Applied Mathematics Theses
http://hdl.handle.net/10023/97
Wed, 21 Mar 2018 14:47:47 GMT2018-03-21T14:47:47ZRelaxation methods in compressible flow
http://hdl.handle.net/10023/12298
Sat, 01 Jan 1949 00:00:00 GMThttp://hdl.handle.net/10023/122981949-01-01T00:00:00ZMitchell, Andrew RonaldNew aspects of particle acceleration in collapsing magnetic traps
http://hdl.handle.net/10023/11954
Collapsing magnetic traps (CMTs) have been suggested as one of the mechanisms that could contribute to particle energisation in solar flares. The basic idea behind CMTs is that charged particles will be trapped on the magnetic field lines below the reconnection region of a flare. This thesis discusses a number of important new aspects in particle energisation processes in CMTs, based on the model by Giuliani et al. (2005). In particular, we extend previous studies of particle acceleration in this CMT model to the relativistic regime and compare our results obtained using relativistic guiding centre theory with results obtained using the non-relativistic guiding centre theory. The similarities and differences found are discussed. We then present a detailed study of the question, what leads to the trapping or escape of particle orbits from CMTs. The answer to this question is investigated by using results from the non-relativistic orbit calculations with guiding centre theory and a number of simple models for particle energy gain in CMTs. We find that there is a critical pitch angle dividing trapped particle orbits from the escaping particle orbits and that this critical pitch angle does not coincide with the initial loss cone angle. Furthermore, we also present a calculation of the time evolution of an anisotropic pressure tensor and of the plasma density under the assumptions that they evolve in line with our kinematic MHD CMT model and that the pressure tensor satisfies the double-adiabatic Chew-Goldburger-Low (CGL) theory.
Finally, we make a first step to introduce Coulomb scattering by a Maxwellian background plasma into our guiding centre equations by changing them into a set of stochastic differential equations. We study the influence of a static background plasma onto selected particle orbits by pitch angle scattering and energy losses, and look at its effect on the particle energy and the trapping conditions.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10023/119542014-01-01T00:00:00ZEradat Oskoui, SolmazCollapsing magnetic traps (CMTs) have been suggested as one of the mechanisms that could contribute to particle energisation in solar flares. The basic idea behind CMTs is that charged particles will be trapped on the magnetic field lines below the reconnection region of a flare. This thesis discusses a number of important new aspects in particle energisation processes in CMTs, based on the model by Giuliani et al. (2005). In particular, we extend previous studies of particle acceleration in this CMT model to the relativistic regime and compare our results obtained using relativistic guiding centre theory with results obtained using the non-relativistic guiding centre theory. The similarities and differences found are discussed. We then present a detailed study of the question, what leads to the trapping or escape of particle orbits from CMTs. The answer to this question is investigated by using results from the non-relativistic orbit calculations with guiding centre theory and a number of simple models for particle energy gain in CMTs. We find that there is a critical pitch angle dividing trapped particle orbits from the escaping particle orbits and that this critical pitch angle does not coincide with the initial loss cone angle. Furthermore, we also present a calculation of the time evolution of an anisotropic pressure tensor and of the plasma density under the assumptions that they evolve in line with our kinematic MHD CMT model and that the pressure tensor satisfies the double-adiabatic Chew-Goldburger-Low (CGL) theory.
Finally, we make a first step to introduce Coulomb scattering by a Maxwellian background plasma into our guiding centre equations by changing them into a set of stochastic differential equations. We study the influence of a static background plasma onto selected particle orbits by pitch angle scattering and energy losses, and look at its effect on the particle energy and the trapping conditions.Theory of one-dimensional Vlasov-Maxwell equilibria: with applications to collisionless current sheets and flux tubes
http://hdl.handle.net/10023/11916
Vlasov-Maxwell equilibria are characterised by the self-consistent descriptions of the steady-states of collisionless plasmas in particle phase-space, and balanced macroscopic forces. We study the theory of Vlasov-Maxwell equilibria in one spatial dimension, as well as its application to current sheet and flux tube models.
The ‘inverse problem’ is that of determining a Vlasov-Maxwell equilibrium distribution function self-consistent with a given magnetic field. We develop the theory of inversion using expansions in Hermite polynomial functions of the canonical momenta. Sufficient conditions for the convergence of a Hermite expansion are found, given a pressure tensor. For large classes of DFs, we prove that non-negativity of the distribution function is contingent on the magnetisation
of the plasma, and make conjectures for all classes.
The inverse problem is considered for nonlinear ‘force-free Harris sheets’. By applying the Hermite method, we construct new models that can describe sub-unity values of the plasma beta (βpl) for the first time. Whilst analytical convergence is proven for all βpl, numerical convergence is attained for βpl = 0.85, and then βpl = 0.05 after a ‘re-gauging’ process.
