aims. By running a series of simulations, we aim to understand under what conditions the destabilisation of a single magnetic thread can also trigger a release of energy in a nearby thread. Methods. The 3D magnetohydrodynamics code, Lare3d, is used to simulate the temporal evolution of coronal magnetic fields during a kink instability and the subsequent relaxation process. We assume that a coronal magnetic loop consists of nonpotential magnetic threads that are initially in an equilibrium state. Results. The nonlinear kink instability in one magnetic thread forms a helical current sheet and initiates magnetic reconnection. The current sheet fragments, and magnetic energy is released throughout that thread. We find that, under certain conditions, this event can destabilise a nearby thread, which is a necessary requirement for starting an avalanche of energy release in magnetic threads. Conclusions. It is possible to initiate an energy release in a nearby, nonpotential magnetic thread, because the energy released from one unstable magnetic thread can trigger energy release in nearby threads, provided that the nearby structures are close to marginal stability.
We acknowledge the financial support of STFC through the Consolidated grant to the University of St Andrews.
20150801T00:00:00Z
Tam, Kuan Vai
Hood, Alan William
Browning, Philippa
Cargill, Peter
Context. Heating the solar corona to several million degrees requires the conversion of magnetic energy into thermal energy. In this paper, we investigate whether an unstable magnetic thread within a coronal loop can destabilise a neighbouring magnetic thread. Aims. By running a series of simulations, we aim to understand under what conditions the destabilisation of a single magnetic thread can also trigger a release of energy in a nearby thread. Methods. The 3D magnetohydrodynamics code, Lare3d, is used to simulate the temporal evolution of coronal magnetic fields during a kink instability and the subsequent relaxation process. We assume that a coronal magnetic loop consists of nonpotential magnetic threads that are initially in an equilibrium state. Results. The nonlinear kink instability in one magnetic thread forms a helical current sheet and initiates magnetic reconnection. The current sheet fragments, and magnetic energy is released throughout that thread. We find that, under certain conditions, this event can destabilise a nearby thread, which is a necessary requirement for starting an avalanche of energy release in magnetic threads. Conclusions. It is possible to initiate an energy release in a nearby, nonpotential magnetic thread, because the energy released from one unstable magnetic thread can trigger energy release in nearby threads, provided that the nearby structures are close to marginal stability.

Are tornadolike magnetic structures able to support solar prominence plasma?
http://hdl.handle.net/10023/7202
Recent highresolution and highcadence observations have surprisingly suggested that prominence barbs exhibit apparent rotating motions suggestive of a tornadolike structure. Additional evidence has been provided by Doppler measurements. The observations reveal opposite velocities for both hot and cool plasma on the two sides of a prominence barb. This motion is persistent for several hours and has been interpreted in terms of rotational motion of prominence feet. Several authors suggest that such barb motions are rotating helical structures around a vertical axis similar to tornadoes on Earth. One of the difficulties of such a proposal is how to support cool prominence plasma in almostvertical structures against gravity. In this work we model analytically a tornadolike structure and try to determine possible mechanisms to support the prominence plasma. We have found that the Lorentz force can indeed support the barb plasma provided the magnetic structure is sufficiently twisted and/or significant poloidal flows are present.
M. Luna and F. MorenoInsertis acknowledge support by the Spanish Ministry of Economy and Competitiveness through projects AYA201124808 and AYA201455078P. M.L. is also grateful to ERC2011StG 277829SPIA. E.R.P. is grateful to the UK STFC and the Leverhulme Trust for financial support.
20150720T00:00:00Z
Luna, M.
MorenoInsertis, F.
Priest, E.
Recent highresolution and highcadence observations have surprisingly suggested that prominence barbs exhibit apparent rotating motions suggestive of a tornadolike structure. Additional evidence has been provided by Doppler measurements. The observations reveal opposite velocities for both hot and cool plasma on the two sides of a prominence barb. This motion is persistent for several hours and has been interpreted in terms of rotational motion of prominence feet. Several authors suggest that such barb motions are rotating helical structures around a vertical axis similar to tornadoes on Earth. One of the difficulties of such a proposal is how to support cool prominence plasma in almostvertical structures against gravity. In this work we model analytically a tornadolike structure and try to determine possible mechanisms to support the prominence plasma. We have found that the Lorentz force can indeed support the barb plasma provided the magnetic structure is sufficiently twisted and/or significant poloidal flows are present.

Effect of Prandtl's ration on balance in geophysical turbulence
http://hdl.handle.net/10023/7201
The fluid dynamics of the atmosphere and oceans are to a large extent controlled by the slow evolution of a scalar field called ‘potential vorticity’, with relatively fast motions such as inertiagravity waves playing only a minor role. This state of affairs is commonly referred to as ‘balance’. Potential vorticity (PV) is a special scalar field which is materially conserved in the absence of diabatic effects and dissipation, effects which are generally weak in the atmosphere and oceans. Moreover, in a balanced flow, PV induces the entire fluid motion and its thermodynamic structure (Hoskins et al. 1985). While exact balance is generally not achievable, it is now well established that balance holds to a high degree of accuracy in rapidly rotating and strongly stratified flows. Such flows are characterised by both a small Rossby number, Ro ≡ ζmax/f, and a small Froude number, Fr ≡ .hmax/N, where ζ and .h are the relative vertical and horizontal vorticity components, while f and N are the Coriolis and buoyancy frequencies. In fact, balance can even be a good approximation when Fr < ∼ Ro ∼ O(1). In this study, we examine how balance depends specifically on Prandtl’s ratio, f/N, in unforced freelyevolving turbulence. We examine a wide variety of turbulent flows, at a mature and complex stage of their evolution, making use of the fully nonhydrostatic equations under the Boussinesq and incompressible approximations. We perform numerical simulations at exceptionally high resolution in order to carefully assess the degree to which balance holds, and to determine when it breaks down. For this purpose, it proves most useful to employ an invariant, PVbased Rossby number ε, together with f/N. For a given ε, our key finding is that — for at least tens of characteristic vortex rotation periods — the flow is insensitive to f/N for all values for which the flow remains statically stable (typically f/N < ∼1). Only the vertical velocity varies in proportion to f/N, in line with quasigeostrophic scaling for which Fr2 ≪ Ro ≪ 1. We also find that as ε increases toward unity, the maximum f/N attainable decreases toward 0. No statically stable flows occur for ε > ∼ 1. For all stable flows, balance is found to hold to a remarkably high degree: as measured by an energy norm, imbalance never exceeds more than a few percent of the balance, even in flows where Ro > 1. The vertical velocity w remains a tiny fraction of the horizontal velocity uh, even when w is dominantly balanced. Finally, typical vertical to horizontal scale ratios H/L remain close to f/N, as found previously in quasigeostrophic turbulence for which Fr ∼ Ro ≪ 1.
Support for this research has come from the UK Engineering and Physical Sciences Research Council (grant no. EP/H001794/1).
20150721T00:00:00Z
Dritschel, David Gerard
McKiver, William Joseph
The fluid dynamics of the atmosphere and oceans are to a large extent controlled by the slow evolution of a scalar field called ‘potential vorticity’, with relatively fast motions such as inertiagravity waves playing only a minor role. This state of affairs is commonly referred to as ‘balance’. Potential vorticity (PV) is a special scalar field which is materially conserved in the absence of diabatic effects and dissipation, effects which are generally weak in the atmosphere and oceans. Moreover, in a balanced flow, PV induces the entire fluid motion and its thermodynamic structure (Hoskins et al. 1985). While exact balance is generally not achievable, it is now well established that balance holds to a high degree of accuracy in rapidly rotating and strongly stratified flows. Such flows are characterised by both a small Rossby number, Ro ≡ ζmax/f, and a small Froude number, Fr ≡ .hmax/N, where ζ and .h are the relative vertical and horizontal vorticity components, while f and N are the Coriolis and buoyancy frequencies. In fact, balance can even be a good approximation when Fr < ∼ Ro ∼ O(1). In this study, we examine how balance depends specifically on Prandtl’s ratio, f/N, in unforced freelyevolving turbulence. We examine a wide variety of turbulent flows, at a mature and complex stage of their evolution, making use of the fully nonhydrostatic equations under the Boussinesq and incompressible approximations. We perform numerical simulations at exceptionally high resolution in order to carefully assess the degree to which balance holds, and to determine when it breaks down. For this purpose, it proves most useful to employ an invariant, PVbased Rossby number ε, together with f/N. For a given ε, our key finding is that — for at least tens of characteristic vortex rotation periods — the flow is insensitive to f/N for all values for which the flow remains statically stable (typically f/N < ∼1). Only the vertical velocity varies in proportion to f/N, in line with quasigeostrophic scaling for which Fr2 ≪ Ro ≪ 1. We also find that as ε increases toward unity, the maximum f/N attainable decreases toward 0. No statically stable flows occur for ε > ∼ 1. For all stable flows, balance is found to hold to a remarkably high degree: as measured by an energy norm, imbalance never exceeds more than a few percent of the balance, even in flows where Ro > 1. The vertical velocity w remains a tiny fraction of the horizontal velocity uh, even when w is dominantly balanced. Finally, typical vertical to horizontal scale ratios H/L remain close to f/N, as found previously in quasigeostrophic turbulence for which Fr ∼ Ro ≪ 1.

On the parallel and perpendicular propagating motions visible in polar plumes : an incubator for (fast) solar wind acceleration?
http://hdl.handle.net/10023/7190
We combine observations of the Coronal Multichannel Polarimeter and the Atmospheric Imaging Assembly on board the Solar Dynamics Observatory to study the characteristic properties of (propagating) Alfvenic motions and quasiperiodic intensity disturbances in polar plumes. This unique combination of instruments highlights the physical richness of the processes taking place at the base of the (fast) solar wind. The (parallel) intensity perturbations with intensity enhancements around 1% have an apparent speed of 120 km s(1) (in both the 171 and 193 angstrom passbands) and a periodicity of 15 minutes, while the (perpendicular) Alfvenic wave motions have a velocity amplitude of 0.5 km s(1), a phase speed of 830 km s(1), and a shorter period of 5 minutes on the same structures. These observations illustrate a scenario where the excited Alfvenic motions are propagating along an inhomogeneously loaded magnetic field structure such that the combination could be a potential progenitor of the magnetohydrodynamic turbulence required to accelerate the fast solar wind.
J.L. was a student visitor at HAO. J.L. acknowledges the financial support for his visit to HAO from the Chinese Scholarship Council (CSC). The authors acknowledge support from NSFC 41131065, 41121003, 973 Key Project 2011CB811403, and CAS Key Research Program KZZDEW014. We also acknowledge support from NASA contracts NNX08BA99G, NNX11AN98G, NNM12AB40P, NNG09FA40C (IRIS), and NNM07AA01C (Hinode). The research leading to these results has also received funding from the European Commission Seventh Framework Programme (FP7/ 20072013) under the grant agreement SOLSPANET (project No. 269299, www.solspanet.eu/solspanet) Date of Acceptance: 25/05/2015
20150620T00:00:00Z
Liu, Jiajia
McIntosh, Scott W.
De Moortel, Ineke
Wang, Yuming
We combine observations of the Coronal Multichannel Polarimeter and the Atmospheric Imaging Assembly on board the Solar Dynamics Observatory to study the characteristic properties of (propagating) Alfvenic motions and quasiperiodic intensity disturbances in polar plumes. This unique combination of instruments highlights the physical richness of the processes taking place at the base of the (fast) solar wind. The (parallel) intensity perturbations with intensity enhancements around 1% have an apparent speed of 120 km s(1) (in both the 171 and 193 angstrom passbands) and a periodicity of 15 minutes, while the (perpendicular) Alfvenic wave motions have a velocity amplitude of 0.5 km s(1), a phase speed of 830 km s(1), and a shorter period of 5 minutes on the same structures. These observations illustrate a scenario where the excited Alfvenic motions are propagating along an inhomogeneously loaded magnetic field structure such that the combination could be a potential progenitor of the magnetohydrodynamic turbulence required to accelerate the fast solar wind.

Uncertainties in polarimetric 3D reconstructions of coronal mass ejections
http://hdl.handle.net/10023/7178
Aims. The aim of this work is to quantify the uncertainties in the threedimensional (3D) reconstruction of the location of coronal mass ejections (CMEs) obtained with the socalled polarization ratio technique. The method takes advantage of the different distributions along the line of sight of total (tB) and polarized (pB) brightnesses emitted by Thomson scattering to estimate the average location of the emitting plasma. This is particularly important to correctly identify of CME propagation angles and unprojected velocities, thus allowing better capabilities for space weather forecastings. Methods. To this end, we assumed two simple electron density distributions along the line of sight (a constant density and Gaussian density profiles) for a plasma blob and synthesized the expected tB and pB for different distances z of the blob from the plane of the sky and different projected altitudes.. Reconstructed locations of the blob along the line of sight were thus compared with the real ones, allowing a precise determination of uncertainties in the method. Results. Results show that, independently of the analytical density profile, when the blob is centered at a small distance from the plane of the sky (i. e. for limb CMEs) the distance from the plane of the sky starts to be significantly overestimated. Polarization ratio technique provides the lineofsight position of the center of mass of what we call folded density distribution, given by reflecting and summing in front of the plane of the sky the fraction of density profile located behind that plane. On the other hand, when the blob is far from the plane of the sky, but with very small projected altitudes (i. e. for halo CMEs, rho< 1.4 Rcircle dot), the inferred distance from that plane is significantly underestimated. Better determination of the real blob position along the line of sight is given for intermediate locations, and in particular when the blob is centered at an angle of 20 degrees from the plane of the sky. Conclusions. These result have important consequences not only for future 3D reconstruction of CMEs with polarization ratio technique, but also for the design of future coronagraphs aimed at providing a continuous monitoring of haloCMEs for space weather prediction purposes.
P.P. acknowledges STFC for financial support. Date of Acceptance: 21/01/2015
20150406T00:00:00Z
Bemporad, A.
Pagano, P.
Aims. The aim of this work is to quantify the uncertainties in the threedimensional (3D) reconstruction of the location of coronal mass ejections (CMEs) obtained with the socalled polarization ratio technique. The method takes advantage of the different distributions along the line of sight of total (tB) and polarized (pB) brightnesses emitted by Thomson scattering to estimate the average location of the emitting plasma. This is particularly important to correctly identify of CME propagation angles and unprojected velocities, thus allowing better capabilities for space weather forecastings. Methods. To this end, we assumed two simple electron density distributions along the line of sight (a constant density and Gaussian density profiles) for a plasma blob and synthesized the expected tB and pB for different distances z of the blob from the plane of the sky and different projected altitudes.. Reconstructed locations of the blob along the line of sight were thus compared with the real ones, allowing a precise determination of uncertainties in the method. Results. Results show that, independently of the analytical density profile, when the blob is centered at a small distance from the plane of the sky (i. e. for limb CMEs) the distance from the plane of the sky starts to be significantly overestimated. Polarization ratio technique provides the lineofsight position of the center of mass of what we call folded density distribution, given by reflecting and summing in front of the plane of the sky the fraction of density profile located behind that plane. On the other hand, when the blob is far from the plane of the sky, but with very small projected altitudes (i. e. for halo CMEs, rho< 1.4 Rcircle dot), the inferred distance from that plane is significantly underestimated. Better determination of the real blob position along the line of sight is given for intermediate locations, and in particular when the blob is centered at an angle of 20 degrees from the plane of the sky. Conclusions. These result have important consequences not only for future 3D reconstruction of CMEs with polarization ratio technique, but also for the design of future coronagraphs aimed at providing a continuous monitoring of haloCMEs for space weather prediction purposes.

NonLTE modelling of prominence fine structures using hydrogen Lymanline profiles
http://hdl.handle.net/10023/7177
Aims. We perform a detailed statistical analysis of the spectral Lymanline observations of the quiescent prominence observed on May 18, 2005. Methods. We used a profiletoprofile comparison of the synthetic Lyman spectra obtained by 2D singlethread prominence finestructure model as a starting point for a full statistical analysis of the observed Lyman spectra. We employed 2D multithread finestructure models with random positions and lineofsight velocities of each thread to obtain a statistically significant set of synthetic Lymanline profiles. We used for the first time multithread models composed of nonidentical threads and viewed at lineofsight angles different from perpendicular to the magnetic field. Results. We investigated the plasma properties of the prominence observed with the SoHO/SUMER spectrograph on May 18, 2005 by comparing the histograms of three statistical parameters characterizing the properties of the synthetic and observed line profiles. In this way, the integrated intensity, Lyman decrement ratio, and the ratio of intensity at the central reversal to the average intensity of peaks provided insight into the column mass and the central temperature of the prominence fine structures.
Date of Acceptance: 10/03/2015
20150508T00:00:00Z
Schwartz, P.
Gunar, S.
Curdt, W.
Aims. We perform a detailed statistical analysis of the spectral Lymanline observations of the quiescent prominence observed on May 18, 2005. Methods. We used a profiletoprofile comparison of the synthetic Lyman spectra obtained by 2D singlethread prominence finestructure model as a starting point for a full statistical analysis of the observed Lyman spectra. We employed 2D multithread finestructure models with random positions and lineofsight velocities of each thread to obtain a statistically significant set of synthetic Lymanline profiles. We used for the first time multithread models composed of nonidentical threads and viewed at lineofsight angles different from perpendicular to the magnetic field. Results. We investigated the plasma properties of the prominence observed with the SoHO/SUMER spectrograph on May 18, 2005 by comparing the histograms of three statistical parameters characterizing the properties of the synthetic and observed line profiles. In this way, the integrated intensity, Lyman decrement ratio, and the ratio of intensity at the central reversal to the average intensity of peaks provided insight into the column mass and the central temperature of the prominence fine structures.

Extreme ultraviolet imaging of threedimensional magnetic reconnection in a solar eruption
http://hdl.handle.net/10023/7025
Magnetic reconnection, a change of magnetic field connectivity, is a fundamental physical process in which magnetic energy is released explosively, and it is responsible for various eruptive phenomena in the universe. However, this process is difficult to observe directly. Here, the magnetic topology associated with a solar reconnection event is studied in three dimensions using the combined perspectives of two spacecraft. The sequence of extreme ultraviolet images clearly shows that two groups of oppositely directed and noncoplanar magnetic loops gradually approach each other, forming a separator or quasiseparator and then reconnecting. The plasma near the reconnection site is subsequently heated from ∼ 1 to ≥ 5MK. Shortly afterwards, warm flare loops (∼3MK) appear underneath the hot plasma. Other observational signatures of reconnection, including plasma inflows and downflows, are unambiguously revealed and quantitatively measured. These observations provide direct evidence of magnetic reconnection in a threedimensional configuration and reveal its origin.
X.C., J.Q.S., M.D.D., Y.G., P.F.C. and C.F. are supported by NSFC through grants 11303016, 11373023, 11203014 and 11025314, and by NKBRSF through grants 2011CB811402 and 2014CB744203. C.E.P. and S.J.E. are supported by the UK STFC. J.Z. is supported by US NSF AGS1249270 and AGS1156120.
20150626T00:00:00Z
Sun, J.Q.
Cheng, X.
Ding, M.D.
Guo, Y.
Priest, E.R.
Parnell, C.E.
Edwards, S.J.
Zhang, J.
Chen, P.F.
Fang, C.
Magnetic reconnection, a change of magnetic field connectivity, is a fundamental physical process in which magnetic energy is released explosively, and it is responsible for various eruptive phenomena in the universe. However, this process is difficult to observe directly. Here, the magnetic topology associated with a solar reconnection event is studied in three dimensions using the combined perspectives of two spacecraft. The sequence of extreme ultraviolet images clearly shows that two groups of oppositely directed and noncoplanar magnetic loops gradually approach each other, forming a separator or quasiseparator and then reconnecting. The plasma near the reconnection site is subsequently heated from ∼ 1 to ≥ 5MK. Shortly afterwards, warm flare loops (∼3MK) appear underneath the hot plasma. Other observational signatures of reconnection, including plasma inflows and downflows, are unambiguously revealed and quantitatively measured. These observations provide direct evidence of magnetic reconnection in a threedimensional configuration and reveal its origin.

Ergodicity and spectral cascades in point vortex flows on the sphere
http://hdl.handle.net/10023/7024
We present results for the equilibrium statistics and dynamic evolution of moderately large [n=O(102103)] numbers of interacting point vortices on the sphere under the constraint of zero mean angular momentum. For systems with equal numbers of positive and negative identical circulations, the density of rescaled energies, p(E), converges rapidly with n to a function with a single maximum with maximum entropy. Ensembleaveraged wavenumber spectra of the nonsingular velocity field induced by the vortices exhibit the expected k1 behavior at small scales for all energies. Spectra at the largest scales vary continuously with the inverse temperature of the system. For positive temperatures, spectra peak at finite intermediate wave numbers; for negative temperatures, spectra decrease everywhere. Comparisons of time and ensemble averages, over a large range of energies, strongly support ergodicity in the dynamics even for highly atypical initial vortex configurations. Crucially, rapid relaxation of spectra toward the microcanonical average implies that the direction of any spectral cascade process depends only on the relative difference between the initial spectrum and the ensemble mean spectrum at that energy, not on the energy, or temperature, of the system.
A.C.P. was supported under DOD (MURI) Grant No. N000141110087 ONR. The computations were supported by the CUNY HPCC under NSF Grants No. CNS0855217 and No. CNS0958379.
20150629T00:00:00Z
Dritschel, D.G.
Lucia, M.
Poje, A.C.
We present results for the equilibrium statistics and dynamic evolution of moderately large [n=O(102103)] numbers of interacting point vortices on the sphere under the constraint of zero mean angular momentum. For systems with equal numbers of positive and negative identical circulations, the density of rescaled energies, p(E), converges rapidly with n to a function with a single maximum with maximum entropy. Ensembleaveraged wavenumber spectra of the nonsingular velocity field induced by the vortices exhibit the expected k1 behavior at small scales for all energies. Spectra at the largest scales vary continuously with the inverse temperature of the system. For positive temperatures, spectra peak at finite intermediate wave numbers; for negative temperatures, spectra decrease everywhere. Comparisons of time and ensemble averages, over a large range of energies, strongly support ergodicity in the dynamics even for highly atypical initial vortex configurations. Crucially, rapid relaxation of spectra toward the microcanonical average implies that the direction of any spectral cascade process depends only on the relative difference between the initial spectrum and the ensemble mean spectrum at that energy, not on the energy, or temperature, of the system.

Fast approximate radiative transfer method for visualizing the fine structure of prominences in the hydrogen Hα line
http://hdl.handle.net/10023/7020
Aims. We present a novel approximate radiative transfer method developed to visualize 3D wholeprominence models with multiple fine structures using the hydrogen Hα spectral line. Methods. This method employs a fast lineofsight synthesis of the Hα line profiles through the whole 3D prominence volume and realistically reflects the basic properties of the Hα line formation in the cool and lowdensity prominence medium. The method can be applied both to prominences seen above the limb and filaments seen against the disk. Results. We provide recipes for the use of this method for visualizing the prominence or filament models that have multiple fine structures. We also perform tests of the method that demonstrate its accuracy under prominence conditions. Conclusions. We demonstrate that this fast approximate radiative transfer method provides realistic synthetic Hα intensities useful for a reliable visualization of prominences and filaments. Such synthetic highresolution images of modeled prominences/filaments can be used for a direct comparison with highresolution observations.
P.H. acknowledges the support from grant 209/12/0906 of the Grant Agency of the Czech Republic. S.G. acknowledges support from the European Commission via the Marie Curie Actions – IntraEuropean Fellowships Project No. 328138. P.H. and S.G. acknowledge support from project RVO:67985815 of the Astronomical Institute of the Czech Academy of Sciences and from the MPA Garching. U.A. thanks for the support from the Astronomical Institute of the Czech Academy of Sciences.
20150701T00:00:00Z
Heinzel, P.
Gunár, S.
Anzer, U.
Aims. We present a novel approximate radiative transfer method developed to visualize 3D wholeprominence models with multiple fine structures using the hydrogen Hα spectral line. Methods. This method employs a fast lineofsight synthesis of the Hα line profiles through the whole 3D prominence volume and realistically reflects the basic properties of the Hα line formation in the cool and lowdensity prominence medium. The method can be applied both to prominences seen above the limb and filaments seen against the disk. Results. We provide recipes for the use of this method for visualizing the prominence or filament models that have multiple fine structures. We also perform tests of the method that demonstrate its accuracy under prominence conditions. Conclusions. We demonstrate that this fast approximate radiative transfer method provides realistic synthetic Hα intensities useful for a reliable visualization of prominences and filaments. Such synthetic highresolution images of modeled prominences/filaments can be used for a direct comparison with highresolution observations.

The use of the Poynting vector in interpreting ULF waves in magnetospheric waveguides
http://hdl.handle.net/10023/6976
We numerically model ultralow frequency (ULF) waves in the magnetosphere assuming an ideal, lowbeta inhomogeneous plasma waveguide. The waveguide is based on the hydromagnetic box model. We develop a novel boundary condition that drives the magnetospheric boundary by pressure perturbations, in order to simulate solar wind dynamic pressure uctuations disturbing the magnetopause. The model is applied to observations from Cluster and THEMIS. Our model is able to reproduce similar wave signatures to those in the data, such as a unidirectional azimuthal Poynting vector, by interpreting the observations in terms of fast waveguide modes. Despite the simplicity of the model, we can shed light on the nature of these modes and the location of the energy source relative to the spacecraft. This is achieved by demonstrating that important information, such as phase shifts between components of the electric and magnetic fields and the balance of radial to azimuthal propagation of energy, may be extracted from a careful analysis of the components of the Poynting vector.
T.E. would like to thank STFC for financial support for a doctoral training grant. Date of Acceptance: 04/12/2014
20150101T00:00:00Z
Elsden, Tom
Wright, Andrew Nicholas
We numerically model ultralow frequency (ULF) waves in the magnetosphere assuming an ideal, lowbeta inhomogeneous plasma waveguide. The waveguide is based on the hydromagnetic box model. We develop a novel boundary condition that drives the magnetospheric boundary by pressure perturbations, in order to simulate solar wind dynamic pressure uctuations disturbing the magnetopause. The model is applied to observations from Cluster and THEMIS. Our model is able to reproduce similar wave signatures to those in the data, such as a unidirectional azimuthal Poynting vector, by interpreting the observations in terms of fast waveguide modes. Despite the simplicity of the model, we can shed light on the nature of these modes and the location of the energy source relative to the spacecraft. This is achieved by demonstrating that important information, such as phase shifts between components of the electric and magnetic fields and the balance of radial to azimuthal propagation of energy, may be extracted from a careful analysis of the components of the Poynting vector.

Ellipsoidal vortices in rotating stratified fluids : beyond the quasigeostrophic approximation
http://hdl.handle.net/10023/6889
We examine the basic properties and stability of isolated vortices having uniform potential vorticity (PV) in a nonhydrostatic rotating stratified fluid, under the Boussinesq approximation. For simplicity, we consider a uniform background rotation and a linear basicstate stratification for which both the Coriolis and buoyancy frequencies, f and N, are constant. Moreover, we take ƒ/N≪1, as typically observed in the Earth’s atmosphere and oceans. In the small Rossby number ‘quasigeostrophic’ (QG) limit, when the flow is weak compared to the background rotation, there exist exact solutions for steadily rotating ellipsoidal volumes of uniform PV in an unbounded flow (Zhmur & Shchepetkin, Izv. Akad. Nauk SSSR Atmos. Ocean. Phys., vol. 27, 1991, pp. 492–503; Meacham, Dyn. Atmos. Oceans, vol. 16, 1992, pp. 189–223). Furthermore, a wide range of these solutions are stable as long as the horizontal and vertical aspect ratios λ and μ do not depart greatly from unity (Dritschel et al.,J. Fluid Mech., vol. 536, 2005, pp. 401–421). In the present study, we examine the behaviour of ellipsoidal vortices at Rossby numbers up to near unity in magnitude. We find that there is a monotonic increase in stability as one varies the Rossby number from nearly −1 (anticyclone) to nearly +1 (cyclone). That is, QG vortices are more stable than anticyclones at finite negative Rossby number, and generally less stable than cyclones at finite positive Rossby number. Ageostrophic effects strengthen both the rotation and the stratification within a cyclone, enhancing its stability. The converse is true for an anticyclone. For all Rossby numbers, stability is reinforced by increasing λ towards unity or decreasing μ. An unstable vortex often restabilises by developing a nearcircular crosssection, typically resulting in a roughly ellipsoidal vortex, but occasionally a binary system is formed. Throughout the nonlinear evolution of a vortex, the emission of inertia–gravity waves (IGWs) is negligible across the entire parameter space investigated. Thus, vortices at small to moderate Rossby numbers, and any associated instabilities, are (ageostrophically) balanced. A manifestation of this balance is that, at finite Rossby number, an anticyclone rotates faster than a cyclone.
Support for this research has come from the UK Engineering and Physical Sciences Research Council (grant number EP/H001794/1).
20150101T00:00:00Z
Tsang, YueKin
Dritschel, David G.
We examine the basic properties and stability of isolated vortices having uniform potential vorticity (PV) in a nonhydrostatic rotating stratified fluid, under the Boussinesq approximation. For simplicity, we consider a uniform background rotation and a linear basicstate stratification for which both the Coriolis and buoyancy frequencies, f and N, are constant. Moreover, we take ƒ/N≪1, as typically observed in the Earth’s atmosphere and oceans. In the small Rossby number ‘quasigeostrophic’ (QG) limit, when the flow is weak compared to the background rotation, there exist exact solutions for steadily rotating ellipsoidal volumes of uniform PV in an unbounded flow (Zhmur & Shchepetkin, Izv. Akad. Nauk SSSR Atmos. Ocean. Phys., vol. 27, 1991, pp. 492–503; Meacham, Dyn. Atmos. Oceans, vol. 16, 1992, pp. 189–223). Furthermore, a wide range of these solutions are stable as long as the horizontal and vertical aspect ratios λ and μ do not depart greatly from unity (Dritschel et al.,J. Fluid Mech., vol. 536, 2005, pp. 401–421). In the present study, we examine the behaviour of ellipsoidal vortices at Rossby numbers up to near unity in magnitude. We find that there is a monotonic increase in stability as one varies the Rossby number from nearly −1 (anticyclone) to nearly +1 (cyclone). That is, QG vortices are more stable than anticyclones at finite negative Rossby number, and generally less stable than cyclones at finite positive Rossby number. Ageostrophic effects strengthen both the rotation and the stratification within a cyclone, enhancing its stability. The converse is true for an anticyclone. For all Rossby numbers, stability is reinforced by increasing λ towards unity or decreasing μ. An unstable vortex often restabilises by developing a nearcircular crosssection, typically resulting in a roughly ellipsoidal vortex, but occasionally a binary system is formed. Throughout the nonlinear evolution of a vortex, the emission of inertia–gravity waves (IGWs) is negligible across the entire parameter space investigated. Thus, vortices at small to moderate Rossby numbers, and any associated instabilities, are (ageostrophically) balanced. A manifestation of this balance is that, at finite Rossby number, an anticyclone rotates faster than a cyclone.

Excitation and damping of broadband kink waves in the solar corona
http://hdl.handle.net/10023/6839
Context. Observations such as those by the Coronal MultiChannel Polarimeter (CoMP) have revealed that broadband kink oscillations are ubiquitous in the solar corona. Aims. We consider footpointdriven kink waves propagating in a low β coronal plasma with a cylindrical density structure. We investigate the excitation and damping of propagating kink waves by a broadband driver, including the effects of different spatial profiles for the driver. Methods. We employ a general spatial damping profile in which the initial stage of the damping envelope is approximated by a Gaussian profile and the asymptotic state by an exponential one. We develop a method of accounting for the presence of these different damping regimes and test it using data from numerical simulations. Results. Strongly damped oscillations in low density coronal loops are more accurately described by a Gaussian spatial damping profile than an exponential profile. The consequences for coronal seismology are investigated and applied to observational data for the ubiquitous broadband waves observed by CoMP. Current data cannot distinguish between the exponential and Gaussian profiles because of the levels of noise. We demonstrate the importance of the spatial profile of the driver on the resulting damping profile. Furthermore, we show that a smallscale turbulent driver is inefficient at exciting propagating kink waves
D.J.P. acknowledges financial support from STFC. I.D.M. acknowledges support of a Royal Society University Research Fellowship. The research leading to these results has also received funding from the European Commission Seventh Framework Programme (FP7/ 2007−2013) under the grant agreement SOLSPANET (project No. 269299, www.solspanet.eu/solspanet).
20150601T00:00:00Z
Pascoe, David James
Wright, Andrew Nicholas
De Moortel, Ineke
Hood, Alan William
Context. Observations such as those by the Coronal MultiChannel Polarimeter (CoMP) have revealed that broadband kink oscillations are ubiquitous in the solar corona. Aims. We consider footpointdriven kink waves propagating in a low β coronal plasma with a cylindrical density structure. We investigate the excitation and damping of propagating kink waves by a broadband driver, including the effects of different spatial profiles for the driver. Methods. We employ a general spatial damping profile in which the initial stage of the damping envelope is approximated by a Gaussian profile and the asymptotic state by an exponential one. We develop a method of accounting for the presence of these different damping regimes and test it using data from numerical simulations. Results. Strongly damped oscillations in low density coronal loops are more accurately described by a Gaussian spatial damping profile than an exponential profile. The consequences for coronal seismology are investigated and applied to observational data for the ubiquitous broadband waves observed by CoMP. Current data cannot distinguish between the exponential and Gaussian profiles because of the levels of noise. We demonstrate the importance of the spatial profile of the driver on the resulting damping profile. Furthermore, we show that a smallscale turbulent driver is inefficient at exciting propagating kink waves

CALIFA, the Calar Alto Legacy Integral Field Area survey : III. Second public data release
http://hdl.handle.net/10023/6664
This paper describes the Second Public Data Release (DR2) of the Calar Alto Legacy Integral Field Area (CALIFA) survey. The data for 200 objects are made public, including the 100 galaxies of the First Public Data Release (DR1). Data were obtained with the integralfield spectrograph PMAS/PPak mounted on the 3.5 m telescope at the Calar Alto observatory. Two different spectral setups are available for each galaxy, (i) a lowresolution V500 setup covering the wavelength range 37457500 Å with a spectral resolution of 6.0 Å (FWHM); and (ii) a mediumresolution V1200 setup covering the wavelength range 36504840 Å with a spectral resolution of 2.3 Å (FWHM). The sample covers a redshift range between 0.005 and 0.03, with a wide range of properties in the colormagnitude diagram, stellar mass, ionization conditions, and morphological types. All the cubes in the data release were reduced with the latest pipeline, which includes improvedspectrophotometric calibration, spatial registration, and spatial resolution. The spectrophotometric calibration is better than 6% and the median spatial resolution is 2.4. In total, the second data release contains over 1.5 million spectra.
J.M.A. acknowledges support from the European Research Council Starting Grant (SEDmorph; P.I. V. Wild). V.W. acknowledges support from the European Research Council Starting Grant (SEDMorph P.I. V. Wild) and European Career Reintegration Grant (PhizEv P.I. V. Wild).
20150401T00:00:00Z
GarcíaBenito, R.
Zibetti, S.
Sánchez, S. F.
Husemann, B.
de Amorim, A. L.
CastilloMorales, A.
Cid Fernandes, R.
Ellis, S. C.
FalcónBarroso, J.
Galbany, L.
Gil de Paz, A.
González Delgado, R. M.
Lacerda, E. A. D.
LópezFernandez, R.
de LorenzoCáceres, A.
Lyubenova, M.
Marino, R. A.
Mast, D.
Mendoza, M. A.
Pérez, E.
Vale Asari, N.
Aguerri, J. A. L.
Ascasibar, Y.
Bekerait*error*ė, S.
BlandHawthorn, J.
BarreraBallesteros, J. K.
Bomans, D. J.
CanoDíaz, M.
CatalánTorrecilla, C.
Cortijo, C.
DelgadoInglada, G.
Demleitner, M.
Dettmar, R.J.
Díaz, A. I.
Florido, E.
Gallazzi, A.
GarcíaLorenzo, B.
Gomes, J. M.
Holmes, L.
IglesiasPáramo, J.
Jahnke, K.
Kalinova, V.
Kehrig, C.
Kennicutt, R. C.
LópezSánchez, Á. R.
Márquez, I.
Masegosa, J.
Meidt, S. E.
MendezAbreu, J.
Mollá, M.
MonrealIbero, A.
Morisset, C.
del Olmo, A.
Papaderos, P.
Pérez, I.
Quirrenbach, A.
RosalesOrtega, F. F.
Roth, M. M.
RuizLara, T.
SánchezBlázquez, P.
SánchezMenguiano, L.
Singh, R.
Spekkens, K.
Stanishev, V.
TorresPapaqui, J. P.
van de Ven, G.
Vilchez, J. M.
Walcher, C. J.
Wild, V.
Wisotzki, L.
Ziegler, B.
Alves, J.
Barrado, D.
Quintana, J. M.
Aceituno, J.
This paper describes the Second Public Data Release (DR2) of the Calar Alto Legacy Integral Field Area (CALIFA) survey. The data for 200 objects are made public, including the 100 galaxies of the First Public Data Release (DR1). Data were obtained with the integralfield spectrograph PMAS/PPak mounted on the 3.5 m telescope at the Calar Alto observatory. Two different spectral setups are available for each galaxy, (i) a lowresolution V500 setup covering the wavelength range 37457500 Å with a spectral resolution of 6.0 Å (FWHM); and (ii) a mediumresolution V1200 setup covering the wavelength range 36504840 Å with a spectral resolution of 2.3 Å (FWHM). The sample covers a redshift range between 0.005 and 0.03, with a wide range of properties in the colormagnitude diagram, stellar mass, ionization conditions, and morphological types. All the cubes in the data release were reduced with the latest pipeline, which includes improvedspectrophotometric calibration, spatial registration, and spatial resolution. The spectrophotometric calibration is better than 6% and the median spatial resolution is 2.4. In total, the second data release contains over 1.5 million spectra.