We consider the properties that a pressure tensor must satisfy to be consistent with ‘asymmetric Harris sheets’, and construct new examples. It is possible to analytically solve the inverse problem in some cases, but others must be tackled numerically. We present new exact Vlasov-Maxwell equilibria for asymmetric current sheets, which can be written as a sum of shifted Maxwellian distributions. This is ideal for implementations in particle-in-cell simulations.
We study the correspondence between the microscopic and macroscopic descriptions of equilibrium in cylindrical geometry, and then attempt to find Vlasov-Maxwell equilibria for the nonlinear force-free ‘Gold-Hoyle’ model. However, it is necessary to include a background field, which can be arbitrarily weak if desired. The equilibrium can be electrically non-neutral, depending on the bulk flows.
Thu, 07 Dec 2017 00:00:00 GMThttp://hdl.handle.net/10023/119162017-12-07T00:00:00ZAllanson, Oliver DouglasVlasov-Maxwell equilibria are characterised by the self-consistent descriptions of the steady-states of collisionless plasmas in particle phase-space, and balanced macroscopic forces. We study the theory of Vlasov-Maxwell equilibria in one spatial dimension, as well as its application to current sheet and flux tube models.
The ‘inverse problem’ is that of determining a Vlasov-Maxwell equilibrium distribution function self-consistent with a given magnetic field. We develop the theory of inversion using expansions in Hermite polynomial functions of the canonical momenta. Sufficient conditions for the convergence of a Hermite expansion are found, given a pressure tensor. For large classes of DFs, we prove that non-negativity of the distribution function is contingent on the magnetisation
of the plasma, and make conjectures for all classes.
The inverse problem is considered for nonlinear ‘force-free Harris sheets’. By applying the Hermite method, we construct new models that can describe sub-unity values of the plasma beta (βpl) for the first time. Whilst analytical convergence is proven for all βpl, numerical convergence is attained for βpl = 0.85, and then βpl = 0.05 after a ‘re-gauging’ process.
We consider the properties that a pressure tensor must satisfy to be consistent with ‘asymmetric Harris sheets’, and construct new examples. It is possible to analytically solve the inverse problem in some cases, but others must be tackled numerically. We present new exact Vlasov-Maxwell equilibria for asymmetric current sheets, which can be written as a sum of shifted Maxwellian distributions. This is ideal for implementations in particle-in-cell simulations.
We study the correspondence between the microscopic and macroscopic descriptions of equilibrium in cylindrical geometry, and then attempt to find Vlasov-Maxwell equilibria for the nonlinear force-free ‘Gold-Hoyle’ model. However, it is necessary to include a background field, which can be arbitrarily weak if desired. The equilibrium can be electrically non-neutral, depending on the bulk flows.Numerical simulations of sunspot rotation driven by magnetic flux emergence
http://hdl.handle.net/10023/10129
Magnetic flux continually emerges from the Sun, rising through the solar interior, emerging at the photosphere in the form of sunspots and expanding into the atmosphere. Observations of sunspot rotations have been reported for over a century and are often accompanied by solar eruptions and flaring activity. In this thesis, we present 3D numerical simulations of the emergence of twisted flux tubes from the uppermost layers of the solar interior, examining the rotational movements of sunspots in the photospheric plane. The basic experiment introduces the mechanism and characteristics of sunspot rotation by a clear calculation of rotation angle, vorticity, magnetic helicity and energy, whereby we find an untwisting of the interior portion of the field, accompanied by an injection of twist into the atmospheric field. We extend this model by altering the initial field strength and twist of the sub-photospheric tube. This comparison reveals the rotation angle, helicity and current show a direct dependence on field strength. An increase in field strength increases the rotation angle, the length of fieldlines extending into the atmosphere, and the magnetic energy transported to the atmosphere. The fieldline length is crucial as we predict the twist per unit length
equilibrates to a lower value on longer fieldlines, and hence possesses a larger rotation angle. No such direct dependence is found when varying the twist but there is a clear ordering in rotation angle, helicity, and energy, with more highly twisted tubes undergoing larger rotation angles. We believe the final angle of rotation is reached when the system achieves a constant degree of twist along the length of fieldlines. By extrapolating the size of the modelled active region, we find rotation angles and rates comparable with those observed. In addition, we explore sunspot rotation caused by sub-photospheric velocities twisting the
footpoints of flux tubes.