Numerical simulations of a flux rope ejection
http://hdl.handle.net/10023/6650
Coronal mass ejections (CMEs) are the most violent phenomena observed on the Sun. One of the most successful models to explain CMEs is the flux rope ejection model, where a magnetic flux rope is expelled from the solar corona after a long phase along which the flux rope stays in equilibrium while magnetic energy is being accumulated. However, still many questions are outstanding on the detailed mechanism of the ejection and observations continuously provide new data to interpret and put in the context. Currently, extreme ultraviolet (EUV) images from the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamic Observatory (SDO) are providing new insights into the early phase of CME evolution. In particular, observations show the ejection of magnetic flux ropes from the solar corona and how they evolve into CMEs. However, these observations are difficult to interpret in terms of basic physical mechanisms and quantities, thus, we need to compare equivalent quantities to test and improve our models. In our work, we intend to bridge the gap between models and observations with our model of flux rope ejection where we consistently describe the full life span of a flux rope from its formation to ejection. This is done by coupling the global nonlinear forcefree model (GNLFFF) built to describe the slow low β formation phase, with a full MHD simulation run with the software MPIAMRVAC, suitable to describe the fast MHD evolution of the flux rope ejection that happens in a heterogeneous β regime. We also explore the parameter space to identify the conditions upon which the ejection is favoured (gravity stratification and magnetic field intensity) and we produce synthesised AIA observations (171 Å and 211 Å). To carry this out, we run 3D MHD simulation in spherical coordinates where we include the role of thermal conduction and radiative losses, both of which are important for determining the temperature distribution of the solar corona during a CME. Our model of flux rope ejection is successful in realistically describing the entire life span of a flux rope and we also set some conditions for the backgroud solar corona to favour the escape of the flux rope, so that it turns into a CME. Furthermore, our MHD simulation reproduces many of the features found in the AIA observations.
20150301T00:00:00Z
Pagano, P.
Mackay, D.H.
Poedts, S.
Coronal mass ejections (CMEs) are the most violent phenomena observed on the Sun. One of the most successful models to explain CMEs is the flux rope ejection model, where a magnetic flux rope is expelled from the solar corona after a long phase along which the flux rope stays in equilibrium while magnetic energy is being accumulated. However, still many questions are outstanding on the detailed mechanism of the ejection and observations continuously provide new data to interpret and put in the context. Currently, extreme ultraviolet (EUV) images from the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamic Observatory (SDO) are providing new insights into the early phase of CME evolution. In particular, observations show the ejection of magnetic flux ropes from the solar corona and how they evolve into CMEs. However, these observations are difficult to interpret in terms of basic physical mechanisms and quantities, thus, we need to compare equivalent quantities to test and improve our models. In our work, we intend to bridge the gap between models and observations with our model of flux rope ejection where we consistently describe the full life span of a flux rope from its formation to ejection. This is done by coupling the global nonlinear forcefree model (GNLFFF) built to describe the slow low β formation phase, with a full MHD simulation run with the software MPIAMRVAC, suitable to describe the fast MHD evolution of the flux rope ejection that happens in a heterogeneous β regime. We also explore the parameter space to identify the conditions upon which the ejection is favoured (gravity stratification and magnetic field intensity) and we produce synthesised AIA observations (171 Å and 211 Å). To carry this out, we run 3D MHD simulation in spherical coordinates where we include the role of thermal conduction and radiative losses, both of which are important for determining the temperature distribution of the solar corona during a CME. Our model of flux rope ejection is successful in realistically describing the entire life span of a flux rope and we also set some conditions for the backgroud solar corona to favour the escape of the flux rope, so that it turns into a CME. Furthermore, our MHD simulation reproduces many of the features found in the AIA observations.

3D wholeprominence fine structure modeling
http://hdl.handle.net/10023/6541
We present the first 3D wholeprominence fine structure model. The model combines a 3D magnetic field configuration of an entire prominence obtained from nonlinear forcefree field simulations, with a detailed description of the prominence plasma. The plasma is located in magnetic dips in hydrostatic equilibrium and is distributed along multiple fine structures within the 3D magnetic model. Through the use of a novel radiative transfer visualization technique for the Halpha line such plasmaloaded magnetic field model produces synthetic images of the modeled prominence comparable with highresolution observations. This allows us for the first time to use a single technique to consistently study, in both emission on the limb and absorption against the solar disk, the fine structures of prominences/filaments produced by a magnetic field model.
20150420T00:00:00Z
Gunar, Stanislav
Mackay, Duncan Hendry
We present the first 3D wholeprominence fine structure model. The model combines a 3D magnetic field configuration of an entire prominence obtained from nonlinear forcefree field simulations, with a detailed description of the prominence plasma. The plasma is located in magnetic dips in hydrostatic equilibrium and is distributed along multiple fine structures within the 3D magnetic model. Through the use of a novel radiative transfer visualization technique for the Halpha line such plasmaloaded magnetic field model produces synthetic images of the modeled prominence comparable with highresolution observations. This allows us for the first time to use a single technique to consistently study, in both emission on the limb and absorption against the solar disk, the fine structures of prominences/filaments produced by a magnetic field model.

Experiments on the structure and stability of mode2 internal solitarylike waves propagating on an offset pycnocline
http://hdl.handle.net/10023/6519
The structure and stability of mode2 internal solitarylike waves is investigated experimentally. A rankordered train of mode2 internal solitary waves is generated using a lock release configuration. The pycnocline is centred either on the middepth of the water column (the 0% offset case) or it is offset in the positive vertical direction by a fraction of 5%, 10% or 20% of the total fluid depth. It is found that offsetting the pycnocline has little effect on the basic wave properties (e.g wave speed, wave amplitude and wave length) but it does significantly affect wave stability. Instability takes the form of small KHlike billows in the rear of the wave and small scale overturning in the core of the wave. In the 0% offset case, instability occurs on both the upper and lower interfaces of the pycnocline and is similar in extent and vigour over the two interfaces. As the offset percentage is increased, however, instability is more pronounced on the lower interface with little or no evidence of instability being observed on the upper interface. In the 20% offset case a mode1 tail is associated with the wave and the wave characteristics resemble qualitatively the recent field observations of Shroyer et al [E. L. Shroyer, J. N. Moum and J. D. Nash, J. Geophys. Res. 115, C07001 (2010)].
20150101T00:00:00Z
Carr, Magda
Davies, Peter
Hoebers, Ruud
The structure and stability of mode2 internal solitarylike waves is investigated experimentally. A rankordered train of mode2 internal solitary waves is generated using a lock release configuration. The pycnocline is centred either on the middepth of the water column (the 0% offset case) or it is offset in the positive vertical direction by a fraction of 5%, 10% or 20% of the total fluid depth. It is found that offsetting the pycnocline has little effect on the basic wave properties (e.g wave speed, wave amplitude and wave length) but it does significantly affect wave stability. Instability takes the form of small KHlike billows in the rear of the wave and small scale overturning in the core of the wave. In the 0% offset case, instability occurs on both the upper and lower interfaces of the pycnocline and is similar in extent and vigour over the two interfaces. As the offset percentage is increased, however, instability is more pronounced on the lower interface with little or no evidence of instability being observed on the upper interface. In the 20% offset case a mode1 tail is associated with the wave and the wave characteristics resemble qualitatively the recent field observations of Shroyer et al [E. L. Shroyer, J. N. Moum and J. D. Nash, J. Geophys. Res. 115, C07001 (2010)].

The motion of point vortices on closed surfaces
http://hdl.handle.net/10023/6297
We develop a mathematical framework for the dynamics of a set of point vortices on a class of differentiable surfaces conformal to the unit sphere. When the sum of the vortex circulations is nonzero, a compensating uniform vorticity field is required to satisfy the Gauss condition (that the integral of the LaplaceBeltrami operator must vanish). On variable Gaussian curvature surfaces, this results in selfinduced vortex motion, a feature entirely absent on the plane, the sphere or the hyperboloid.We derive explicit equations of motion for vortices on surfaces of revolution and compute their solutions for a variety of surfaces. We also apply these equations to study the linear stability of a ring of vortices on any surface of revolution. On an ellipsoid of revolution, as few as 2 vortices can be unstable on oblate surfaces or sufficiently prolate ones. This extends known results for the plane, where 7 vortices are marginally unstable [1,2], and the sphere, where 4 vortices may be unstable if sufficiently close to the equator [3].
Date of Acceptance: 29/01/2015
20150225T00:00:00Z
Dritschel, David Gerard
Boatto, S
We develop a mathematical framework for the dynamics of a set of point vortices on a class of differentiable surfaces conformal to the unit sphere. When the sum of the vortex circulations is nonzero, a compensating uniform vorticity field is required to satisfy the Gauss condition (that the integral of the LaplaceBeltrami operator must vanish). On variable Gaussian curvature surfaces, this results in selfinduced vortex motion, a feature entirely absent on the plane, the sphere or the hyperboloid.We derive explicit equations of motion for vortices on surfaces of revolution and compute their solutions for a variety of surfaces. We also apply these equations to study the linear stability of a ring of vortices on any surface of revolution. On an ellipsoid of revolution, as few as 2 vortices can be unstable on oblate surfaces or sufficiently prolate ones. This extends known results for the plane, where 7 vortices are marginally unstable [1,2], and the sphere, where 4 vortices may be unstable if sufficiently close to the equator [3].

Understanding the Mg II and Hα spectra in a highly dynamical solar prominence
http://hdl.handle.net/10023/6190
Mg ii h and k and Hα spectra in a dynamical prominence have been obtained along the slit of the Interface Region Imaging Spectrograph (IRIS) and with the Meudon Multichannel Subtractive Double Pass spectrograph on 2013 September 24, respectively. Single Mg ii line profiles are not much reversed, while at some positions along the IRIS slit the profiles show several discrete peaks that are Dopplershifted. The intensity of these peaks is generally decreasing with their increasing Doppler shift. We interpret this unusual behavior as being due to the Doppler dimming effect. We discuss the possibility to interpret the unreversed single profiles by using a twodimensional (2D) model of the entire prominence body with specific radiative boundary conditions. We have performed new 2D isothermal–isobaric modeling of both Hα and Mg ii lines and show the ability of such models to account for the line profile variations as observed. However, the Mg ii linecenter intensities require the model with a temperature increase toward the prominence boundary. We show that even simple onedimensional (1D) models with a prominencetocorona transition region (PCTR) fit the observed Mg ii and Hα lines quite well, while the isothermal–isobaric models (1D or 2D) are inconsistent with simultaneous observations in the Mg ii h and k and Hα lines, meaning that the Hα line provides a strong additional constraint on the modeling. IRIS farUV detection of the C ii lines in this prominence seems to provide a direct constraint on the PCTR part of the model.
Date of Acceptance: 08/01/2015
20150210T00:00:00Z
Heinzel, P.
Schmieder, B.
Mein, N.
Gunar, S.
Mg ii h and k and Hα spectra in a dynamical prominence have been obtained along the slit of the Interface Region Imaging Spectrograph (IRIS) and with the Meudon Multichannel Subtractive Double Pass spectrograph on 2013 September 24, respectively. Single Mg ii line profiles are not much reversed, while at some positions along the IRIS slit the profiles show several discrete peaks that are Dopplershifted. The intensity of these peaks is generally decreasing with their increasing Doppler shift. We interpret this unusual behavior as being due to the Doppler dimming effect. We discuss the possibility to interpret the unreversed single profiles by using a twodimensional (2D) model of the entire prominence body with specific radiative boundary conditions. We have performed new 2D isothermal–isobaric modeling of both Hα and Mg ii lines and show the ability of such models to account for the line profile variations as observed. However, the Mg ii linecenter intensities require the model with a temperature increase toward the prominence boundary. We show that even simple onedimensional (1D) models with a prominencetocorona transition region (PCTR) fit the observed Mg ii and Hα lines quite well, while the isothermal–isobaric models (1D or 2D) are inconsistent with simultaneous observations in the Mg ii h and k and Hα lines, meaning that the Hα line provides a strong additional constraint on the modeling. IRIS farUV detection of the C ii lines in this prominence seems to provide a direct constraint on the PCTR part of the model.

Simplyconnected vortexpatch shallowwater quasiequilibria
http://hdl.handle.net/10023/6179
We examine the form, properties, stability and evolution of simplyconnected vortexpatch relative quasiequilibria in the singlelayer ƒplane shallowwater model of geophysical fluid dynamics. We examine the effects of the size, shape and strength of vortices in this system, represented by three distinct parameters completely describing the families of the quasiequilibria. Namely, these are the ratio γ=L/LD between the horizontal size of the vortices and the Rossby deformation length; the aspect ratio λ between the minor to major axes of the vortex; and a potential vorticity (PV)based Rossby number Ro=q′/ƒ, the ratio of the PV anomaly q′ within the vortex to the Coriolis frequency ƒ. By defining an appropriate steadiness parameter, we find that the quasiequilibria remain steady for long times, enabling us to determine the boundary of stability λc=λc(γ, Ro), for 0.25≤γ≤6 and Ro≤1. By calling two states which share γ,Ro and λ ‘equivalent’, we find a clear asymmetry in the stability of cyclonic (Ro>0) and anticyclonic (Ro<0) equilibria, with cyclones being able to sustain greater deformations than anticyclones before experiencing an instability. We find that ageostrophic motions stabilise cyclones and destabilise anticyclones. Both types of vortices undergo the same main types of unstable evolution, albeit in different ranges of the parameter space, (a) vacillations for largeγ, largeRo states, (b) filamentation for smallγ states and (c) vortex splitting, asymmetric for intermediateγ and symmetric for largeγ states.
This work is supported by a UK Natural Environment Research Council studentship
20140305T00:00:00Z
Plotka, Hanna
Dritschel, David Gerard
We examine the form, properties, stability and evolution of simplyconnected vortexpatch relative quasiequilibria in the singlelayer ƒplane shallowwater model of geophysical fluid dynamics. We examine the effects of the size, shape and strength of vortices in this system, represented by three distinct parameters completely describing the families of the quasiequilibria. Namely, these are the ratio γ=L/LD between the horizontal size of the vortices and the Rossby deformation length; the aspect ratio λ between the minor to major axes of the vortex; and a potential vorticity (PV)based Rossby number Ro=q′/ƒ, the ratio of the PV anomaly q′ within the vortex to the Coriolis frequency ƒ. By defining an appropriate steadiness parameter, we find that the quasiequilibria remain steady for long times, enabling us to determine the boundary of stability λc=λc(γ, Ro), for 0.25≤γ≤6 and Ro≤1. By calling two states which share γ,Ro and λ ‘equivalent’, we find a clear asymmetry in the stability of cyclonic (Ro>0) and anticyclonic (Ro<0) equilibria, with cyclones being able to sustain greater deformations than anticyclones before experiencing an instability. We find that ageostrophic motions stabilise cyclones and destabilise anticyclones. Both types of vortices undergo the same main types of unstable evolution, albeit in different ranges of the parameter space, (a) vacillations for largeγ, largeRo states, (b) filamentation for smallγ states and (c) vortex splitting, asymmetric for intermediateγ and symmetric for largeγ states.

The formation and stability of Petschek reconnection
http://hdl.handle.net/10023/6100
A combined analytical and numerical study of magnetic reconnection in twodimensional resistive magnetohydrodynamics is carried out by using different explicit spatial variations of the resistivity. A special emphasis on the existence of stable/unstable Petschek's solutions is taken, comparing with the recent analytical model given by Forbes et al. [Phys. Plasmas 20, 052902 (2013)]. Our results show good quantitative agreement between the analytical theory and the numerical solutions for a Petschektype solution to within an accuracy of about 10% or better. Our simulations also show that if the resistivity profile is relatively flat near the Xpoint, one of two possible asymmetric solutions will occur. Which solution occurs depends on small random perturbations of the initial conditions. The existence of two possible asymmetric solutions, in a system which is otherwise symmetric, constitutes an example of spontaneous symmetry breaking.
E. R. Priest is grateful to the Leverhulme Trust. T. G. Forbes received support from NASA grant NNX10AC04G to the University of New Hampshire. H. Baty acknowledges support by French National Research Agency (ANR) through Grant ANR13JS05000301 (Project EMPERE).
20141101T00:00:00Z
Baty, H.
Forbes, T.G.
Priest, E.R.
A combined analytical and numerical study of magnetic reconnection in twodimensional resistive magnetohydrodynamics is carried out by using different explicit spatial variations of the resistivity. A special emphasis on the existence of stable/unstable Petschek's solutions is taken, comparing with the recent analytical model given by Forbes et al. [Phys. Plasmas 20, 052902 (2013)]. Our results show good quantitative agreement between the analytical theory and the numerical solutions for a Petschektype solution to within an accuracy of about 10% or better. Our simulations also show that if the resistivity profile is relatively flat near the Xpoint, one of two possible asymmetric solutions will occur. Which solution occurs depends on small random perturbations of the initial conditions. The existence of two possible asymmetric solutions, in a system which is otherwise symmetric, constitutes an example of spontaneous symmetry breaking.

Helical blowout jets in the sun : untwisting and propagation of waves
http://hdl.handle.net/10023/6097
We report on a numerical experiment of the recurrent onset of helical "blowout" jets in an emerging flux region. We find that these jets are running with velocities of ∼100250 km s1 and they transfer a vast amount of heavy plasma into the outer solar atmosphere. During their emission, they undergo an untwisting motion as a result of reconnection between the twisted emerging and the nontwisted preexisting magnetic field in the solar atmosphere. For the first time in the context of blowout jets, we provide direct evidence that their untwisting motion is associated with the propagation of torsional Alfvén waves in the corona.
20150101T00:00:00Z
Lee, E.J.
Archontis, V.
Hood, A.W.
We report on a numerical experiment of the recurrent onset of helical "blowout" jets in an emerging flux region. We find that these jets are running with velocities of ∼100250 km s1 and they transfer a vast amount of heavy plasma into the outer solar atmosphere. During their emission, they undergo an untwisting motion as a result of reconnection between the twisted emerging and the nontwisted preexisting magnetic field in the solar atmosphere. For the first time in the context of blowout jets, we provide direct evidence that their untwisting motion is associated with the propagation of torsional Alfvén waves in the corona.

Statistical evidence for the existence of Alfvénic turbulence in solar coronal loops
http://hdl.handle.net/10023/5987
Recent observations have demonstrated that waves capable of carrying large amounts of energy are ubiquitous throughout the solar corona. However, the question of how this wave energy is dissipated (on which timescales and length scales) and released into the plasma remains largely unanswered. Both analytic and numerical models have previously shown that Alfvénic turbulence may play a key role not only in the generation of the fast solar wind, but in the heating of coronal loops. In an effort to bridge the gap between theory and observations, we expand on a recent study by analyzing 37 clearly isolated coronal loops using data from the Coronal Multichannel Polarimeter instrument.We observe Alfvénic perturbations with phase speeds which range from 250 to 750 km s1 and periods from 140 to 270 s for the chosen loops. While excesses of highfrequency wave power are observed near the apex of some loops (tentatively supporting the onset of Alfvénic turbulence), we show that this excess depends on loop length and the wavelength of the observed oscillations. In deriving a proportional relationship between the loop length/wavelength ratio and the enhanced wave power at the loop apex, and from the analysis of the line widths associated with these loops, our findings are supportive of the existence of Alfvénic turbulence in coronal loops.
The authors acknowledge support from NASA contracts NNX08BA99G, NNX11AN98G, NNM12AB40P, NNG09FA40C (IRIS), and NNM07AA01C (Hinode). The research leading to these results has also received funding from the European Commission Seventh Framework Programme (FP7/ 20072013) under the grant agreement SOLSPANET (project No. 269299, www.solspanet.eu/solspanet).
20141210T00:00:00Z
Liu, J.
Mcintosh, S.W.
De Moortel, I.
Threlfall, J.
Bethge, C.
Recent observations have demonstrated that waves capable of carrying large amounts of energy are ubiquitous throughout the solar corona. However, the question of how this wave energy is dissipated (on which timescales and length scales) and released into the plasma remains largely unanswered. Both analytic and numerical models have previously shown that Alfvénic turbulence may play a key role not only in the generation of the fast solar wind, but in the heating of coronal loops. In an effort to bridge the gap between theory and observations, we expand on a recent study by analyzing 37 clearly isolated coronal loops using data from the Coronal Multichannel Polarimeter instrument.We observe Alfvénic perturbations with phase speeds which range from 250 to 750 km s1 and periods from 140 to 270 s for the chosen loops. While excesses of highfrequency wave power are observed near the apex of some loops (tentatively supporting the onset of Alfvénic turbulence), we show that this excess depends on loop length and the wavelength of the observed oscillations. In deriving a proportional relationship between the loop length/wavelength ratio and the enhanced wave power at the loop apex, and from the analysis of the line widths associated with these loops, our findings are supportive of the existence of Alfvénic turbulence in coronal loops.

Validation of the magnetic energy vs. helicity scaling in solar magnetic structures
http://hdl.handle.net/10023/5872
Aims. We assess the validity of the free magnetic energy – relative magnetic helicity diagram for solar magnetic structures. Methods. We used two different methods of calculating the free magnetic energy and the relative magnetic helicity budgets: a classical, volumecalculation nonlinear forcefree (NLFF) method applied to finite coronal magnetic structures and a surfacecalculation NLFF derivation that relies on a single photospheric or chromospheric vector magnetogram. Both methods were applied to two different data sets, namely synthetic activeregion cases obtained by threedimensional magnetohydrodynamic (MHD) simulations and observed activeregion cases, which include both eruptive and noneruptive magnetic structures. Results. The derived energyhelicity diagram shows a consistent monotonic scaling between relative helicity and free energy with a scaling index 0.84 ± 0.05 for both data sets and calculation methods. It also confirms the segregation between noneruptive and eruptive active regions and the existence of thresholds in both free energy and relative helicity for active regions to enter eruptive territory. Conclusions. We consider the previously reported energyhelicity diagram of solar magnetic structures as adequately validated and envision a significant role of the uncovered scaling in future studies of solar magnetism.
V.A. acknowledges support by the Royal Society. This work was supported from the EU’s Seventh Framework Program under grant agreement n° PIRG07GA2010268245. It has been also cofinanced by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) – Research Funding Program: Thales.
20141001T00:00:00Z
Tziotziou, K.
Moraitis, K.
Georgoulis, M.K.
Archontis, V.
Aims. We assess the validity of the free magnetic energy – relative magnetic helicity diagram for solar magnetic structures. Methods. We used two different methods of calculating the free magnetic energy and the relative magnetic helicity budgets: a classical, volumecalculation nonlinear forcefree (NLFF) method applied to finite coronal magnetic structures and a surfacecalculation NLFF derivation that relies on a single photospheric or chromospheric vector magnetogram. Both methods were applied to two different data sets, namely synthetic activeregion cases obtained by threedimensional magnetohydrodynamic (MHD) simulations and observed activeregion cases, which include both eruptive and noneruptive magnetic structures. Results. The derived energyhelicity diagram shows a consistent monotonic scaling between relative helicity and free energy with a scaling index 0.84 ± 0.05 for both data sets and calculation methods. It also confirms the segregation between noneruptive and eruptive active regions and the existence of thresholds in both free energy and relative helicity for active regions to enter eruptive territory. Conclusions. We consider the previously reported energyhelicity diagram of solar magnetic structures as adequately validated and envision a significant role of the uncovered scaling in future studies of solar magnetism.

Simulating AIA observations of a flux rope ejection
http://hdl.handle.net/10023/5821
Context. Coronal mass ejections (CMEs) are the most violent phenomena observed on the Sun. Currently, extreme ultraviolet (EUV) images from the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamic Observatory (SDO) are providing new insights into the early phase of CME evolution. In particular, observations now show the ejection of magnetic flux ropes from the solar corona and how they evolve into CMEs. While this is the case, these observations are difficult to interpret in terms of basic physical mechanisms and quantities. To fully understand CMEs we need to compare equivalent quantities derived from both observations and theoretical models. This will aid in bridging the gap between observations and models. Aims: To this end, we aim to produce synthesised AIA observations from simulations of a flux rope ejection. To carry this out we include the role of thermal conduction and radiative losses, both of which are important for determining the temperature distribution of the solar corona during a CME. Methods: We perform a simulation where a flux rope is ejected from the solar corona. From the density and temperature of the plasma in the simulation we synthesise AIA observations. The emission is then integrated along the line of sight using the instrumental response function of AIA. Results: We sythesise observations of AIA in the channels at 304 Å, 171 Å, 335 Å, and 94 Å. The synthesised observations show a number of features similar to actual observations and in particular reproduce the general development of CMEs in the low corona as observed by AIA. In particular we reproduce an erupting and expanding arcade in the 304 Å and 171 Å channels with a high density core. Conclusions: The ejection of a flux rope reproduces many of the features found in the AIA observations. This work is therefore a step forward in bridging the gap between observations and models, and can lead to more direct interpretations of EUV observations in terms of flux rope ejections. We plan to improve the model in future studies in order to perform a more quantitative comparison. Movies associated with Figs. 3, 9, and 10 are available in electronic form at http://www.aanda.org
D.H.M. would like to thank STFC, the Leverhulme Trust and the European Commission’s Seventh Framework Programme (FP7/20072013) for their financial support. P.P. would like to thank the European Commission’s Seventh Framework Programme (FP7/20072013) under grant agreement SWIFF (project 263340, http://www.swiff.eu) and STFC for financial support. These results were obtained in the framework of the projects GOA/2009009 (KU Leuven), G.0729.11 (FWOVlaanderen) and C 90347 (ESA Prodex 9). The research leading to these results has also received funding from the European Commission’s Seventh Framework Programme (FP7/20072013) under the grant agreements SOLSPANET (project No. 269299, http:// www.solspanet.eu), SPACECAST (project No. 262468, fp7spacecast.eu), eHeroes (project n 284461, http://www.eheroes.eu). The computational work for this paper was carried out on the joint STFC and SFC (SRIF) funded cluster at the University of St Andrews (Scotland, UK).
20140801T00:00:00Z
Pagano, Paolo
Mackay, Duncan Hendry
Poedts, Stephan
Context. Coronal mass ejections (CMEs) are the most violent phenomena observed on the Sun. Currently, extreme ultraviolet (EUV) images from the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamic Observatory (SDO) are providing new insights into the early phase of CME evolution. In particular, observations now show the ejection of magnetic flux ropes from the solar corona and how they evolve into CMEs. While this is the case, these observations are difficult to interpret in terms of basic physical mechanisms and quantities. To fully understand CMEs we need to compare equivalent quantities derived from both observations and theoretical models. This will aid in bridging the gap between observations and models. Aims: To this end, we aim to produce synthesised AIA observations from simulations of a flux rope ejection. To carry this out we include the role of thermal conduction and radiative losses, both of which are important for determining the temperature distribution of the solar corona during a CME. Methods: We perform a simulation where a flux rope is ejected from the solar corona. From the density and temperature of the plasma in the simulation we synthesise AIA observations. The emission is then integrated along the line of sight using the instrumental response function of AIA. Results: We sythesise observations of AIA in the channels at 304 Å, 171 Å, 335 Å, and 94 Å. The synthesised observations show a number of features similar to actual observations and in particular reproduce the general development of CMEs in the low corona as observed by AIA. In particular we reproduce an erupting and expanding arcade in the 304 Å and 171 Å channels with a high density core. Conclusions: The ejection of a flux rope reproduces many of the features found in the AIA observations. This work is therefore a step forward in bridging the gap between observations and models, and can lead to more direct interpretations of EUV observations in terms of flux rope ejections. We plan to improve the model in future studies in order to perform a more quantitative comparison. Movies associated with Figs. 3, 9, and 10 are available in electronic form at http://www.aanda.org

Stellar differential rotation and coronal timescales
http://hdl.handle.net/10023/5820
We investigate the timescales of evolution of stellar coronae in response to surface differential rotation and diffusion. To quantify this, we study both the formation time and lifetime of a magnetic flux rope in a decaying bipolar active region. We apply a magnetic flux transport model to prescribe the evolution of the stellar photospheric field, and use this to drive the evolution of the coronal magnetic field via a magnetofrictional technique. Increasing the differential rotation (i.e. decreasing the equatorpole lap time) decreases the flux rope formation time. We find that the formation time is dependent upon the lap time and the surface diffusion timescale through the relation tau_Form ∝ &surd;{tau_Laptau_Diff}. In contrast, the lifetimes of flux ropes are proportional to the lap time (tauLife∝tauLap). With this, flux ropes on stars with a differential rotation of more than eight times the solar value have a lifetime of less than 2 d. As a consequence, we propose that features such as solarlike quiescent prominences may not be easily observable on such stars, as the lifetimes of the flux ropes which host the cool plasma are very short. We conclude that such high differential rotation stars may have very dynamical coronae.
GPSG would like to thank the STFC for financial support. DHM would like to thank the STFC and the Leverhulme Trust for financial support.
20141001T00:00:00Z
Gibb, Gordon Peter Samuel
Jardine, Moira Mary
Mackay, Duncan Hendry
We investigate the timescales of evolution of stellar coronae in response to surface differential rotation and diffusion. To quantify this, we study both the formation time and lifetime of a magnetic flux rope in a decaying bipolar active region. We apply a magnetic flux transport model to prescribe the evolution of the stellar photospheric field, and use this to drive the evolution of the coronal magnetic field via a magnetofrictional technique. Increasing the differential rotation (i.e. decreasing the equatorpole lap time) decreases the flux rope formation time. We find that the formation time is dependent upon the lap time and the surface diffusion timescale through the relation tau_Form ∝ &surd;{tau_Laptau_Diff}. In contrast, the lifetimes of flux ropes are proportional to the lap time (tauLife∝tauLap). With this, flux ropes on stars with a differential rotation of more than eight times the solar value have a lifetime of less than 2 d. As a consequence, we propose that features such as solarlike quiescent prominences may not be easily observable on such stars, as the lifetimes of the flux ropes which host the cool plasma are very short. We conclude that such high differential rotation stars may have very dynamical coronae.

Backward wave cyclotronmaser emission in the auroral magnetosphere
http://hdl.handle.net/10023/5802
In this Letter, we present theory and particleincell simulations describing cyclotron radio emission from Earth's auroral region and similar phenomena in other astrophysical environments. In particular, we find that the radiation, generated by a downgoing electron horseshoe distribution is due to a backward wave cyclotronmaser emission process. The backward wave nature of the radiation contributes to upward refraction of the radiation that is also enhanced by a density inhomogeneity. We also show that the radiation is preferentially amplified along the auroral oval rather than transversely. The results are in agreement with recent Cluster observations.
This work was supported by EPSRC Grant No. EP/G04239X/1.
20141007T00:00:00Z
Speirs, D. C.
Bingham, R.
Cairns, R. A.
Vorgul, I.
Kellett, B. J.
Phelps, A. D. R.
Ronald, K.
In this Letter, we present theory and particleincell simulations describing cyclotron radio emission from Earth's auroral region and similar phenomena in other astrophysical environments. In particular, we find that the radiation, generated by a downgoing electron horseshoe distribution is due to a backward wave cyclotronmaser emission process. The backward wave nature of the radiation contributes to upward refraction of the radiation that is also enhanced by a density inhomogeneity. We also show that the radiation is preferentially amplified along the auroral oval rather than transversely. The results are in agreement with recent Cluster observations.

The nature of separator current layers in MHS equilibria I. Current parallel to the separator
http://hdl.handle.net/10023/5785
Separators, which are in many ways the threedimensional equivalent to twodimensional nulls, are important sites for magnetic reconnection. Magnetic reconnection occurs in strong current layers which have very short length scales. The aim of this work is to explore the nature of current layers around separators. A separator is a special field line which lies along the intersection of two separatrix surfaces and forms the boundary between four topologically distinct flux domains. In particular, here the current layer about a separator that joins two 3D nulls and lies along the intersection of their separatrix surfaces is investigated. A magnetic configuration containing a single separator embedded in a uniform plasma with a uniform electric current parallel to the separator is considered. This initial magnetic setup, which is not in equilibrium, relaxes in a nonresistive manner to form an equilibrium. The relaxation is achieved using the 3D MHD code, Lare3d, with resistivity set to zero. A series of experiments with varying initial current are run to investigate the characteristics of the resulting current layers present in the final (quasi) equilibrium states. In each experiment, the separator collapses and a current layer forms along it. The dimensions and strength of the current layer increase with initial current. It is found that separator current layers formed from current parallel to the separator are twisted. Also the collapse of the separator is a process that evolves like an infinitetime singularity where the length, width and peak current in the layer grow slowly whilst the depth of the current layer decreases.
JEHS would like to thank STFC for financial support during her Ph.D and CEP acknowledges support from the STFC consolidated grant.
20150101T00:00:00Z
Stevenson, Julie Elizabeth Helen
Parnell, Clare Elizabeth
Priest, Eric Ronald
Haynes, Andrew Lewis
Separators, which are in many ways the threedimensional equivalent to twodimensional nulls, are important sites for magnetic reconnection. Magnetic reconnection occurs in strong current layers which have very short length scales. The aim of this work is to explore the nature of current layers around separators. A separator is a special field line which lies along the intersection of two separatrix surfaces and forms the boundary between four topologically distinct flux domains. In particular, here the current layer about a separator that joins two 3D nulls and lies along the intersection of their separatrix surfaces is investigated. A magnetic configuration containing a single separator embedded in a uniform plasma with a uniform electric current parallel to the separator is considered. This initial magnetic setup, which is not in equilibrium, relaxes in a nonresistive manner to form an equilibrium. The relaxation is achieved using the 3D MHD code, Lare3d, with resistivity set to zero. A series of experiments with varying initial current are run to investigate the characteristics of the resulting current layers present in the final (quasi) equilibrium states. In each experiment, the separator collapses and a current layer forms along it. The dimensions and strength of the current layer increase with initial current. It is found that separator current layers formed from current parallel to the separator are twisted. Also the collapse of the separator is a process that evolves like an infinitetime singularity where the length, width and peak current in the layer grow slowly whilst the depth of the current layer decreases.

Particle acceleration at a reconnecting magnetic separator
http://hdl.handle.net/10023/5782
While the exact acceleration mechanism of energetic particles during solar flares is (as yet) unknown, magnetic reconnection plays a key role both in the release of stored magnetic energy of the solar corona and the magnetic restructuring during a flare. Recent work has shown that special field lines, called separators, are common sites of reconnection in 3D numerical experiments. To date, 3D separator reconnection sites have received little attention as particle accelerators. We investigate the effectiveness of separator reconnection as a particle acceleration mechanism for electrons and protons. We study the particle acceleration using a relativistic guidingcentre particle code in a timedependent kinematic model of magnetic reconnection at a separator. The effect upon particle behaviour of initial position, pitch angle and initial kinetic energy are examined in detail, both for specific (single) particle examples and for large distributions of initial conditions. The separator reconnection model contains several free parameters and we study the effect of changing these parameters upon particle acceleration, in particular in view of the final particle energy ranges which agree with observed energy spectra.
20150201T00:00:00Z
Threlfall, J.
Neukirch, T.
Parnell, Clare Elizabeth
Eradat Oskoui, S.
While the exact acceleration mechanism of energetic particles during solar flares is (as yet) unknown, magnetic reconnection plays a key role both in the release of stored magnetic energy of the solar corona and the magnetic restructuring during a flare. Recent work has shown that special field lines, called separators, are common sites of reconnection in 3D numerical experiments. To date, 3D separator reconnection sites have received little attention as particle accelerators. We investigate the effectiveness of separator reconnection as a particle acceleration mechanism for electrons and protons. We study the particle acceleration using a relativistic guidingcentre particle code in a timedependent kinematic model of magnetic reconnection at a separator. The effect upon particle behaviour of initial position, pitch angle and initial kinetic energy are examined in detail, both for specific (single) particle examples and for large distributions of initial conditions. The separator reconnection model contains several free parameters and we study the effect of changing these parameters upon particle acceleration, in particular in view of the final particle energy ranges which agree with observed energy spectra.

Alfvén wave boundary condition for responsive magnetosphere ionosphere coupling
http://hdl.handle.net/10023/5660
The solution of electric fields and currents in a heightresolved ionosphere is traditionally solved as an elliptic equation with Dirichlet or Neumann boundary condition in which the magnetosphere is represented as an unresponsive (prescribed) voltage generator or current source. In this paper we derive an alternative boundary condition based upon Alfvén waves in which only the Alfvén wave from the magnetosphere that is incident upon the ionosphere (E) is prescribed. For a uniform magnetosphere the new boundary condition reduces to ∂φ/∂z=(∂2φ/ ∂x2+2∂Exi/∂x)/(μ0VAσ≥) and is evaluated at the magnetosphereionosphere interface. The resulting solution is interpreted as a responsive magnetosphere and establishes a key stage in the full selfconsistent and nonlinear coupling of the magnetosphere and ionosphere.
A.J.B.R. thanks STFC for present support through consolidated grant ST/K000993/1 and gratefully acknowledges a Royal Commission for the Exhibition of 1851 Research Fellowship that also assisted this work.
20140501T00:00:00Z
Wright, A.N.
Russell, A.J.B.
The solution of electric fields and currents in a heightresolved ionosphere is traditionally solved as an elliptic equation with Dirichlet or Neumann boundary condition in which the magnetosphere is represented as an unresponsive (prescribed) voltage generator or current source. In this paper we derive an alternative boundary condition based upon Alfvén waves in which only the Alfvén wave from the magnetosphere that is incident upon the ionosphere (E) is prescribed. For a uniform magnetosphere the new boundary condition reduces to ∂φ/∂z=(∂2φ/ ∂x2+2∂Exi/∂x)/(μ0VAσ≥) and is evaluated at the magnetosphereionosphere interface. The resulting solution is interpreted as a responsive magnetosphere and establishes a key stage in the full selfconsistent and nonlinear coupling of the magnetosphere and ionosphere.

Ultraviolet and extremeultraviolet emissions at the flare footpoints observed by atmosphere imaging assembly
http://hdl.handle.net/10023/5499
A solar flare is composed of impulsive energy release events by magnetic reconnection, which forms and heats flare loops. Recent studies have revealed a twophase evolution pattern of UV 1600 Å emission at the feet of these loops: a rapid pulse lasting for a few seconds to a few minutes, followed by a gradual decay on timescales of a few tens of minutes. Multiple band EUV observations by the Atmosphere Imaging Assembly further reveal very similar signatures. These two phases represent different but related signatures of an impulsive energy release in the corona. The rapid pulse is an immediate response of the lower atmosphere to an intense thermal conduction flux resulting from the sudden heating of the corona to high temperatures (we rule out energetic particles due to a lack of significant hard Xray emission). The gradual phase is associated with the cooling of hot plasma that has been evaporated into the corona. The observed footpoint emission is again powered by thermal conduction (and enthalpy), but now during a period when approximate steadystate conditions are established in the loop. UV and EUV light curves of individual pixels may therefore be separated into contributions from two distinct physical mechanisms to shed light on the nature of energy transport in a flare. We demonstrate this technique using coordinated, spatially resolved observations of UV and EUV emissions from the footpoints of a C3.2 thermal flare.
20130901T00:00:00Z
Qiu, J.
Sturrock, Z.
Longcope, D.W.
Klimchuk, J.A.
Liu, W.J.
A solar flare is composed of impulsive energy release events by magnetic reconnection, which forms and heats flare loops. Recent studies have revealed a twophase evolution pattern of UV 1600 Å emission at the feet of these loops: a rapid pulse lasting for a few seconds to a few minutes, followed by a gradual decay on timescales of a few tens of minutes. Multiple band EUV observations by the Atmosphere Imaging Assembly further reveal very similar signatures. These two phases represent different but related signatures of an impulsive energy release in the corona. The rapid pulse is an immediate response of the lower atmosphere to an intense thermal conduction flux resulting from the sudden heating of the corona to high temperatures (we rule out energetic particles due to a lack of significant hard Xray emission). The gradual phase is associated with the cooling of hot plasma that has been evaporated into the corona. The observed footpoint emission is again powered by thermal conduction (and enthalpy), but now during a period when approximate steadystate conditions are established in the loop. UV and EUV light curves of individual pixels may therefore be separated into contributions from two distinct physical mechanisms to shed light on the nature of energy transport in a flare. We demonstrate this technique using coordinated, spatially resolved observations of UV and EUV emissions from the footpoints of a C3.2 thermal flare.