Fri, 23 Jun 2017 00:00:00 GMThttp://hdl.handle.net/10023/101292017-06-23T00:00:00ZSturrock, ZoeMagnetic flux continually emerges from the Sun, rising through the solar interior, emerging at the photosphere in the form of sunspots and expanding into the atmosphere. Observations of sunspot rotations have been reported for over a century and are often accompanied by solar eruptions and flaring activity. In this thesis, we present 3D numerical simulations of the emergence of twisted flux tubes from the uppermost layers of the solar interior, examining the rotational movements of sunspots in the photospheric plane. The basic experiment introduces the mechanism and characteristics of sunspot rotation by a clear calculation of rotation angle, vorticity, magnetic helicity and energy, whereby we find an untwisting of the interior portion of the field, accompanied by an injection of twist into the atmospheric field. We extend this model by altering the initial field strength and twist of the sub-photospheric tube. This comparison reveals the rotation angle, helicity and current show a direct dependence on field strength. An increase in field strength increases the rotation angle, the length of fieldlines extending into the atmosphere, and the magnetic energy transported to the atmosphere. The fieldline length is crucial as we predict the twist per unit length
equilibrates to a lower value on longer fieldlines, and hence possesses a larger rotation angle. No such direct dependence is found when varying the twist but there is a clear ordering in rotation angle, helicity, and energy, with more highly twisted tubes undergoing larger rotation angles. We believe the final angle of rotation is reached when the system achieves a constant degree of twist along the length of fieldlines. By extrapolating the size of the modelled active region, we find rotation angles and rates comparable with those observed. In addition, we explore sunspot rotation caused by sub-photospheric velocities twisting the
footpoints of flux tubes.On the theory of symmetric MHD equilibria with anisotropic pressure
http://hdl.handle.net/10023/9908
In this thesis we discuss the theory of symmetric MHD equilibria with anisotropic pressure. More
specifically, we focus on gyrotropic pressures, where the pressure tensor can be split into components along
and across the magnetic field. We first explore 2D solutions, which can be found using total field type
formalisms. These formalisms rely on treating quantities as functions of both the magnetic flux function
and the magnetic field strength, and reduce the equilibrium equations to a single Grad-Shafranov equation
that can be solved to find the magnetic flux function. However, these formalisms are not appropriate
when one includes a shear field component of magnetic flux, since they lead to a set of equations which
are implicitly coupled. Therefore, in order to solve the equilibrium problem with a magnetic shear field
component, we introduce the poloidal formalism. This new formalism considers quantities as functions
of the poloidal magnetic field strength (instead of the total magnetic field strength), and yields a set
of two equations which are not coupled, and can be solved to find the magnetic flux function and the
shear field. There are some situations where the poloidal formalism is difficult to use, however, such as
in rotationally symmetric systems. Thus we require a further formalism, which we call the combined
approach, which allows a more general use of the poloidal formalism. One finds that the combined
formalism leads to multi-valued functions, which must be dealt with appropriately. Finally, we present
some numerical examples of MHD equilibria, which have been found using each of the three formalisms
mentioned above.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10023/99082016-01-01T00:00:00ZHodgson, Jonathan David BrockieIn this thesis we discuss the theory of symmetric MHD equilibria with anisotropic pressure. More
specifically, we focus on gyrotropic pressures, where the pressure tensor can be split into components along
and across the magnetic field. We first explore 2D solutions, which can be found using total field type
formalisms. These formalisms rely on treating quantities as functions of both the magnetic flux function
and the magnetic field strength, and reduce the equilibrium equations to a single Grad-Shafranov equation
that can be solved to find the magnetic flux function. However, these formalisms are not appropriate
when one includes a shear field component of magnetic flux, since they lead to a set of equations which
are implicitly coupled. Therefore, in order to solve the equilibrium problem with a magnetic shear field
component, we introduce the poloidal formalism. This new formalism considers quantities as functions
of the poloidal magnetic field strength (instead of the total magnetic field strength), and yields a set
of two equations which are not coupled, and can be solved to find the magnetic flux function and the
shear field. There are some situations where the poloidal formalism is difficult to use, however, such as
in rotationally symmetric systems. Thus we require a further formalism, which we call the combined
approach, which allows a more general use of the poloidal formalism. One finds that the combined
formalism leads to multi-valued functions, which must be dealt with appropriately. Finally, we present
some numerical examples of MHD equilibria, which have been found using each of the three formalisms
mentioned above.Numerical simulations of footpoint driven coronal heating
http://hdl.handle.net/10023/8871
Magnetic field permeates the solar atmosphere and plays a crucial role in the dynamics, energetics and structures observed. In particular, magnetic flux tubes provide the structure for coronal loops that extend from the solar surface into the corona. In this thesis, we present 3D numerical simulations examining the heating produced by reconnection between flux tubes driven by rotational footpoint motions. The basic model consists of two, initially aligned, flux tubes that are forced to interact by rotational driving velocities on the flux concentrations on the boundaries. A single, twisted current layer is created in the centre of the domain and strong, localised heating is produced. We extend this model by altering the number, distribution and strength of the sources, while maintaining the same total magnetic flux on the boundaries. The dynamical evolution and the resultant magnitude, distribution and timing of the heating events are examined for the different flux distributions. In all cases, the magnetic field is stressed by the boundary motions and a current grows within the domain. A comparison of cases with two and four sources shows that there are more locations of current concentrations, but with reduced maximum current density values, for the four source case. This produces weaker reconnection and less efficient heating. In addition, for the case with two sources, we also consider the effect of splitting up one of the sources into many smaller flux fragments. The evolution and heating are shown to be very similar to the two source case. The impact of increasing the strength of the background field between the flux tubes is also examined and we find that it delays and increases the strength of the heating, although by how much depends on the distribution of the flux sources.