Nonlinear forcefree magnetic dip models of quiescent prominence fine structures
http://hdl.handle.net/10023/5476
Aims. We use 3D nonlinear forcefree magnetic field modeling of prominence/filament magnetic fields to develop the first 2D models of individual prominence fine structures based on the 3D configuration of the magnetic field of the whole prominence. Methods. We use an iterative technique to fill the magnetic dips produced by the 3D modeling with realistic prominence plasma in hydrostatic equilibrium and with a temperature structure that contains the prominencecorona transition region. With this welldefined plasma structure the radiative transfer can be treated in detail in 2D and the resulting synthetic emission can be compared with prominence/filament observations. Results. Newly developed nonlinear forcefree magnetic dip models are able to produce synthetic hydrogen Lyman spectra in a qualitative agreement with a range of quiescent prominence observations. Moreover, the plasma structure of these models agrees with the gravity induced prominence fine structure models which have already been shown to produce synthetic spectra in good qualitative agreement with several observed prominences. Conclusions. We describe in detail the iterative technique which can be used to produce realistic plasma models of prominence fine structures located in prominence magnetic field configurations containing dips, obtained using any kind of magnetic field modeling.
S.G. and P.H. acknowledge the support from grant 209/12/0906 of the Grant Agency of the Czech Republic. P.H. acknowledges the support from grant P209/10/1680 of the Grant Agency of the Czech Republic. S.G. and P.H. acknowledge the support from the MPA Garching; U.A. thanks for support from the Ondřejov Observatory. S.G. acknowledges the support from St Andrews University. Work of S.G. and P.H. was supported by the project RVO: 67985815. DHM acknowledges financial support from the STFC and the Leverhulme Trust. In addition research leading to these results has received funding from the European Commission’s Seventh Framework Programme (FP7/20072013) under the grant agreement SWIFF (project N° 263340, http://www.swiff.eu).
20130301T00:00:00Z
Gunar, Stanislav
Mackay, Duncan Hendry
Anzer, U
Heinzel, Petr
Aims. We use 3D nonlinear forcefree magnetic field modeling of prominence/filament magnetic fields to develop the first 2D models of individual prominence fine structures based on the 3D configuration of the magnetic field of the whole prominence. Methods. We use an iterative technique to fill the magnetic dips produced by the 3D modeling with realistic prominence plasma in hydrostatic equilibrium and with a temperature structure that contains the prominencecorona transition region. With this welldefined plasma structure the radiative transfer can be treated in detail in 2D and the resulting synthetic emission can be compared with prominence/filament observations. Results. Newly developed nonlinear forcefree magnetic dip models are able to produce synthetic hydrogen Lyman spectra in a qualitative agreement with a range of quiescent prominence observations. Moreover, the plasma structure of these models agrees with the gravity induced prominence fine structure models which have already been shown to produce synthetic spectra in good qualitative agreement with several observed prominences. Conclusions. We describe in detail the iterative technique which can be used to produce realistic plasma models of prominence fine structures located in prominence magnetic field configurations containing dips, obtained using any kind of magnetic field modeling.

Effects of M dwarf magnetic fields on potentially habitable planets
http://hdl.handle.net/10023/5462
We investigate the effect of the magnetic fields of M dwarf (dM) stars on potentially habitable Earthlike planets. These fields can reduce the size of planetary magnetospheres to such an extent that a significant fraction of the planet’s atmosphere may be exposed to erosion by the stellar wind. We used a sample of 15 active dM stars, for which surface magneticfield maps were reconstructed, to determine the magnetic pressure at the planet orbit and hence the largest size of its magnetosphere, which would only be decreased by considering the stellar wind. Our method provides a fast means to assess which planets are most affected by the stellar magnetic field, which can be used as a first study to be followed by more sophisticated models. We show that hypothetical Earthlike planets with similar terrestrial magnetisation (~1 G) orbiting at the inner (outer) edge of the habitable zone of these stars would present magnetospheres that extend at most up to 6 (11.7) planetary radii. To be able to sustain an Earthsized magnetosphere, with the exception of only a few cases, the terrestrial planet would either (1) need to orbit significantly farther out than the traditional limits of the habitable zone; or else, (2) if it were orbiting within the habitable zone, it would require at least a magnetic field ranging from a few G to up to a few thousand G. By assuming a magnetospheric size that is more appropriate for the youngEarth (3.4 Gyr ago), the required planetary magnetic fields are one order of magnitude weaker. However, in this case, the polarcap area of the planet, which is unprotected from transport of particles to/from interplanetary space, is twice as large. At present, we do not know how small the smallest area of the planetary surface is that could be exposed and would still not affect the potential for formation and development of life in a planet. As the star becomes older and, therefore, its rotation rate and magnetic field reduce, the interplanetary magnetic pressure decreases and the magnetosphere of planets probably expands. Using an empirically derived rotationactivity/magnetism relation, we provide an analytical expression for estimating the shortest stellar rotation period for which an Earthanalogue in the habitable zone could sustain an Earthsized magnetosphere. We find that the required rotation rate of the early and middM stars (with periods ≳37–202 days) is slower than the solar one, and even slower for the latedM stars (≳63–263 days). Planets orbiting in the habitable zone of dM stars that rotate faster than this have smaller magnetospheric sizes than that of the Earth magnetosphere. Because many latedM stars are fast rotators, conditions for terrestrial planets to harbour Earthsized magnetospheres are more easily achieved for planets orbiting slowly rotating early and middM stars.
A.A.V. acknowledges support from the Royal Astronomical Society through a postdoctoral fellowship. J.M. acknowledges support from a fellowship of the Alexander von Humboldt foundation. P.L. acknowledges funding from a STFC scholarship. AJBR is a Research Fellow of the Royal Commission for the Exhibition of 1851.
20130902T00:00:00Z
Vidotto, A.A.
Jardine, M.
Morin, J.
Donati, J.F.
Lang, P.
Russell, A.J.B.
We investigate the effect of the magnetic fields of M dwarf (dM) stars on potentially habitable Earthlike planets. These fields can reduce the size of planetary magnetospheres to such an extent that a significant fraction of the planet’s atmosphere may be exposed to erosion by the stellar wind. We used a sample of 15 active dM stars, for which surface magneticfield maps were reconstructed, to determine the magnetic pressure at the planet orbit and hence the largest size of its magnetosphere, which would only be decreased by considering the stellar wind. Our method provides a fast means to assess which planets are most affected by the stellar magnetic field, which can be used as a first study to be followed by more sophisticated models. We show that hypothetical Earthlike planets with similar terrestrial magnetisation (~1 G) orbiting at the inner (outer) edge of the habitable zone of these stars would present magnetospheres that extend at most up to 6 (11.7) planetary radii. To be able to sustain an Earthsized magnetosphere, with the exception of only a few cases, the terrestrial planet would either (1) need to orbit significantly farther out than the traditional limits of the habitable zone; or else, (2) if it were orbiting within the habitable zone, it would require at least a magnetic field ranging from a few G to up to a few thousand G. By assuming a magnetospheric size that is more appropriate for the youngEarth (3.4 Gyr ago), the required planetary magnetic fields are one order of magnitude weaker. However, in this case, the polarcap area of the planet, which is unprotected from transport of particles to/from interplanetary space, is twice as large. At present, we do not know how small the smallest area of the planetary surface is that could be exposed and would still not affect the potential for formation and development of life in a planet. As the star becomes older and, therefore, its rotation rate and magnetic field reduce, the interplanetary magnetic pressure decreases and the magnetosphere of planets probably expands. Using an empirically derived rotationactivity/magnetism relation, we provide an analytical expression for estimating the shortest stellar rotation period for which an Earthanalogue in the habitable zone could sustain an Earthsized magnetosphere. We find that the required rotation rate of the early and middM stars (with periods ≳37–202 days) is slower than the solar one, and even slower for the latedM stars (≳63–263 days). Planets orbiting in the habitable zone of dM stars that rotate faster than this have smaller magnetospheric sizes than that of the Earth magnetosphere. Because many latedM stars are fast rotators, conditions for terrestrial planets to harbour Earthsized magnetospheres are more easily achieved for planets orbiting slowly rotating early and middM stars.

Numerical simulation of a selfsimilar cascade of filament instabilities in the surface quasigeostrophic system
http://hdl.handle.net/10023/5436
We provide numerical evidence for the existence of a cascade of filament instabilities in the surface quasigeostrophic system for rotating, stratified flow near a horizontal boundary. The cascade involves geometrically shrinking spatial and temporal scales and implies the singular collapse of the filament width to zero in a finite time. The numerical method is both spatially and temporally adaptive, permitting the accurate simulation of the evolution over an unprecedented range of spatial scales spanning over ten orders of magnitude. It provides the first convincing demonstration of the cascade, in which the large separation of scales between subsequent instabilities has made previous numerical simulation difficult.
20140411T00:00:00Z
Scott, R. K.
Dritschel, D. G.
We provide numerical evidence for the existence of a cascade of filament instabilities in the surface quasigeostrophic system for rotating, stratified flow near a horizontal boundary. The cascade involves geometrically shrinking spatial and temporal scales and implies the singular collapse of the filament width to zero in a finite time. The numerical method is both spatially and temporally adaptive, permitting the accurate simulation of the evolution over an unprecedented range of spatial scales spanning over ten orders of magnitude. It provides the first convincing demonstration of the cascade, in which the large separation of scales between subsequent instabilities has made previous numerical simulation difficult.

Inertialrange dynamics and scaling laws of twodimensional magnetic turbulence in the weakfield regime
http://hdl.handle.net/10023/5358
We study inertialrange dynamics and scaling laws in unforced twodimensional magnetohydrodynamic turbulence in the regime of moderately small and small initial magnetictokinetic energy ratio $r_0$, with an emphasis on the latter. The regime of small $r_0$ corresponds to a relatively weak field and strong magnetic stretching, whereby the turbulence is characterized by an intense conversion of kinetic into magnetic energy (dynamo action in the threedimensional context). This conversion is an inertialrange phenomenon and, upon becoming quasisaturated, deposits the converted energy within the inertial range rather than transferring it to the small scales. As a result, the magnetic energy spectrum $E_\b(k)$ in the inertial range can become quite shallow and may not be adequately explained or understood in terms of conventional cascade theories. It is demonstrated by numerical simulations at high Reynolds numbers (and unity magnetic Prandtl number) that the energetics and inertialrange scaling depend strongly on $r_0$. In particular, for fully developed turbulence with $r_0$ in the range $[1/4,1/4096]$, $E_\b(k)$ is found to scale as $k^{\alpha}$, where $\alpha\gtrsim1$, including $\alpha>0$. The extent of such a shallow spectrum is limited, becoming broader as $r_0$ is decreased. The slope $\alpha$ increases as $r_0$ is decreased, appearing to tend to $+1$ in the limit of small $r_0$. This implies equipartition of magnetic energy among the Fourier modes of the inertial range and the scaling $k^{1}$ of the magnetic potential variance, whose flux is direct rather than inverse. This behavior of the potential resembles that of a passive scalar. However, unlike a passive scalar whose variance dissipation rate slowly vanishes in the diffusionless limit, the dissipation rate of the magnetic potential variance scales linearly with the diffusivity in that limit. Meanwhile, the kinetic energy spectrum is relatively steep, followed by a much shallower tail due to strong antidynamo excitation. This gives rise to a total energy spectrum poorly obeying a powerlaw scaling.
The work reported here was partially supported by an EPSRC postgraduate studentship to L.A.K.B. L.A.K.B. was further supported by an EPSRC doctoral prize.
20140821T00:00:00Z
Blackbourn, Luke Austen Kazimierz
Tran, Chuong Van
We study inertialrange dynamics and scaling laws in unforced twodimensional magnetohydrodynamic turbulence in the regime of moderately small and small initial magnetictokinetic energy ratio $r_0$, with an emphasis on the latter. The regime of small $r_0$ corresponds to a relatively weak field and strong magnetic stretching, whereby the turbulence is characterized by an intense conversion of kinetic into magnetic energy (dynamo action in the threedimensional context). This conversion is an inertialrange phenomenon and, upon becoming quasisaturated, deposits the converted energy within the inertial range rather than transferring it to the small scales. As a result, the magnetic energy spectrum $E_\b(k)$ in the inertial range can become quite shallow and may not be adequately explained or understood in terms of conventional cascade theories. It is demonstrated by numerical simulations at high Reynolds numbers (and unity magnetic Prandtl number) that the energetics and inertialrange scaling depend strongly on $r_0$. In particular, for fully developed turbulence with $r_0$ in the range $[1/4,1/4096]$, $E_\b(k)$ is found to scale as $k^{\alpha}$, where $\alpha\gtrsim1$, including $\alpha>0$. The extent of such a shallow spectrum is limited, becoming broader as $r_0$ is decreased. The slope $\alpha$ increases as $r_0$ is decreased, appearing to tend to $+1$ in the limit of small $r_0$. This implies equipartition of magnetic energy among the Fourier modes of the inertial range and the scaling $k^{1}$ of the magnetic potential variance, whose flux is direct rather than inverse. This behavior of the potential resembles that of a passive scalar. However, unlike a passive scalar whose variance dissipation rate slowly vanishes in the diffusionless limit, the dissipation rate of the magnetic potential variance scales linearly with the diffusivity in that limit. Meanwhile, the kinetic energy spectrum is relatively steep, followed by a much shallower tail due to strong antidynamo excitation. This gives rise to a total energy spectrum poorly obeying a powerlaw scaling.

Distribution of electric currents in solar active regions
http://hdl.handle.net/10023/5322
There has been a longstanding debate on the question of whether or not electric currents in solar active regions are neutralized. That is, whether or not the main (or direct) coronal currents connecting the active region polarities are surrounded by shielding (or return) currents of equal total value and opposite direction. Both theory and observations are not yet fully conclusive regarding this question, and numerical simulations have, surprisingly, barely been used to address it. Here we quantify the evolution of electric currents during the formation of a bipolar active region by considering a threedimensional magnetohydrodynamic simulation of the emergence of a subphotospheric, currentneutralized magnetic flux rope into the solar atmosphere. We find that a strong deviation from current neutralization develops simultaneously with the onset of significant flux emergence into the corona, accompanied by the development of substantial magnetic shear along the active region's polarity inversion line. After the region has formed and flux emergence has ceased, the strong magnetic fields in the region's center are connected solely by direct currents, and the total direct current is several times larger than the total return current. These results suggest that active regions, the main sources of coronal mass ejections and flares, are born with substantial net currents, in agreement with recent observations. Furthermore, they support eruption models that employ preeruption magnetic fields containing such currents.
The contributions of T.T., V.S.T., and Z.M. were supported by NASA's HTP, LWS, and SR&T programs. J.E.L and M.G.L. were supported by NASA/LWS. M.G.L. received support also from the ONR 6.1 program. The simulation was performed under grant of computer time from the D.o.D. HPC Program. B.K. was supported by the DFG. V.A. acknowledges support through the IEF272549 grant.
20140210T00:00:00Z
Török, T.
Leake, J.E.
Titov, V.S.
Archontis, V.
Mikić, Z.
Linton, M.G.
Dalmasse, K.
Aulanier, G.
Kliem, B.
There has been a longstanding debate on the question of whether or not electric currents in solar active regions are neutralized. That is, whether or not the main (or direct) coronal currents connecting the active region polarities are surrounded by shielding (or return) currents of equal total value and opposite direction. Both theory and observations are not yet fully conclusive regarding this question, and numerical simulations have, surprisingly, barely been used to address it. Here we quantify the evolution of electric currents during the formation of a bipolar active region by considering a threedimensional magnetohydrodynamic simulation of the emergence of a subphotospheric, currentneutralized magnetic flux rope into the solar atmosphere. We find that a strong deviation from current neutralization develops simultaneously with the onset of significant flux emergence into the corona, accompanied by the development of substantial magnetic shear along the active region's polarity inversion line. After the region has formed and flux emergence has ceased, the strong magnetic fields in the region's center are connected solely by direct currents, and the total direct current is several times larger than the total return current. These results suggest that active regions, the main sources of coronal mass ejections and flares, are born with substantial net currents, in agreement with recent observations. Furthermore, they support eruption models that employ preeruption magnetic fields containing such currents.

Recurrent explosive eruptions and the "sigmoidtoarcade" transformation in the Sun driven by dynamical magnetic flux emergence
http://hdl.handle.net/10023/5319
We report on threedimensional MHD simulations of recurrent mini coronal mass ejection (CME)like eruptions in a small active region (AR), which is formed by the dynamical emergence of a twisted (not kink unstable) flux tube from the solar interior. The eruptions develop as a result of the repeated formation and expulsion of new flux ropes due to continuous emergence and reconnection of sheared field lines along the polarity inversion line of the AR. The acceleration of the eruptions is triggered by tethercutting reconnection at the current sheet underneath the erupting field. We find that each explosive eruption is followed by reformation of a sigmoidal structure and a subsequent "sigmoidtoflare arcade" transformation in the AR. These results might have implications for recurrent CMEs and eruptive sigmoids/flares observations and theoretical studies.
The authors acknowledge support by EU (IEF272549 grant) and the Royal Society.
20140510T00:00:00Z
Archontis, V.
Hood, A.W.
Tsinganos, K.
We report on threedimensional MHD simulations of recurrent mini coronal mass ejection (CME)like eruptions in a small active region (AR), which is formed by the dynamical emergence of a twisted (not kink unstable) flux tube from the solar interior. The eruptions develop as a result of the repeated formation and expulsion of new flux ropes due to continuous emergence and reconnection of sheared field lines along the polarity inversion line of the AR. The acceleration of the eruptions is triggered by tethercutting reconnection at the current sheet underneath the erupting field. We find that each explosive eruption is followed by reformation of a sigmoidal structure and a subsequent "sigmoidtoflare arcade" transformation in the AR. These results might have implications for recurrent CMEs and eruptive sigmoids/flares observations and theoretical studies.

Observations of a hybrid doublestreamer/pseudostreamer in the solar corona
http://hdl.handle.net/10023/5318
We report on the first observation of a single hybrid magnetic structure that contains both a pseudostreamer and a double streamer. This structure was originally observed by the SWAP instrument on board the PROBA2 satellite between 2013 May 5 and 10. It consists of a pair of filament channels near the south pole of the Sun. On the western edge of the structure, the magnetic morphology above the filaments is that of a sidebyside double streamer, with open field between the two channels. On the eastern edge, the magnetic morphology is that of a coronal pseudostreamer without the central open field. We investigated this structure with multiple observations and modeling techniques. We describe the topology and dynamic consequences of such a unified structure.
D.B.S. and L.A.R. acknowledge support from the Belgian Federal Science Policy Office (BELSPO) through the ESAPRODEX program, grant No. 4000103240. S.J.P. acknowledges the financial support of the Isle of Man Government.
20140520T00:00:00Z
Rachmeler, L.A.
Platten, S.J.
Bethge, C.
Seaton, D.B.
Yeates, A.R.
We report on the first observation of a single hybrid magnetic structure that contains both a pseudostreamer and a double streamer. This structure was originally observed by the SWAP instrument on board the PROBA2 satellite between 2013 May 5 and 10. It consists of a pair of filament channels near the south pole of the Sun. On the western edge of the structure, the magnetic morphology above the filaments is that of a sidebyside double streamer, with open field between the two channels. On the eastern edge, the magnetic morphology is that of a coronal pseudostreamer without the central open field. We investigated this structure with multiple observations and modeling techniques. We describe the topology and dynamic consequences of such a unified structure.

Clusters of small eruptive flares produced by magnetic reconnection in the Sun
http://hdl.handle.net/10023/5316
We report on the formation of small solar flares produced by patchy magnetic reconnection between interacting magnetic loops. A threedimensional (3D) magnetohydrodynamic (MHD) numerical experiment was performed, where a uniform magnetic flux sheet was injected into a fully developed convective layer. The gradual emergence of the field into the solar atmosphere results in a network of magnetic loops, which interact dynamically forming current layers at their interfaces. The formation and ejection of plasmoids out of the current layers leads to patchy reconnection and the spontaneous formation of several small (size ≈12 Mm) flares. We find that these flares are shortlived (30 s3 minutes) bursts of energy in the range O(10251027) erg, which is basically the nanoflaremicroflare range. Their persistent formation and cooperative action and evolution leads to recurrent emission of fast EUV/Xray jets and considerable plasma heating in the active corona.
This research was supported by the Research Council of Norway through the grant "Solar Atmospheric Modelling" and through grants of computing time from the Programme for Supercomputing, by the European Research Council under the European Union's Seventh Framework Programme (FP7/20072013)/ERC Grant agreement No. 291058 and by computing project s1061 from the High End Computing Division of NASA. The authors acknowledge support by the EU (IEF272549 grant) and the Royal Society.
20140610T00:00:00Z
Archontis, V.
Hansteen, V.
We report on the formation of small solar flares produced by patchy magnetic reconnection between interacting magnetic loops. A threedimensional (3D) magnetohydrodynamic (MHD) numerical experiment was performed, where a uniform magnetic flux sheet was injected into a fully developed convective layer. The gradual emergence of the field into the solar atmosphere results in a network of magnetic loops, which interact dynamically forming current layers at their interfaces. The formation and ejection of plasmoids out of the current layers leads to patchy reconnection and the spontaneous formation of several small (size ≈12 Mm) flares. We find that these flares are shortlived (30 s3 minutes) bursts of energy in the range O(10251027) erg, which is basically the nanoflaremicroflare range. Their persistent formation and cooperative action and evolution leads to recurrent emission of fast EUV/Xray jets and considerable plasma heating in the active corona.

Loss cone evolution and particle escape in collapsing magnetic trap models in solar flares
http://hdl.handle.net/10023/5275
Context. Collapsing magnetic traps (CMTs) have been suggested as one possible mechanism responsible for the acceleration of highenergy particles during solar flares. An important question regarding the CMT acceleration mechanism is which particle orbits escape and which are trapped during the time evolution of a CMT. While some models predict the escape of the majority of particle orbits, other more sophisticated CMT models show that, in particular, the highestenergy particles remain trapped at all times. The exact prediction is not straightforward because both the loss cone angle and the particle orbit pitch angle evolve in time in a CMT. Aims. Our aim is to gain a better understanding of the conditions leading to either particle orbit escape or trapping in CMTs. Methods. We present a detailed investigation of the time evolution of particle orbit pitch angles in the CMT model of Giuliani and collaborators and compare this with the time evolution of the loss cone angle. The nonrelativistic guiding centre approximation is used to calculate the particle orbits. We also use simplified models to corroborate the findings of the particle orbit calculations. Results. We find that there is a critical initial pitch angle for each field line of a CMT that divides trapped and escaping particle orbits. This critical initial pitch angle is greater than the initial loss cone angle, but smaller than the asymptotic (final) loss cone angle for that field line. As the final loss cone angle in CMTs is larger than the initial loss cone angle, particle orbits with pitch angles that cross into the loss cone during their time evolution will escape whereas all other particle orbits are trapped. We find that in realistic CMT models, Fermi acceleration will only dominate in the initial phase of the CMT evolution and, in this case, can reduce the pitch angle, but that betatron acceleration will dominate for later stages of the CMT evolution leading to a systematic increase of the pitch angle. Whether a particle escapes or remains trapped depends critically on the relative importance of the two acceleration mechanisms, which cannot be decoupled in more sophisticated CMT models.
This work was financially supported by the UK’s Science and Technology Facilities Council.
20140312T00:00:00Z
Neukirch, Thomas
Eradat Oskoui, Solmaz
Grady, Keith James
Context. Collapsing magnetic traps (CMTs) have been suggested as one possible mechanism responsible for the acceleration of highenergy particles during solar flares. An important question regarding the CMT acceleration mechanism is which particle orbits escape and which are trapped during the time evolution of a CMT. While some models predict the escape of the majority of particle orbits, other more sophisticated CMT models show that, in particular, the highestenergy particles remain trapped at all times. The exact prediction is not straightforward because both the loss cone angle and the particle orbit pitch angle evolve in time in a CMT. Aims. Our aim is to gain a better understanding of the conditions leading to either particle orbit escape or trapping in CMTs. Methods. We present a detailed investigation of the time evolution of particle orbit pitch angles in the CMT model of Giuliani and collaborators and compare this with the time evolution of the loss cone angle. The nonrelativistic guiding centre approximation is used to calculate the particle orbits. We also use simplified models to corroborate the findings of the particle orbit calculations. Results. We find that there is a critical initial pitch angle for each field line of a CMT that divides trapped and escaping particle orbits. This critical initial pitch angle is greater than the initial loss cone angle, but smaller than the asymptotic (final) loss cone angle for that field line. As the final loss cone angle in CMTs is larger than the initial loss cone angle, particle orbits with pitch angles that cross into the loss cone during their time evolution will escape whereas all other particle orbits are trapped. We find that in realistic CMT models, Fermi acceleration will only dominate in the initial phase of the CMT evolution and, in this case, can reduce the pitch angle, but that betatron acceleration will dominate for later stages of the CMT evolution leading to a systematic increase of the pitch angle. Whether a particle escapes or remains trapped depends critically on the relative importance of the two acceleration mechanisms, which cannot be decoupled in more sophisticated CMT models.

The solar cycle variation of topological structures in the global solar corona
http://hdl.handle.net/10023/5271
Context. The complicated distribution of magnetic flux across the solar photosphere results in a complex web of coronal magnetic field structures. To understand this complexity, the magnetic skeleton of the coronal field can be calculated. The skeleton highlights the (separatrix) surfaces that divide the field into topologically distinct regions, allowing openfield regions on the solar surface to be located. Furthermore, separatrix surfaces and their intersections with other separatrix surfaces (i.e., separators) are important likely energy release sites. Aims. The aim of this paper is to investigate, throughout the solar cycle, the nature of coronal magneticfield topologies that arise under the potentialfield sourcesurface approximation. In particular, we characterise the typical global fields at solar maximum and minimum. Methods. Global magnetic fields are extrapolated from observed Kitt Peak and SOLIS synoptic magnetograms, from Carrington rotations 1645 to 2144, using the potentialfield sourcesurface model. This allows the variations in the coronal skeleton to be studied over three solar cycles. Results. The main building blocks which make up magnetic fields are identified and classified according to the nature of their separatrix surfaces. The magnetic skeleton reveals that, at solar maximum, the global coronal field involves a multitude of topological structures at all latitudes crisscrossing throughout the atmosphere. Many openfield regions exist originating anywhere on the photosphere. At solar minimum, the coronal topology is heavily influenced by the solar magnetic dipole. A strong dipole results in a simple largescale structure involving just two large polar openfield regions, but, at short radial distances between ± 60° latitude, the smallscale topology is complex. If the solar magnetic dipole if weak, as in the recent minimum, then the lowlatitude quietsun magnetic fields may be globally significant enough to create many disconnected openfield regions between ± 60° latitude, in addition to the two polar openfield regions.
S.J.P. acknowledges financial support from the Isle of Man Government. E.R.P. is grateful to the Leverhulme Trust for his emeritus fellowship. The research leading to these results has received funding from the European Commission’s Seventh Framework Programme (FP7/20072013) under the grant agreement SWIFF (project No. 263340, www.swiff.eu).
20140501T00:00:00Z
Platten, S.J.
Parnell, C.E.
Haynes, A.L.
Priest, E.R.
MacKay, D.H.
Context. The complicated distribution of magnetic flux across the solar photosphere results in a complex web of coronal magnetic field structures. To understand this complexity, the magnetic skeleton of the coronal field can be calculated. The skeleton highlights the (separatrix) surfaces that divide the field into topologically distinct regions, allowing openfield regions on the solar surface to be located. Furthermore, separatrix surfaces and their intersections with other separatrix surfaces (i.e., separators) are important likely energy release sites. Aims. The aim of this paper is to investigate, throughout the solar cycle, the nature of coronal magneticfield topologies that arise under the potentialfield sourcesurface approximation. In particular, we characterise the typical global fields at solar maximum and minimum. Methods. Global magnetic fields are extrapolated from observed Kitt Peak and SOLIS synoptic magnetograms, from Carrington rotations 1645 to 2144, using the potentialfield sourcesurface model. This allows the variations in the coronal skeleton to be studied over three solar cycles. Results. The main building blocks which make up magnetic fields are identified and classified according to the nature of their separatrix surfaces. The magnetic skeleton reveals that, at solar maximum, the global coronal field involves a multitude of topological structures at all latitudes crisscrossing throughout the atmosphere. Many openfield regions exist originating anywhere on the photosphere. At solar minimum, the coronal topology is heavily influenced by the solar magnetic dipole. A strong dipole results in a simple largescale structure involving just two large polar openfield regions, but, at short radial distances between ± 60° latitude, the smallscale topology is complex. If the solar magnetic dipole if weak, as in the recent minimum, then the lowlatitude quietsun magnetic fields may be globally significant enough to create many disconnected openfield regions between ± 60° latitude, in addition to the two polar openfield regions.

Dynamic properties of bright points in an active region
http://hdl.handle.net/10023/5264
Context. Bright points (BPs) are smallscale, magnetic features ubiquitous across the solar surface. Previously, we have observed and noted their properties for quiet Sun regions. Here, we determine the dynamic properties of BPs using simultaneous quiet Sun and active region data. Aims. The aim of this paper is to compare the properties of BPs in both active and quiet Sun regions and to determine any difference in the dynamics and general properties of BPs as a result of the varying magnetic activity within these two regions. Methods. High spatial and temporal resolution Gband observations of active region AR11372 were obtained with the Rapid Oscillations in the Solar Atmosphere instrument at the Dunn Solar Telescope. Three subfields of varying polarity and magnetic flux density were selected with the aid of magnetograms obtained from the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory. Bright points within these subfields were subsequently tracked and analysed. Results. It is found that BPs within active regions display attenuated velocity distributions with an average horizontal velocity of ∼0.6 km s1, compared to the quiet region which had an average velocity of 0.9 km s1. Active region BPs are also ∼21% larger than quiet region BPs and have longer average lifetimes (∼132 s) than their quiet region counterparts (88 s). No preferential flow directions are observed within the active region subfields. The diffusion index (γ) is estimated at ∼1.2 for the three regions. Conclusions. We confirm that the dynamic properties of BPs arise predominately from convective motions. The presence of stronger field strengths within active regions is the likely reason behind the varying properties observed. We believe that larger amounts of magnetic flux will attenuate BP velocities by a combination of restricting motion within the intergranular lanes and by increasing the number of stagnation points produced by inhibited convection. Larger BPs are found in regions of higher magnetic flux density and we believe that lifetimes increase in active regions as the magnetic flux stabilises the BPs.
This work has been supported by the UK Science and Technology Facilities Council (STFC). Observations were obtained at the National Solar Observatory, operated by the Association of Universities for Research in Astronomy, Inc. (AURA), under cooperative agreement with the National Science Foundation. D.B.J. would like to thank the STFC for an Ernest Rutherford Fellowship. We are also grateful for support sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF under grant number FA86550913085.
20140620T00:00:00Z
Keys, P.H.
Mathioudakis, M.
Jess, D.B.
MacKay, D.H.
Keenan, F.P.
Context. Bright points (BPs) are smallscale, magnetic features ubiquitous across the solar surface. Previously, we have observed and noted their properties for quiet Sun regions. Here, we determine the dynamic properties of BPs using simultaneous quiet Sun and active region data. Aims. The aim of this paper is to compare the properties of BPs in both active and quiet Sun regions and to determine any difference in the dynamics and general properties of BPs as a result of the varying magnetic activity within these two regions. Methods. High spatial and temporal resolution Gband observations of active region AR11372 were obtained with the Rapid Oscillations in the Solar Atmosphere instrument at the Dunn Solar Telescope. Three subfields of varying polarity and magnetic flux density were selected with the aid of magnetograms obtained from the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory. Bright points within these subfields were subsequently tracked and analysed. Results. It is found that BPs within active regions display attenuated velocity distributions with an average horizontal velocity of ∼0.6 km s1, compared to the quiet region which had an average velocity of 0.9 km s1. Active region BPs are also ∼21% larger than quiet region BPs and have longer average lifetimes (∼132 s) than their quiet region counterparts (88 s). No preferential flow directions are observed within the active region subfields. The diffusion index (γ) is estimated at ∼1.2 for the three regions. Conclusions. We confirm that the dynamic properties of BPs arise predominately from convective motions. The presence of stronger field strengths within active regions is the likely reason behind the varying properties observed. We believe that larger amounts of magnetic flux will attenuate BP velocities by a combination of restricting motion within the intergranular lanes and by increasing the number of stagnation points produced by inhibited convection. Larger BPs are found in regions of higher magnetic flux density and we believe that lifetimes increase in active regions as the magnetic flux stabilises the BPs.

Vortical control of forced twodimensional turbulence
http://hdl.handle.net/10023/5236
A new numerical technique for the simulation of forced twodimensional turbulence[D. Dritschel and J. Fontane, “The combined Lagrangian advection method,” J. Comput. Phys.229, 5408–5417 (Year: 2010)10.1016/j.jcp.2010.03.048] is used to examine the validity of KraichnanBatchelor scaling laws at higher Reynolds number than previously accessible with classical pseudospectral methods, making use of large simulation ensembles to allow a detailed consideration of the inverse cascade in a quasisteady state. Our results support the recent finding of Scott [R. Scott, “Nonrobustness of the twodimensional turbulent inverse cascade,” Phys. Rev. E75, 046301 (Year: 2007)10.1103/PhysRevE.75.046301], namely that when a direct enstrophy cascading range is wellrepresented numerically, a steeper energy spectrum proportional to k−2 is obtained in place of the classical k −5/3 prediction. It is further shown that this steep spectrum is associated with a faster growth of energy at large scales, scaling like t −1 rather than Kraichnan's prediction of t −3/2. The deviation from Kraichnan's theory is related to the emergence of a population of vortices that dominate the distribution of energy across scales, and whose number density and vorticity distribution with respect to vortex area are related to the shape of the enstrophy spectrum. An analytical model is proposed which closely matches the numerical spectra between the large scales and the forcing scale.
Jérôme Fontane is supported by the European Community in the framework of the CONVECT project under Grant No. PIEFGA2008221003.
20130114T00:00:00Z
Fontane, Jerome Jacob Louis
Dritschel, David Gerard
Scott, Richard Kirkness
A new numerical technique for the simulation of forced twodimensional turbulence[D. Dritschel and J. Fontane, “The combined Lagrangian advection method,” J. Comput. Phys.229, 5408–5417 (Year: 2010)10.1016/j.jcp.2010.03.048] is used to examine the validity of KraichnanBatchelor scaling laws at higher Reynolds number than previously accessible with classical pseudospectral methods, making use of large simulation ensembles to allow a detailed consideration of the inverse cascade in a quasisteady state. Our results support the recent finding of Scott [R. Scott, “Nonrobustness of the twodimensional turbulent inverse cascade,” Phys. Rev. E75, 046301 (Year: 2007)10.1103/PhysRevE.75.046301], namely that when a direct enstrophy cascading range is wellrepresented numerically, a steeper energy spectrum proportional to k−2 is obtained in place of the classical k −5/3 prediction. It is further shown that this steep spectrum is associated with a faster growth of energy at large scales, scaling like t −1 rather than Kraichnan's prediction of t −3/2. The deviation from Kraichnan's theory is related to the emergence of a population of vortices that dominate the distribution of energy across scales, and whose number density and vorticity distribution with respect to vortex area are related to the shape of the enstrophy spectrum. An analytical model is proposed which closely matches the numerical spectra between the large scales and the forcing scale.

Resistive magnetohydrodynamic reconnection : resolving longterm, chaotic dynamics
http://hdl.handle.net/10023/5233
In this paper, we address the longterm evolution of an idealised double current system entering reconnection regimes where chaotic behavior plays a prominent role. Our aim is to quantify the energetics in high magnetic Reynolds number evolutions, enriched by secondary tearing events, multiple magnetic island coalescence, and compressive versus resistive heating scenarios. Our study will pay particular attention to the required numerical resolutions achievable by modern (gridadaptive) computations, and comment on the challenge associated with resolving chaotic island formation and interaction. We will use shockcapturing, conservative, gridadaptive simulations for investigating trends dominated by both physical (resistivity) and numerical (resolution) parameters, and confront them with (visco)resistive magnetohydrodynamic simulations performed with very different, but equally widely used discretization schemes. This will allow us to comment on the obtained evolutions in a manner irrespective of the adopted discretization strategy. Our findings demonstrate that all schemes used (finite volume based shockcapturing, high order finite differences, and particle in celllike methods) qualitatively agree on the various evolutionary stages, and that resistivity values of order 0.001 already can lead to chaotic island appearance. However, none of the methods exploited demonstrates convergence in the strong sense in these chaotic regimes. At the same time, nonperturbed tests for showing convergence over long time scales in ideal to resistive regimes are provided as well, where all methods are shown to agree. Both the advantages and disadvantages of specific discretizations as applied to this challenging problem are discussed.
We acknowledge financial support from the EC FP7/20072013 Grant Agreement SWIFF (No. 263340) and from project GOA/2009/009 (KU Leuven). This research has been funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office (IAP P7/08 CHARM). Part of the simulations used the infrastructure of the VSCFlemish Supercomputer Center, funded by the Hercules Foundation and the Flemish GovernmentDepartment EWI. Another part of the simulations was done at the former Danish Center for Scientific Computing at Copenhagen University which is now part of DeIC Danish eInfrastructure Cooperation.
20130913T00:00:00Z
Keppens, R.
Porth, O.
Galsgaard, K.
Frederiksen, J.T.
Restante, A.L.
Lapenta, G.
Parnell, C.
In this paper, we address the longterm evolution of an idealised double current system entering reconnection regimes where chaotic behavior plays a prominent role. Our aim is to quantify the energetics in high magnetic Reynolds number evolutions, enriched by secondary tearing events, multiple magnetic island coalescence, and compressive versus resistive heating scenarios. Our study will pay particular attention to the required numerical resolutions achievable by modern (gridadaptive) computations, and comment on the challenge associated with resolving chaotic island formation and interaction. We will use shockcapturing, conservative, gridadaptive simulations for investigating trends dominated by both physical (resistivity) and numerical (resolution) parameters, and confront them with (visco)resistive magnetohydrodynamic simulations performed with very different, but equally widely used discretization schemes. This will allow us to comment on the obtained evolutions in a manner irrespective of the adopted discretization strategy. Our findings demonstrate that all schemes used (finite volume based shockcapturing, high order finite differences, and particle in celllike methods) qualitatively agree on the various evolutionary stages, and that resistivity values of order 0.001 already can lead to chaotic island appearance. However, none of the methods exploited demonstrates convergence in the strong sense in these chaotic regimes. At the same time, nonperturbed tests for showing convergence over long time scales in ideal to resistive regimes are provided as well, where all methods are shown to agree. Both the advantages and disadvantages of specific discretizations as applied to this challenging problem are discussed.

The effect of slip length on vortex rebound from a rigid boundary
http://hdl.handle.net/10023/5232
The problem of a dipole incident normally on a rigid boundary, for moderate to large Reynolds numbers, has recently been treated numerically using a volume penalisation method by Nguyen van yen, Farge, and Schneider [Phys. Rev. Lett.106, 184502 (2011)]. Their results indicate that energy dissipating structures persist in the inviscid limit. They found that the use of penalisation methods intrinsically introduces some slip at the boundary wall, where the slip approaches zero as the Reynolds number goes to infinity, so reducing to the noslip case in this limit. We study the same problem, for both noslip and partial slip cases, using compact differences on a Chebyshev grid in the direction normal to the wall and Fourier methods in the direction along the wall. We find that for the noslip case there is no indication of the persistence of energy dissipating structures in the limit as viscosity approaches zero and that this also holds for any fixed slip length. However, when the slip length is taken to vary inversely with Reynolds number then the results of Nguyen van yen et al. are regained. It therefore appears that the prediction that energy dissipating structures persist in the inviscid limit follows from the two limits of wall slip length going to zero, and viscosity going to zero, not being treated independently in their use of the volume penalisation method.
20130923T00:00:00Z
Sutherland, D.
Macaskill, C.
Dritschel, D.G.
The problem of a dipole incident normally on a rigid boundary, for moderate to large Reynolds numbers, has recently been treated numerically using a volume penalisation method by Nguyen van yen, Farge, and Schneider [Phys. Rev. Lett.106, 184502 (2011)]. Their results indicate that energy dissipating structures persist in the inviscid limit. They found that the use of penalisation methods intrinsically introduces some slip at the boundary wall, where the slip approaches zero as the Reynolds number goes to infinity, so reducing to the noslip case in this limit. We study the same problem, for both noslip and partial slip cases, using compact differences on a Chebyshev grid in the direction normal to the wall and Fourier methods in the direction along the wall. We find that for the noslip case there is no indication of the persistence of energy dissipating structures in the limit as viscosity approaches zero and that this also holds for any fixed slip length. However, when the slip length is taken to vary inversely with Reynolds number then the results of Nguyen van yen et al. are regained. It therefore appears that the prediction that energy dissipating structures persist in the inviscid limit follows from the two limits of wall slip length going to zero, and viscosity going to zero, not being treated independently in their use of the volume penalisation method.