Fri, 24 Jun 2016 00:00:00 GMThttp://hdl.handle.net/10023/88712016-06-24T00:00:00ZO'Hara, JenniferMagnetic field permeates the solar atmosphere and plays a crucial role in the dynamics, energetics and structures observed. In particular, magnetic flux tubes provide the structure for coronal loops that extend from the solar surface into the corona. In this thesis, we present 3D numerical simulations examining the heating produced by reconnection between flux tubes driven by rotational footpoint motions. The basic model consists of two, initially aligned, flux tubes that are forced to interact by rotational driving velocities on the flux concentrations on the boundaries. A single, twisted current layer is created in the centre of the domain and strong, localised heating is produced. We extend this model by altering the number, distribution and strength of the sources, while maintaining the same total magnetic flux on the boundaries. The dynamical evolution and the resultant magnitude, distribution and timing of the heating events are examined for the different flux distributions. In all cases, the magnetic field is stressed by the boundary motions and a current grows within the domain. A comparison of cases with two and four sources shows that there are more locations of current concentrations, but with reduced maximum current density values, for the four source case. This produces weaker reconnection and less efficient heating. In addition, for the case with two sources, we also consider the effect of splitting up one of the sources into many smaller flux fragments. The evolution and heating are shown to be very similar to the two source case. The impact of increasing the strength of the background field between the flux tubes is also examined and we find that it delays and increases the strength of the heating, although by how much depends on the distribution of the flux sources.The formation and eruption of magnetic flux ropes in solar and stellar coronae
http://hdl.handle.net/10023/7069
Flux ropes are magnetic structures commonly found in the solar corona. They are thought to play an important role in solar flares and coronal mass ejections. Understanding their formation and eruption is of paramount importance for our understanding of space weather. In this thesis the magnetofrictional method is applied to simulate the formation of flux ropes and track their evolution up to eruption both in solar and stellar coronae.
Initially, the coronal magnetic field of a solar active region is simulated using observed magnetograms to drive the coronal evolution. From the sequence of magnetograms the formation of a flux rope is simulated, and compared with coronal observations.
Secondly a procedure to produce proxy SOLIS synoptic magnetograms from SDO/HMI and SOHO/MDI magnetograms is presented. This procedure allows SOLIS-like synoptic magnetograms to be produced during times when SOLIS magnetograms are not available.
Thirdly, a series of scaling laws for the formation and life-times of flux ropes in stellar coronae are determined as a function of stellar differential rotation and surface diffusion. These scaling laws can be used to infer the response of stellar coronae to the transport of magnetic fields at their surface.
Finally, global long-term simulations of stellar corona are carried out to determine the coronal response to flux emergence and differential rotation. A bipole emergence model is developed and is used in conjunction with a surface flux transport model in order to drive the global coronal evolution. These global simulations allow the flux, energy and flux rope distributions to be studied as a function of a star’s differential rotation and flux emergence rate.
Mon, 30 Nov 2015 00:00:00 GMThttp://hdl.handle.net/10023/70692015-11-30T00:00:00ZGibb, Gordon Peter SamuelFlux ropes are magnetic structures commonly found in the solar corona. They are thought to play an important role in solar flares and coronal mass ejections. Understanding their formation and eruption is of paramount importance for our understanding of space weather. In this thesis the magnetofrictional method is applied to simulate the formation of flux ropes and track their evolution up to eruption both in solar and stellar coronae.
Initially, the coronal magnetic field of a solar active region is simulated using observed magnetograms to drive the coronal evolution. From the sequence of magnetograms the formation of a flux rope is simulated, and compared with coronal observations.
Secondly a procedure to produce proxy SOLIS synoptic magnetograms from SDO/HMI and SOHO/MDI magnetograms is presented. This procedure allows SOLIS-like synoptic magnetograms to be produced during times when SOLIS magnetograms are not available.
Thirdly, a series of scaling laws for the formation and life-times of flux ropes in stellar coronae are determined as a function of stellar differential rotation and surface diffusion. These scaling laws can be used to infer the response of stellar coronae to the transport of magnetic fields at their surface.