Progress towards numerical and experimental simulations of fusion relevant beam instabilities
http://hdl.handle.net/10023/5186
In certain plasmas, nonthermal electron distributions can produce instabilities. These instabilities may be useful or potentially disruptive. Therefore the study of these instabilities is of importance in a variety of fields including fusion science and astrophysics. Following on from previous work conducted at the University of Strathclyde on the cyclotron resonance maser instability that was relevant to astrophysical radiowave generation, further instabilities are being investigated. Particular instabilities of interest are the anomalous Doppler instability which can occur in magnetic confinement fusion plasmas and the twostream instability that is of importance in fastignition inertial confinement fusion. To this end, computational simulations have been undertaken to investigate the behaviour of both the anomalous Doppler and twostream instabilities with the goal of designing an experiment to observe these behaviours in a laboratory.
20140507T00:00:00Z
King, M.
Bryson, R.
Ronald, K.
Cairns, R. A.
McConville, S. L.
Speirs, D. C.
Phelps, A. D. R.
Bingham, R.
Gillespie, K. M.
Cross, A. W.
Vorgul, I.
Trines, R.
In certain plasmas, nonthermal electron distributions can produce instabilities. These instabilities may be useful or potentially disruptive. Therefore the study of these instabilities is of importance in a variety of fields including fusion science and astrophysics. Following on from previous work conducted at the University of Strathclyde on the cyclotron resonance maser instability that was relevant to astrophysical radiowave generation, further instabilities are being investigated. Particular instabilities of interest are the anomalous Doppler instability which can occur in magnetic confinement fusion plasmas and the twostream instability that is of importance in fastignition inertial confinement fusion. To this end, computational simulations have been undertaken to investigate the behaviour of both the anomalous Doppler and twostream instabilities with the goal of designing an experiment to observe these behaviours in a laboratory.

Scaled Experiment to Investigate Auroral Kilometric Radiation Mechanisms in the Presence of Background Electrons
http://hdl.handle.net/10023/5185
Auroral Kilometric Radiation (AKR) emissions occur at frequencies similar to 300kHz polarised in the Xmode with efficiencies similar to 12% [1,2] in the auroral density cavity in the polar regions of the Earth's magnetosphere, a region of low density plasma similar to 3200km above the Earth's surface, where electrons are accelerated down towards the Earth whilst undergoing magnetic compression. As a result of this magnetic compression the electrons acquire a horseshoe distribution function in velocity space. Previous theoretical studies have predicted that this distribution is capable of driving the cyclotron maser instability. To test this theory a scaled laboratory experiment was constructed to replicate this phenomenon in a controlled environment, [35] whilst 2D and 3D simulations are also being conducted to predict the experimental radiation power and mode, [69]. The experiment operates in the microwave frequency regime and incorporates a region of increasing magnetic field as found at the Earth's pole using magnet solenoids to encase the cylindrical interaction waveguide through which an initially rectilinear electron beam (12A) was accelerated by a 75keV pulse. Experimental results showed evidence of the formation of the horseshoe distribution function. The radiation was produced in the near cutoff TE01 mode, comparable with Xmode characteristics, at 4.42GHz. Peak microwave output power was measured similar to 35kW and peak efficiency of emission similar to 2%, [3]. A Penning trap was constructed and inserted into the interaction waveguide to enable generation of a background plasma which would lead to closer comparisons with the magnetospheric conditions. Initial design and measurements are presented showing the principle features of the new geometry.
20140507T00:00:00Z
McConville, S. L.
Ronald, K.
Speirs, D. C.
Gillespie, K. M.
Phelps, A. D. R.
Cross, A. W.
Bingham, R.
Robertson, C. W.
Whyte, C. G.
He, W.
King, M.
Bryson, R.
Vorgul, I.
Cairns, R. A.
Kellett, B. J.
Auroral Kilometric Radiation (AKR) emissions occur at frequencies similar to 300kHz polarised in the Xmode with efficiencies similar to 12% [1,2] in the auroral density cavity in the polar regions of the Earth's magnetosphere, a region of low density plasma similar to 3200km above the Earth's surface, where electrons are accelerated down towards the Earth whilst undergoing magnetic compression. As a result of this magnetic compression the electrons acquire a horseshoe distribution function in velocity space. Previous theoretical studies have predicted that this distribution is capable of driving the cyclotron maser instability. To test this theory a scaled laboratory experiment was constructed to replicate this phenomenon in a controlled environment, [35] whilst 2D and 3D simulations are also being conducted to predict the experimental radiation power and mode, [69]. The experiment operates in the microwave frequency regime and incorporates a region of increasing magnetic field as found at the Earth's pole using magnet solenoids to encase the cylindrical interaction waveguide through which an initially rectilinear electron beam (12A) was accelerated by a 75keV pulse. Experimental results showed evidence of the formation of the horseshoe distribution function. The radiation was produced in the near cutoff TE01 mode, comparable with Xmode characteristics, at 4.42GHz. Peak microwave output power was measured similar to 35kW and peak efficiency of emission similar to 2%, [3]. A Penning trap was constructed and inserted into the interaction waveguide to enable generation of a background plasma which would lead to closer comparisons with the magnetospheric conditions. Initial design and measurements are presented showing the principle features of the new geometry.

3D PiC code investigations of Auroral Kilometric Radiation mechanisms
http://hdl.handle.net/10023/5184
Efficient (similar to 1%) electron cyclotron radio emissions are known to originate in the X mode from regions of locally depleted plasma in the Earths polar magnetosphere. These emissions are commonly referred to as the Auroral Kilometric Radiation (AKR). AKR occurs naturally in these polar regions where electrons are accelerated by electric fields into the increasing planetary magnetic dipole. Here conservation of the magnetic moment converts axial to rotational momentum forming a horseshoe distribution in velocity phase space. This distribution is unstable to cyclotron emission with radiation emitted in the Xmode. Initial studies were conducted in the form of 2D PiC code simulations [1] and a scaled laboratory experiment that was constructed to reproduce the mechanism of AKR. As studies progressed, 3D PiC code simulations were conducted to enable complete investigation of the complex interaction dimensions. A maximum efficiency of 1.25% is predicted from these simulations in the same mode and frequency as measured in the experiment. This is also consistent with geophysical observations and the predictions of theory.
20140101T00:00:00Z
Gillespie, K. M.
McConville, S. L.
Speirs, D. C.
Ronald, K.
Phelps, A. D. R.
Bingham, R.
Cross, A. W.
Robertson, C. W.
Whyte, C. G.
He, W.
Vorgul, I.
Cairns, R. A.
Kellett, B. J.
Efficient (similar to 1%) electron cyclotron radio emissions are known to originate in the X mode from regions of locally depleted plasma in the Earths polar magnetosphere. These emissions are commonly referred to as the Auroral Kilometric Radiation (AKR). AKR occurs naturally in these polar regions where electrons are accelerated by electric fields into the increasing planetary magnetic dipole. Here conservation of the magnetic moment converts axial to rotational momentum forming a horseshoe distribution in velocity phase space. This distribution is unstable to cyclotron emission with radiation emitted in the Xmode. Initial studies were conducted in the form of 2D PiC code simulations [1] and a scaled laboratory experiment that was constructed to reproduce the mechanism of AKR. As studies progressed, 3D PiC code simulations were conducted to enable complete investigation of the complex interaction dimensions. A maximum efficiency of 1.25% is predicted from these simulations in the same mode and frequency as measured in the experiment. This is also consistent with geophysical observations and the predictions of theory.

Numerical simulation of unconstrained cyclotron resonant maser emission
http://hdl.handle.net/10023/5183
When a mainly rectilinear electron beam is subject to significant magnetic compression, conservation of magnetic moment results in the formation of a horseshoe shaped velocity distribution. It has been shown that such a distribution is unstable to cyclotron emission and may be responsible for the generation of Auroral Kilometric Radiation (AKR) an intense rf emission sourced at high altitudes in the terrestrial auroral magnetosphere. PiC code simulations have been undertaken to investigate the dynamics of the cyclotron emission process in the absence of cavity boundaries with particular consideration of the spatial growth rate, spectral output and rf conversion efficiency. Computations reveal that a welldefined cyclotron emission process occurs albeit with a low spatial growth rate compared to waveguide bounded simulations. The rf output is near perpendicular to the electron beam with a slight backwardwave character reflected in the spectral output with a well defined peak at 2.68GHz, just below the relativistic electron cyclotron frequency. The corresponding rf conversion efficiency of 1.1% is comparable to waveguide bounded simulations and consistent with the predictions of kinetic theory that suggest efficient, spectrally well defined radiation emission can be obtained from an electron horseshoe distribution in the absence of radiation boundaries.
20140507T00:00:00Z
Speirs, D. C.
Gillespie, K. M.
Ronald, K.
McConville, S. L.
Phelps, A. D. R.
Cross, A. W.
Bingham, R.
Kellett, B. J.
Cairns, R. A.
Vorgul, I.
When a mainly rectilinear electron beam is subject to significant magnetic compression, conservation of magnetic moment results in the formation of a horseshoe shaped velocity distribution. It has been shown that such a distribution is unstable to cyclotron emission and may be responsible for the generation of Auroral Kilometric Radiation (AKR) an intense rf emission sourced at high altitudes in the terrestrial auroral magnetosphere. PiC code simulations have been undertaken to investigate the dynamics of the cyclotron emission process in the absence of cavity boundaries with particular consideration of the spatial growth rate, spectral output and rf conversion efficiency. Computations reveal that a welldefined cyclotron emission process occurs albeit with a low spatial growth rate compared to waveguide bounded simulations. The rf output is near perpendicular to the electron beam with a slight backwardwave character reflected in the spectral output with a well defined peak at 2.68GHz, just below the relativistic electron cyclotron frequency. The corresponding rf conversion efficiency of 1.1% is comparable to waveguide bounded simulations and consistent with the predictions of kinetic theory that suggest efficient, spectrally well defined radiation emission can be obtained from an electron horseshoe distribution in the absence of radiation boundaries.

Laminar shocks in high power laser plasma interactions
http://hdl.handle.net/10023/5180
We propose a theory to describe laminar ion sound structures in a collisionless plasma. Reflection of a small fraction of the upstream ions converts the well known ion acoustic soliton into a structure with a steep potential gradient upstream and with downstream oscillations. The theory provides a simple interpretation of results dating back more than forty years but, more importantly, is shown to provide an explanation for recent observations on laser produced plasmas relevant to inertial fusion and to ion acceleration. (C) 2014 AIP Publishing LLC.
20140201T00:00:00Z
Cairns, R. A.
Bingham, R.
Norreys, P.
Trines, R.
We propose a theory to describe laminar ion sound structures in a collisionless plasma. Reflection of a small fraction of the upstream ions converts the well known ion acoustic soliton into a structure with a steep potential gradient upstream and with downstream oscillations. The theory provides a simple interpretation of results dating back more than forty years but, more importantly, is shown to provide an explanation for recent observations on laser produced plasmas relevant to inertial fusion and to ion acceleration. (C) 2014 AIP Publishing LLC.

Effect of collisions on amplification of laser beams by Brillouin scattering in plasmas
http://hdl.handle.net/10023/5173
We report on particle in cell simulations of energy transfer between a laser pump beam and a counterpropagating seed beam using the Brillouin scattering process in uniform plasma including collisions. The results presented show that the ion acoustic waves excited through naturally occurring Brillouin scattering of the pump field are preferentially damped without affecting the driven Brillouin scattering process resulting from the beating of the pump and seed fields together. We find that collisions, including the effects of Landau damping, allow for a more efficient transfer of energy between the laser beams, and a significant reduction in the amount of seed prepulse produced.
Authors KH, RT, DCS, RAC, RB were supported by EPSRC grant EP/G04239X/1.
20131001T00:00:00Z
Humphrey, K. A.
Trines, R. M. G. M.
Fiuza, F.
Speirs, D. C.
Norreys, P.
Cairns, R. A.
Silva, L. O.
Bingham, R.
We report on particle in cell simulations of energy transfer between a laser pump beam and a counterpropagating seed beam using the Brillouin scattering process in uniform plasma including collisions. The results presented show that the ion acoustic waves excited through naturally occurring Brillouin scattering of the pump field are preferentially damped without affecting the driven Brillouin scattering process resulting from the beating of the pump and seed fields together. We find that collisions, including the effects of Landau damping, allow for a more efficient transfer of energy between the laser beams, and a significant reduction in the amount of seed prepulse produced.

Quasigeostrophic shallowwater doublyconnected vortex equilibria and their stability
http://hdl.handle.net/10023/5172
We examine the form, properties, stability and evolution of doublyconnected (twovortex) relative equilibria in the singlelayer ƒplane quasigeostrophic shallowwater model of geophysical fluid dynamics. Three parameters completely describe families of equilibria in this system: the ratio γ =L/LD between the horizontal size of the vortices and the Rossby deformation length; the area ratio α of the smaller to the larger vortex; and the minimum distance δ between the two vortices. We vary 0 < γ ≤ 10 and 0.1 ≤ α ≤ 1.0, determining the boundary of stability δ = δC(γ,α). We also examine the nonlinear development of the instabilities and the transitions to other nearequilibrium configurations. Two modes of instability occur when δ < δC: a small γ asymmetric (wave 3) mode, which is absent for α ≳ 0.6; and a large γ mode. In general, major structural changes take place during the nonlinear evolution of the vortices, which near δC may be classified as follows: (i) vacillations about equilibrium for γ ≳ 2.5; (ii) partial straining out, associated with the small γ mode, where either one or both of the vortices get smaller for γ ≲ 2.5 and α ≲ 0.6; (iii) partial merger, occurring at the transition region between the two modes of instability, where one of the vortices gets bigger, and (iv) complete merger, associated with the largeγ mode. We also find that although conservative inviscid transitions to equilibria with the same energy, angular momentum and circulation are possible, they are not the preferred evolutionary path.
H.P. acknowledges the support of a NERC studentship. D.G.D. received support for this research from the UK Engineering and Physical Sciences Research Council (grant EP/H001794/1).
20130501T00:00:00Z
Plotka, Hanna
Dritschel, David Gerard
We examine the form, properties, stability and evolution of doublyconnected (twovortex) relative equilibria in the singlelayer ƒplane quasigeostrophic shallowwater model of geophysical fluid dynamics. Three parameters completely describe families of equilibria in this system: the ratio γ =L/LD between the horizontal size of the vortices and the Rossby deformation length; the area ratio α of the smaller to the larger vortex; and the minimum distance δ between the two vortices. We vary 0 < γ ≤ 10 and 0.1 ≤ α ≤ 1.0, determining the boundary of stability δ = δC(γ,α). We also examine the nonlinear development of the instabilities and the transitions to other nearequilibrium configurations. Two modes of instability occur when δ < δC: a small γ asymmetric (wave 3) mode, which is absent for α ≳ 0.6; and a large γ mode. In general, major structural changes take place during the nonlinear evolution of the vortices, which near δC may be classified as follows: (i) vacillations about equilibrium for γ ≳ 2.5; (ii) partial straining out, associated with the small γ mode, where either one or both of the vortices get smaller for γ ≲ 2.5 and α ≲ 0.6; (iii) partial merger, occurring at the transition region between the two modes of instability, where one of the vortices gets bigger, and (iv) complete merger, associated with the largeγ mode. We also find that although conservative inviscid transitions to equilibria with the same energy, angular momentum and circulation are possible, they are not the preferred evolutionary path.

The influence of the magnetic field on running penumbral waves in the solar chromosphere
http://hdl.handle.net/10023/5155
We use images of high spatial and temporal resolution, obtained using both ground and spacebased instrumentation, to investigate the role magnetic field inclination angles play in the propagation characteristics of running penumbral waves in the solar chromosphere. Analysis of a nearcircular sunspot, close to the center of the solar disk, reveals a smooth rise in oscillatory period as a function of distance from the umbral barycenter. However, in one directional quadrant, corresponding to the north direction, a pronounced kink in the perioddistance diagram is found. Utilizing a combination of the inversion of magnetic Stokes vectors and forcefree field extrapolations, we attribute this behavior to the cutoff frequency imposed by the magnetic field geometry in this location. A rapid, localized inclination of the magnetic field lines in the north direction results in a faster increase in the dominant periodicity due to an accelerated reduction in the cutoff frequency. For the first time, we reveal how the spatial distribution of dominant wave periods, obtained with one of the highest resolution solar instruments currently available, directly reflects the magnetic geometry of the underlying sunspot, thus opening up a wealth of possibilities in future magnetohydrodynamic seismology studies. In addition, the intrinsic relationships we find between the underlying magnetic field geometries connecting the photosphere to the chromosphere, and the characteristics of running penumbral waves observed in the upper chromosphere, directly supports the interpretation that running penumbral wave phenomena are the chromospheric signature of upwardly propagating magnetoacoustic waves generated in the photosphere.
D.B.J. acknowledges the European Commission and the Fonds Wetenschappelijk Onderzoek (FWO) for the award of a Marie Curie Pegasus Fellowship during which this work was initiated, in addition to the UK Science and Technology Facilities Council (STFC) for the award of an Ernest Rutherford Fellowship which allowed the completion of this project. The research carried out by V.E.R. is partly supported by grant MC FP7PEOPLE2011IRSES295272. T.V.D. acknowledges funding from the Odysseus Programme of the FWO Vlaanderen and from the EU's 7th Framework Programme as an ERG with grant number 276808. P.H.K. and D.H.M. are grateful to STFC for research support. This research has been funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office (IAP P7/08 CHARM).
20131203T00:00:00Z
Jess, David
Reznikova, V
Van Doorsselaere, Tom
Mackay, Duncan Hendry
Keys, Peter
We use images of high spatial and temporal resolution, obtained using both ground and spacebased instrumentation, to investigate the role magnetic field inclination angles play in the propagation characteristics of running penumbral waves in the solar chromosphere. Analysis of a nearcircular sunspot, close to the center of the solar disk, reveals a smooth rise in oscillatory period as a function of distance from the umbral barycenter. However, in one directional quadrant, corresponding to the north direction, a pronounced kink in the perioddistance diagram is found. Utilizing a combination of the inversion of magnetic Stokes vectors and forcefree field extrapolations, we attribute this behavior to the cutoff frequency imposed by the magnetic field geometry in this location. A rapid, localized inclination of the magnetic field lines in the north direction results in a faster increase in the dominant periodicity due to an accelerated reduction in the cutoff frequency. For the first time, we reveal how the spatial distribution of dominant wave periods, obtained with one of the highest resolution solar instruments currently available, directly reflects the magnetic geometry of the underlying sunspot, thus opening up a wealth of possibilities in future magnetohydrodynamic seismology studies. In addition, the intrinsic relationships we find between the underlying magnetic field geometries connecting the photosphere to the chromosphere, and the characteristics of running penumbral waves observed in the upper chromosphere, directly supports the interpretation that running penumbral wave phenomena are the chromospheric signature of upwardly propagating magnetoacoustic waves generated in the photosphere.

Simulating the formation of a sigmoidal flux rope in AR10977 from SOHO/MDI magnetograms
http://hdl.handle.net/10023/5154
The modeling technique of Mackay et al. is applied to simulate the coronal magnetic field of NOAA active region AR10977 over a seven day period (2007 December 210). The simulation is driven with a sequence of lineofsight component magnetograms from SOHO/MDI and evolves the coronal magnetic field though a continuous series of nonlinear forcefree states. Upon comparison with Hinode/XRT observations, results show that the simulation reproduces many features of the active region's evolution. In particular, it describes the formation of a flux rope across the polarity inversion line during flux cancellation. The flux rope forms at the same location as an observed Xray sigmoid. After five days of evolution, the free magnetic energy contained within the flux rope was found to be 3.9 × 1030 erg. This value is more than sufficient to account for the B1.4 GOES flare observed from the active region on 2007 December 7. At the time of the observed eruption, the flux rope was found to contain 20% of the active region flux. We conclude that the modeling technique proposed in Mackay et al.—which directly uses observed magnetograms to energize the coronal field—is a viable method to simulate the evolution of the coronal magnetic field.
G.P.S.G. acknowledges STFC for financial support. D.H.M. acknowledges the STFC, the Leverhulme Trust, and the EU FP7 funded project "SWIFF" (263340) for financial support. L.M.G. acknowledges to the Royal Society for a University Research Fellowship. K.A.M. acknowledges the Leverhulme Trust for financial support. Simulations were carried out on a STFC/SRIF funded UKMHD cluster at St Andrews.
20140220T00:00:00Z
Gibb, Gordon Peter Samuel
Mackay, Duncan Hendry
Green, Lucie
Meyer, Karen Alison
The modeling technique of Mackay et al. is applied to simulate the coronal magnetic field of NOAA active region AR10977 over a seven day period (2007 December 210). The simulation is driven with a sequence of lineofsight component magnetograms from SOHO/MDI and evolves the coronal magnetic field though a continuous series of nonlinear forcefree states. Upon comparison with Hinode/XRT observations, results show that the simulation reproduces many features of the active region's evolution. In particular, it describes the formation of a flux rope across the polarity inversion line during flux cancellation. The flux rope forms at the same location as an observed Xray sigmoid. After five days of evolution, the free magnetic energy contained within the flux rope was found to be 3.9 × 1030 erg. This value is more than sufficient to account for the B1.4 GOES flare observed from the active region on 2007 December 7. At the time of the observed eruption, the flux rope was found to contain 20% of the active region flux. We conclude that the modeling technique proposed in Mackay et al.—which directly uses observed magnetograms to energize the coronal field—is a viable method to simulate the evolution of the coronal magnetic field.

First comparison of wave observations from CoMP and AIA/SDO
http://hdl.handle.net/10023/5153
Context. Waves have long been thought to contribute to the heating of the solar corona and the generation of the solar wind. Recent observations have demonstrated evidence of quasiperiodic longitudinal disturbances and ubiquitous transverse wave propagation in many different coronal environments. Aims. This paper investigates signatures of different types of oscillatory behaviour, both above the solar limb and ondisk, by comparing findings from the Coronal Multichannel Polarimeter (CoMP) and the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO) for the same active region. Methods. We study both transverse and longitudinal motion by comparing and contrasting timedistance images of parallel and perpendicular cuts along/across active region fan loops. Comparisons between parallel spacetime diagram features in CoMP Doppler velocity and transverse oscillations in AIA images are made, together with spacetime analysis of propagating quasiperiodic intensity features seen near the base of loops in AIA. Results. Signatures of transverse motions are observed along the same magnetic structure using CoMP Doppler velocity (vphase = 600 → 750 km s1, P = 3 → 6 min) and in AIA/SDO above the limb (P = 3 → 8 min). Quasiperiodic intensity features (vphase = 100 → 200 km s1, P = 6 → 11 min) also travel along the base of the same structure. On the disk, signatures of both transverse and longitudinal intensity features were observed by AIA, and both show similar properties to signatures found along structures anchored in the same active region three days earlier above the limb. Correlated features are recovered by spacetime analysis of neighbouring tracks over perpendicular distances of ≲2.6 Mm.
I.D.M. acknowledges support of a Royal Society University Research Fellowship. The research leading to these results has also received funding from the European Commission Seventh Framework Programme (FP7/20072013) under the grant agreements SOLSPANET (project No. 269299, www.solspanet.eu/solspanet).
20130801T00:00:00Z
Threlfall, James William
De Moortel, Ineke
McIntosh, Scott
Bethge, Christian
Context. Waves have long been thought to contribute to the heating of the solar corona and the generation of the solar wind. Recent observations have demonstrated evidence of quasiperiodic longitudinal disturbances and ubiquitous transverse wave propagation in many different coronal environments. Aims. This paper investigates signatures of different types of oscillatory behaviour, both above the solar limb and ondisk, by comparing findings from the Coronal Multichannel Polarimeter (CoMP) and the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO) for the same active region. Methods. We study both transverse and longitudinal motion by comparing and contrasting timedistance images of parallel and perpendicular cuts along/across active region fan loops. Comparisons between parallel spacetime diagram features in CoMP Doppler velocity and transverse oscillations in AIA images are made, together with spacetime analysis of propagating quasiperiodic intensity features seen near the base of loops in AIA. Results. Signatures of transverse motions are observed along the same magnetic structure using CoMP Doppler velocity (vphase = 600 → 750 km s1, P = 3 → 6 min) and in AIA/SDO above the limb (P = 3 → 8 min). Quasiperiodic intensity features (vphase = 100 → 200 km s1, P = 6 → 11 min) also travel along the base of the same structure. On the disk, signatures of both transverse and longitudinal intensity features were observed by AIA, and both show similar properties to signatures found along structures anchored in the same active region three days earlier above the limb. Correlated features are recovered by spacetime analysis of neighbouring tracks over perpendicular distances of ≲2.6 Mm.

Erratum : "a numerical model of standard to blowout jets" (2013, ApJL, 769, L21)
http://hdl.handle.net/10023/5152
20130610T00:00:00Z
Archontis, Vasilis
Hood, Alan William

The emergence of weakly twisted magnetic fields in the Sun
http://hdl.handle.net/10023/5151
We have studied the emergence of a weakly twisted magnetic flux tube from the upper convection zone into the solar atmosphere. It is found that the rising magnetized plasma does not undergo the classical, single Ωshaped loop emergence, but it becomes unstable in two places, forming two magnetic lobes that are anchored in smallscale bipolar structures at the photosphere, between the two main flux concentrations. The two magnetic lobes rise and expand into the corona, forming an overall undulating magnetic flux system. The dynamical interaction of the lobes results in the triggering of highspeed and hot jets and the formation of successive cool and hot loops that coexist in the emerging flux region. Although the initial emerging field is weakly twisted, a highly twisted magnetic flux rope is formed at the low atmosphere, due to shearing and reconnection. The new flux rope (hereafter postemergence flux rope) does not erupt. It remains confined by the overlying field. Although there is no ejective eruption of the postemergence rope, it is found that a considerable amount of axial and azimuthal flux is transferred into the solar atmosphere during the emergence of the magnetic field.
The simulations were performed on the STFC and SRIF funded UKMHD cluster, at the University of St Andrews. K.T. and V.A. acknowledge EU support (IEF272549 grant).
20131101T00:00:00Z
Archontis, Vasilis
Hood, Alan William
Tsinganos, K
We have studied the emergence of a weakly twisted magnetic flux tube from the upper convection zone into the solar atmosphere. It is found that the rising magnetized plasma does not undergo the classical, single Ωshaped loop emergence, but it becomes unstable in two places, forming two magnetic lobes that are anchored in smallscale bipolar structures at the photosphere, between the two main flux concentrations. The two magnetic lobes rise and expand into the corona, forming an overall undulating magnetic flux system. The dynamical interaction of the lobes results in the triggering of highspeed and hot jets and the formation of successive cool and hot loops that coexist in the emerging flux region. Although the initial emerging field is weakly twisted, a highly twisted magnetic flux rope is formed at the low atmosphere, due to shearing and reconnection. The new flux rope (hereafter postemergence flux rope) does not erupt. It remains confined by the overlying field. Although there is no ejective eruption of the postemergence rope, it is found that a considerable amount of axial and azimuthal flux is transferred into the solar atmosphere during the emergence of the magnetic field.

Production of smallscale Alfvén waves by ionospheric depletion, nonlinear magnetosphereionosphere coupling and phase mixing
http://hdl.handle.net/10023/5150
Rockets and satellites have previously observed smallscale Alfven waves inside largescale downward fieldaligned currents, and numerical simulations have associated their formation with selfconsistent magnetosphereionosphere coupling. The origin of these waves was previously attributed to ionospheric feedback instability; however, we show that they arise in numerical experiments in which the instability is excluded. A new interpretation is proposed in which strong ionospheric depletion and associated current broadening (a nonlinear steepening/wavebreaking process) form magnetosphereionosphere waves inside a downward current region and these oscillations drive upgoing inertial Alfven waves in the overlying plasma. The resulting waves are governed by characteristic periods, which are a good match to previously observed periods for reasonable assumed conditions. Meanwhile, wavelengths perpendicular to the magnetic field initially map to an ionospheric scale comparable to the electron inertial length for the lowaltitude magnetosphere, but become shorter with time due to frequencybased phase mixing of boundary waves (a new manifestation of phase mixing). Under suitable conditions, these could act as seeds for the ionospheric feedback instability.
The authors acknowledge the International Space Science Institute (Switzerland) for funding the program that inspired this work. AJBR is grateful to the Royal Commission for the Exhibition of 1851 for present support and acknowledges an STFC studentship that funded part of this work.
20130403T00:00:00Z
Russell, A. J. B.
Wright, Andrew Nicholas
Streltsov, A. V.
Rockets and satellites have previously observed smallscale Alfven waves inside largescale downward fieldaligned currents, and numerical simulations have associated their formation with selfconsistent magnetosphereionosphere coupling. The origin of these waves was previously attributed to ionospheric feedback instability; however, we show that they arise in numerical experiments in which the instability is excluded. A new interpretation is proposed in which strong ionospheric depletion and associated current broadening (a nonlinear steepening/wavebreaking process) form magnetosphereionosphere waves inside a downward current region and these oscillations drive upgoing inertial Alfven waves in the overlying plasma. The resulting waves are governed by characteristic periods, which are a good match to previously observed periods for reasonable assumed conditions. Meanwhile, wavelengths perpendicular to the magnetic field initially map to an ionospheric scale comparable to the electron inertial length for the lowaltitude magnetosphere, but become shorter with time due to frequencybased phase mixing of boundary waves (a new manifestation of phase mixing). Under suitable conditions, these could act as seeds for the ionospheric feedback instability.

A numerical model of standard to blowout jets
http://hdl.handle.net/10023/5140
We report on threedimensional (3D) MHD simulations of the formation of jets produced during the emergence and eruption of solar magnetic fields. The interaction between an emerging and an ambient magnetic field in the solar atmosphere leads to (external) reconnection and the formation of "standard" jets with an inverse Yshaped configuration. Eventually, lowatmosphere (internal) reconnection of sheared fieldlines in the emerging flux region produces an erupting magnetic flux rope and a reconnection jet underneath it. The erupting plasma blows out the ambient field and, moreover, it unwinds as it is ejected into the outer solar atmosphere. The fast emission of the cool material that erupts together with the hot outflows due to external/internal reconnection form a wider "blowout" jet. We show the transition from "standard" to "blowout" jets and report on their 3D structure. The physical plasma properties of the jets are consistent with observational studies.
20130509T00:00:00Z
Archontis, Vasilis
Hood, A. W.
We report on threedimensional (3D) MHD simulations of the formation of jets produced during the emergence and eruption of solar magnetic fields. The interaction between an emerging and an ambient magnetic field in the solar atmosphere leads to (external) reconnection and the formation of "standard" jets with an inverse Yshaped configuration. Eventually, lowatmosphere (internal) reconnection of sheared fieldlines in the emerging flux region produces an erupting magnetic flux rope and a reconnection jet underneath it. The erupting plasma blows out the ambient field and, moreover, it unwinds as it is ejected into the outer solar atmosphere. The fast emission of the cool material that erupts together with the hot outflows due to external/internal reconnection form a wider "blowout" jet. We show the transition from "standard" to "blowout" jets and report on their 3D structure. The physical plasma properties of the jets are consistent with observational studies.

SWIFF : space weather integrated forecasting framework
http://hdl.handle.net/10023/5049
SWIFF is a project funded by the Seventh Framework Programme of the European Commission to study the mathematicalphysics models that form the basis for space weather forecasting. The phenomena of space weather span a tremendous scale of densities and temperature with scales ranging 10 orders of magnitude in space and time. Additionally even in local regions there are concurrent processes developing at the electron, ion and global scales strongly interacting with each other. The fundamental challenge in modelling space weather is the need to address multiple physics and multiple scales. Here we present our approach to take existing expertise in fluid and kinetic models to produce an integrated mathematical approach and software infrastructure that allows fluid and kinetic processes to be modelled together. SWIFF aims also at using this new infrastructure to model specific coupled processes at the Solar Corona, in the interplanetary space and in the interaction at the Earth magnetosphere.
This research has received funding from the European Commission’s FP7 Program with the grant agreement SWIFF (Project No. 2633430, swiff.eu). The KU Leuven simulations were conducted on the computational resources provided by the PRACE Tier0 Project No. 2011050747 (Curie supercomputer) and by the Flemish Supercomputer Center (VIC3). Additional computational support is provided at KU Leuven by the NASA NCCS (Discover) and NAS (Pleiades) Divisons, as part of the support to the NASA MMS Mission. UNIPI acknowledges the HPC resources of CINECA made available within the Distributed European Computing Initiative by the PRACE2IP, receiving funding from the European Community’s Seventh Framework Programme (FP7/ 20072013) under Grant Agreement No. nRI283493. Work at UNIPI was supported by the Italian Supercomputing Center – CINECA under the ISCRA initiative. Work at UNIPI was supported by the HPCEUROPA2 project (Project No. 228398) with the support of the European Commission – Capacities Area – Research Infrastructures. Work performed at IAP, ASCR was supported also by the Project RVO: 68378289.
20130218T00:00:00Z
Lapenta, Giovanni
Pierrard, Viviane
Keppens, Rony
Markidis, Stefano
Poedts, Stefaan
Šebek, Ondřej
Trávníček, Pavel M
Henri, Pierre
Califano, Francesco
Pegoraro, Francesco
Faganello, Matteo
Olshevsky, Vyacheslav
Restante, Anna Lisa
Nordlund, Åke
Trier Frederiksen, Jacob
Mackay, Duncan Hendry
Parnell, Clare Elizabeth
Bemporad, Alessandro
Susino, Roberto
Borremans, Kris
SWIFF is a project funded by the Seventh Framework Programme of the European Commission to study the mathematicalphysics models that form the basis for space weather forecasting. The phenomena of space weather span a tremendous scale of densities and temperature with scales ranging 10 orders of magnitude in space and time. Additionally even in local regions there are concurrent processes developing at the electron, ion and global scales strongly interacting with each other. The fundamental challenge in modelling space weather is the need to address multiple physics and multiple scales. Here we present our approach to take existing expertise in fluid and kinetic models to produce an integrated mathematical approach and software infrastructure that allows fluid and kinetic processes to be modelled together. SWIFF aims also at using this new infrastructure to model specific coupled processes at the Solar Corona, in the interplanetary space and in the interaction at the Earth magnetosphere.

Frequency of behavior witnessed and conformity in an everyday social context
http://hdl.handle.net/10023/5024
Conformity is thought to be an important force in human evolution because it has the potential to stabilize cultural homogeneity within groups and cultural diversity between groups. However, the effects of such conformity on cultural and biological evolution will depend much on the particular way in which individuals are influenced by the frequency of alternative behavioral options they witness. In a previous study we found that in a natural situation people displayed a tendency to be 'linearconformist'. When visitors to a Zoo exhibit were invited to write or draw answers to questions on cards to win a small prize and we manipulated the proportion of text versus drawings on display, we found a strong and significant effect of the proportion of text displayed on the proportion of text in the answers, a conformist effect that was largely linear with a small nonlinear component. However, although this overall effect is important to understand cultural evolution, it might mask a greater diversity of behavioral responses shaped by variables such as age, sex, social environment and attention of the participants. Accordingly we performed a further study explicitly to analyze the effects of these variables, together with the quality of the information participants' responses made available to further visitors. Results again showed a largely linear conformity effect that varied little with the variables analyzed. © 2014 Claidière et al.
20140620T00:00:00Z
Claidière, N.
Bowler, M.
Brookes, S.
Brown, R.
Whiten, A.
Conformity is thought to be an important force in human evolution because it has the potential to stabilize cultural homogeneity within groups and cultural diversity between groups. However, the effects of such conformity on cultural and biological evolution will depend much on the particular way in which individuals are influenced by the frequency of alternative behavioral options they witness. In a previous study we found that in a natural situation people displayed a tendency to be 'linearconformist'. When visitors to a Zoo exhibit were invited to write or draw answers to questions on cards to win a small prize and we manipulated the proportion of text versus drawings on display, we found a strong and significant effect of the proportion of text displayed on the proportion of text in the answers, a conformist effect that was largely linear with a small nonlinear component. However, although this overall effect is important to understand cultural evolution, it might mask a greater diversity of behavioral responses shaped by variables such as age, sex, social environment and attention of the participants. Accordingly we performed a further study explicitly to analyze the effects of these variables, together with the quality of the information participants' responses made available to further visitors. Results again showed a largely linear conformity effect that varied little with the variables analyzed. © 2014 Claidière et al.

The transterminator ion flow at Venus at solar minimum
http://hdl.handle.net/10023/4795
The transterminator ion flow in the Venusian ionosphere is observed at solar minimum for the first time. Such a flow, which transports ions from the day to the nightside, has been observed previously around solar maximum. At solar minimum this transport process is severely inhibited by the lower altitude of the ionopause. The observations presented were those made of the Venusian ionospheric plasma by the ASPERA4 experiment onboard the Venus Express spacecraft, and which constitute the first extensive insitu measurements of the plasma near solar minimum. Observations near the terminator of the energies of ions of ionospheric origin showed asymmetry between the noon and midnight sectors, which indicated an antisunward ion flow with a velocity of (2.5 +/ 1.5) km s(1). It is suggested that this ion flow contributes to maintaining the nightside ionosphere near the terminator region at solar minimum. The interpretation of the result was reinforced by observed asymmetries in the ion number counts. The observed dawndusk asymmetry was consistent with a nightward transport of ions while the noonmidnight observations indicated that the flow was highly variable but could contribute to the maintenance of the nightside ionosphere.
Financial support for this paper was provided by the UK Science and Technology Facilities Council under grant PP/E001157/1.
20121201T00:00:00Z
Wood, A. G.
Pryse, S. E.
Grande, M.
Whittaker, I. C.
Coates, A. J.
Husband, K.
Baumjohann, W.
Zhang, T. L.
Mazelle, C.
Kallio, E.
Fraenz, M.
McKennaLawlor, S.
Wurz, P.
The transterminator ion flow in the Venusian ionosphere is observed at solar minimum for the first time. Such a flow, which transports ions from the day to the nightside, has been observed previously around solar maximum. At solar minimum this transport process is severely inhibited by the lower altitude of the ionopause. The observations presented were those made of the Venusian ionospheric plasma by the ASPERA4 experiment onboard the Venus Express spacecraft, and which constitute the first extensive insitu measurements of the plasma near solar minimum. Observations near the terminator of the energies of ions of ionospheric origin showed asymmetry between the noon and midnight sectors, which indicated an antisunward ion flow with a velocity of (2.5 +/ 1.5) km s(1). It is suggested that this ion flow contributes to maintaining the nightside ionosphere near the terminator region at solar minimum. The interpretation of the result was reinforced by observed asymmetries in the ion number counts. The observed dawndusk asymmetry was consistent with a nightward transport of ions while the noonmidnight observations indicated that the flow was highly variable but could contribute to the maintenance of the nightside ionosphere.