Finally, global long-term simulations of stellar corona are carried out to determine the coronal response to flux emergence and differential rotation. A bipole emergence model is developed and is used in conjunction with a surface flux transport model in order to drive the global coronal evolution. These global simulations allow the flux, energy and flux rope distributions to be studied as a function of a star’s differential rotation and flux emergence rate.Propagation and damping of MHD waves in the solar atmosphere
http://hdl.handle.net/10023/7054
Quasi-periodic disturbances have been observed in the outer solar atmosphere for many years. Although first interpreted as upflows (Schrijver et al. (1999)), they have been widely regarded as slow magneto-acoustic waves, due to their observed velocities and periods. Here we conduct a detailed analysis of the velocities of these disturbances across several wavelengths using the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamics Observatory (SDO). We analysed 41 examples, including both sunspot and non-sunspot regions of the Sun. We found that the velocities of propagating disturbances (PDs) located at sunspots are more likely to be temperature dependent, whereas the velocities of PDs at non-sunspot locations do not show a clear temperature dependence. This suggests an interpretation in terms of slow magneto-acoustic waves in sunspots but the nature of PDs in non-sunspot (plage) regions remains unclear. Finally, we found that removing the contribution due to the cooler ions in the 193 wavelength suggests that a substantial part of the 193 emission of sunspot PDs can be attributed to the cool component of 193. Phase mixing is a well known and studied phenomenon in the solar corona, to enhance the dissipation of Alfvén waves (Heyvaerts and Priest (1982)). In this study we run numerical simulations of a continuously driven Alfvén wave in a low beta plasma along a uniform magnetic field. We model phase mixing by introducing a density inhomogeneity. Thermal conduction is then added into the model in the form of Braginskii thermal conduction. This acts to transport heat along the magnetic field. A parameter study will be carried out to investigate how changing the density structure and other parameters changes the results. We go on to consider the effect of wave reflection on phase mixing. We found that wave reflection has no effect on the damping of Alfvén waves but increases the heat in the system. We also consider a more realistic experiment where we drive both boundaries and study how the loop is heated in this situation. We also study what effect changing the frequency of one of the drivers so there is a small difference between the frequencies (10%) and a large difference (50%). We find the general behaviour is similar, but the heat is tilted.
We have investigated basic phase mixing model which incorporates the mass exchange between the corona and the chromosphere. Chromospheric evaporation is approximated by using a non dimensional version of the RTV (Rosner et al. (1978)) scaling laws, relating heating (by phase mixing of Alfvén waves), density and temperature. By combining this scaling law with our numerical MHD model for phase mixing of Alfvén waves, we investigate the modification of the density profile through the mass up flow. We find a rapid modification of the density profile, leading to drifting of the heating layers. We also find that similar results are own seen in the propagating Alfvén wave case when we incorporate the effects of reflection.
Fri, 27 Jun 2014 00:00:00 GMThttp://hdl.handle.net/10023/70542014-06-27T00:00:00ZKiddie, GregQuasi-periodic disturbances have been observed in the outer solar atmosphere for many years. Although first interpreted as upflows (Schrijver et al. (1999)), they have been widely regarded as slow magneto-acoustic waves, due to their observed velocities and periods. Here we conduct a detailed analysis of the velocities of these disturbances across several wavelengths using the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamics Observatory (SDO). We analysed 41 examples, including both sunspot and non-sunspot regions of the Sun. We found that the velocities of propagating disturbances (PDs) located at sunspots are more likely to be temperature dependent, whereas the velocities of PDs at non-sunspot locations do not show a clear temperature dependence. This suggests an interpretation in terms of slow magneto-acoustic waves in sunspots but the nature of PDs in non-sunspot (plage) regions remains unclear. Finally, we found that removing the contribution due to the cooler ions in the 193 wavelength suggests that a substantial part of the 193 emission of sunspot PDs can be attributed to the cool component of 193. Phase mixing is a well known and studied phenomenon in the solar corona, to enhance the dissipation of Alfvén waves (Heyvaerts and Priest (1982)). In this study we run numerical simulations of a continuously driven Alfvén wave in a low beta plasma along a uniform magnetic field. We model phase mixing by introducing a density inhomogeneity. Thermal conduction is then added into the model in the form of Braginskii thermal conduction. This acts to transport heat along the magnetic field. A parameter study will be carried out to investigate how changing the density structure and other parameters changes the results. We go on to consider the effect of wave reflection on phase mixing. We found that wave reflection has no effect on the damping of Alfvén waves but increases the heat in the system. We also consider a more realistic experiment where we drive both boundaries and study how the loop is heated in this situation. We also study what effect changing the frequency of one of the drivers so there is a small difference between the frequencies (10%) and a large difference (50%). We find the general behaviour is similar, but the heat is tilted.