Shallowwater vortex equilibria and their stability
http://hdl.handle.net/10023/4762
We first describe the equilibrium form and stability of steadilyrotating simplyconnected vortex patches in the singlelayer quasigeostrophic model of geophysical fluid dynamics. This model, valid for rotating shallowwater flow in the limit of small Rossby and Froude numbers, has an intrinsic length scale L called the "Rossby deformation length" relating the strength of stratification to that of the background rotation rate. Specifically, L = c/f where c = √gH is a characteristic gravitywave speed, g is gravity (or "reduced" gravity in a twolayer context where one layer is infinitely deep), H is the mean active layer depth, and f is the Coriolis frequency (here constant). We next introduce ageostrophic effects by using the full shallowwater model to generate what we call "quasiequilibria". These equilibria are not strictly steady, but radiate such weak gravity waves that they are steady for all practical purposes. Through an artificial ramping procedure, we ramp up the potential vorticity anomaly of the fluid particles in our quasigeostrophic equilibria to obtain shallowwater quasiequilibria at finite Rossby number. We show a few examples of these states in this paper.
20110101T00:00:00Z
Płotka, H.
Dritschel, D.G.
We first describe the equilibrium form and stability of steadilyrotating simplyconnected vortex patches in the singlelayer quasigeostrophic model of geophysical fluid dynamics. This model, valid for rotating shallowwater flow in the limit of small Rossby and Froude numbers, has an intrinsic length scale L called the "Rossby deformation length" relating the strength of stratification to that of the background rotation rate. Specifically, L = c/f where c = √gH is a characteristic gravitywave speed, g is gravity (or "reduced" gravity in a twolayer context where one layer is infinitely deep), H is the mean active layer depth, and f is the Coriolis frequency (here constant). We next introduce ageostrophic effects by using the full shallowwater model to generate what we call "quasiequilibria". These equilibria are not strictly steady, but radiate such weak gravity waves that they are steady for all practical purposes. Through an artificial ramping procedure, we ramp up the potential vorticity anomaly of the fluid particles in our quasigeostrophic equilibria to obtain shallowwater quasiequilibria at finite Rossby number. We show a few examples of these states in this paper.

Propagating coupled Alfvén and kink oscillations in an arbitrary inhomogeneous corona
http://hdl.handle.net/10023/4755
Observations have revealed ubiquitous transverse velocity perturbation waves propagating in the solar corona. We perform threedimensional numerical simulations of footpointdriven transverse waves propagating in a low β plasma. We consider the cases of distorted cylindrical flux tubes and a randomly generated inhomogeneous medium. When density structuring is present, mode coupling in inhomogeneous regions leads to the coupling of the kink mode to the Alfvén mode. The decay of the propagating kink wave is observed as energy is transferred to the local Alfvén mode. In all cases considered, modest changes in density were capable of efficiently converting energy from the driving footpoint motion to localized Alfv´en modes. We have demonstrated that mode coupling efficiently couples propagating kink perturbations to Alfvén modes in an arbitrary inhomogeneous medium. This has the consequence that transverse footpoint motions at the base of the corona will deposit energy to Alfvén modes in the corona.
D.J.P. acknowledges financial support from STFC. I.D.M. acknowledges support of a Royal Society University Research Fellowship.
20110410T00:00:00Z
Pascoe, David James
Wright, Andrew Nicholas
De Moortel, Ineke
Observations have revealed ubiquitous transverse velocity perturbation waves propagating in the solar corona. We perform threedimensional numerical simulations of footpointdriven transverse waves propagating in a low β plasma. We consider the cases of distorted cylindrical flux tubes and a randomly generated inhomogeneous medium. When density structuring is present, mode coupling in inhomogeneous regions leads to the coupling of the kink mode to the Alfvén mode. The decay of the propagating kink wave is observed as energy is transferred to the local Alfvén mode. In all cases considered, modest changes in density were capable of efficiently converting energy from the driving footpoint motion to localized Alfv´en modes. We have demonstrated that mode coupling efficiently couples propagating kink perturbations to Alfvén modes in an arbitrary inhomogeneous medium. This has the consequence that transverse footpoint motions at the base of the corona will deposit energy to Alfvén modes in the corona.

Modeling the dispersal of an active region : quantifying energy input into the corona
http://hdl.handle.net/10023/4754
In this paper, a new technique for modeling nonlinear forcefree fields directly from lineofsight magnetogram observations is presented. The technique uses sequences of magnetograms directly as lower boundary conditions to drive the evolution of coronal magnetic fields between successive forcefree equilibria over long periods of time. It is illustrated by applying it to SOHO: MDI observations of a decaying active region, NOAA AR 8005. The active region is modeled during a fourday period around its central meridian passage. Over this time, the dispersal of the active region is dominated by random motions due to smallscale convective cells. Through studying the buildup of magnetic energy in the model, it is found that such smallscale motions may inject anywhere from (2.53) × 1025 erg s1 of free magnetic energy into the coronal field. Most of this energy is stored within the center of the active region in the low corona, below 30 Mm. After four days, the buildup of free energy is 10% that of the corresponding potential field. This energy buildup is sufficient to explain the radiative losses at coronal temperatures within the active region. Smallscale convective motions therefore play an integral part in the energy balance of the corona. This new technique has wide ranging applications with the new highresolution, highcadence observations from the SDO:HMI and SDO:AIA instruments.
Funding: UK STFC. Royal Society Research Grants Scheme.
20110310T00:00:00Z
Mackay, Duncan Hendry
Green, Lucie
van Ballegooijen, Aad
In this paper, a new technique for modeling nonlinear forcefree fields directly from lineofsight magnetogram observations is presented. The technique uses sequences of magnetograms directly as lower boundary conditions to drive the evolution of coronal magnetic fields between successive forcefree equilibria over long periods of time. It is illustrated by applying it to SOHO: MDI observations of a decaying active region, NOAA AR 8005. The active region is modeled during a fourday period around its central meridian passage. Over this time, the dispersal of the active region is dominated by random motions due to smallscale convective cells. Through studying the buildup of magnetic energy in the model, it is found that such smallscale motions may inject anywhere from (2.53) × 1025 erg s1 of free magnetic energy into the coronal field. Most of this energy is stored within the center of the active region in the low corona, below 30 Mm. After four days, the buildup of free energy is 10% that of the corresponding potential field. This energy buildup is sufficient to explain the radiative losses at coronal temperatures within the active region. Smallscale convective motions therefore play an integral part in the energy balance of the corona. This new technique has wide ranging applications with the new highresolution, highcadence observations from the SDO:HMI and SDO:AIA instruments.

The effects of lineofsight integration on multistrand coronal loop oscillations
http://hdl.handle.net/10023/4752
IDM acknowledges support of a Royal Society University Research Fellowship.
20120210T00:00:00Z
De Moortel, Ineke
Pascoe, David James

Standing kink modes in threedimensional coronal loops
http://hdl.handle.net/10023/4745
So far, the straight flux tube model proposed by Edwin & Roberts is the most commonly used tool in practical coronal seismology, in particular, to infer values of the (coronal) magnetic field from observed, standing kink mode oscillations. In this paper, we compare the period predicted by this basic model with threedimensional (3D) numerical simulations of standing kink mode oscillations, as the period is a crucial parameter in the seismological inversion to determine the magnetic field. We perform numerical simulations of standing kink modes in both straight and curved 3D coronal loops and consider excitation by internal and external drivers. The period of oscillation for the displacement of dense coronal loops is determined by the loop length and the kink speed, in agreement with the estimate based on analytical theory for straight flux tubes. For curved coronal loops embedded in a magnetic arcade and excited by an external driver, a secondary mode with a period determined by the loop length and external Alfvén speed is also present. When a low number of oscillations is considered, these two periods can result in a single, nonresolved (broad) peak in the power spectrum, particularly for low values of the density contrast for which the two periods will be relatively similar. In that case (and for this particular geometry), the presence of this additional mode would lead to ambiguous seismological estimates of the magnetic field strength.
I.D.M. acknowledges support from a Royal Society University Research Fellowship. The computational work for this paper was carried out at the joint STFC and SFC (SRIF)fundedclusterattheUniversityofStAndrews(UK). The research leading to these results has also received funding from the European Commissions Seventh Framework Programme (FP7/20072013) under the grant agreement SOLSPANET (project No. 269299;www.solspanet.eu/solspanet).
20140311T00:00:00Z
De Moortel, Ineke
Pascoe, David James
So far, the straight flux tube model proposed by Edwin & Roberts is the most commonly used tool in practical coronal seismology, in particular, to infer values of the (coronal) magnetic field from observed, standing kink mode oscillations. In this paper, we compare the period predicted by this basic model with threedimensional (3D) numerical simulations of standing kink mode oscillations, as the period is a crucial parameter in the seismological inversion to determine the magnetic field. We perform numerical simulations of standing kink modes in both straight and curved 3D coronal loops and consider excitation by internal and external drivers. The period of oscillation for the displacement of dense coronal loops is determined by the loop length and the kink speed, in agreement with the estimate based on analytical theory for straight flux tubes. For curved coronal loops embedded in a magnetic arcade and excited by an external driver, a secondary mode with a period determined by the loop length and external Alfvén speed is also present. When a low number of oscillations is considered, these two periods can result in a single, nonresolved (broad) peak in the power spectrum, particularly for low values of the density contrast for which the two periods will be relatively similar. In that case (and for this particular geometry), the presence of this additional mode would lead to ambiguous seismological estimates of the magnetic field strength.

Simulating the "Sliding Doors" Effect Through Magnetic Flux Emergence
http://hdl.handle.net/10023/4742
Recent Hinode photospheric vector magnetogram observations have shown that the opposite polarities of a long arcade structure move apart and then come together. In addition to this "sliding doors" effect, orientations of horizontal magnetic fields along the polarity inversion line on the photosphere evolve from a normalpolarity configuration to an inverse one. To explain this behavior, a simple model by Okamoto et al. suggested that it is the result of the emergence of a twisted flux rope. Here, we model this scenario using a threedimensional megnatohydrodynamic simulation of a twisted flux rope emerging into a preexisting overlying arcade. We construct magnetograms from the simulation and compare them with the observations. The model produces the two signatures mentioned above. However, the cause of the "sliding doors" effect differs from the previous model.
D.M. acknowledges financial assistance from STFC. The computational work for this Letter was carried out on the joint STFC and SFC (SRIF) funded cluster at the University of St. Andrews. D.M. and A.W.H. acknowledge financial support form the European Commission through the SOLAIRE Network (MTRNCT2006035484).
20100604T00:00:00Z
MacTaggart, David
Hood, Alan William
Recent Hinode photospheric vector magnetogram observations have shown that the opposite polarities of a long arcade structure move apart and then come together. In addition to this "sliding doors" effect, orientations of horizontal magnetic fields along the polarity inversion line on the photosphere evolve from a normalpolarity configuration to an inverse one. To explain this behavior, a simple model by Okamoto et al. suggested that it is the result of the emergence of a twisted flux rope. Here, we model this scenario using a threedimensional megnatohydrodynamic simulation of a twisted flux rope emerging into a preexisting overlying arcade. We construct magnetograms from the simulation and compare them with the observations. The model produces the two signatures mentioned above. However, the cause of the "sliding doors" effect differs from the previous model.

The storage and dissipation of magnetic energy in the quiet sun corona determined from SDO/HMI magnetograms
http://hdl.handle.net/10023/4741
In recent years, higher cadence, higher resolution observations have revealed the quietSun photosphere to be complex and rapidly evolving. Since magnetic fields anchored in the photosphere extend up into the solar corona, it is expected that the smallscale coronal magnetic field exhibits similar complexity. For the first time, the quietSun coronal magnetic field is continuously evolved through a series of nonpotential, quasistatic equilibria, deduced from magnetograms observed by the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory, where the photospheric boundary condition which drives the coronal evolution exactly reproduces the observed magnetograms. The buildup, storage, and dissipation of magnetic energy within the simulations is studied. We find that the free magnetic energy built up and stored within the field is sufficient to explain smallscale, impulsive events such as nanoflares. On comparing with coronal images of the same region, the energy storage and dissipation visually reproduces many of the observed features. The results indicate that the complex smallscale magnetic evolution of a large number of magnetic features is a key element in explaining the nature of the solar corona.
2013ApJ...770L..18M
20130530T00:00:00Z
Meyer, Karen Alison
Sabol, Juraj
Mackay, Duncan Hendry
van Ballegooijen, Aad
In recent years, higher cadence, higher resolution observations have revealed the quietSun photosphere to be complex and rapidly evolving. Since magnetic fields anchored in the photosphere extend up into the solar corona, it is expected that the smallscale coronal magnetic field exhibits similar complexity. For the first time, the quietSun coronal magnetic field is continuously evolved through a series of nonpotential, quasistatic equilibria, deduced from magnetograms observed by the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory, where the photospheric boundary condition which drives the coronal evolution exactly reproduces the observed magnetograms. The buildup, storage, and dissipation of magnetic energy within the simulations is studied. We find that the free magnetic energy built up and stored within the field is sufficient to explain smallscale, impulsive events such as nanoflares. On comparing with coronal images of the same region, the energy storage and dissipation visually reproduces many of the observed features. The results indicate that the complex smallscale magnetic evolution of a large number of magnetic features is a key element in explaining the nature of the solar corona.

Potential Evidence for the Onset of Alfvénic Turbulence in Transequatorial Coronal Loops
http://hdl.handle.net/10023/4740
This study investigates Coronal Multichannel Polarimeter Dopplershift observations of a large, offlimb, transequatorial loop system observed on 2012 April 1011. Dopplershift oscillations with a broad range of frequencies are found to propagate along the loop with a speed of about 500 km s–1. The power spectrum of perturbations travelling up from both loop footpoints is remarkably symmetric, probably due to the almost perfect northsouth alignment of the loop system. Compared to the power spectrum at the footpoints of the loop, the Fourier power at the apex appears to be higher in the highfrequency part of the spectrum than expected from theoretical models. We suggest this excess highfrequency power could be tentative evidence for the onset of a cascade of the lowtomid frequency waves into (Alfvénic) turbulence.
20140210T00:00:00Z
De Moortel, Ineke
McIntosh, Scott
Threlfall, James William
Bethge, Christian
Liu, J
This study investigates Coronal Multichannel Polarimeter Dopplershift observations of a large, offlimb, transequatorial loop system observed on 2012 April 1011. Dopplershift oscillations with a broad range of frequencies are found to propagate along the loop with a speed of about 500 km s–1. The power spectrum of perturbations travelling up from both loop footpoints is remarkably symmetric, probably due to the almost perfect northsouth alignment of the loop system. Compared to the power spectrum at the footpoints of the loop, the Fourier power at the apex appears to be higher in the highfrequency part of the spectrum than expected from theoretical models. We suggest this excess highfrequency power could be tentative evidence for the onset of a cascade of the lowtomid frequency waves into (Alfvénic) turbulence.

The detection of numerous magnetic separators in a threedimensional magnetohydrodynamic model of solar emerging flux
http://hdl.handle.net/10023/4739
Magnetic separators in threedimensional (3D) magnetic fields are believed to be often associated with locations of magnetic reconnection. In this preliminary study, we investigate this relationship using data from a numerical resistive 3D MHD experiment of a solar flux emergence event. For the first time separators are detected in complex magnetic fields resulting from a 3D resistive MHD model of flux emergence. Two snapshots of the model, taken from different stages of its evolution, are analyzed. Numerous separators are found in both snapshots, and their properties, including their geometry, length, relationship to the magnetic null points, and integrated parallel electric field are studied. The separators reside at the junctions between the emerging flux, the overlying field, and two other flux domains that are newly formed by reconnection. The long separators, which connect clusters of nulls that lie either side of the emerging flux, pass through spatially localized regions of high parallel electric field and correspond to local maxima in integrated parallel electric field. These factors indicate that strong magnetic reconnection takes place along many of the separators, and that separators play a key role during the interaction of emerging and overlying flux.
20101220T00:00:00Z
Parnell, Clare Elizabeth
Maclean, Rhona Claire
Haynes, Andrew Lewis
Magnetic separators in threedimensional (3D) magnetic fields are believed to be often associated with locations of magnetic reconnection. In this preliminary study, we investigate this relationship using data from a numerical resistive 3D MHD experiment of a solar flux emergence event. For the first time separators are detected in complex magnetic fields resulting from a 3D resistive MHD model of flux emergence. Two snapshots of the model, taken from different stages of its evolution, are analyzed. Numerous separators are found in both snapshots, and their properties, including their geometry, length, relationship to the magnetic null points, and integrated parallel electric field are studied. The separators reside at the junctions between the emerging flux, the overlying field, and two other flux domains that are newly formed by reconnection. The long separators, which connect clusters of nulls that lie either side of the emerging flux, pass through spatially localized regions of high parallel electric field and correspond to local maxima in integrated parallel electric field. These factors indicate that strong magnetic reconnection takes place along many of the separators, and that separators play a key role during the interaction of emerging and overlying flux.

Globalscale consequences of magnetichelicity injection and condensation on the sun
http://hdl.handle.net/10023/4735
In the recent paper of Antiochos, a new concept for the injection of magnetic helicity into the solar corona by smallscale convective motions and its condensation onto polarity inversion lines (PILs) has been developed. We investigate this concept through global simulations of the Sun’s photospheric and coronal magnetic fields, and compare the results with the hemispheric pattern of solar filaments. Assuming that the vorticity of the cells is predominately counterclockwise/clockwise in the northern/southern hemisphere, the convective motions inject negative/positive helicity into each hemisphere. The simulations show that: (1) on a north–south oriented PIL, both differential rotation and convective motions inject the same sign of helicity, which matches that required to reproduce the hemispheric pattern of filaments. (2) On a highlatitude east–west oriented polar crown or subpolar crown PIL, the vorticity of the cells has to be approximately 2–3 times greater than the local differentialrotation gradient in order to overcome the incorrect sign of helicity injection from differential rotation. (3) In the declining phase of the cycle, as a bipole interacts with the polar field, in some cases, helicity condensation can reverse the effect of differential rotation along the east–west lead arm but not in all cases. The results show that this newly developed concept of magnetic helicity injection and condensation, in conjunction with the mechanisms used in Yeates et al., is a viable explanation for the hemispheric pattern of filaments. Future observational studies should focus on examining the vorticity component within convective motions to determine both its magnitude and latitudinal variation relative to the differentialrotation gradient on the Sun.
20140401T00:00:00Z
Mackay, Duncan Hendry
DeVore, Rick
Antiochos, Spiro
In the recent paper of Antiochos, a new concept for the injection of magnetic helicity into the solar corona by smallscale convective motions and its condensation onto polarity inversion lines (PILs) has been developed. We investigate this concept through global simulations of the Sun’s photospheric and coronal magnetic fields, and compare the results with the hemispheric pattern of solar filaments. Assuming that the vorticity of the cells is predominately counterclockwise/clockwise in the northern/southern hemisphere, the convective motions inject negative/positive helicity into each hemisphere. The simulations show that: (1) on a north–south oriented PIL, both differential rotation and convective motions inject the same sign of helicity, which matches that required to reproduce the hemispheric pattern of filaments. (2) On a highlatitude east–west oriented polar crown or subpolar crown PIL, the vorticity of the cells has to be approximately 2–3 times greater than the local differentialrotation gradient in order to overcome the incorrect sign of helicity injection from differential rotation. (3) In the declining phase of the cycle, as a bipole interacts with the polar field, in some cases, helicity condensation can reverse the effect of differential rotation along the east–west lead arm but not in all cases. The results show that this newly developed concept of magnetic helicity injection and condensation, in conjunction with the mechanisms used in Yeates et al., is a viable explanation for the hemispheric pattern of filaments. Future observational studies should focus on examining the vorticity component within convective motions to determine both its magnitude and latitudinal variation relative to the differentialrotation gradient on the Sun.

The Sun's global photospheric and coronal magnetic fields : observations and models
http://hdl.handle.net/10023/4714
In this review, our present day understanding of the Sun's global photospheric and coronal magnetic fields is discussed from both observational and theoretical viewpoints. Firstly, the largescale properties of photospheric magnetic fields are described, along with recent advances in photospheric magnetic flux transport models. Following this, the wide variety of theoretical models used to simulate global coronal magnetic fields are described. From this, the combined application of both magnetic flux transport simulations and coronal modeling techniques to describe the phenomena of coronal holes, the Sun's open magnetic flux and the hemispheric pattern of solar filaments is discussed. Finally, recent advances in noneruptive global MHD models are described. While the review focuses mainly on solar magnetic fields, recent advances in measuring and modeling stellar magnetic fields are described where appropriate. In the final section key areas of future research are identified.
2012LRSP....9....6M Funding: STFC, the Leverhulme Trust and European Commission’s Seventh Framework Programme (FP7/20072013) under the grant agreement SWIFF (project no. 263340, http://www.swiff.eu).
20121101T00:00:00Z
Mackay, Duncan Hendry
Yeates, Anthony Robinson
In this review, our present day understanding of the Sun's global photospheric and coronal magnetic fields is discussed from both observational and theoretical viewpoints. Firstly, the largescale properties of photospheric magnetic fields are described, along with recent advances in photospheric magnetic flux transport models. Following this, the wide variety of theoretical models used to simulate global coronal magnetic fields are described. From this, the combined application of both magnetic flux transport simulations and coronal modeling techniques to describe the phenomena of coronal holes, the Sun's open magnetic flux and the hemispheric pattern of solar filaments is discussed. Finally, recent advances in noneruptive global MHD models are described. While the review focuses mainly on solar magnetic fields, recent advances in measuring and modeling stellar magnetic fields are described where appropriate. In the final section key areas of future research are identified.

Laboratory astrophysics : investigation of planetary and astrophysical maser emission
http://hdl.handle.net/10023/4494
This paper describes a model for cyclotron maser emission applicable to planetary auroral radio emission, the stars UV Ceti and CU Virginus, blazar jets and astrophysical shocks. These emissions may be attributed to energetic electrons moving into convergent magnetic fields that are typically found in association with dipole like planetary magnetospheres or shocks. It is found that magnetic compression leads to the formation of a velocity distribution having a horseshoe shape as a result of conservation of the electron magnetic moment. Under certain plasma conditions where the local electron plasma frequency ωpe is much less than the cyclotron frequency ωce the distribution is found to be unstable to maser type radiation emission. We have established a laboratorybased facility that has verified many of the details of our original theoretical description and agrees well with numerical simulations. The experiment has demonstrated that the horseshoe distribution produces cyclotron emission at a frequency just below the local electron cyclotron frequency, with polarisation close to Xmode and propagating nearly perpendicularly to the electron beam motion. We discuss recent developments in the theory and simulation of the instability including addressing radiation escape problems, and relate these to the laboratory, space, and astrophysical observations. The experiments showed strong narrow band EM emissions at frequencies just below the coldplasma cyclotron frequency as predicted by the theory. Measurements of the conversion efficiency, mode and spectral content were in close agreement with the predictions of numerical simulations undertaken using a particleincell code and also with satellite observations confirming the horseshoe maser as an important emission mechanism in geophysical/astrophysical plasmas. In each case we address how the radiation can escape the plasma without suffering strong absorption at the second harmonic layer.
20130101T00:00:00Z
Speirs, David
Cairns, R Alan
Kellett, Barry
Vorgul, Irena
McConville, Sandra
Cross, Adrian
Phelps, Alan
Ronald, Kevin
Bingham, Robert
This paper describes a model for cyclotron maser emission applicable to planetary auroral radio emission, the stars UV Ceti and CU Virginus, blazar jets and astrophysical shocks. These emissions may be attributed to energetic electrons moving into convergent magnetic fields that are typically found in association with dipole like planetary magnetospheres or shocks. It is found that magnetic compression leads to the formation of a velocity distribution having a horseshoe shape as a result of conservation of the electron magnetic moment. Under certain plasma conditions where the local electron plasma frequency ωpe is much less than the cyclotron frequency ωce the distribution is found to be unstable to maser type radiation emission. We have established a laboratorybased facility that has verified many of the details of our original theoretical description and agrees well with numerical simulations. The experiment has demonstrated that the horseshoe distribution produces cyclotron emission at a frequency just below the local electron cyclotron frequency, with polarisation close to Xmode and propagating nearly perpendicularly to the electron beam motion. We discuss recent developments in the theory and simulation of the instability including addressing radiation escape problems, and relate these to the laboratory, space, and astrophysical observations. The experiments showed strong narrow band EM emissions at frequencies just below the coldplasma cyclotron frequency as predicted by the theory. Measurements of the conversion efficiency, mode and spectral content were in close agreement with the predictions of numerical simulations undertaken using a particleincell code and also with satellite observations confirming the horseshoe maser as an important emission mechanism in geophysical/astrophysical plasmas. In each case we address how the radiation can escape the plasma without suffering strong absorption at the second harmonic layer.

Magnetohydrodynamics dynamical relaxation of coronal magnetic fields : I. Parallel untwisted magnetic fields in 2D
http://hdl.handle.net/10023/4378
Context. For the last thirty years, most of the studies on the relaxation of stressed magnetic fields in the solar environment have only considered the Lorentz force, neglecting plasma contributions, and therefore, limiting every equilibrium to that of a forcefree field. Aims: Here we begin a study of the nonresistive evolution of finite beta plasmas and their relaxation to magnetohydrostatic states, where magnetic forces are balanced by plasmapressure gradients, by using a simple 2D scenario involving a hydromagnetic disturbance to a uniform magnetic field. The final equilibrium state is predicted as a function of the initial disturbances, with aims to demonstrate what happens to the plasma during the relaxation process and to see what effects it has on the final equilibrium state. Methods: A set of numerical experiments are run using a full MHD code, with the relaxation driven by magnetoacoustic waves damped by viscous effects. The numerical results are compared with analytical calculations made within the linear regime, in which the whole process must remain adiabatic. Particular attention is paid to the thermodynamic behaviour of the plasma during the relaxation. Results: The analytical predictions for the final non forcefree equilibrium depend only on the initial perturbations and the total pressure of the system. It is found that these predictions hold surprisingly well even for amplitudes of the perturbation far outside the linear regime. Conclusions: Including the effects of a finite plasma beta in relaxation experiments leads to significant differences from the forcefree case.
20100501T00:00:00Z
Fuentes Fernandez, Jorge
Parnell, Clare Elizabeth
Hood, Alan William
Context. For the last thirty years, most of the studies on the relaxation of stressed magnetic fields in the solar environment have only considered the Lorentz force, neglecting plasma contributions, and therefore, limiting every equilibrium to that of a forcefree field. Aims: Here we begin a study of the nonresistive evolution of finite beta plasmas and their relaxation to magnetohydrostatic states, where magnetic forces are balanced by plasmapressure gradients, by using a simple 2D scenario involving a hydromagnetic disturbance to a uniform magnetic field. The final equilibrium state is predicted as a function of the initial disturbances, with aims to demonstrate what happens to the plasma during the relaxation process and to see what effects it has on the final equilibrium state. Methods: A set of numerical experiments are run using a full MHD code, with the relaxation driven by magnetoacoustic waves damped by viscous effects. The numerical results are compared with analytical calculations made within the linear regime, in which the whole process must remain adiabatic. Particular attention is paid to the thermodynamic behaviour of the plasma during the relaxation. Results: The analytical predictions for the final non forcefree equilibrium depend only on the initial perturbations and the total pressure of the system. It is found that these predictions hold surprisingly well even for amplitudes of the perturbation far outside the linear regime. Conclusions: Including the effects of a finite plasma beta in relaxation experiments leads to significant differences from the forcefree case.

Flux emergence and coronal eruption
http://hdl.handle.net/10023/4376
Aims. Our aim is to study the photospheric flux distribution of a twisted flux tube that emerges from the solar interior. We also report on the eruption of a new flux rope when the emerging tube rises into a preexisting magnetic field in the corona. Methods. To study the evolution, we use 3D numerical simulations by solving the timedependent and resistive MHD equations. We qualitatively compare our numerical results with MDI magnetograms of emerging flux at the solar surface. Results. We find that the photospheric magnetic flux distribution consists of two regions of opposite polarities and elongated magnetic tails on the two sides of the polarity inversion line (PIL), depending on the azimuthal nature of the emerging field lines and the initial field strength of the rising tube. Their shape is progressively deformed due to plasma motions towards the PIL. Our results are in qualitative agreement with observational studies of magnetic flux emergence in active regions (ARs). Moreover, if the initial twist of the emerging tube is small, the photospheric magnetic field develops an undulating shape and does not possess tails. In all cases, we find that a new flux rope is formed above the original axis of the emerging tube that may erupt into the corona, depending on the strength of the ambient field.
20100501T00:00:00Z
Archontis, Vasilis
Hood, Alan William
Aims. Our aim is to study the photospheric flux distribution of a twisted flux tube that emerges from the solar interior. We also report on the eruption of a new flux rope when the emerging tube rises into a preexisting magnetic field in the corona. Methods. To study the evolution, we use 3D numerical simulations by solving the timedependent and resistive MHD equations. We qualitatively compare our numerical results with MDI magnetograms of emerging flux at the solar surface. Results. We find that the photospheric magnetic flux distribution consists of two regions of opposite polarities and elongated magnetic tails on the two sides of the polarity inversion line (PIL), depending on the azimuthal nature of the emerging field lines and the initial field strength of the rising tube. Their shape is progressively deformed due to plasma motions towards the PIL. Our results are in qualitative agreement with observational studies of magnetic flux emergence in active regions (ARs). Moreover, if the initial twist of the emerging tube is small, the photospheric magnetic field develops an undulating shape and does not possess tails. In all cases, we find that a new flux rope is formed above the original axis of the emerging tube that may erupt into the corona, depending on the strength of the ambient field.

Magnetohydrodynamic kink waves in twodimensional nonuniform prominence threads
http://hdl.handle.net/10023/4374
Aims. We analyse the oscillatory properties of resonantly damped transverse kink oscillations in twodimensional prominence threads. Methods. The fine structures are modelled as cylindrically symmetric magnetic flux tubes with a dense central part with prominence plasma properties and an evacuated part, both surrounded by coronal plasma. The equilibrium density is allowed to vary nonuniformly in both the transverse and the longitudinal directions. We examine the influence of longitudinal density structuring on periods, damping times, and damping rates for transverse kink modes computed by numerically solving the linear resistive magnetohydrodynamic (MHD) equations. Results. The relevant parameters are the length of the thread and the density in the evacuated part of the tube, two quantities that are difficult to directly estimate from observations. We find that both of them strongly influence the oscillatory periods and damping times, and to a lesser extent the damping ratios. The analysis of the spatial distribution of perturbations and of the energy flux into the resonances allows us to explain the obtained damping times. Conclusions. Implications for prominence seismology, the physics of resonantly damped kink modes in twodimensional magnetic flux tubes, and the heating of prominence plasmas are discussed.
20110901T00:00:00Z
Arregui, I
Soler, R
Ballester, J.
Wright, Andrew Nicholas
Aims. We analyse the oscillatory properties of resonantly damped transverse kink oscillations in twodimensional prominence threads. Methods. The fine structures are modelled as cylindrically symmetric magnetic flux tubes with a dense central part with prominence plasma properties and an evacuated part, both surrounded by coronal plasma. The equilibrium density is allowed to vary nonuniformly in both the transverse and the longitudinal directions. We examine the influence of longitudinal density structuring on periods, damping times, and damping rates for transverse kink modes computed by numerically solving the linear resistive magnetohydrodynamic (MHD) equations. Results. The relevant parameters are the length of the thread and the density in the evacuated part of the tube, two quantities that are difficult to directly estimate from observations. We find that both of them strongly influence the oscillatory periods and damping times, and to a lesser extent the damping ratios. The analysis of the spatial distribution of perturbations and of the energy flux into the resonances allows us to explain the obtained damping times. Conclusions. Implications for prominence seismology, the physics of resonantly damped kink modes in twodimensional magnetic flux tubes, and the heating of prominence plasmas are discussed.

Thermal conduction effects on the kink instability in coronal loops
http://hdl.handle.net/10023/4373
Context. Heating of the solar corona by nanoflares, which are small transient events in which stored magnetic energy is dissipated by magnetic reconnection, may occur as the result of the nonlinear phase of the kink instability (Hood et al. 2009). Because of the high temperatures reached through these reconnection events, thermal conduction cannot be ignored in the evolution of the kink instability. Aims. To study the effect of thermal conduction on the nonlinear evolution of the kink instability of a coronal loop. To assess the efficiency of loop heating and the role of thermal conduction, both during the kink instability and for the long time evolution of the loop. Methods. Numerically solve the 3D nonlinear magnetohydrodynamic equations to simulate the evolution of a coronal loop that is initially in an unstable equilibrium. The initial state has zero net current. A comparison is made of the time evolution of the loop with thermal conduction and without thermal conduction. Results. Thermal conduction along magnetic field lines reduces the local temperature. This leads to temperatures that are an order of magnitude lower than those obtained in the absence of thermal conductivity. Consequently, different spectral lines are activated with and without the inclusion of thermal conduction, which have consequences for observations of solar corona loops. The conduction process is also important on the timescale of the fast magnetohydrodynamic phenomena. It reduces the kinetic energy released by an order of magnitude. Conclusions. Thermal conduction plays an essential role in the kink instability of coronal loops and cannot be ignored in the forward modelling of such loops.
20110101T00:00:00Z
Botha, G. J. J.
Arber, T. D.
Hood, A. W.
Context. Heating of the solar corona by nanoflares, which are small transient events in which stored magnetic energy is dissipated by magnetic reconnection, may occur as the result of the nonlinear phase of the kink instability (Hood et al. 2009). Because of the high temperatures reached through these reconnection events, thermal conduction cannot be ignored in the evolution of the kink instability. Aims. To study the effect of thermal conduction on the nonlinear evolution of the kink instability of a coronal loop. To assess the efficiency of loop heating and the role of thermal conduction, both during the kink instability and for the long time evolution of the loop. Methods. Numerically solve the 3D nonlinear magnetohydrodynamic equations to simulate the evolution of a coronal loop that is initially in an unstable equilibrium. The initial state has zero net current. A comparison is made of the time evolution of the loop with thermal conduction and without thermal conduction. Results. Thermal conduction along magnetic field lines reduces the local temperature. This leads to temperatures that are an order of magnitude lower than those obtained in the absence of thermal conductivity. Consequently, different spectral lines are activated with and without the inclusion of thermal conduction, which have consequences for observations of solar corona loops. The conduction process is also important on the timescale of the fast magnetohydrodynamic phenomena. It reduces the kinetic energy released by an order of magnitude. Conclusions. Thermal conduction plays an essential role in the kink instability of coronal loops and cannot be ignored in the forward modelling of such loops.

Alfven wave phasemixing and damping in the ion cyclotron range of frequencies
http://hdl.handle.net/10023/4372
Aims. We determine the effect of the Hall term in the generalised Ohm's law on the damping and phase mixing of Alfven waves in the ion cyclotron range of frequencies in uniform and nonuniform equilibrium plasmas. Methods. Wave damping in a uniform plasma is treated analytically, whilst a Lagrangian remap code (Lare2d) is used to study Hall effects on damping and phase mixing in the presence of an equilibrium density gradient. Results. The magnetic energy associated with an initially Gaussian field perturbation in a uniform resistive plasma is shown to decay algebraically at a rate that is unaffected by the Hall term to leading order in k(2)delta(2)(i) where k is wavenumber and delta(i) is ion skin depth. A similar algebraic decay law applies to whistler perturbations in the limit k(2)delta(2)(i) >> 1. In a nonuniform plasma it is found that the spatiallyintegrated damping rate due to phase mixing is lower in Hall MHD than it is in MHD, but the reduction in the damping rate, which can be attributed to the effects of wave dispersion, tends to zero in both the weak and strong phase mixing limits.
20110101T00:00:00Z
Threlfall, J.
McClements, K. G.
De Moortel, I.
Aims. We determine the effect of the Hall term in the generalised Ohm's law on the damping and phase mixing of Alfven waves in the ion cyclotron range of frequencies in uniform and nonuniform equilibrium plasmas. Methods. Wave damping in a uniform plasma is treated analytically, whilst a Lagrangian remap code (Lare2d) is used to study Hall effects on damping and phase mixing in the presence of an equilibrium density gradient. Results. The magnetic energy associated with an initially Gaussian field perturbation in a uniform resistive plasma is shown to decay algebraically at a rate that is unaffected by the Hall term to leading order in k(2)delta(2)(i) where k is wavenumber and delta(i) is ion skin depth. A similar algebraic decay law applies to whistler perturbations in the limit k(2)delta(2)(i) >> 1. In a nonuniform plasma it is found that the spatiallyintegrated damping rate due to phase mixing is lower in Hall MHD than it is in MHD, but the reduction in the damping rate, which can be attributed to the effects of wave dispersion, tends to zero in both the weak and strong phase mixing limits.

Nonlinear wave propagation and reconnection at magnetic Xpoints in the Hall MHD regime
http://hdl.handle.net/10023/4368
Context: The highly dynamical, complex nature of the solar atmosphere naturally implies the presence of waves in a topologically varied magnetic environment. Here, the interaction of waves with topological features such as null points is inevitable and potentially important for energetics. The low resistivity of the solar coronal plasma implies that nonmagnetohydrodynamic (MHD) effects should be considered in studies of magnetic energy release in this environment. Aims: This paper investigates the role of the Hall term in the propagation and dissipation of waves, their interaction with 2D magnetic Xpoints and the nature of the resulting reconnection. Methods: A Lagrangian remap shockcapturing code (Lare2d) was used to study the evolution of an initial fast magnetoacoustic wave annulus for a range of values of the ion skin depth (δi) in resistive Hall MHD. A magnetic nullpoint finding algorithm was also used to locate and track the evolution of the multiple nullpoints that are formed in the system. Results: Depending on the ratio of ion skin depth to system size, our model demonstrates that Hall effects can play a key role in the wavenull interaction. In particular, the initial fastwave pulse now consists of whistler and ioncyclotron components; the dispersive nature of the whistler wave leads to (i) earlier interaction with the null; (ii) the creation of multiple additional, transient nulls and, hence, an increased number of energy release sites. In the Hall regime, the relevant timescales (such as the onset of reconnection and the period of the oscillatory relaxation) of the system are reduced significantly, and the reconnection rate is enhanced.
20120701T00:00:00Z
Threlfall, James William
Parnell, Clare Elizabeth
De Moortel, Ineke
McClements, Ken
Arber, Tony D.
Context: The highly dynamical, complex nature of the solar atmosphere naturally implies the presence of waves in a topologically varied magnetic environment. Here, the interaction of waves with topological features such as null points is inevitable and potentially important for energetics. The low resistivity of the solar coronal plasma implies that nonmagnetohydrodynamic (MHD) effects should be considered in studies of magnetic energy release in this environment. Aims: This paper investigates the role of the Hall term in the propagation and dissipation of waves, their interaction with 2D magnetic Xpoints and the nature of the resulting reconnection. Methods: A Lagrangian remap shockcapturing code (Lare2d) was used to study the evolution of an initial fast magnetoacoustic wave annulus for a range of values of the ion skin depth (δi) in resistive Hall MHD. A magnetic nullpoint finding algorithm was also used to locate and track the evolution of the multiple nullpoints that are formed in the system. Results: Depending on the ratio of ion skin depth to system size, our model demonstrates that Hall effects can play a key role in the wavenull interaction. In particular, the initial fastwave pulse now consists of whistler and ioncyclotron components; the dispersive nature of the whistler wave leads to (i) earlier interaction with the null; (ii) the creation of multiple additional, transient nulls and, hence, an increased number of energy release sites. In the Hall regime, the relevant timescales (such as the onset of reconnection and the period of the oscillatory relaxation) of the system are reduced significantly, and the reconnection rate is enhanced.

Phase mixing of nonlinear viscoresistive Alfvén waves
http://hdl.handle.net/10023/4367
Aims: We investigate the behaviour of nonlinear, nonideal Alfvén wave propagation within an inhomogeneous magnetic environment. Methods: The governing MHD equations are solved in 1D and 2D using both analytical techniques and numerical simulations. Results: We find clear evidence for the ponderomotive effect and viscoresistive heating. The ponderomotive effect generates a longitudinal component to the transverse Alfvén wave, with a frequency twice that of the driving frequency. Analytical work shows the addition of resistive heating. This leads to a substantial increase in the local temperature and thus gas pressure of the plasma, resulting in material being pushed along the magnetic field. In 2D, our system exhibits phase mixing and we observe an evolution in the location of the maximum heating, i.e. we find a drifting of the heating layer. Conclusions: Considering Alfvén wave propagation in 2D with an inhomogeneous density gradient, we find that the equilibrium density profile is significantly modified by both the flow of density due to viscoresistive heating and the nonlinear response to the localised heating through phase mixing.
20110201T00:00:00Z
McLaughlin, James Alexander
De Moortel, Ineke
Hood, Alan William
Aims: We investigate the behaviour of nonlinear, nonideal Alfvén wave propagation within an inhomogeneous magnetic environment. Methods: The governing MHD equations are solved in 1D and 2D using both analytical techniques and numerical simulations. Results: We find clear evidence for the ponderomotive effect and viscoresistive heating. The ponderomotive effect generates a longitudinal component to the transverse Alfvén wave, with a frequency twice that of the driving frequency. Analytical work shows the addition of resistive heating. This leads to a substantial increase in the local temperature and thus gas pressure of the plasma, resulting in material being pushed along the magnetic field. In 2D, our system exhibits phase mixing and we observe an evolution in the location of the maximum heating, i.e. we find a drifting of the heating layer. Conclusions: Considering Alfvén wave propagation in 2D with an inhomogeneous density gradient, we find that the equilibrium density profile is significantly modified by both the flow of density due to viscoresistive heating and the nonlinear response to the localised heating through phase mixing.

The period ratio for kink and sausage modes in a magnetic slab
http://hdl.handle.net/10023/4366
Aims. Increasing observational evidence of wave modes in the solar corona brings us to a closer understanding of that medium. Coronal seismology allows us to combine wave observations and theory to determine otherwise unknown parameters. The period ratio, P1/2P(2), between the period P1 of the fundamental mode and twice the period P2 of its first overtone, is one such tool of coronal seismology and its departure from unity provides information about the structure of the corona. Methods. We consider analytically the period ratio for the fast kink and sausage modes of a magnetic slab, discussing both an Epstein density profile and a simple step function profile. Results. Transverse density structuring in the form of an Epstein profile or a step function profile may contribute to the shift of the period ratio for long thin slablike structures.
A75 article number
20110201T00:00:00Z
Macnamara, C. K.
Roberts, B.
Aims. Increasing observational evidence of wave modes in the solar corona brings us to a closer understanding of that medium. Coronal seismology allows us to combine wave observations and theory to determine otherwise unknown parameters. The period ratio, P1/2P(2), between the period P1 of the fundamental mode and twice the period P2 of its first overtone, is one such tool of coronal seismology and its departure from unity provides information about the structure of the corona. Methods. We consider analytically the period ratio for the fast kink and sausage modes of a magnetic slab, discussing both an Epstein density profile and a simple step function profile. Results. Transverse density structuring in the form of an Epstein profile or a step function profile may contribute to the shift of the period ratio for long thin slablike structures.

Coronal heating and nanoflares : current sheet formation and heating
http://hdl.handle.net/10023/4364
Aims: Solar photospheric footpoint motions can produce strong, localised currents in the corona. A detailed understanding of the formation process and the resulting heating is important in modelling nanoflares, as a mechanism for heating the solar corona. Methods: A 3D MHD simulation is described in which an initially straight magnetic field is sheared in two directions. Grid resolutions up to 5123 were used and two boundary drivers were considered; one where the boundaries are continuously driven and one where the driving is switched off once a current layer is formed. Results: For both drivers a twisted current layer is formed. After a long time we see that, when the boundary driving has been switched off, the system relaxes towards a lower energy equilibrium. For the driver which continuously shears the magnetic field we see a repeating cycle of strong current structures forming, fragmenting and decreasing in magnitude and then building up again. Realistic coronal temperatures are obtained.
20131201T00:00:00Z
Bowness, Ruth
Hood, Alan William
Parnell, Clare Elizabeth
Aims: Solar photospheric footpoint motions can produce strong, localised currents in the corona. A detailed understanding of the formation process and the resulting heating is important in modelling nanoflares, as a mechanism for heating the solar corona. Methods: A 3D MHD simulation is described in which an initially straight magnetic field is sheared in two directions. Grid resolutions up to 5123 were used and two boundary drivers were considered; one where the boundaries are continuously driven and one where the driving is switched off once a current layer is formed. Results: For both drivers a twisted current layer is formed. After a long time we see that, when the boundary driving has been switched off, the system relaxes towards a lower energy equilibrium. For the driver which continuously shears the magnetic field we see a repeating cycle of strong current structures forming, fragmenting and decreasing in magnitude and then building up again. Realistic coronal temperatures are obtained.

Damping of kink waves by mode coupling. II. Parametric study and seismology
http://hdl.handle.net/10023/4363
Context: Recent observations of the corona reveal ubiquitous transverse velocity perturbations that undergo strong damping as they propagate. These can be understood in terms of propagating kink waves that undergo mode coupling in inhomogeneous regions. Aims: The use of these propagating waves as a seismological tool for the investigation of the solar corona depends upon an accurate understanding of how the mode coupling behaviour is determined by local plasma parameters. Our previous work suggests the exponential spatial damping profile provides a poor description of the behaviour of strongly damped kink waves. We aim to investigate the spatial damping profile in detail and provide a guide to the approximations most suitable for performing seismological inversions. Methods: We propose a general spatial damping profile based on analytical results that accounts for the initial Gaussian stage of damped kink waves as well as the asymptotic exponential stage considered by previous authors. The applicability of this profile is demonstrated by a full parametric study of the relevant physical parameters. The implication of this profile for seismological inversions is investigated. Results: The Gaussian damping profile is found to be most suitable for application as a seismological tool for observations of oscillations in loops with a low density contrast. This profile also provides accurate estimates for data in which only a few wavelengths or periods are observed.
20130201T00:00:00Z
Pascoe, David James
Hood, Alan William
De Moortel, Ineke
Wright, Andrew Nicholas
Context: Recent observations of the corona reveal ubiquitous transverse velocity perturbations that undergo strong damping as they propagate. These can be understood in terms of propagating kink waves that undergo mode coupling in inhomogeneous regions. Aims: The use of these propagating waves as a seismological tool for the investigation of the solar corona depends upon an accurate understanding of how the mode coupling behaviour is determined by local plasma parameters. Our previous work suggests the exponential spatial damping profile provides a poor description of the behaviour of strongly damped kink waves. We aim to investigate the spatial damping profile in detail and provide a guide to the approximations most suitable for performing seismological inversions. Methods: We propose a general spatial damping profile based on analytical results that accounts for the initial Gaussian stage of damped kink waves as well as the asymptotic exponential stage considered by previous authors. The applicability of this profile is demonstrated by a full parametric study of the relevant physical parameters. The implication of this profile for seismological inversions is investigated. Results: The Gaussian damping profile is found to be most suitable for application as a seismological tool for observations of oscillations in loops with a low density contrast. This profile also provides accurate estimates for data in which only a few wavelengths or periods are observed.

Cyclotron maser radiation from inhomogeneous plasmas
http://hdl.handle.net/10023/4335
Cyclotron maser instabilities are important in space, astrophysical, and laboratory plasmas. While extensive work has been done on these instabilities, most of it deals with homogeneous plasmas with uniform magnetic fields while in practice, of course, the systems are generally inhomogeneous. Here we expand on our previous work [R. A. Cairns, I. Vorgul, and R. Bingham, Phys. Rev. Lett. 101, 215003 (2008)] in which we showed that localized regions of instability can exist in an inhomogeneous plasma and that the way in which waves propagate away from this region is not necessarily obvious from the homogeneous plasma dispersion relation. While we consider only a simple ring distribution in velocity space, because of its tractability, the ideas may point toward understanding the behavior in the presence of more realistic distributions. The main object of the present work is to move away from consideration of the local dispersion relation and show how global growing eigenmodes can be constructed.
20110201T00:00:00Z
Cairns, R Alan
Vorgul, I.
Bingham, Robert
Ronald, K.
Speirs, D. C.
McConville, S. L.
Gillespie, K. M.
Bryson, R.
Phelps, A. D. R.
Kellett, B. J.
Cross, A. W.
Roberston, C. W.
Whyte, C. G.
He, W.
Cyclotron maser instabilities are important in space, astrophysical, and laboratory plasmas. While extensive work has been done on these instabilities, most of it deals with homogeneous plasmas with uniform magnetic fields while in practice, of course, the systems are generally inhomogeneous. Here we expand on our previous work [R. A. Cairns, I. Vorgul, and R. Bingham, Phys. Rev. Lett. 101, 215003 (2008)] in which we showed that localized regions of instability can exist in an inhomogeneous plasma and that the way in which waves propagate away from this region is not necessarily obvious from the homogeneous plasma dispersion relation. While we consider only a simple ring distribution in velocity space, because of its tractability, the ideas may point toward understanding the behavior in the presence of more realistic distributions. The main object of the present work is to move away from consideration of the local dispersion relation and show how global growing eigenmodes can be constructed.

Cyclotron maser emission : Stars, planets, and laboratory
http://hdl.handle.net/10023/4334
This paper is a review of results by the group over the past decade on auroral kilometric radiation and similar cyclotron emissions from stars and planets. These emissions are often attributed to a horseshoe or crescent shaped momentum distribution of energetic electrons moving into the convergent magnetic field which exists around polar regions of dipoletype stars and planets. We have established a laboratorybased facility that has verified many of the details of our original theoretical description and agrees well with numerical simulations. The experiment has demonstrated that the horseshoe distribution does indeed produce cyclotron emission at a frequency just below the local cyclotron frequency, with polarization close to Xmode and propagating nearly perpendicularly to the beam motion. We discuss recent developments in the theory and simulation of the instability including addressing a radiation escape problem and the effect of competing instabilities, relating these to the laboratory, space, and astrophysical observations.
20110501T00:00:00Z
Vorgul, I.
Kellett, B. J.
Cairns, R Alan
Bingham, Robert
Ronald, K.
Speirs, D.C.
McConville, S. L.
Gillespie, K. M.
Phelps, A. D. R.
This paper is a review of results by the group over the past decade on auroral kilometric radiation and similar cyclotron emissions from stars and planets. These emissions are often attributed to a horseshoe or crescent shaped momentum distribution of energetic electrons moving into the convergent magnetic field which exists around polar regions of dipoletype stars and planets. We have established a laboratorybased facility that has verified many of the details of our original theoretical description and agrees well with numerical simulations. The experiment has demonstrated that the horseshoe distribution does indeed produce cyclotron emission at a frequency just below the local cyclotron frequency, with polarization close to Xmode and propagating nearly perpendicularly to the beam motion. We discuss recent developments in the theory and simulation of the instability including addressing a radiation escape problem and the effect of competing instabilities, relating these to the laboratory, space, and astrophysical observations.

Energy dissipation and resolution of steep gradients in onedimensional Burgers flows
http://hdl.handle.net/10023/4333
Travelingwave solutions of the inviscid Burgers equation having smooth initial wave profiles of suitable shapes are known to develop shocks (infinite gradients) in finite times. Such singular solutions are characterized by energy spectra that scale with the wave number k as k−2. In the presence of viscosity ν>0, no shocks can develop, and smooth solutions remain so for all times t>0, eventually decaying to zero as t→∞. At peak energy dissipation, say t = t∗, the spectrum of such a smooth solution extends to a finite dissipation wave number kν and falls off more rapidly, presumably exponentially, for k>kν. The number N of Fourier modes within the socalled inertial range is proportional to kν. This represents the number of modes necessary to resolve the dissipation scale and can be thought of as the system’s number of degrees of freedom. The peak energy dissipation rate ϵ remains positive and becomes independent of ν in the inviscid limit. In this study, we carry out an analysis which verifies the dynamical features described above and derive upper bounds for ϵ and N. It is found that ϵ satisfies ϵ ≤ ν2α−1‖u∗‖∞2(1−α)‖(−Δ)α/2u∗‖2, where α<1 and u∗ = u(x,t∗) is the velocity field at t = t∗. Given ϵ>0 in the limit ν→0, this implies that the energy spectrum remains no steeper than k−2 in that limit. For the critical k−2 scaling, the bound for ϵ reduces to ϵ ≤ k0‖u0‖∞‖u0‖2, where k0 marks the lower end of the inertial range and u0 = u(x,0). This implies N ≤ L‖u0‖∞/ν, where L is the domain size, which is shown to coincide with a rigorous estimate for the number of degrees of freedom defined in terms of local Lyapunov exponents. We demonstrate both analytically and numerically an instance, where the k−2 scaling is uniquely realizable. The numerics also return ϵ and t∗, consistent with analytic values derived from the corresponding limiting weak solution.
20100301T00:00:00Z
Tran, Chuong Van
Dritschel, David Gerard
Travelingwave solutions of the inviscid Burgers equation having smooth initial wave profiles of suitable shapes are known to develop shocks (infinite gradients) in finite times. Such singular solutions are characterized by energy spectra that scale with the wave number k as k−2. In the presence of viscosity ν>0, no shocks can develop, and smooth solutions remain so for all times t>0, eventually decaying to zero as t→∞. At peak energy dissipation, say t = t∗, the spectrum of such a smooth solution extends to a finite dissipation wave number kν and falls off more rapidly, presumably exponentially, for k>kν. The number N of Fourier modes within the socalled inertial range is proportional to kν. This represents the number of modes necessary to resolve the dissipation scale and can be thought of as the system’s number of degrees of freedom. The peak energy dissipation rate ϵ remains positive and becomes independent of ν in the inviscid limit. In this study, we carry out an analysis which verifies the dynamical features described above and derive upper bounds for ϵ and N. It is found that ϵ satisfies ϵ ≤ ν2α−1‖u∗‖∞2(1−α)‖(−Δ)α/2u∗‖2, where α<1 and u∗ = u(x,t∗) is the velocity field at t = t∗. Given ϵ>0 in the limit ν→0, this implies that the energy spectrum remains no steeper than k−2 in that limit. For the critical k−2 scaling, the bound for ϵ reduces to ϵ ≤ k0‖u0‖∞‖u0‖2, where k0 marks the lower end of the inertial range and u0 = u(x,0). This implies N ≤ L‖u0‖∞/ν, where L is the domain size, which is shown to coincide with a rigorous estimate for the number of degrees of freedom defined in terms of local Lyapunov exponents. We demonstrate both analytically and numerically an instance, where the k−2 scaling is uniquely realizable. The numerics also return ϵ and t∗, consistent with analytic values derived from the corresponding limiting weak solution.

Boundary layer flow beneath an internal solitary wave of elevation
http://hdl.handle.net/10023/4331
The waveinduced flow over a fixed bottom boundary beneath an internal solitary wave of elevation propagating in an unsheared, twolayer, stably stratified fluid is investigated experimentally. Measurements of the velocity field close to the bottom boundary are presented to illustrate that in the lower layer the fluid velocity near the bottom reverses direction as the wave decelerates while higher in the water column the fluid velocity is in the same direction as the wave propagation. The observation is similar in nature to that for waveinduced flow beneath a surface solitary wave. Contrary to theoretical predictions for internal solitary waves, no evidence for either boundary layer separation or vortex formation is found beneath the front half of the wave in the adverse pressure gradient region of the flow.
20100201T00:00:00Z
Carr, Magda
Davies, P A
The waveinduced flow over a fixed bottom boundary beneath an internal solitary wave of elevation propagating in an unsheared, twolayer, stably stratified fluid is investigated experimentally. Measurements of the velocity field close to the bottom boundary are presented to illustrate that in the lower layer the fluid velocity near the bottom reverses direction as the wave decelerates while higher in the water column the fluid velocity is in the same direction as the wave propagation. The observation is similar in nature to that for waveinduced flow beneath a surface solitary wave. Contrary to theoretical predictions for internal solitary waves, no evidence for either boundary layer separation or vortex formation is found beneath the front half of the wave in the adverse pressure gradient region of the flow.

Effect of gravitational stratification on the propagation of a CME
http://hdl.handle.net/10023/4244
Our aim is to study the role of gravitational stratification on the propagation of CMEs. In particular, we assess how it influences the speed and shape of CMEs and under what conditions the flux rope ejection becomes a CME or when it is quenched. We ran a set of MHD simulations that adopt an eruptive initial magnetic configuration that has already been shown to be suitable for a flux rope ejection. We varied the temperature of the backgroud corona and the intensity of the initial magnetic field to tune the gravitational stratification and the amount of ejected magnetic flux. We used an automatic technique to track the expansion and the propagation of the magnetic flux rope in the MHD simulations. From the analysis of the parameter space, we evaluate the role of gravitational stratification on the CME speed and expansion. Our study shows that gravitational stratification plays a significant role in determining whether the flux rope ejection will turn into a full CME or whether the magnetic flux rope will stop in the corona. The CME speed is affected by the background corona where it travels faster when the corona is colder and when the initial magnetic field is more intense. The fastest CME we reproduce in our parameter space travels at 850 km/s. Moreover, the background gravitational stratification plays a role in the side expansion of the CME, and we find that when the background temperature is higher, the resulting shape of the CME is flattened more. Our study shows that although the initiation mechanisms of the CME are purely magnetic, the background coronal plasma plays a key role in the CME propagation, and full MHD models should be applied when one focusses especially on the production of a CME from a flux rope ejection.
20131202T00:00:00Z
Pagano, Paolo
Mackay, Duncan Hendry
Poedts, Stefaan
Our aim is to study the role of gravitational stratification on the propagation of CMEs. In particular, we assess how it influences the speed and shape of CMEs and under what conditions the flux rope ejection becomes a CME or when it is quenched. We ran a set of MHD simulations that adopt an eruptive initial magnetic configuration that has already been shown to be suitable for a flux rope ejection. We varied the temperature of the backgroud corona and the intensity of the initial magnetic field to tune the gravitational stratification and the amount of ejected magnetic flux. We used an automatic technique to track the expansion and the propagation of the magnetic flux rope in the MHD simulations. From the analysis of the parameter space, we evaluate the role of gravitational stratification on the CME speed and expansion. Our study shows that gravitational stratification plays a significant role in determining whether the flux rope ejection will turn into a full CME or whether the magnetic flux rope will stop in the corona. The CME speed is affected by the background corona where it travels faster when the corona is colder and when the initial magnetic field is more intense. The fastest CME we reproduce in our parameter space travels at 850 km/s. Moreover, the background gravitational stratification plays a role in the side expansion of the CME, and we find that when the background temperature is higher, the resulting shape of the CME is flattened more. Our study shows that although the initiation mechanisms of the CME are purely magnetic, the background coronal plasma plays a key role in the CME propagation, and full MHD models should be applied when one focusses especially on the production of a CME from a flux rope ejection.

Magnetohydrodynamics dynamical relaxation of coronal magnetic fields : IV. 3D tilted nulls
http://hdl.handle.net/10023/4084
In this paper we study current accumulations in 3D "tilted" nulls formed by a folding of the spine and fan. A nonzero component of current parallel to the fan is required such that the null's fan plane and spine are not perpendicular. Our aims are to provide valid magnetohydrostatic equilibria and to describe the current accumulations in various cases involving finite plasma pressure.To create our equilibrium current structures we use a full, nonresistive, magnetohydrodynamic (MHD) code so that no reconnection is allowed. A series of experiments are performed in which a perturbed 3D tilted null relaxes towards an equilibrium via real, viscous damping forces. Changes to the initial plasma pressure and to magnetic parameters are investigated systematically.An initially tilted fan is associated with a nonzero Lorentz force that drives the fan and spine to collapse towards each other, in a similar manner to the collapse of a 2D Xpoint. In the final equilibrium state for an initially radial null with only the current perpendicular to the spine, the current concentrates along the tilt axis of the fan and in a layer about the null point with a sharp peak at the null itself. The continued growth of this peak indicates that the system is in an asymptotic regime involving an infinite time singularity at the null. When the initial tilt disturbance (current perpendicular to the spine) is combined with a spiraltype disturbance (current parallel to the spine), the final current density concentrates in three regions: one on the fan along its tilt axis and two around the spine, above and below the fan. The increased area of current accumulation leads to a weakening of the singularity formed at the null. The 3D spinefan collapse with generic current studied here provides the ideal setup for nonsteady reconnection studies.
20130912T00:00:00Z
FuentesFernandez, Jorge
Parnell, Clare E.
In this paper we study current accumulations in 3D "tilted" nulls formed by a folding of the spine and fan. A nonzero component of current parallel to the fan is required such that the null's fan plane and spine are not perpendicular. Our aims are to provide valid magnetohydrostatic equilibria and to describe the current accumulations in various cases involving finite plasma pressure.To create our equilibrium current structures we use a full, nonresistive, magnetohydrodynamic (MHD) code so that no reconnection is allowed. A series of experiments are performed in which a perturbed 3D tilted null relaxes towards an equilibrium via real, viscous damping forces. Changes to the initial plasma pressure and to magnetic parameters are investigated systematically.An initially tilted fan is associated with a nonzero Lorentz force that drives the fan and spine to collapse towards each other, in a similar manner to the collapse of a 2D Xpoint. In the final equilibrium state for an initially radial null with only the current perpendicular to the spine, the current concentrates along the tilt axis of the fan and in a layer about the null point with a sharp peak at the null itself. The continued growth of this peak indicates that the system is in an asymptotic regime involving an infinite time singularity at the null. When the initial tilt disturbance (current perpendicular to the spine) is combined with a spiraltype disturbance (current parallel to the spine), the final current density concentrates in three regions: one on the fan along its tilt axis and two around the spine, above and below the fan. The increased area of current accumulation leads to a weakening of the singularity formed at the null. The 3D spinefan collapse with generic current studied here provides the ideal setup for nonsteady reconnection studies.

The structure of zonal jets in geostrophic turbulence
http://hdl.handle.net/10023/4064
The structure of zonal jets arising in forceddissipative, twodimensional turbulent flow on the βplane is investigated using highresolution, longtime numerical integrations, with particular emphasis on the latetime distribution of potential vorticity. The structure of the jets is found to depend in a simple way on a single nondimensional parameter, which may be conveniently expressed as the ratio LRh/Lg, where LRh = √U/β and Lg = (ε/β3)1/5 are two natural length scales arising in the problem; here U may be taken as the r.m.s. velocity, β is the background gradient of potential vorticity in the north–south direction, and ε is the rate of energy input by the forcing. It is shown that jet strength increases with LRh/Lg, with the limiting case of the potential vorticity staircase, comprising a monotonic, piecewiseconstant profile in the north–south direction, being approached for LRh/Lg ∼ 0(10). At lower values, eddies created by the forcing become sufficiently intense to continually disrupt the steepening of potential vorticity gradients in the jet cores, preventing strong jets from developing. Although detailed features such as the regularity of jet spacing and intensity are found to depend on the spectral distribution of the forcing, the approach of the staircase limit with increasing LRh/Lg is robust across a variety of different forcing types considered.
20121101T00:00:00Z
Scott, Richard Kirkness
Dritschel, David Gerard
The structure of zonal jets arising in forceddissipative, twodimensional turbulent flow on the βplane is investigated using highresolution, longtime numerical integrations, with particular emphasis on the latetime distribution of potential vorticity. The structure of the jets is found to depend in a simple way on a single nondimensional parameter, which may be conveniently expressed as the ratio LRh/Lg, where LRh = √U/β and Lg = (ε/β3)1/5 are two natural length scales arising in the problem; here U may be taken as the r.m.s. velocity, β is the background gradient of potential vorticity in the north–south direction, and ε is the rate of energy input by the forcing. It is shown that jet strength increases with LRh/Lg, with the limiting case of the potential vorticity staircase, comprising a monotonic, piecewiseconstant profile in the north–south direction, being approached for LRh/Lg ∼ 0(10). At lower values, eddies created by the forcing become sufficiently intense to continually disrupt the steepening of potential vorticity gradients in the jet cores, preventing strong jets from developing. Although detailed features such as the regularity of jet spacing and intensity are found to depend on the spectral distribution of the forcing, the approach of the staircase limit with increasing LRh/Lg is robust across a variety of different forcing types considered.

Consequences of spontaneous reconnection at a twodimensional nonforcefree current layer
http://hdl.handle.net/10023/4007
Magnetic neutral points, where the magnitude of the magnetic field vanishes locally, are potential locations for energy conversion in the solar corona. The fact that the magnetic field is identically zero at these points suggests that for the study of current sheet formation and of any subsequent resistive dissipation phase, a finite beta plasma should be considered, rather than neglecting the plasma pressure as has often been the case in the past. The rapid dissipation of a finite current layer in nonforcefree equilibrium is investigated numerically, after the sudden onset of an anomalous resistivity. The aim of this study is to determine how the energy is redistributed during the initial diffusion phase, and what is the nature of the outward transmission of information and energy. The resistivity rapidly diffuses the current at the null point. The presence of a plasma pressure allows the vast majority of the free energy to be transferred into internal energy. Most of the converted energy is used in direct heating of the surrounding plasma, and only about 3% is converted into kinetic energy, causing a perturbation in the magnetic field and the plasma which propagates away from the null at the local fast magnetoacoustic speed. The propagating pulses show a complex structure due to the highly nonuniform initial state. It is shown that this perturbation carries no net current as it propagates away from the null. The fact that, under the assumptions taken in this paper, most of the magnetic energy released in the reconnection converts internal energy of the plasma, may be highly important for the chromospheric and coronal heating problem.
20120201T00:00:00Z
Fuentes Fernandez, Jorge
Parnell, Clare Elizabeth
Hood, Alan William
Priest, Eric Ronald
Longcope, Dana
Magnetic neutral points, where the magnitude of the magnetic field vanishes locally, are potential locations for energy conversion in the solar corona. The fact that the magnetic field is identically zero at these points suggests that for the study of current sheet formation and of any subsequent resistive dissipation phase, a finite beta plasma should be considered, rather than neglecting the plasma pressure as has often been the case in the past. The rapid dissipation of a finite current layer in nonforcefree equilibrium is investigated numerically, after the sudden onset of an anomalous resistivity. The aim of this study is to determine how the energy is redistributed during the initial diffusion phase, and what is the nature of the outward transmission of information and energy. The resistivity rapidly diffuses the current at the null point. The presence of a plasma pressure allows the vast majority of the free energy to be transferred into internal energy. Most of the converted energy is used in direct heating of the surrounding plasma, and only about 3% is converted into kinetic energy, causing a perturbation in the magnetic field and the plasma which propagates away from the null at the local fast magnetoacoustic speed. The propagating pulses show a complex structure due to the highly nonuniform initial state. It is shown that this perturbation carries no net current as it propagates away from the null. The fact that, under the assumptions taken in this paper, most of the magnetic energy released in the reconnection converts internal energy of the plasma, may be highly important for the chromospheric and coronal heating problem.

Magnetohydrodynamics dynamical relaxation of coronal magnetic fields : III. 3D spiral nulls
http://hdl.handle.net/10023/3978
Context: The majority of studies on stressed 3D magnetic null points consider magnetic reconnection driven by an external perturbation, but the formation of a genuine current sheet equilibrium remains poorly understood. This problem has been considered more extensively in twodimensions, but lacks a generalization into 3D fields. Aims: 3D magnetic nulls are more complex than 2D nulls and the field can take a greater range of magnetic geometries local to the null. Here, we focus on one type and consider the dynamical nonresistive relaxation of 3D spiral nulls with initial spinealigned current. We aim to provide a valid magnetohydrostatic equilibrium, and describe the electric current accumulations in various cases, involving a finite plasma pressure. Methods: A full MHD code is used, with the resistivity set to zero so that reconnection is not allowed, to run a series of experiments in which a perturbed spiral 3D null point is allowed to relax towards an equilibrium, via real, viscous damping forces. Changes to the initial plasma pressure and other magnetic parameters are investigated systematically. Results: For the axisymmetric case, the evolution of the field and the plasma is such that it concentrates the current density in two coneshaped regions along the spine, thus concentrating the twist of the magnetic field around the spine, leaving a radial configuration in the fan plane. The plasma pressure redistributes in order to maintain the current density accumulations. However, it is found that changes in the initial plasma pressure do not modify the final state significantly. In the cases where the initial magnetic field is not axisymmetric, a infinitetime singularity of current perpendicular to the fan is found at the location of the null.
20120601T00:00:00Z
FuentesFernandez, Jorge
Parnell, Clare E.
Context: The majority of studies on stressed 3D magnetic null points consider magnetic reconnection driven by an external perturbation, but the formation of a genuine current sheet equilibrium remains poorly understood. This problem has been considered more extensively in twodimensions, but lacks a generalization into 3D fields. Aims: 3D magnetic nulls are more complex than 2D nulls and the field can take a greater range of magnetic geometries local to the null. Here, we focus on one type and consider the dynamical nonresistive relaxation of 3D spiral nulls with initial spinealigned current. We aim to provide a valid magnetohydrostatic equilibrium, and describe the electric current accumulations in various cases, involving a finite plasma pressure. Methods: A full MHD code is used, with the resistivity set to zero so that reconnection is not allowed, to run a series of experiments in which a perturbed spiral 3D null point is allowed to relax towards an equilibrium, via real, viscous damping forces. Changes to the initial plasma pressure and other magnetic parameters are investigated systematically. Results: For the axisymmetric case, the evolution of the field and the plasma is such that it concentrates the current density in two coneshaped regions along the spine, thus concentrating the twist of the magnetic field around the spine, leaving a radial configuration in the fan plane. The plasma pressure redistributes in order to maintain the current density accumulations. However, it is found that changes in the initial plasma pressure do not modify the final state significantly. In the cases where the initial magnetic field is not axisymmetric, a infinitetime singularity of current perpendicular to the fan is found at the location of the null.

The onset of impulsive bursty reconnection at a twodimensional current layer
http://hdl.handle.net/10023/3977
The sudden reconnection of a nonforce free 2D current layer, embedded in a lowbeta plasma, triggered by the onset of an anomalous resistivity, is studied in detail. The resulting behaviour consists of two main phases. Firstly, a transient reconnection phase, in which the current in the layer is rapidly dispersed and some flux is reconnected. This dispersal of current launches a family of small amplitude magnetic and plasma perturbations, which propagate away from the null at the local fast and slow magnetosonic speeds. The vast majority of the magnetic energy released in this phase goes into internal energy of the plasma, and only a tiny amount is converted into kinetic energy. In the wake of the outwards propagating pulses, an imbalance of Lorentz and pressure forces creates a stagnation flow which drives a regime of impulsive bursty reconnection, in which fast reconnection is turned on and off in a turbulent manner as the current density exceeds and falls below a critical value. During this phase, the null current density is continuously built up above a certain critical level, then dissipated very rapidly, and built up again, in a stochastic manner. Interestingly, the magnetic energy converted during this quasisteady phase is greater than that converted during the initial transient reconnection phase. Again essentially all the energy converted during this phase goes directly to internal energy. These results are of potential importance for solar flares and coronal heating, and set a conceptually important reference for future 3D studies.
20120509T00:00:00Z
FuentesFernández, J.
Parnell, Clare Elizabeth
Priest, Eric Ronald
The sudden reconnection of a nonforce free 2D current layer, embedded in a lowbeta plasma, triggered by the onset of an anomalous resistivity, is studied in detail. The resulting behaviour consists of two main phases. Firstly, a transient reconnection phase, in which the current in the layer is rapidly dispersed and some flux is reconnected. This dispersal of current launches a family of small amplitude magnetic and plasma perturbations, which propagate away from the null at the local fast and slow magnetosonic speeds. The vast majority of the magnetic energy released in this phase goes into internal energy of the plasma, and only a tiny amount is converted into kinetic energy. In the wake of the outwards propagating pulses, an imbalance of Lorentz and pressure forces creates a stagnation flow which drives a regime of impulsive bursty reconnection, in which fast reconnection is turned on and off in a turbulent manner as the current density exceeds and falls below a critical value. During this phase, the null current density is continuously built up above a certain critical level, then dissipated very rapidly, and built up again, in a stochastic manner. Interestingly, the magnetic energy converted during this quasisteady phase is greater than that converted during the initial transient reconnection phase. Again essentially all the energy converted during this phase goes directly to internal energy. These results are of potential importance for solar flares and coronal heating, and set a conceptually important reference for future 3D studies.

Magnetohydodynamics dynamical relaxation of coronal magnetic fields : II. 2D Magnetic XPoints
http://hdl.handle.net/10023/3976
Context. Magnetic neutral points are potential locations for energy conversion in the solar corona. 2D Xpoints have been widely studied in the past, but only a few of those studies have taken finite plasma beta effects into consideration, and none of them look at the dynamical evolution of the system. At the moment there exists no description of the formation of a nonforcefree equilibrium around a twodimensional Xpoint. Aims. Our aim is to provide a valid magnetohydrostatic equilibrium from the collapse of a 2D Xpoint in the presence of a finite plasma pressure, in which the current density is not simply concentrated in an infinitesimally thin, onedimensional current sheet, as found in forcefree solutions. In particular, we wish to determine if a finite pressure current sheet will still involve a singular current, and if so, what is the nature of the singularity. Methods. We use a full MHD code, with the resistivity set to zero, so that reconnection is not allowed, to run a series of experiments in which an Xpoint is perturbed and then is allowed to relax towards an equilibrium, via real, viscous damping forces. Changes to the magnitude of the perturbation and the initial plasma pressure are investigated systematically. Results. The final state found in our experiments is a “quasistatic” equilibrium where the viscous relaxation has completely ended, but the peak current density at the null increases very slowly following an asymptotic regime towards an infinite time singularity. Using a high grid resolution allows us to resolve the current structures in this state both in width and length. In comparison with the well known pressureless studies, the system does not evolve towards a thin current sheet, but concentrates the current at the null and the separatrices. The growth rate of the singularity is found to be tD, with 0 < D < 1. This rate depends directly on the initial plasma pressure, and decreases as the pressure is increased. At the end of our study, we present an analytical description of the system in a quasistatic nonsingular equilibrium at a given time, in which a finite thick current layer has formed at the null. The dynamical evolution of the system and the dependence of the final state on the initial plasma and magnetic quantities is discussed, as are the energetic consequences.
20111201T00:00:00Z
FuentesFernández, Jorge
Parnell, Clare Elizabeth
Hood, Alan William
Context. Magnetic neutral points are potential locations for energy conversion in the solar corona. 2D Xpoints have been widely studied in the past, but only a few of those studies have taken finite plasma beta effects into consideration, and none of them look at the dynamical evolution of the system. At the moment there exists no description of the formation of a nonforcefree equilibrium around a twodimensional Xpoint. Aims. Our aim is to provide a valid magnetohydrostatic equilibrium from the collapse of a 2D Xpoint in the presence of a finite plasma pressure, in which the current density is not simply concentrated in an infinitesimally thin, onedimensional current sheet, as found in forcefree solutions. In particular, we wish to determine if a finite pressure current sheet will still involve a singular current, and if so, what is the nature of the singularity. Methods. We use a full MHD code, with the resistivity set to zero, so that reconnection is not allowed, to run a series of experiments in which an Xpoint is perturbed and then is allowed to relax towards an equilibrium, via real, viscous damping forces. Changes to the magnitude of the perturbation and the initial plasma pressure are investigated systematically. Results. The final state found in our experiments is a “quasistatic” equilibrium where the viscous relaxation has completely ended, but the peak current density at the null increases very slowly following an asymptotic regime towards an infinite time singularity. Using a high grid resolution allows us to resolve the current structures in this state both in width and length. In comparison with the well known pressureless studies, the system does not evolve towards a thin current sheet, but concentrates the current at the null and the separatrices. The growth rate of the singularity is found to be tD, with 0 < D < 1. This rate depends directly on the initial plasma pressure, and decreases as the pressure is increased. At the end of our study, we present an analytical description of the system in a quasistatic nonsingular equilibrium at a given time, in which a finite thick current layer has formed at the null. The dynamical evolution of the system and the dependence of the final state on the initial plasma and magnetic quantities is discussed, as are the energetic consequences.

Magnetohydrodynamic simulations of the ejection of a magnetic flux rope
http://hdl.handle.net/10023/3855
Context. Coronal mass ejections (CME’s) are one of the most violent phenomena found on the Sun. One model to explain their occurrence is the flux rope ejection model. In this model, magnetic flux ropes form slowly over time periods of days to weeks. They then lose equilibrium and are ejected from the solar corona over a few hours. The contrasting time scales of formation and ejection pose a serious problem for numerical simulations. Aims. We simulate the whole life span of a flux rope from slow formation to rapid ejection and investigate whether magnetic flux ropes formed from a continuous magnetic field distribution, during a quasistatic evolution, can erupt to produce a CME. Methods. To model the full life span of magnetic flux ropes we couple two models. The global nonlinear forcefree field (GNLFFF) evolution model is used to follow the quasistatic formation of a flux rope. The MHD code ARMVAC is used to simulate the production of a CME through the loss of equilibrium and ejection of this flux rope. Results. We show that the two distinct models may be successfully coupled and that the flux rope is ejected out of our simulation box, where the outer boundary is placed at 2.5 R⊙. The plasma expelled during the flux rope ejection travels outward at a speed of 100 km s1, which is consistent with the observed speed of CMEs in the low corona. Conclusions. Our work shows that flux ropes formed in the GNLFFF can lead to the ejection of a mass loaded magnetic flux rope in full MHD simulations. Coupling the two distinct models opens up a new avenue of research to investigate phenomena where different phases of their evolution occur on drastically different time scales.
20130601T00:00:00Z
Pagano, Paolo
Mackay, Duncan Hendry
Poedts, Stefaan
Context. Coronal mass ejections (CME’s) are one of the most violent phenomena found on the Sun. One model to explain their occurrence is the flux rope ejection model. In this model, magnetic flux ropes form slowly over time periods of days to weeks. They then lose equilibrium and are ejected from the solar corona over a few hours. The contrasting time scales of formation and ejection pose a serious problem for numerical simulations. Aims. We simulate the whole life span of a flux rope from slow formation to rapid ejection and investigate whether magnetic flux ropes formed from a continuous magnetic field distribution, during a quasistatic evolution, can erupt to produce a CME. Methods. To model the full life span of magnetic flux ropes we couple two models. The global nonlinear forcefree field (GNLFFF) evolution model is used to follow the quasistatic formation of a flux rope. The MHD code ARMVAC is used to simulate the production of a CME through the loss of equilibrium and ejection of this flux rope. Results. We show that the two distinct models may be successfully coupled and that the flux rope is ejected out of our simulation box, where the outer boundary is placed at 2.5 R⊙. The plasma expelled during the flux rope ejection travels outward at a speed of 100 km s1, which is consistent with the observed speed of CMEs in the low corona. Conclusions. Our work shows that flux ropes formed in the GNLFFF can lead to the ejection of a mass loaded magnetic flux rope in full MHD simulations. Coupling the two distinct models opens up a new avenue of research to investigate phenomena where different phases of their evolution occur on drastically different time scales.

Twodimensional magnetohydrodynamic turbulence in the small magnetic Prandtl number limit
http://hdl.handle.net/10023/3698
In this paper we introduce a new method for computations of twodimensional magnetohydrodynamic (MHD) turbulence at low magnetic Prandtl number $\Pra=\nu/\eta$. When $\Pra \ll 1$, the magnetic field dissipates at a scale much larger than the velocity field. The method we utilise is a novel hybrid contourspectral method, the ``Combined Lagrangian Advection Method'', formally to integrate the equations with zero viscous dissipation. The method is compared with a standard pseudospectral method for decreasing $\Pra$ for the problem of decaying twodimensional MHD turbulence. The method is shown to agree well for a wide range of imposed magnetic field strengths. Examples of problems for which such a method may prove invaluable are also given.
20120701T00:00:00Z
Dritschel, David Gerard
Tobias, Steve
In this paper we introduce a new method for computations of twodimensional magnetohydrodynamic (MHD) turbulence at low magnetic Prandtl number $\Pra=\nu/\eta$. When $\Pra \ll 1$, the magnetic field dissipates at a scale much larger than the velocity field. The method we utilise is a novel hybrid contourspectral method, the ``Combined Lagrangian Advection Method'', formally to integrate the equations with zero viscous dissipation. The method is compared with a standard pseudospectral method for decreasing $\Pra$ for the problem of decaying twodimensional MHD turbulence. The method is shown to agree well for a wide range of imposed magnetic field strengths. Examples of problems for which such a method may prove invaluable are also given.