We have investigated basic phase mixing model which incorporates the mass exchange between the corona and the chromosphere. Chromospheric evaporation is approximated by using a non dimensional version of the RTV (Rosner et al. (1978)) scaling laws, relating heating (by phase mixing of Alfvén waves), density and temperature. By combining this scaling law with our numerical MHD model for phase mixing of Alfvén waves, we investigate the modification of the density profile through the mass up flow. We find a rapid modification of the density profile, leading to drifting of the heating layers. We also find that similar results are own seen in the propagating Alfvén wave case when we incorporate the effects of reflection.On the properties of single-separator MHS equilibria and the nature of separator reconnection
http://hdl.handle.net/10023/6678
This thesis considers the properties of MHS equilibria formed through non-resistive MHD relaxation of analytical non-potential magnetic field models, which contain two null points connected by a generic separator. Four types of analytical magnetic fields are formulated, with different forms of current. The magnetic field model which has a uniform current directed along the separator, is used through the rest of this thesis to form MHS equilibria and to study reconnection.
This magnetic field, which is not force-free, embedded in a high-beta plasma, relaxes non-resistively using a 3D MHD code. The relaxation causes the field about the separator to collapse leading to a twisted current layer forming along the separator. The MHS equilibrium current layer slowly becomes stronger, longer, wider and thinner with time. Its properties, and the properties of the plasma, are found to depend on the initial parameters of the magnetic field, which control the geometry of the magnetic configuration.
Such a MHS equilibria is used in a high plasma-beta reconnection experiment. An anomalous resistivity ensures that only the central strong current in the separator current layer is dissipated. The reconnection occurs in two phases characterised by fast and slow reconnection, respectively. Waves, launched from the diffusion site, communicate the loss of force balance at the current layer and set up flows in the system. The energy transport in this system is dominated by Ohmic dissipation.
Several methods are presented which allow a low plasma-beta value to be approached in the single-separator model. One method is chosen and this model is relaxed non-resistively to form a MHS equilibrium. A twisted current layer grows along the separator, containing stronger current than in the high plasma-beta experiments, and has a local enhancement in pressure inside it. The growth rate of this current layer is similar to that found in the high plasma-beta experiments, however, the current layer becomes thinner and narrower over time.
Fri, 26 Jun 2015 00:00:00 GMThttp://hdl.handle.net/10023/66782015-06-26T00:00:00ZStevenson, Julie E. H.This thesis considers the properties of MHS equilibria formed through non-resistive MHD relaxation of analytical non-potential magnetic field models, which contain two null points connected by a generic separator. Four types of analytical magnetic fields are formulated, with different forms of current. The magnetic field model which has a uniform current directed along the separator, is used through the rest of this thesis to form MHS equilibria and to study reconnection.
This magnetic field, which is not force-free, embedded in a high-beta plasma, relaxes non-resistively using a 3D MHD code. The relaxation causes the field about the separator to collapse leading to a twisted current layer forming along the separator. The MHS equilibrium current layer slowly becomes stronger, longer, wider and thinner with time. Its properties, and the properties of the plasma, are found to depend on the initial parameters of the magnetic field, which control the geometry of the magnetic configuration.
Such a MHS equilibria is used in a high plasma-beta reconnection experiment. An anomalous resistivity ensures that only the central strong current in the separator current layer is dissipated. The reconnection occurs in two phases characterised by fast and slow reconnection, respectively. Waves, launched from the diffusion site, communicate the loss of force balance at the current layer and set up flows in the system. The energy transport in this system is dominated by Ohmic dissipation.
Several methods are presented which allow a low plasma-beta value to be approached in the single-separator model. One method is chosen and this model is relaxed non-resistively to form a MHS equilibrium. A twisted current layer grows along the separator, containing stronger current than in the high plasma-beta experiments, and has a local enhancement in pressure inside it. The growth rate of this current layer is similar to that found in the high plasma-beta experiments, however, the current layer becomes thinner and narrower over time.MHD simulations of coronal heating
http://hdl.handle.net/10023/6373
The problem of heating the solar corona requires the conversion of magnetic energy into thermal energy. Presently, there are two promising mechanisms for heating the solar corona: wave heating and nanoflare heating. In this thesis, we consider nanoflare heating only. Previous modelling has shown that the kink instability can trigger energy release and heating in large scale loops, as the field rapidly relaxes to a lower energy state under the Taylor relaxation theory. Two distinct experiments were developed to understand the coronal heating problem: the avalanche effect within a multiple loop system, and the importance of thermal conduction and optically thin radiation during the evolution of the kinked-unstable coronal magnetic field.