On energetics and inertialrange scaling laws of twodimensional magnetohydrodynamic turbulence
http://hdl.handle.net/10023/3668
We study twodimensional magnetohydrodynamic turbulence, with an emphasis on its energetics and inertial range scaling laws. A detailed spectral analysis shows that dynamo triads (those converting kinetic into magnetic energy) are associated with a direct magnetic energy flux while antidynamo triads (those converting magnetic into kinetic energy) are associated with an inverse magnetic energy flux. As both dynamo and antidynamo interacting triads are integral parts of the direct energy transfer, the antidynamo inverse flux partially neutralizes the dynamo direct flux, arguably resulting in relatively weak direct energy transfer and giving rise to dynamo saturation. This result is consistent with a qualitative prediction of energy transfer reduction owing to Alfv\'en wave effects by the IroshnikovKraichnan theory (which was originally formulated for magnetohydrodynamic turbulence in three dimensions). We numerically confirm the correlation between dynamo action and direct magnetic energy flux and investigate the applicability of quantitative aspects of the IroshnikovKraichnan theory to the present case, particularly its predictions of energy equipartition and $k^{3/2}$ spectra in the energy inertial range. It is found that for turbulence satisfying the Kraichnan condition of magnetic energy at large scales exceeding total energy in the inertial range, the kinetic energy spectrum, which is significantly shallower than $k^{3/2}$, is shallower than its magnetic counterpart. This result suggests no energy equipartition. The total energy spectrum appears to depend on the energy composition of the turbulence but is clearly shallower than $k^{3/2}$ for $r\approx2$, even at moderate resolutions. Here $r\approx2$ is the magnetictokinetic energy ratio during the stage when the turbulence can be considered fully developed. The implication of the present findings is discussed in conjunction with further numerical results on the dependence of the energy dissipation rate on resolution.
L. Blackbourn was supported by an EPSRC postgraduate studentship.
20120701T00:00:00Z
Blackbourn, Luke Austen Kazimierz
Tran, Chuong Van
We study twodimensional magnetohydrodynamic turbulence, with an emphasis on its energetics and inertial range scaling laws. A detailed spectral analysis shows that dynamo triads (those converting kinetic into magnetic energy) are associated with a direct magnetic energy flux while antidynamo triads (those converting magnetic into kinetic energy) are associated with an inverse magnetic energy flux. As both dynamo and antidynamo interacting triads are integral parts of the direct energy transfer, the antidynamo inverse flux partially neutralizes the dynamo direct flux, arguably resulting in relatively weak direct energy transfer and giving rise to dynamo saturation. This result is consistent with a qualitative prediction of energy transfer reduction owing to Alfv\'en wave effects by the IroshnikovKraichnan theory (which was originally formulated for magnetohydrodynamic turbulence in three dimensions). We numerically confirm the correlation between dynamo action and direct magnetic energy flux and investigate the applicability of quantitative aspects of the IroshnikovKraichnan theory to the present case, particularly its predictions of energy equipartition and $k^{3/2}$ spectra in the energy inertial range. It is found that for turbulence satisfying the Kraichnan condition of magnetic energy at large scales exceeding total energy in the inertial range, the kinetic energy spectrum, which is significantly shallower than $k^{3/2}$, is shallower than its magnetic counterpart. This result suggests no energy equipartition. The total energy spectrum appears to depend on the energy composition of the turbulence but is clearly shallower than $k^{3/2}$ for $r\approx2$, even at moderate resolutions. Here $r\approx2$ is the magnetictokinetic energy ratio during the stage when the turbulence can be considered fully developed. The implication of the present findings is discussed in conjunction with further numerical results on the dependence of the energy dissipation rate on resolution.

Twodimensional magnetohydrodynamic turbulence in the limits of infinite and vanishing magnetic Prandtl number
http://hdl.handle.net/10023/3539
We study both theoretically and numerically twodimensional magnetohydrodynamic turbulence at infinite and zero magnetic Prandtl number $Pm$ (and the limits thereof), with an emphasis on solution regularity. For $Pm=0$, both $\norm{\omega}^2$ and $\norm{j}^2$, where $\omega$ and $j$ are, respectively, the vorticity and current, are uniformly bounded. Furthermore, $\norm{\nabla j}^2$ is integrable over $[0,\infty)$. The uniform boundedness of $\norm{\omega}^2$ implies that in the presence of vanishingly small viscosity $\nu$ (i.e. in the limit $Pm\to0$), the kinetic energy dissipation rate $\nu\norm{\omega}^2$ vanishes for all times $t$, including $t=\infty$. Furthermore, for sufficiently small $Pm$, this rate decreases linearly with $Pm$. This linear behaviour of $\nu\norm{\omega}^2$ is investigated and confirmed by highresolution simulations with $Pm$ in the range $[1/64,1]$. Several criteria for solution regularity are established and numerically tested. As $Pm$ is decreased from unity, the ratio $\norm{\omega}_\infty/\norm{\omega}$ is observed to increase relatively slowly. This, together with the integrability of $\norm{\nabla j}^2$, suggests global regularity for $Pm=0$. When $Pm=\infty$, global regularity is secured when either $\norm{\nabla\u}_\infty/\norm{\omega}$, where $\u$ is the fluid velocity, or $\norm{j}_\infty/\norm{j}$ is bounded. The former is plausible given the presence of viscous effects for this case. Numerical results over the range $Pm\in[1,64]$ show that $\norm{\nabla\u}_\infty/\norm{\omega}$ varies slightly (with similar behaviour for $\norm{j}_\infty/\norm{j}$), thereby lending strong support for the possibility $\norm{\nabla\u}_\infty/\norm{\omega}<\infty$ in the limit $Pm\to\infty$. The peak of the magnetic energy dissipation rate $\mu\norm{j}^2$ is observed to decrease rapidly as $Pm$ is increased. This result suggests the possibility $\norm{j}^2<\infty$ in the limit $Pm\to\infty$. We discuss further evidence for the boundedness of the ratios $\norm{\omega}_\infty/\norm{\omega}$, $\norm{\nabla\u}_\infty/\norm{\omega}$ and $\norm{j}_\infty/\norm{j}$ in conjunction with observation on the density of filamentary structures in the vorticity, velocity gradient and current fields.
LAKB was supported by an EPSRC postgraduate studentship.
20130601T00:00:00Z
Tran, Chuong Van
Yu, Xinwei
Blackbourn, Luke Austen Kazimierz
We study both theoretically and numerically twodimensional magnetohydrodynamic turbulence at infinite and zero magnetic Prandtl number $Pm$ (and the limits thereof), with an emphasis on solution regularity. For $Pm=0$, both $\norm{\omega}^2$ and $\norm{j}^2$, where $\omega$ and $j$ are, respectively, the vorticity and current, are uniformly bounded. Furthermore, $\norm{\nabla j}^2$ is integrable over $[0,\infty)$. The uniform boundedness of $\norm{\omega}^2$ implies that in the presence of vanishingly small viscosity $\nu$ (i.e. in the limit $Pm\to0$), the kinetic energy dissipation rate $\nu\norm{\omega}^2$ vanishes for all times $t$, including $t=\infty$. Furthermore, for sufficiently small $Pm$, this rate decreases linearly with $Pm$. This linear behaviour of $\nu\norm{\omega}^2$ is investigated and confirmed by highresolution simulations with $Pm$ in the range $[1/64,1]$. Several criteria for solution regularity are established and numerically tested. As $Pm$ is decreased from unity, the ratio $\norm{\omega}_\infty/\norm{\omega}$ is observed to increase relatively slowly. This, together with the integrability of $\norm{\nabla j}^2$, suggests global regularity for $Pm=0$. When $Pm=\infty$, global regularity is secured when either $\norm{\nabla\u}_\infty/\norm{\omega}$, where $\u$ is the fluid velocity, or $\norm{j}_\infty/\norm{j}$ is bounded. The former is plausible given the presence of viscous effects for this case. Numerical results over the range $Pm\in[1,64]$ show that $\norm{\nabla\u}_\infty/\norm{\omega}$ varies slightly (with similar behaviour for $\norm{j}_\infty/\norm{j}$), thereby lending strong support for the possibility $\norm{\nabla\u}_\infty/\norm{\omega}<\infty$ in the limit $Pm\to\infty$. The peak of the magnetic energy dissipation rate $\mu\norm{j}^2$ is observed to decrease rapidly as $Pm$ is increased. This result suggests the possibility $\norm{j}^2<\infty$ in the limit $Pm\to\infty$. We discuss further evidence for the boundedness of the ratios $\norm{\omega}_\infty/\norm{\omega}$, $\norm{\nabla\u}_\infty/\norm{\omega}$ and $\norm{j}_\infty/\norm{j}$ in conjunction with observation on the density of filamentary structures in the vorticity, velocity gradient and current fields.

Note on solution regularity of the generalized magnetohydrodynamic equations with partial dissipation
http://hdl.handle.net/10023/3538
In this brief note we study the ndimensional magnetohydrodynamic equations with hyperviscosity and zero resistivity. We prove global regularity of solutions when the hyperviscosity is sufficiently strong.
20130701T00:00:00Z
Tran, Chuong Van
Yu, Xinwei
Zhai, Zhichun
In this brief note we study the ndimensional magnetohydrodynamic equations with hyperviscosity and zero resistivity. We prove global regularity of solutions when the hyperviscosity is sufficiently strong.

Solar magnetic carpet III : coronal modelling of synthetic magnetograms
http://hdl.handle.net/10023/3536
20130901T00:00:00Z
Meyer, Karen Alison
Mackay, Duncan Hendry
van Ballegooijen, Aad
Parnell, Clare Elizabeth

On global regularity of 2D generalized magnetohydrodynamic equations
http://hdl.handle.net/10023/3401
In this article we study the global regularity of 2D generalized magnetohydrodynamic equations (2D GMHD), in which the dissipation terms are –ν(–Δ)αu and –κ(–Δ)βb. We show that smooth solutions are global in the following three cases: α≥1/2, β≥1; 0≤α≤1/2, 2α+β>2; α≥2, β=0. We also show that in the inviscid case ν=0, if β>1, then smooth solutions are global as long as the direction of the magnetic field remains smooth enough.
20130515T00:00:00Z
Tran, Chuong Van
Yu, Xinwei
Zhai, Zhichun
In this article we study the global regularity of 2D generalized magnetohydrodynamic equations (2D GMHD), in which the dissipation terms are –ν(–Δ)αu and –κ(–Δ)βb. We show that smooth solutions are global in the following three cases: α≥1/2, β≥1; 0≤α≤1/2, 2α+β>2; α≥2, β=0. We also show that in the inviscid case ν=0, if β>1, then smooth solutions are global as long as the direction of the magnetic field remains smooth enough.

Sharp global nonlinear stability for a fluid overlying a highly porous material
http://hdl.handle.net/10023/3399
The stability of convection in a twolayer system in which a layer of fluid with a temperaturedependent viscosity overlies and saturates a highly porous material is studied. Owing to the difficulties associated with incorporating the nonlinear advection term in the NavierStokes equations into a stability analysis, previous literature on fluid/porous thermal convection has modelled the fluid using the linear Stokes equations. This paper derives global stability for the full nonlinear system, by utilizing a model proposed by Ladyzhenskaya. The nonlinear stability boundaries are shown to be sharp when compared with the linear instability thresholds.
20100108T00:00:00Z
Hill, Antony A.
Carr, Magda
The stability of convection in a twolayer system in which a layer of fluid with a temperaturedependent viscosity overlies and saturates a highly porous material is studied. Owing to the difficulties associated with incorporating the nonlinear advection term in the NavierStokes equations into a stability analysis, previous literature on fluid/porous thermal convection has modelled the fluid using the linear Stokes equations. This paper derives global stability for the full nonlinear system, by utilizing a model proposed by Ladyzhenskaya. The nonlinear stability boundaries are shown to be sharp when compared with the linear instability thresholds.

Nonlinear stability of the onedomain approach to modelling convection in superposed fluid and porous layers
http://hdl.handle.net/10023/3398
Studies of the nonlinear stability of fluid/porous systems have been developed very recently. A twodomain modelling approach has been adopted in previous works, but was restricted to specific configurations. The extension to the more general case of a Navier–Stokes modelled fluid over a porous material was not achieved for the twodomain approach owing to the difficulties associated with handling the interfacial boundary conditions. This paper addresses this issue by adopting a onedomain approach, where the governing equations for both regions are combined into a unique set of equations that are valid for the entire domain. It is shown that the nonlinear stability bound, in the onedomain approach, is very sharp and hence excludes the possibility of subcritical instabilities. Moreover, the onedomain approach is compared with an equivalent twodomain approach, and excellent agreement is found between the two.
20100901T00:00:00Z
Hill, A A
Carr, Magda
Studies of the nonlinear stability of fluid/porous systems have been developed very recently. A twodomain modelling approach has been adopted in previous works, but was restricted to specific configurations. The extension to the more general case of a Navier–Stokes modelled fluid over a porous material was not achieved for the twodomain approach owing to the difficulties associated with handling the interfacial boundary conditions. This paper addresses this issue by adopting a onedomain approach, where the governing equations for both regions are combined into a unique set of equations that are valid for the entire domain. It is shown that the nonlinear stability bound, in the onedomain approach, is very sharp and hence excludes the possibility of subcritical instabilities. Moreover, the onedomain approach is compared with an equivalent twodomain approach, and excellent agreement is found between the two.

Instability in internal solitary waves with trapped cores
http://hdl.handle.net/10023/3397
A numerical method that employs a combination of contour advection and pseudospectral techniques is used to investigate instability in internal solitary waves with trapped cores. A threelayer configuration for the background stratification in which the top two layers are linearly stratified and the lower layer is homogeneous is considered throughout. The strength of the stratification in the very top layer is chosen to be sufficient so that waves of depression with trapped cores can be generated. The flow is assumed to satisfy the DubrielJacotinLong equation both inside and outside of the core region. The BruntVaisala frequency is modelled such that it varies from a constant value outside of the core to zero inside the core over a sharp but continuous transition length. This results in a stagnant core in which the vorticity is zero and the density is homogeneous and approximately equal to that at the core boundary. The time dependent simulations show that instability occurs on the boundary of the core. The instability takes the form of KelvinHelmholtz billows. If the instability in the vorticity field is energetic enough, disturbance in the buoyancy field is also seen and fluid exchange takes place across the core boundary. Occurrence of the KelvinHelmholtz billows is attributed to the sharp change in the vorticity field at the boundary between the core and the pycnocline. The numerical scheme is not limited by small Richardson number unlike the other alternatives currently available in the literature which appear to be.
20120101T00:00:00Z
Carr, Magda
King, Stuart Edward
Dritschel, David Gerard
A numerical method that employs a combination of contour advection and pseudospectral techniques is used to investigate instability in internal solitary waves with trapped cores. A threelayer configuration for the background stratification in which the top two layers are linearly stratified and the lower layer is homogeneous is considered throughout. The strength of the stratification in the very top layer is chosen to be sufficient so that waves of depression with trapped cores can be generated. The flow is assumed to satisfy the DubrielJacotinLong equation both inside and outside of the core region. The BruntVaisala frequency is modelled such that it varies from a constant value outside of the core to zero inside the core over a sharp but continuous transition length. This results in a stagnant core in which the vorticity is zero and the density is homogeneous and approximately equal to that at the core boundary. The time dependent simulations show that instability occurs on the boundary of the core. The instability takes the form of KelvinHelmholtz billows. If the instability in the vorticity field is energetic enough, disturbance in the buoyancy field is also seen and fluid exchange takes place across the core boundary. Occurrence of the KelvinHelmholtz billows is attributed to the sharp change in the vorticity field at the boundary between the core and the pycnocline. The numerical scheme is not limited by small Richardson number unlike the other alternatives currently available in the literature which appear to be.

Shear induced breaking of large internal solitary waves
http://hdl.handle.net/10023/3396
The stability properties of 24 experimentally generated internal solitary waves (ISWs) of extremely large amplitude, all with minimum Richardson number less than 1/4, are investigated. The study is supplemented by fully nonlinear calculations in a threelayer fluid. The waves move along a linearly stratified pycnocline (depth h2) sandwiched between a thin upper layer (depth h1) and a deep lower layer (depth h3), both homogeneous. In particular, the waveinduced velocity profile through the pycnocline is measured by particle image velocimetry (PIV) and obtained in computation. Breaking ISWs were found to have amplitudes (a1) in the range a1>2.24 √h1h2(1+h2/h1), while stable waves were on or below this limit. Breaking ISWs were investigated for 0.27 < h2/h1 < 1 and 4.14 < h3/(h1 + h2) < 7.14 and stable waves for 0.36 < h2/h1 < 3.67 and 3.22 < h3/(h1 + h2) < 7.25. Kelvin–Helmholtzlike billows were observed in the breaking cases. They had a length of 7.9h2 and a propagation speed 0.09 times the wave speed. These measured values compared well with predicted values from a stability analysis, assuming steady shear flow with U(z) and ρ(z) taken at the wave maximum (U(z) horizontal velocity profile, ρ(z) density along the vertical z). Only unstable modes in waves of sufficient strength have the chance to grow sufficiently fast to develop breaking: the waves that broke had an estimated growth (of unstable modes) more than 3.3–3.7 times than in the strongest stable case. Evaluation of the minimum Richardson number (Rimin, in the pycnocline), the horizontal length of a pocket of possible instability, with waveinduced Ri < 14, (Lx) and the wavelength (λ), showed that all measurements fall within the range Rimin = −0.23Lx/λ + 0.298 ± 0.016 in the (Lx/λ, Rimin)plane. Breaking ISWs were found for Lx/λ > 0.86 and stable waves for Lx/λ < 0.86. The results show a sort of thresholdlike behaviour in terms of Lx/λ. The results demonstrate that the breaking threshold of Lx/λ = 0.86 was sharper than one based on a minimum Richardson number and reveal that the Richardson number was found to become almost antisymmetric across relatively thick pycnoclines, with the minimum occurring towards the top part of the pycnocline
20090201T00:00:00Z
Fructus, D
Carr, Magda
Grue, J
Jensen, A
Davies, P A
The stability properties of 24 experimentally generated internal solitary waves (ISWs) of extremely large amplitude, all with minimum Richardson number less than 1/4, are investigated. The study is supplemented by fully nonlinear calculations in a threelayer fluid. The waves move along a linearly stratified pycnocline (depth h2) sandwiched between a thin upper layer (depth h1) and a deep lower layer (depth h3), both homogeneous. In particular, the waveinduced velocity profile through the pycnocline is measured by particle image velocimetry (PIV) and obtained in computation. Breaking ISWs were found to have amplitudes (a1) in the range a1>2.24 √h1h2(1+h2/h1), while stable waves were on or below this limit. Breaking ISWs were investigated for 0.27 < h2/h1 < 1 and 4.14 < h3/(h1 + h2) < 7.14 and stable waves for 0.36 < h2/h1 < 3.67 and 3.22 < h3/(h1 + h2) < 7.25. Kelvin–Helmholtzlike billows were observed in the breaking cases. They had a length of 7.9h2 and a propagation speed 0.09 times the wave speed. These measured values compared well with predicted values from a stability analysis, assuming steady shear flow with U(z) and ρ(z) taken at the wave maximum (U(z) horizontal velocity profile, ρ(z) density along the vertical z). Only unstable modes in waves of sufficient strength have the chance to grow sufficiently fast to develop breaking: the waves that broke had an estimated growth (of unstable modes) more than 3.3–3.7 times than in the strongest stable case. Evaluation of the minimum Richardson number (Rimin, in the pycnocline), the horizontal length of a pocket of possible instability, with waveinduced Ri < 14, (Lx) and the wavelength (λ), showed that all measurements fall within the range Rimin = −0.23Lx/λ + 0.298 ± 0.016 in the (Lx/λ, Rimin)plane. Breaking ISWs were found for Lx/λ > 0.86 and stable waves for Lx/λ < 0.86. The results show a sort of thresholdlike behaviour in terms of Lx/λ. The results demonstrate that the breaking threshold of Lx/λ = 0.86 was sharper than one based on a minimum Richardson number and reveal that the Richardson number was found to become almost antisymmetric across relatively thick pycnoclines, with the minimum occurring towards the top part of the pycnocline

Convectively induced shear instability in large amplitude internal solitary waves
http://hdl.handle.net/10023/3395
Laboratory study has been carried out to investigate the instability of an internal solitary wave of depression in a shallow stratified fluid system. The experimental campaign has been supported by theoretical computations and has focused on a two layered stratification consisting of a homogeneous dense layer below a linearly stratified top layer. The initial background stratification has been varied and it is found that the onset and intensity of breaking are affected dramatically by changes in the background stratification. Manifestations of a combination of shear and convective instability are seen on the leading face of the wave. It is shown that there is an interplay between the two instability types and convective instability induces shear by enhancing isopycnal compression. Variation in the upper boundary condition is also found to have an effect on stability. In particular, the implications for convective instability are shown to be profound and a dramatic increase in wave amplitude is seen for a fixed (as opposed to free) upper boundary condition.
Partially funded by grant no. GR/S27368/01 from EPSRC
20081201T00:00:00Z
Carr, Magda
Fructus, D
Grue, J
Jensen, A
Davies, P A
Laboratory study has been carried out to investigate the instability of an internal solitary wave of depression in a shallow stratified fluid system. The experimental campaign has been supported by theoretical computations and has focused on a two layered stratification consisting of a homogeneous dense layer below a linearly stratified top layer. The initial background stratification has been varied and it is found that the onset and intensity of breaking are affected dramatically by changes in the background stratification. Manifestations of a combination of shear and convective instability are seen on the leading face of the wave. It is shown that there is an interplay between the two instability types and convective instability induces shear by enhancing isopycnal compression. Variation in the upper boundary condition is also found to have an effect on stability. In particular, the implications for convective instability are shown to be profound and a dramatic increase in wave amplitude is seen for a fixed (as opposed to free) upper boundary condition.

Number of degrees of freedom and energy spectrum of surface quasigeostrophic turbulence
http://hdl.handle.net/10023/3377
We study both theoretically and numerically surface quasigeostrophic turbulence regularized by the usual molecular viscosity, with an emphasis on a number of classical predictions. It is found that the system's number of degrees of freedom N, which is defined in terms of local Lyapunov exponents, scales as Re3/2, where R e is the Reynolds number expressible in terms of the viscosity, energy dissipation rate and system's integral scale. For general powerlaw energy spectra k(alpha), a comparison of N with the number of dynamically active Fourier modes, i.e. the modes within the energy inertial range, yields alpha = 5/3. This comparison further renders the scaling Re1/2 for the exponential dissipation rate at the dissipation wavenumber. These results have been predicted on the basis of Kolmogorov's theory. Our approach thus recovers these classical predictions and is an analytic alternative to the traditional phenomenological method. The implications of the present findings are discussed in conjunction with related results in the literature. Support for the analytic results is provided through a series of direct numerical simulations.
L.A.K.B. was supported by an EPSRC postgraduate studentship.
20111001T00:00:00Z
Tran, Chuong Van
Blackbourn, Luke Austen Kazimierz
Scott, Richard Kirkness
We study both theoretically and numerically surface quasigeostrophic turbulence regularized by the usual molecular viscosity, with an emphasis on a number of classical predictions. It is found that the system's number of degrees of freedom N, which is defined in terms of local Lyapunov exponents, scales as Re3/2, where R e is the Reynolds number expressible in terms of the viscosity, energy dissipation rate and system's integral scale. For general powerlaw energy spectra k(alpha), a comparison of N with the number of dynamically active Fourier modes, i.e. the modes within the energy inertial range, yields alpha = 5/3. This comparison further renders the scaling Re1/2 for the exponential dissipation rate at the dissipation wavenumber. These results have been predicted on the basis of Kolmogorov's theory. Our approach thus recovers these classical predictions and is an analytic alternative to the traditional phenomenological method. The implications of the present findings are discussed in conjunction with related results in the literature. Support for the analytic results is provided through a series of direct numerical simulations.

Coronal heating by the partial relaxation of twisted loops
http://hdl.handle.net/10023/3373
Context: Relaxation theory offers a straightforward method for estimating the energy that is released when a magnetic field becomes unstable, as a result of continual convective driving. Aims: We present new results obtained from nonlinear magnetohydrodynamic (MHD) simulations of idealised coronal loops. The purpose of this work is to determine whether or not the simulation results agree with Taylor relaxation, which will require a modified version of relaxation theory applicable to unbounded field configurations. Methods: A threedimensional (3D) MHD Lagrangianremap code is used to simulate the evolution of a linetied cylindrical coronal loop model. This model comprises three concentric layers surrounded by a potential envelope; hence, being twisted locally, each loop configuration is distinguished by a piecewiseconstant current profile. Initially, all configurations carry zeronetcurrent fields and are in ideally unstable equilibrium. The simulation results are compared with the predictions of helicity conserving relaxation theory. Results: For all simulations, the change in helicity is no more than 2% of the initial value; also, the numerical helicities match the analyticallydetermined values. Magnetic energy dissipation predominantly occurs via shock heating associated with magnetic reconnection in distributed current sheets. The energy release and final field profiles produced by the numerical simulations are in agreement with the predictions given by a new model of partial relaxation theory: the relaxed field is close to a linear force free state; however, the extent of the relaxation region is limited, while the loop undergoes some radial expansion. Conclusions: The results presented here support the use of partial relaxation theory, specifically, when calculating the heatingevent distributions produced by ensembles of kinkunstable loops.
20130201T00:00:00Z
Bareford, Michael
Hood, Alan
Browning, Philippa
Context: Relaxation theory offers a straightforward method for estimating the energy that is released when a magnetic field becomes unstable, as a result of continual convective driving. Aims: We present new results obtained from nonlinear magnetohydrodynamic (MHD) simulations of idealised coronal loops. The purpose of this work is to determine whether or not the simulation results agree with Taylor relaxation, which will require a modified version of relaxation theory applicable to unbounded field configurations. Methods: A threedimensional (3D) MHD Lagrangianremap code is used to simulate the evolution of a linetied cylindrical coronal loop model. This model comprises three concentric layers surrounded by a potential envelope; hence, being twisted locally, each loop configuration is distinguished by a piecewiseconstant current profile. Initially, all configurations carry zeronetcurrent fields and are in ideally unstable equilibrium. The simulation results are compared with the predictions of helicity conserving relaxation theory. Results: For all simulations, the change in helicity is no more than 2% of the initial value; also, the numerical helicities match the analyticallydetermined values. Magnetic energy dissipation predominantly occurs via shock heating associated with magnetic reconnection in distributed current sheets. The energy release and final field profiles produced by the numerical simulations are in agreement with the predictions given by a new model of partial relaxation theory: the relaxed field is close to a linear force free state; however, the extent of the relaxation region is limited, while the loop undergoes some radial expansion. Conclusions: The results presented here support the use of partial relaxation theory, specifically, when calculating the heatingevent distributions produced by ensembles of kinkunstable loops.

Damping of kink waves by mode coupling : I. Analytical treatment
http://hdl.handle.net/10023/3340
Aims. To investigate the spatial damping of propagating kink waves in an inhomogeneous plasma. In the limit of a thin tube surrounded by a thin transition layer, an analytical formulation for kink waves driven in from the bottom boundary of the corona is presented. Methods. The spatial form for the damping of the kink mode was investigated using various analytical approximations. When the density ratio between the internal density and the external density is not too large, a simple di.erentialintegral equation was used. Approximate analytical solutions to this equation are presented. Results. For the first time, the form of the spatial damping of the kink mode is shown analytically to be Gaussian in nature near the driven boundary. For several wavelengths, the amplitude of the kink mode is proportional to (1 + exp(z2 /L2 g))/2, where L2g = 16/ǫκ2 k2 . Although the actual value of 16 in Lg depends on the particular form of the driver, this form is very general and its dependence on the other parameters does not change. For large distances, the damping profile appears to be roughly linear exponential decay. This is shown analytically by a series expansion when the inhomogeneous layer width is small enough.
20130301T00:00:00Z
Hood, Alan William
Ruderman, Michael
Pascoe, David James
De Moortel, Ineke
Terradas, Jaume
Wright, Andrew Nicholas
Aims. To investigate the spatial damping of propagating kink waves in an inhomogeneous plasma. In the limit of a thin tube surrounded by a thin transition layer, an analytical formulation for kink waves driven in from the bottom boundary of the corona is presented. Methods. The spatial form for the damping of the kink mode was investigated using various analytical approximations. When the density ratio between the internal density and the external density is not too large, a simple di.erentialintegral equation was used. Approximate analytical solutions to this equation are presented. Results. For the first time, the form of the spatial damping of the kink mode is shown analytically to be Gaussian in nature near the driven boundary. For several wavelengths, the amplitude of the kink mode is proportional to (1 + exp(z2 /L2 g))/2, where L2g = 16/ǫκ2 k2 . Although the actual value of 16 in Lg depends on the particular form of the driver, this form is very general and its dependence on the other parameters does not change. For large distances, the damping profile appears to be roughly linear exponential decay. This is shown analytically by a series expansion when the inhomogeneous layer width is small enough.

The influence of a fluidporous interface on solar pond stability
http://hdl.handle.net/10023/3338
The linear instability of the gradient zone of a solar pond containing a fluidporous interface is investigated. It is found that the gradient zone can retain the same stability for lower values of the solute Rayleigh number with the introduction of a porous material compared with a purely fluid layer, whilst maintaining the same lower convective zone temperature. Interestingly, it is also shown that for certain parameter values the penetration of a porous medium into the gradient zone can cause the temperature of the lower convective zone to rise. However, for certain parameter ranges, when the fluidporous interface is towards the top of the gradient zone, the solar pond can become highly unstable.
20130201T00:00:00Z
Hill, A. A
Carr, Magda
The linear instability of the gradient zone of a solar pond containing a fluidporous interface is investigated. It is found that the gradient zone can retain the same stability for lower values of the solute Rayleigh number with the introduction of a porous material compared with a purely fluid layer, whilst maintaining the same lower convective zone temperature. Interestingly, it is also shown that for certain parameter values the penetration of a porous medium into the gradient zone can cause the temperature of the lower convective zone to rise. However, for certain parameter ranges, when the fluidporous interface is towards the top of the gradient zone, the solar pond can become highly unstable.

A Bayesian approach to fitting Gibbs processes with temporal random effects
http://hdl.handle.net/10023/3305
We consider spatial point pattern data that have been observed repeatedly over a period of time in an inhomogeneous environment. Each spatial point pattern can be regarded as a “snapshot” of the underlying point process at a series of times. Thus, the number of points and corresponding locations of points differ for each snapshot. Each snapshot can be analyzed independently, but in many cases there may be little information in the data relating to model parameters, particularly parameters relating to the interaction between points. Thus, we develop an integrated approach, simultaneously analyzing all snapshots within a single robust and consistent analysis. We assume that sufficient time has passed between observation dates so that the spatial point patterns can be regarded as independent replicates, given spatial covariates. We develop a joint mixed effects Gibbs point process model for the replicates of spatial point patterns by considering environmental covariates in the analysis as fixed effects, to model the heterogeneous environment, with a random effects (or hierarchical) component to account for the different observation days for the intensity function. We demonstrate how the model can be fitted within a Bayesian framework using an auxiliary variable approach to deal with the issue of the random effects component. We apply the methods to a data set of musk oxen herds and demonstrate the increased precision of the parameter estimates when considering all available data within a single integrated analysis.
This work is partially supported by Research Councils UK
20121201T00:00:00Z
King, Ruth
Illian, Janine Baerbel
King, Stuart Edward
Nightingale, Glenna Faith
Hendrichsen, Ditte
We consider spatial point pattern data that have been observed repeatedly over a period of time in an inhomogeneous environment. Each spatial point pattern can be regarded as a “snapshot” of the underlying point process at a series of times. Thus, the number of points and corresponding locations of points differ for each snapshot. Each snapshot can be analyzed independently, but in many cases there may be little information in the data relating to model parameters, particularly parameters relating to the interaction between points. Thus, we develop an integrated approach, simultaneously analyzing all snapshots within a single robust and consistent analysis. We assume that sufficient time has passed between observation dates so that the spatial point patterns can be regarded as independent replicates, given spatial covariates. We develop a joint mixed effects Gibbs point process model for the replicates of spatial point patterns by considering environmental covariates in the analysis as fixed effects, to model the heterogeneous environment, with a random effects (or hierarchical) component to account for the different observation days for the intensity function. We demonstrate how the model can be fitted within a Bayesian framework using an auxiliary variable approach to deal with the issue of the random effects component. We apply the methods to a data set of musk oxen herds and demonstrate the increased precision of the parameter estimates when considering all available data within a single integrated analysis.

Collisionless distribution function for the relativistic forcefree Harris sheet
http://hdl.handle.net/10023/3154
A selfconsistent collisionless distribution function for the relativistic analogue of the forcefree Harris sheet is presented. This distribution function is the relativistic generalization of the distribution function for the nonrelativistic collisionless forcefree Harris sheet recently found by Harrison and Neukirch [Phys. Rev. Lett. 102, 135003 (2009)], as it has the same dependence on the particle energy and canonical momenta. We present a detailed calculation which shows that the proposed distribution function generates the required current density profile (and thus magnetic field profile) in a frame of reference in which the electric potential vanishes identically. The connection between the parameters of the distribution function and the macroscopic parameters such as the current sheet thickness is discussed. (C) 2012 American Institute of Physics. [doi: 10.1063/1.3677268]
20120101T00:00:00Z
Stark, C. R.
Neukirch, T.
A selfconsistent collisionless distribution function for the relativistic analogue of the forcefree Harris sheet is presented. This distribution function is the relativistic generalization of the distribution function for the nonrelativistic collisionless forcefree Harris sheet recently found by Harrison and Neukirch [Phys. Rev. Lett. 102, 135003 (2009)], as it has the same dependence on the particle energy and canonical momenta. We present a detailed calculation which shows that the proposed distribution function generates the required current density profile (and thus magnetic field profile) in a frame of reference in which the electric potential vanishes identically. The connection between the parameters of the distribution function and the macroscopic parameters such as the current sheet thickness is discussed. (C) 2012 American Institute of Physics. [doi: 10.1063/1.3677268]

Numerical simulation of shearinduced instabilities in internal solitary waves
http://hdl.handle.net/10023/3054
A numerical method that employs a combination of contour advection and pseudospectral techniques is used to simulate shearinduced instabilities in an internal solitary wave (ISW). A threelayer configuration for the background stratification, in which a linearly stratified intermediate layer is sandwiched between two homogeneous ones, is considered throughout. The flow is assumed to satisfy the inviscid, incompressible, Oberbeck–Boussinesq equations in two dimensions. Simulations are initialized by fully nonlinear, steadystate, ISWs. The results of the simulations show that the instability takes place in the pycnocline and manifests itself as Kelvin–Helmholtz billows. The billows form near the trough of the wave, subsequently grow and disturb the tail. Both the critical Richardson number (Ric) and the critical amplitude required for instability are found to be functions of the ratio of the undisturbed layer thicknesses. It is shown, therefore, that the constant, critical bound for instability in ISWs given in Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61–95), namely Ric = 0.1 ± 0.01 , is not a sufficient condition for instability. It is also shown that the critical value of Lx/λ required for instability, where Lx is the length of the region in a wave in which Ri < 1/4 and λ is the halfwidth of the wave, is sensitive to the ratio of the layer thicknesses. Similarly, a linear stability analysis reveals that δiTw (where δi is the growth rate of the instability averaged over Tw, the period in which parcels of fluid are subjected to Ri < 1/4) is very sensitive to the transition between the undisturbed pycnocline and the homogeneous layers, and the amplitude of the wave. Therefore, the alternative tests for instability presented in Fructus et al. (J. Fluid Mech., vol. 620, 2009, pp. 1–29) and Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61–95), respectively, namely Lx/λ ≥ 0.86 and δiTw > 5 , are shown to be valid only for a limited parameter range.
This work was supported by the UK Engineering and Physical Sciences Research Council [grant number EP/F030622/1]
20110925T00:00:00Z
Carr, Magda
King, Stuart Edward
Dritschel, David Gerard
A numerical method that employs a combination of contour advection and pseudospectral techniques is used to simulate shearinduced instabilities in an internal solitary wave (ISW). A threelayer configuration for the background stratification, in which a linearly stratified intermediate layer is sandwiched between two homogeneous ones, is considered throughout. The flow is assumed to satisfy the inviscid, incompressible, Oberbeck–Boussinesq equations in two dimensions. Simulations are initialized by fully nonlinear, steadystate, ISWs. The results of the simulations show that the instability takes place in the pycnocline and manifests itself as Kelvin–Helmholtz billows. The billows form near the trough of the wave, subsequently grow and disturb the tail. Both the critical Richardson number (Ric) and the critical amplitude required for instability are found to be functions of the ratio of the undisturbed layer thicknesses. It is shown, therefore, that the constant, critical bound for instability in ISWs given in Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61–95), namely Ric = 0.1 ± 0.01 , is not a sufficient condition for instability. It is also shown that the critical value of Lx/λ required for instability, where Lx is the length of the region in a wave in which Ri < 1/4 and λ is the halfwidth of the wave, is sensitive to the ratio of the layer thicknesses. Similarly, a linear stability analysis reveals that δiTw (where δi is the growth rate of the instability averaged over Tw, the period in which parcels of fluid are subjected to Ri < 1/4) is very sensitive to the transition between the undisturbed pycnocline and the homogeneous layers, and the amplitude of the wave. Therefore, the alternative tests for instability presented in Fructus et al. (J. Fluid Mech., vol. 620, 2009, pp. 1–29) and Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61–95), respectively, namely Lx/λ ≥ 0.86 and δiTw > 5 , are shown to be valid only for a limited parameter range.

Behind and beyond a theorem on groups related to trivalent graphs
http://hdl.handle.net/10023/2462
In 2006 we completed the proof of a fivepart conjecture that was made in 1977 about a family of groups related to trivalent graphs. This family covers all 2generator, 2relator groups where one relator specifies that a generator is an involution and the other relator has three syllables. Our proof relies upon detailed but general computations in the groups under question. The proof is theoretical, but based upon explicit proofs produced by machine for individual cases. Here we explain how we derived the general proofs from specific cases. The conjecture essentially addressed only the finite groups in the family. Here we extend the results to infinite groups, effectively determining when members of this family of finitely presented groups are simply isomorphic to a specific quotient.
20081201T00:00:00Z
Havas, George
Robertson, Edmund F.
Sutherland, Dale C.
In 2006 we completed the proof of a fivepart conjecture that was made in 1977 about a family of groups related to trivalent graphs. This family covers all 2generator, 2relator groups where one relator specifies that a generator is an involution and the other relator has three syllables. Our proof relies upon detailed but general computations in the groups under question. The proof is theoretical, but based upon explicit proofs produced by machine for individual cases. Here we explain how we derived the general proofs from specific cases. The conjecture essentially addressed only the finite groups in the family. Here we extend the results to infinite groups, effectively determining when members of this family of finitely presented groups are simply isomorphic to a specific quotient.