The first experiment showed that a kink-unstable thread can also destabilise nearby threads under some conditions. The second experiment showed that the inclusion of thermal conduction and optically thin radiation causes significant change to the internal energy of the coronal loop. After the initial instability occurs, there is continual heating throughout the relaxation process. Our simulation results show that the data is consistent with observation values, and the relaxation process can take over 200 seconds to reach the final relaxed state. The inclusion of both effects perhaps provides a more realistic and rapid heating experiment compared to previous investigations.
Mon, 01 Dec 2014 00:00:00 GMThttp://hdl.handle.net/10023/63732014-12-01T00:00:00ZTam, Kuan V.The problem of heating the solar corona requires the conversion of magnetic energy into thermal energy. Presently, there are two promising mechanisms for heating the solar corona: wave heating and nanoflare heating. In this thesis, we consider nanoflare heating only. Previous modelling has shown that the kink instability can trigger energy release and heating in large scale loops, as the field rapidly relaxes to a lower energy state under the Taylor relaxation theory. Two distinct experiments were developed to understand the coronal heating problem: the avalanche effect within a multiple loop system, and the importance of thermal conduction and optically thin radiation during the evolution of the kinked-unstable coronal magnetic field.
The first experiment showed that a kink-unstable thread can also destabilise nearby threads under some conditions. The second experiment showed that the inclusion of thermal conduction and optically thin radiation causes significant change to the internal energy of the coronal loop. After the initial instability occurs, there is continual heating throughout the relaxation process. Our simulation results show that the data is consistent with observation values, and the relaxation process can take over 200 seconds to reach the final relaxed state. The inclusion of both effects perhaps provides a more realistic and rapid heating experiment compared to previous investigations.On the topology of global coronal magnetic fields
http://hdl.handle.net/10023/5896
This thesis considers the magnetic topology of the global solar corona. To understand the magnetic topology we use the magnetic skeleton which provides us with a robust description of the magnetic field. To do this we use a Potential Field model extrapolated from observations of the photospheric magnetic field. Various measurements of the photospheric magnetic field are used from both ground-based observatories (Kitt-Peak and SOLIS) and space-based observatories (MDI and HMI).
Using the magnetic skeleton we characterise particular topological structures and discuss their variations throughout the solar cycle. We find that, from the topology, there are two types of solar minimum magnetic field and one type of solar maximum. The global structure of the coronal magnetic field depends on the relative strengths of the polar fields and the low-latitude fields. During a strong solar dipole minimum the heliospheric current sheet sits near the equator and the heliospheric current sheet curtains enclose a large amount of mixed polarity field which is associated with many low-altitude null points. In a weak solar dipole minimum the heliospheric current sheet becomes warped and large scale topological features can form that are associated with weak magnetic field regions. At solar maximum the heliospheric current sheet is highly warped and there are more null points at high altitudes than at solar minimum.
The number of null points in a magnetic field can be seen as a measure of the complexity of the field so this is investigated. We find that the number of nulls above 10Mm falls off with height as a power law whose slope depends on the phase of the solar cycle.
We compare the magnetic topology we found at particular times with observations of the Doppler velocity and intensity around particular active regions to see if it is possible to determine whether plasma upflows at the edge of active regions are linked to open field regions.
Mon, 01 Dec 2014 00:00:00 GMThttp://hdl.handle.net/10023/58962014-12-01T00:00:00ZEdwards, Sarah J.This thesis considers the magnetic topology of the global solar corona. To understand the magnetic topology we use the magnetic skeleton which provides us with a robust description of the magnetic field. To do this we use a Potential Field model extrapolated from observations of the photospheric magnetic field. Various measurements of the photospheric magnetic field are used from both ground-based observatories (Kitt-Peak and SOLIS) and space-based observatories (MDI and HMI).
Using the magnetic skeleton we characterise particular topological structures and discuss their variations throughout the solar cycle. We find that, from the topology, there are two types of solar minimum magnetic field and one type of solar maximum. The global structure of the coronal magnetic field depends on the relative strengths of the polar fields and the low-latitude fields. During a strong solar dipole minimum the heliospheric current sheet sits near the equator and the heliospheric current sheet curtains enclose a large amount of mixed polarity field which is associated with many low-altitude null points. In a weak solar dipole minimum the heliospheric current sheet becomes warped and large scale topological features can form that are associated with weak magnetic field regions. At solar maximum the heliospheric current sheet is highly warped and there are more null points at high altitudes than at solar minimum.
The number of null points in a magnetic field can be seen as a measure of the complexity of the field so this is investigated. We find that the number of nulls above 10Mm falls off with height as a power law whose slope depends on the phase of the solar cycle.