Lowerhybrid waves generated by anomalous Doppler resonance in auroral plasmas
http://hdl.handle.net/10023/2457
This paper describes sonic aspects of lowerhybrid wave activity in space plasmas. Lowerhybrid waves are particularly important since they can transfer energy efficiently between electrons and ions in a collisionless magnetized plasma. We consider the 'fan' or anomalous Doppler resonance instability driven by energetic electron tails and show that it is responsible for the generation of lowerhybrid waves. We also demonstrate that observations of their intensity are sufficient to drive the modulational instability.
20100801T00:00:00Z
Bingham, Robert
Cairns, R Alan
Vorgul, I.
Shapiro, V. D.
This paper describes sonic aspects of lowerhybrid wave activity in space plasmas. Lowerhybrid waves are particularly important since they can transfer energy efficiently between electrons and ions in a collisionless magnetized plasma. We consider the 'fan' or anomalous Doppler resonance instability driven by energetic electron tails and show that it is responsible for the generation of lowerhybrid waves. We also demonstrate that observations of their intensity are sufficient to drive the modulational instability.

Falling towards forgetfulness : synaptic decay prevents spontaneous recovery of memory
http://hdl.handle.net/10023/2455
Long after a new language has been learned and forgotten, relearning a few words seems to trigger the recall of other words. This "freelunch learning'' (FLL) effect has been demonstrated both in humans and in neural network models. Specifically, previous work proved that linear networks that learn a set of associations, then partially forget them all, and finally relearn some of the associations, show improved performance on the remaining (i.e., nonrelearned) associations. Here, we prove that relearning forgotten associations decreases performance on nonrelearned associations; an effect we call negative freelunch learning. The difference between freelunch learning and the negative freelunch learning presented here is due to the particular method used to induce forgetting. Specifically, if forgetting is induced by isotropic drifting of weight vectors (i.e., by adding isotropic noise), then freelunch learning is observed. However, as proved here, if forgetting is induced by weight values that simply decay or fall towards zero, then negative freelunch learning is observed. From a biological perspective, and assuming that nervous systems are analogous to the networks used here, this suggests that evolution may have selected physiological mechanisms that involve forgetting using a form of synaptic drift rather than synaptic decay, because synaptic drift, but not synaptic decay, yields freelunch learning.
No funding was received for this work.
20080822T00:00:00Z
Stone, James V.
Jupp, Peter Edmund
Long after a new language has been learned and forgotten, relearning a few words seems to trigger the recall of other words. This "freelunch learning'' (FLL) effect has been demonstrated both in humans and in neural network models. Specifically, previous work proved that linear networks that learn a set of associations, then partially forget them all, and finally relearn some of the associations, show improved performance on the remaining (i.e., nonrelearned) associations. Here, we prove that relearning forgotten associations decreases performance on nonrelearned associations; an effect we call negative freelunch learning. The difference between freelunch learning and the negative freelunch learning presented here is due to the particular method used to induce forgetting. Specifically, if forgetting is induced by isotropic drifting of weight vectors (i.e., by adding isotropic noise), then freelunch learning is observed. However, as proved here, if forgetting is induced by weight values that simply decay or fall towards zero, then negative freelunch learning is observed. From a biological perspective, and assuming that nervous systems are analogous to the networks used here, this suggests that evolution may have selected physiological mechanisms that involve forgetting using a form of synaptic drift rather than synaptic decay, because synaptic drift, but not synaptic decay, yields freelunch learning.

On the relationship between equilibrium bifurcations and ideal MHD instabilities for linetied coronal loops
http://hdl.handle.net/10023/2268
For axisymmetric models for coronal loops the relationship between the bifurcation points of magnetohydrodynamic (MHD) equilibrium sequences and the points of linear ideal MHD instability is investigated, imposing linetied boundary conditions. Using a wellstudied example based on the Gold aEuro parts per thousand Hoyle equilibrium, it is demonstrated that if the equilibrium sequence is calculated using the Grad aEuro parts per thousand Shafranov equation, the instability corresponds to the second bifurcation point and not the first bifurcation point, because the equilibrium boundary conditions allow for modes which are excluded from the linear ideal stability analysis. This is shown by calculating the bifurcating equilibrium branches and comparing the spatial structure of the solutions close to the bifurcation point with the spatial structure of the unstable mode. If the equilibrium sequence is calculated using Euler potentials, the first bifurcation point of the Grad aEuro parts per thousand Shafranov case is not found, and the first bifurcation point of the Euler potential description coincides with the ideal instability threshold. An explanation of this results in terms of linear bifurcation theory is given and the implications for the use of MHD equilibrium bifurcations to explain eruptive phenomena is briefly discussed.
20100101T00:00:00Z
Neukirch, T.
Romeou, Z.
For axisymmetric models for coronal loops the relationship between the bifurcation points of magnetohydrodynamic (MHD) equilibrium sequences and the points of linear ideal MHD instability is investigated, imposing linetied boundary conditions. Using a wellstudied example based on the Gold aEuro parts per thousand Hoyle equilibrium, it is demonstrated that if the equilibrium sequence is calculated using the Grad aEuro parts per thousand Shafranov equation, the instability corresponds to the second bifurcation point and not the first bifurcation point, because the equilibrium boundary conditions allow for modes which are excluded from the linear ideal stability analysis. This is shown by calculating the bifurcating equilibrium branches and comparing the spatial structure of the solutions close to the bifurcation point with the spatial structure of the unstable mode. If the equilibrium sequence is calculated using Euler potentials, the first bifurcation point of the Grad aEuro parts per thousand Shafranov case is not found, and the first bifurcation point of the Euler potential description coincides with the ideal instability threshold. An explanation of this results in terms of linear bifurcation theory is given and the implications for the use of MHD equilibrium bifurcations to explain eruptive phenomena is briefly discussed.

Automatic presentations and semigroup constructions
http://hdl.handle.net/10023/2148
An automatic presentation for a relational structure is, informally, an abstract representation of the elements of that structure by means of a regular language such that the relations can all be recognized by finite automata. A structure admitting an automatic presentation is said to be FApresentable. This paper studies the interaction of automatic presentations and certain semigroup constructions, namely: direct products, free products, finite Rees index extensions and subsemigroups, strong semilattices of semigroups, Rees matrix semigroups, BruckReilly extensions, zerodirect unions, semidirect products, wreath products, ideals, and quotient semigroups. For each case, the closure of the class of FApresentable semigroups under that construction is considered, as is the question of whether the FApresentability of the semigroup obtained from such a construction implies the FApresentability of the original semigroup[s]. Classifications are also given of the FApresentable finitely generated Clifford semigroups, completely simple semigroups, and completely 0simple semigroups.
20100801T00:00:00Z
Cain, Alan J.
Oliver, Graham
Ruskuc, Nik
Thomas, Richard M.
An automatic presentation for a relational structure is, informally, an abstract representation of the elements of that structure by means of a regular language such that the relations can all be recognized by finite automata. A structure admitting an automatic presentation is said to be FApresentable. This paper studies the interaction of automatic presentations and certain semigroup constructions, namely: direct products, free products, finite Rees index extensions and subsemigroups, strong semilattices of semigroups, Rees matrix semigroups, BruckReilly extensions, zerodirect unions, semidirect products, wreath products, ideals, and quotient semigroups. For each case, the closure of the class of FApresentable semigroups under that construction is considered, as is the question of whether the FApresentability of the semigroup obtained from such a construction implies the FApresentability of the original semigroup[s]. Classifications are also given of the FApresentable finitely generated Clifford semigroups, completely simple semigroups, and completely 0simple semigroups.

Cancellative and Malcev presentations for finite Rees index subsemigroups and extensions
http://hdl.handle.net/10023/2138
It is known that, for semigroups, the property of admitting a finite presentation is preserved on passing to subsemigroups and extensions of finite Rees index. The present paper shows that the same holds true for Malcev, cancellative, leftcancellative and rightcancellative presentations. (A Malcev (respectively, cancellative, leftcancellative, rightcancellative) presentation is a presentation of a special type that can be used to define any groupembeddable (respectively, cancellative, leftcancellative, rightcancellative) semigroup.).
20080201T00:00:00Z
Cain, Alan James
Robertson, Edmund Frederick
Ruskuc, Nik
It is known that, for semigroups, the property of admitting a finite presentation is preserved on passing to subsemigroups and extensions of finite Rees index. The present paper shows that the same holds true for Malcev, cancellative, leftcancellative and rightcancellative presentations. (A Malcev (respectively, cancellative, leftcancellative, rightcancellative) presentation is a presentation of a special type that can be used to define any groupembeddable (respectively, cancellative, leftcancellative, rightcancellative) semigroup.).

The steadystate form of largeamplitude internal solitary waves
http://hdl.handle.net/10023/2084
A new numerical scheme for obtaining the steadystate form of an internal solitary wave of large amplitude is presented. A stratified inviscid twodimensional fluid under the Boussinesq approximation flowing between horizontal rigid boundaries is considered. The stratification is stable, and buoyancy is continuously differentiable throughout the domain of the flow. Solutions are obtained by tracing the buoyancy frequency along streamlines from the undisturbed far field. From this the vorticity field can be constructed and the streamfunction may then be obtained by inversion of Laplace's operator. The scheme is presented as an iterative solver, where the inversion of Laplace's operator is performed spectrally. The solutions agree well with previous results for stratification in which the buoyancy frequency is a discontinuous function. The new numerical scheme allows significantly larger amplitude waves to be computed than have been presented before and it is shown that waves with Richardson numbers as low as 0.062 can be computed straightforwardly. The method is also extended to deal in a novel way with closed streamlines when they occur in the domain. The new solutions are tested in independent fully nonlinear timedependent simulations and are verified to be steady. Waves with regions of recirculation are also discussed.
20110110T00:00:00Z
King, Stuart Edward
Carr, Magda
Dritschel, David Gerard
A new numerical scheme for obtaining the steadystate form of an internal solitary wave of large amplitude is presented. A stratified inviscid twodimensional fluid under the Boussinesq approximation flowing between horizontal rigid boundaries is considered. The stratification is stable, and buoyancy is continuously differentiable throughout the domain of the flow. Solutions are obtained by tracing the buoyancy frequency along streamlines from the undisturbed far field. From this the vorticity field can be constructed and the streamfunction may then be obtained by inversion of Laplace's operator. The scheme is presented as an iterative solver, where the inversion of Laplace's operator is performed spectrally. The solutions agree well with previous results for stratification in which the buoyancy frequency is a discontinuous function. The new numerical scheme allows significantly larger amplitude waves to be computed than have been presented before and it is shown that waves with Richardson numbers as low as 0.062 can be computed straightforwardly. The method is also extended to deal in a novel way with closed streamlines when they occur in the domain. The new solutions are tested in independent fully nonlinear timedependent simulations and are verified to be steady. Waves with regions of recirculation are also discussed.

Impeded inverse energy transfer in the CharneyHasegawaMima model of quasigeostrophic flows
http://hdl.handle.net/10023/1565
The behaviour of turbulent flows within the singlelayer quasigeostrophic (CharneyHasegawaMima) model is shown to be strongly dependent on the Rossby deformation wavenumber lambda (or freesurface elasticity). Herein, we derive a bound oil the inverse energy transfer, specifically on the growth rate dl/dt of the characteristic length scale e representing the energy centroid. It is found that dl/dt <= 2 parallel to q parallel to(infinity)/(l(s)lambda(2)), where parallel to q parallel to(infinity) is the supremum of the potential vorticity and l(s) represents the potential enstrophy centroid of the reservoir, both invariant. This result implies that in the potentialenergydominated regime (l >= l(s) >> lambda(1)) the inverse energy transfer is strongly impeded, in the sense that under the usual time scale no significant transfer of energy to larger scales occurs. The physical implication is that the elasticity of the free surface impedes turbulent energy transfer in wavenumber space, effectively rendering largescale vortices longlived and inactive. Results from numerical simulations of forceddissipative turbulence confirm this prediction.
20060325T00:00:00Z
Tran, Chuong Van
Dritschel, David Gerard
The behaviour of turbulent flows within the singlelayer quasigeostrophic (CharneyHasegawaMima) model is shown to be strongly dependent on the Rossby deformation wavenumber lambda (or freesurface elasticity). Herein, we derive a bound oil the inverse energy transfer, specifically on the growth rate dl/dt of the characteristic length scale e representing the energy centroid. It is found that dl/dt <= 2 parallel to q parallel to(infinity)/(l(s)lambda(2)), where parallel to q parallel to(infinity) is the supremum of the potential vorticity and l(s) represents the potential enstrophy centroid of the reservoir, both invariant. This result implies that in the potentialenergydominated regime (l >= l(s) >> lambda(1)) the inverse energy transfer is strongly impeded, in the sense that under the usual time scale no significant transfer of energy to larger scales occurs. The physical implication is that the elasticity of the free surface impedes turbulent energy transfer in wavenumber space, effectively rendering largescale vortices longlived and inactive. Results from numerical simulations of forceddissipative turbulence confirm this prediction.

Vanishing enstrophy dissipation in twodimensional NavierStokes turbulence in the inviscid limit
http://hdl.handle.net/10023/1564
Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) developed a theory of twodimensional turbulence based on the assumption that the dissipation of enstrophy (meansquare vorticity) tends to a finite nonzero constant in the limit of infinite Reynolds number Re. Here, by assuming powerlaw spectra, including the one predicted by Batchelor's theory, we prove that the maximum dissipation of enstrophy is in fact zero in this limit. Specifically, as Re > infinity, the dissipation approaches zero no slower than (ln Re)(1/2). The physical reason behind this result is that the decrease of viscosity enhances the production of both palinstrophy (meansquare vorticity gradients) and its dissipation  but in such a way that the net growth of palinstrophy is less rapid than the decrease of viscosity, resulting in vanishing enstrophy dissipation. This result generalizes to a rich class of quasigeostrophic models as well as to the case of a passive tracer in layerwisetwodimensional turbulent flows having bounded enstrophy.
20060725T00:00:00Z
Tran, Chuong Van
Dritschel, David Gerard
Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) developed a theory of twodimensional turbulence based on the assumption that the dissipation of enstrophy (meansquare vorticity) tends to a finite nonzero constant in the limit of infinite Reynolds number Re. Here, by assuming powerlaw spectra, including the one predicted by Batchelor's theory, we prove that the maximum dissipation of enstrophy is in fact zero in this limit. Specifically, as Re > infinity, the dissipation approaches zero no slower than (ln Re)(1/2). The physical reason behind this result is that the decrease of viscosity enhances the production of both palinstrophy (meansquare vorticity gradients) and its dissipation  but in such a way that the net growth of palinstrophy is less rapid than the decrease of viscosity, resulting in vanishing enstrophy dissipation. This result generalizes to a rich class of quasigeostrophic models as well as to the case of a passive tracer in layerwisetwodimensional turbulent flows having bounded enstrophy.

Quasigeostrophic vortices in compressible atmospheres
http://hdl.handle.net/10023/1562
This paper considers the effect of an exponential variation in the background density field (as exists in compressible atmospheres) on the structure and dynamics of the quasigeostrophic system, and compares the results with the corresponding Boussinesq limit in which background density variations are assumed small. The behaviour of the compressible system is understood via a closedform analytic expression for the Green's function of the inversion operator relating potential vorticity and streamfunction. This expression makes explicit the anisotropy of the Green's function, inherited from the density profile, which has a slow, algebraic decay directly above the source and an exponential decay in all other directions. An immediate consequence for finitevolume vortices is a differential rotation of upper and lower levels that results in counterintuitive behaviour during the nonlinear evolution of ellipsoidal vortices, in which vortex destruction is confined to the lower vortex and wave activity is seen to propagate downwards. This is in contrast to the Boussinesq limit, which exhibits symmetric destruction of the upper and lower vortex, and in contrast to naive expectations based on a consideration of the mass distribution alone, which would lead to greater destruction of the upper vortex. Finally, the presence of a horizontal lower boundary introduces a strong barotropic component that is absent in the unbounded case (the presence of an upper boundary has almost no effect). The lower boundary also alters the differential rotation in the lower vortex with important consequences for the nonlinear evolution: for very small separation between the lower boundary and the vortex, the differential rotation is reversed leading to strong deformations of the middle vortex; for a critical separation, the vortex is stabilized by the reduction of the differential rotation, and remains coherent over remarkably long times.
20050510T00:00:00Z
Scott, Richard Kirkness
Dritschel, David Gerard
This paper considers the effect of an exponential variation in the background density field (as exists in compressible atmospheres) on the structure and dynamics of the quasigeostrophic system, and compares the results with the corresponding Boussinesq limit in which background density variations are assumed small. The behaviour of the compressible system is understood via a closedform analytic expression for the Green's function of the inversion operator relating potential vorticity and streamfunction. This expression makes explicit the anisotropy of the Green's function, inherited from the density profile, which has a slow, algebraic decay directly above the source and an exponential decay in all other directions. An immediate consequence for finitevolume vortices is a differential rotation of upper and lower levels that results in counterintuitive behaviour during the nonlinear evolution of ellipsoidal vortices, in which vortex destruction is confined to the lower vortex and wave activity is seen to propagate downwards. This is in contrast to the Boussinesq limit, which exhibits symmetric destruction of the upper and lower vortex, and in contrast to naive expectations based on a consideration of the mass distribution alone, which would lead to greater destruction of the upper vortex. Finally, the presence of a horizontal lower boundary introduces a strong barotropic component that is absent in the unbounded case (the presence of an upper boundary has almost no effect). The lower boundary also alters the differential rotation in the lower vortex with important consequences for the nonlinear evolution: for very small separation between the lower boundary and the vortex, the differential rotation is reversed leading to strong deformations of the middle vortex; for a critical separation, the vortex is stabilized by the reduction of the differential rotation, and remains coherent over remarkably long times.

Subsemigroups of virtually free groups : finite Malcev presentations and testing for freeness
http://hdl.handle.net/10023/1561
This paper shows that, given a finite subset X of a finitely generated virtually free group F, the freeness of the subsemigroup of F generated by X can be tested algorithmically. (A group is virtually free if it contains a free subgroup of finite index.) It is then shown that every finitely generated subsemigroup, of F has a finite Malcev presentation (a type of semigroup presentation which can be used to define any semigroup that embeds in a group), and that such a presentation can be effectively found from any finite generating set.
20060701T00:00:00Z
Cain, AJ
Robertson, Edmund Frederick
Ruskuc, Nikola
This paper shows that, given a finite subset X of a finitely generated virtually free group F, the freeness of the subsemigroup of F generated by X can be tested algorithmically. (A group is virtually free if it contains a free subgroup of finite index.) It is then shown that every finitely generated subsemigroup, of F has a finite Malcev presentation (a type of semigroup presentation which can be used to define any semigroup that embeds in a group), and that such a presentation can be effectively found from any finite generating set.

The critical merger distance between two corotating quasigeostrophic vortices
http://hdl.handle.net/10023/1558
This paper examines the critical merger or strong interaction distance between two equalpotentialvorticity quasigeostrophic vortices. The interaction between the two vortices depends on five parameters: their volume ratio, their heighttowidth aspect ratios and their vertical and horizontal offsets. Due to the size of the parameter space, a direct investigation solving the full quasigeostrophic equations is impossible. We instead determine the critical merger distance approximately using an asymptotic approach. We associate the merger distance with the margin of stability for a family of equilibrium states having prescribed aspect and volume ratios, and vertical offset. The equilibrium states are obtained using an asymptotic solution method which models vortices by ellipsoids. The margin itself is determined by a linear stability analysis. We focus on the interaction between oblate to moderately prolate vortices, the shapes most commonly found in turbulence. Here, a new unexpected instability is found and discussed for prolate vortices which is manifested by the tilting of vortices toward each other. It implies than tall vortices may merge starting from greater separation distances than previously thought.
20050110T00:00:00Z
Reinaud, Jean Noel
Dritschel, David Gerard
This paper examines the critical merger or strong interaction distance between two equalpotentialvorticity quasigeostrophic vortices. The interaction between the two vortices depends on five parameters: their volume ratio, their heighttowidth aspect ratios and their vertical and horizontal offsets. Due to the size of the parameter space, a direct investigation solving the full quasigeostrophic equations is impossible. We instead determine the critical merger distance approximately using an asymptotic approach. We associate the merger distance with the margin of stability for a family of equilibrium states having prescribed aspect and volume ratios, and vertical offset. The equilibrium states are obtained using an asymptotic solution method which models vortices by ellipsoids. The margin itself is determined by a linear stability analysis. We focus on the interaction between oblate to moderately prolate vortices, the shapes most commonly found in turbulence. Here, a new unexpected instability is found and discussed for prolate vortices which is manifested by the tilting of vortices toward each other. It implies than tall vortices may merge starting from greater separation distances than previously thought.

The shape of vortices in quasigeostrophic turbulence
http://hdl.handle.net/10023/1557
The present work discusses the most commonly occurring shape of the coherent vortical structures in rapidly rotating stably stratified turbulence, under the quasigeostrophic approximation. In decaying turbulence, these vorticescoherent regions of the materiallyinvariant potential vorticitydominate the flow evolution, and indeed the flow evolution is governed by their interactions. An analysis of several exceptionally highresolution simulations of quasigeostrophic turbulence is performed. The results indicate that the population of vortices exhibits a mean heighttowidth aspect ratio less than unity, in fact close to 0.8. This finding is justified here by a simple model, in which vortices are taken to be ellipsoids of uniform potential vorticity. The model focuses on steady ellipsoids within a uniform background strain flow. This background flow approximates the effects of surrounding vortices in a turbulent flow on a given vortex. It is argued that the vortices which are able to withstand the highest levels of strain are those most likely to be found in the actual turbulent flow. Our calculations confirm that the optimal heighttowidth aspect ratio is close to 0.8 for a wide range of background straining flows.
Partially supported by the UK EPSRC (Grant GR/N11711)
20030110T00:00:00Z
Reinaud, Jean Noel
Dritschel, David Gerard
Koudella, CR
The present work discusses the most commonly occurring shape of the coherent vortical structures in rapidly rotating stably stratified turbulence, under the quasigeostrophic approximation. In decaying turbulence, these vorticescoherent regions of the materiallyinvariant potential vorticitydominate the flow evolution, and indeed the flow evolution is governed by their interactions. An analysis of several exceptionally highresolution simulations of quasigeostrophic turbulence is performed. The results indicate that the population of vortices exhibits a mean heighttowidth aspect ratio less than unity, in fact close to 0.8. This finding is justified here by a simple model, in which vortices are taken to be ellipsoids of uniform potential vorticity. The model focuses on steady ellipsoids within a uniform background strain flow. This background flow approximates the effects of surrounding vortices in a turbulent flow on a given vortex. It is argued that the vortices which are able to withstand the highest levels of strain are those most likely to be found in the actual turbulent flow. Our calculations confirm that the optimal heighttowidth aspect ratio is close to 0.8 for a wide range of background straining flows.

The quasigeostrophic ellipsoidal vortex model
http://hdl.handle.net/10023/1556
We present a simple approximate model for studying general aspects of vortex interactions in a rotating stablystratified fluid. The model idealizes vortices by ellipsoidal volumes of uniform potential vorticity, a materially conserved quantity in an inviscid, adiabatic fluid. Each vortex thus possesses 9 degrees of freedom, 3 for the centroid and 6 for the shape and orientation. Here, we develop equations for the time evolution of these quantities for a general system of interacting vortices. An isolated ellipsoidal vortex is well known to remain ellipsoidal in a fluid with constant background rotation and uniform stratification, as considered here. However, the interaction between any two ellipsoids in general induces weak nonellipsoidal perturbations. We develop a unique projection method, which follows directly from the Hamiltonian structure of the system, that effectively retains just the part of the interaction which preserves ellipsoidal shapes. This method does not use a moment expansion, e.g. local expansions of the flow in a Taylor series. It is in fact more general, and consequently more accurate. Comparisons of the new model with the full equations of motion prove remarkably close.
20040425T00:00:00Z
Dritschel, David Gerard
Reinaud, Jean Noel
McKiver, William J
We present a simple approximate model for studying general aspects of vortex interactions in a rotating stablystratified fluid. The model idealizes vortices by ellipsoidal volumes of uniform potential vorticity, a materially conserved quantity in an inviscid, adiabatic fluid. Each vortex thus possesses 9 degrees of freedom, 3 for the centroid and 6 for the shape and orientation. Here, we develop equations for the time evolution of these quantities for a general system of interacting vortices. An isolated ellipsoidal vortex is well known to remain ellipsoidal in a fluid with constant background rotation and uniform stratification, as considered here. However, the interaction between any two ellipsoids in general induces weak nonellipsoidal perturbations. We develop a unique projection method, which follows directly from the Hamiltonian structure of the system, that effectively retains just the part of the interaction which preserves ellipsoidal shapes. This method does not use a moment expansion, e.g. local expansions of the flow in a Taylor series. It is in fact more general, and consequently more accurate. Comparisons of the new model with the full equations of motion prove remarkably close.

The merger of vertically offset quasigeostrophic vortices
http://hdl.handle.net/10023/1555
We examine the critical merging distance between two equalvolume, equalpotentialvorticity quasigeostrophic vortices. We focus on how this distance depends on the vertical offset between the two vortices, each having a unit mean heighttowidth aspect ratio. The vertical direction is special in the quasigeostrophic model (used to capture the leadingorder dynamical features of stably stratified and rapidly rotating geophysical flows) since vertical advection is absent. Nevertheless vortex merger may still occur by horizontal advection. In this paper, we first investigate the equilibrium states for the two vortices as a function of their vertical and horizontal separation. We examine their basic properties together with their linear stability. These findings are next compared to numerical simulations of the nonlinear evolution of two spheres of potential vorticity. Three different regimes of interaction are identified, depending on the vertical offset. For a small offset, the interaction differs little from the case when the two vortices are horizontally aligned. On the other hand, when the vertical offset is comparable to the mean vortex radius, strong interaction occurs for greater horizontal gaps than in the horizontally aligned case, and therefore at significantly greater full separation distances. This perhaps surprising result is consistent with the linear stability analysis and appears to be a consequence of the anisotropy of the quasigeostrophic equations. Finally, for large vertical offsets, vortex merger results in the formation of a metastable tilted dumbbell vortex.
20021025T00:00:00Z
Reinaud, Jean Noel
Dritschel, David Gerard
We examine the critical merging distance between two equalvolume, equalpotentialvorticity quasigeostrophic vortices. We focus on how this distance depends on the vertical offset between the two vortices, each having a unit mean heighttowidth aspect ratio. The vertical direction is special in the quasigeostrophic model (used to capture the leadingorder dynamical features of stably stratified and rapidly rotating geophysical flows) since vertical advection is absent. Nevertheless vortex merger may still occur by horizontal advection. In this paper, we first investigate the equilibrium states for the two vortices as a function of their vertical and horizontal separation. We examine their basic properties together with their linear stability. These findings are next compared to numerical simulations of the nonlinear evolution of two spheres of potential vorticity. Three different regimes of interaction are identified, depending on the vertical offset. For a small offset, the interaction differs little from the case when the two vortices are horizontally aligned. On the other hand, when the vertical offset is comparable to the mean vortex radius, strong interaction occurs for greater horizontal gaps than in the horizontally aligned case, and therefore at significantly greater full separation distances. This perhaps surprising result is consistent with the linear stability analysis and appears to be a consequence of the anisotropy of the quasigeostrophic equations. Finally, for large vertical offsets, vortex merger results in the formation of a metastable tilted dumbbell vortex.

The persistence of balance in geophysical flows
http://hdl.handle.net/10023/1496
Rotating stably stratified geophysical flows can exhibit a near 'balanced' evolution controlled by the conservative advection of a single scalar quantity, the potential vorticity (PV). This occurs frequently in the Earth's atmosphere and oceans where motions tend to be weak compared with the background planetary rotation and where stratification greatly inhibits vertical motion. Under these circumstances, both highfrequency acoustic waves and lowerfrequency inertiagravity waves (IGWs) contribute little to the flow evolution compared with the evenlowerfrequency advection of PV. Moreover, this 'slow' PVcontrolled balanced evolution appears unable to excite these higherfrequency waves in any significant wayi.e. balance persists. The present work pushes the limits of balance by systematically exploring the evolution of a range of highly nonlinear flows in which motions are comparable with the background rotation. These flows do not possess a frequency separation between PV advection and IGWs. Nonetheless, the flows exhibit a remarkable persistence of balance. Even when flows are not initialized to minimize the amount of IGWs initially present, and indeed even when flows are deliberately seeded with significant IGW amplitudes, the flow evolutionover many inertial periods (days)remains strongly controlled by PV advection.
This paper introduces a novel, powerful way to understand the why geophysical flows are largely under the control of a single scalar field, the potential vorticity, a materially conserved dynamical tracer in the absence of viscous and diabatic effects.
20070110T00:00:00Z
Dritschel, David Gerard
Viudez, A
Rotating stably stratified geophysical flows can exhibit a near 'balanced' evolution controlled by the conservative advection of a single scalar quantity, the potential vorticity (PV). This occurs frequently in the Earth's atmosphere and oceans where motions tend to be weak compared with the background planetary rotation and where stratification greatly inhibits vertical motion. Under these circumstances, both highfrequency acoustic waves and lowerfrequency inertiagravity waves (IGWs) contribute little to the flow evolution compared with the evenlowerfrequency advection of PV. Moreover, this 'slow' PVcontrolled balanced evolution appears unable to excite these higherfrequency waves in any significant wayi.e. balance persists. The present work pushes the limits of balance by systematically exploring the evolution of a range of highly nonlinear flows in which motions are comparable with the background rotation. These flows do not possess a frequency separation between PV advection and IGWs. Nonetheless, the flows exhibit a remarkable persistence of balance. Even when flows are not initialized to minimize the amount of IGWs initially present, and indeed even when flows are deliberately seeded with significant IGW amplitudes, the flow evolutionover many inertial periods (days)remains strongly controlled by PV advection.

Bending and twisting instabilities of columnar elliptical vortices in a rotating strongly stratified fluid
http://hdl.handle.net/10023/1495
In this paper, we investigate the threedimensional stability of the MooreSaffman elliptical vortex in a rotating stratified fluid. By means of an asymptotic analysis for long vertical wavelength perturbations and small Froude number, we study the effects of Rossby number, external strain, and ellipticity of the vortex on the stability of azimuthal modes m = 1 (corresponding to a bending instability) and m = 2 (corresponding to a twisting instability). In the case of a quasigeostrophic fluid (small Rossby number), the asymptotic results are in striking agreement with previous numerical stability analyses even for vertical wavelengths of order one. For arbitrary Rossby number, the key finding is that the Rossby number has no effect on the domains of longwavelength instability of these two modes: the twodimensional or threedimensional nature of the instabilities is controlled only by the background strain rate gamma and by the rotation rate Omega of the principal axes of the elliptical vortex relative to the rotating frame of reference. For the m = 1 mode, it is shown that when Omega < gamma, the vortex is stable to any longwavelength disturbances, when gamma < Omega less than or similar to 0, twodimensional perturbations are most unstable, when 0 less than or similar to Omega < gamma, longwavelength threedimensional disturbances are the most unstable, and finally when gamma < Omega, shortwavelength threedimensional perturbations are the most unstable. Similarly, the m = 2 instability is twodimensional or threedimensional depending only on gamma and Omega, independent of the Rossby number. This means that if a longwavelength threedimensional instability exists for a given elliptical vortex in a quasigeostrophic fluid, a similar instability should be observed for any other Rossby number, in particular for infinite Rossby number (strongly stratified fluids). This implies that the planetary rotation plays a minor role in the nature of the instabilities observed in rotating strongly stratified fluids. The present results for the azimuthal mode m = 1 suggest that the vortexbending instabilities observed previously in quasigeostrophic fluids (tallcolumn instability) and in strongly stratified fluids (zigzag instability) are fundamentally related.
This is a comprehensive analysis of the linear stability of columnar elliptical vortices subject to twodimensional strain in a rotating, stratified fluid. It is the culmination of two lines of research, one started by Dritschel involving the tallcolumn instability, and another started by Billant and Chomaz involving the zigzag instability. Our joint work unifies these instabilities, and shows that they exist over a vast parameter space. This work represents over 7 years of collaborative effort.
20060825T00:00:00Z
Billant, Paul
Dritschel, David Gerard
Chomaz, JeanMarc
In this paper, we investigate the threedimensional stability of the MooreSaffman elliptical vortex in a rotating stratified fluid. By means of an asymptotic analysis for long vertical wavelength perturbations and small Froude number, we study the effects of Rossby number, external strain, and ellipticity of the vortex on the stability of azimuthal modes m = 1 (corresponding to a bending instability) and m = 2 (corresponding to a twisting instability). In the case of a quasigeostrophic fluid (small Rossby number), the asymptotic results are in striking agreement with previous numerical stability analyses even for vertical wavelengths of order one. For arbitrary Rossby number, the key finding is that the Rossby number has no effect on the domains of longwavelength instability of these two modes: the twodimensional or threedimensional nature of the instabilities is controlled only by the background strain rate gamma and by the rotation rate Omega of the principal axes of the elliptical vortex relative to the rotating frame of reference. For the m = 1 mode, it is shown that when Omega < gamma, the vortex is stable to any longwavelength disturbances, when gamma < Omega less than or similar to 0, twodimensional perturbations are most unstable, when 0 less than or similar to Omega < gamma, longwavelength threedimensional disturbances are the most unstable, and finally when gamma < Omega, shortwavelength threedimensional perturbations are the most unstable. Similarly, the m = 2 instability is twodimensional or threedimensional depending only on gamma and Omega, independent of the Rossby number. This means that if a longwavelength threedimensional instability exists for a given elliptical vortex in a quasigeostrophic fluid, a similar instability should be observed for any other Rossby number, in particular for infinite Rossby number (strongly stratified fluids). This implies that the planetary rotation plays a minor role in the nature of the instabilities observed in rotating strongly stratified fluids. The present results for the azimuthal mode m = 1 suggest that the vortexbending instabilities observed previously in quasigeostrophic fluids (tallcolumn instability) and in strongly stratified fluids (zigzag instability) are fundamentally related.

Revisiting Batchelor's theory of twodimensional turbulence
http://hdl.handle.net/10023/1494
Recent mathematical results have shown that a central assumption in the theory of twodimensional turbulence proposed by Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) is false. That theory, which predicts a X2/3 k(1) enstrophy spectrum in the inertial range of freelydecaying turbulence, and which has evidently been successful in describing certain aspects of numerical simulations at high Reynolds numbers Re, assumes that there is a finite, nonzero enstrophy dissipation X in the limit of infinite Re. This, however, is not true for flows having finite vorticity. The enstrophy dissipation in fact vanishes. We revisit Batchelor's theory and propose a simple modification of it to ensure vanishing X in the limit Re > infinity. Our proposal is supported by high Reynolds number simulations which confirm that X decays like 1/ln Re, and which, following the time of peak enstrophy dissipation, exhibit enstrophy spectra containing an increasing proportion of the total enstrophy (omega(2))/2 in the inertial range as Re increases. Together with the mathematical analysis of vanishing X, these observations motivate a straightforward and, indeed, alarmingly simple modification of Batchelor's theory: just replace Batchelor's enstrophy spectrum X(2/3)k(1) with (omega(2))k(1)(In Re)(1).
20071125T00:00:00Z
Dritschel, David Gerard
Tran, Chuong Van
Scott, Richard Kirkness
Recent mathematical results have shown that a central assumption in the theory of twodimensional turbulence proposed by Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) is false. That theory, which predicts a X2/3 k(1) enstrophy spectrum in the inertial range of freelydecaying turbulence, and which has evidently been successful in describing certain aspects of numerical simulations at high Reynolds numbers Re, assumes that there is a finite, nonzero enstrophy dissipation X in the limit of infinite Re. This, however, is not true for flows having finite vorticity. The enstrophy dissipation in fact vanishes. We revisit Batchelor's theory and propose a simple modification of it to ensure vanishing X in the limit Re > infinity. Our proposal is supported by high Reynolds number simulations which confirm that X decays like 1/ln Re, and which, following the time of peak enstrophy dissipation, exhibit enstrophy spectra containing an increasing proportion of the total enstrophy (omega(2))/2 in the inertial range as Re increases. Together with the mathematical analysis of vanishing X, these observations motivate a straightforward and, indeed, alarmingly simple modification of Batchelor's theory: just replace Batchelor's enstrophy spectrum X(2/3)k(1) with (omega(2))k(1)(In Re)(1).

A balanced approach to modelling rotating stably stratified geophysical flows
http://hdl.handle.net/10023/1493
We describe a new approach to modelling threedimensional rotating stratified flows under the Boussinesq approximation. This approach is based on the explicit conservation of potential vorticity, and exploits the underlying leadingorder geostrophic and hydrostratic balances inherent in these equations in the limit of small Froude and Rossby numbers. These balances are not imposed, but instead are used to motivate the use of a pair of new variables expressing the departure from geostrophic and hydrostratic balance. These new variables are the ageostrophic horizontal vorticity components, i.e. the vorticity not directly associated with the displacement of isopycnal surfaces. The use of potential vorticity and ageostrophic horizontal vorticity, rather than the usual primitive variables of velocity and density, reveals a deep mathematical structure and appears to have advantages numerically. This change of variables results in a diagnostic equation, of MongeAmp re type, for one component of a vector potential phi, and two Poisson equations for the other two components. The curl of phi gives the velocity field while the divergence of phi is proportional to the displacement of isopycnal surfaces. This diagnostic equation makes transparent the conditions for both static and inertial stability, and may change form from (spatially) elliptic to (spatially) hyperbolic even when the flow is statically and inertially stable. A numerical method based on these new variables is developed and used to examine the instability of a horizontal elliptical shear zone (modelling a jet streak). The basicstate flow is in exact geostrophic and hydrostratic balance. Given a small perturbation however, the shear zone destabilizes by rolling up into a street of vortices and radiating inertiagravity waves.
This work was the first to show how one can rewrite the equations for a rotating stratified fluid in a way which makes potential vorticity conservation explicit. Potential vorticity is linked closely to balance, a state void of highfrequency gravity waves. The mathematical transformation reveals a deep underlying mathematical structure, including explicit conditions for inertial and static stability as well as a new double MongeAmpere equation. This work forms the cornerstone of much subsequent research into the fundamental nature of rotating stratified fluids.
20030810T00:00:00Z
Dritschel, David Gerard
Viúdez, Alvaro
We describe a new approach to modelling threedimensional rotating stratified flows under the Boussinesq approximation. This approach is based on the explicit conservation of potential vorticity, and exploits the underlying leadingorder geostrophic and hydrostratic balances inherent in these equations in the limit of small Froude and Rossby numbers. These balances are not imposed, but instead are used to motivate the use of a pair of new variables expressing the departure from geostrophic and hydrostratic balance. These new variables are the ageostrophic horizontal vorticity components, i.e. the vorticity not directly associated with the displacement of isopycnal surfaces. The use of potential vorticity and ageostrophic horizontal vorticity, rather than the usual primitive variables of velocity and density, reveals a deep mathematical structure and appears to have advantages numerically. This change of variables results in a diagnostic equation, of MongeAmp re type, for one component of a vector potential phi, and two Poisson equations for the other two components. The curl of phi gives the velocity field while the divergence of phi is proportional to the displacement of isopycnal surfaces. This diagnostic equation makes transparent the conditions for both static and inertial stability, and may change form from (spatially) elliptic to (spatially) hyperbolic even when the flow is statically and inertially stable. A numerical method based on these new variables is developed and used to examine the instability of a horizontal elliptical shear zone (modelling a jet streak). The basicstate flow is in exact geostrophic and hydrostratic balance. Given a small perturbation however, the shear zone destabilizes by rolling up into a street of vortices and radiating inertiagravity waves.

Penetrative convection in a superposed porousmedium–fluid layer via internal heating
http://hdl.handle.net/10023/1467
Supported by a research studentship by EPRSC
20040601T00:00:00Z
Carr, Magda