We compare the magnetic topology we found at particular times with observations of the Doppler velocity and intensity around particular active regions to see if it is possible to determine whether plasma upflows at the edge of active regions are linked to open field regions.An analytical, phenomenological and numerical study of geophysical and magnetohydrodynamic turbulence in two dimensions
http://hdl.handle.net/10023/4291
In this thesis I study a variety of two-dimensional turbulent systems using a
mixed analytical, phenomenological and numerical approach. The systems under
consideration are governed by the two-dimensional Navier-Stokes (2DNS),
surface quasigeostrophic (SQG), alpha-turbulence and magnetohydrodynamic (MHD)
equations. The main analytical focus is on the number of degrees of freedom
of a given system, defined as the least value $N$ such that all
$n$-dimensional ($n$ ≥ $N$) volume elements along a given trajectory contract
during the course of evolution. By equating $N$ with the number of active
Fourier-space modes, that is the number of modes in the inertial range, and
assuming power-law spectra in the inertial range, the scaling of $N$ with the
Reynolds number $Re$ allows bounds to be put on the exponent of the spectrum.
This allows the recovery of analytic results that have until now only been
derived phenomenologically, such as the $k$[superscript(-5/3)] energy spectrum in the
energy inertial range in SQG turbulence. Phenomenologically I study the modal
interactions that control the transfer of various conserved quantities. Among
other results I show that in MHD dynamo triads (those converting kinetic into
magnetic energy) are associated with a direct magnetic energy flux while
anti-dynamo triads (those converting magnetic into kinetic energy) are
associated with an inverse magnetic energy flux. As both dynamo and anti-dynamo
interacting triads are integral parts of the direct energy transfer, the
anti-dynamo inverse flux partially neutralises the dynamo direct flux, arguably
resulting in relatively weak direct energy transfer and giving rise to dynamo
saturation. These theoretical results are backed up by high resolution
numerical simulations, out of which have emerged some new results such as the
suggestion that for alpha turbulence the generalised enstrophy spectra are not
closely approximated by those that have been derived phenomenologically, and
new theories may be needed in order to explain them.
Fri, 29 Nov 2013 00:00:00 GMThttp://hdl.handle.net/10023/42912013-11-29T00:00:00ZBlackbourn, Luke A. K.In this thesis I study a variety of two-dimensional turbulent systems using a
mixed analytical, phenomenological and numerical approach. The systems under
consideration are governed by the two-dimensional Navier-Stokes (2DNS),
surface quasigeostrophic (SQG), alpha-turbulence and magnetohydrodynamic (MHD)
equations. The main analytical focus is on the number of degrees of freedom
of a given system, defined as the least value $N$ such that all
$n$-dimensional ($n$ ≥ $N$) volume elements along a given trajectory contract
during the course of evolution. By equating $N$ with the number of active
Fourier-space modes, that is the number of modes in the inertial range, and
assuming power-law spectra in the inertial range, the scaling of $N$ with the
Reynolds number $Re$ allows bounds to be put on the exponent of the spectrum.
This allows the recovery of analytic results that have until now only been
derived phenomenologically, such as the $k$[superscript(-5/3)] energy spectrum in the
energy inertial range in SQG turbulence. Phenomenologically I study the modal
interactions that control the transfer of various conserved quantities. Among
other results I show that in MHD dynamo triads (those converting kinetic into
magnetic energy) are associated with a direct magnetic energy flux while
anti-dynamo triads (those converting magnetic into kinetic energy) are
associated with an inverse magnetic energy flux. As both dynamo and anti-dynamo
interacting triads are integral parts of the direct energy transfer, the
anti-dynamo inverse flux partially neutralises the dynamo direct flux, arguably
resulting in relatively weak direct energy transfer and giving rise to dynamo
saturation. These theoretical results are backed up by high resolution
numerical simulations, out of which have emerged some new results such as the
suggestion that for alpha turbulence the generalised enstrophy spectra are not
closely approximated by those that have been derived phenomenologically, and
new theories may be needed in order to explain them.Inhomogeneous magnetic fields in the solar atmosphere
http://hdl.handle.net/10023/3830
The magnetic field in the solar atmosphere is highly inhomogeneous. In the photosphere, the field is concentrated into intense flux tubes and the coronal magnetic field consists of many loops and regions of open field. This thesis investigates some of the basic properties of inhomogeneous solar magnetic fields.
First of all, the equilibrium properties of untwisted flux tubes, confined by a spatially varying external pressure distribution, are investigated. The behaviour of thick flux tubes, including the effects of a transverse field component and a variation in the field across the tube, is compared with slender flux tube theory. It is shown that slender tube theory is accurate for tubes which are approximately slender, but that completely misleading results can be obtained by applying slender tube theory if the pressure distribution is not slowly varying.
Twisted flux tubes are then studied, with the aim of finding how twisting affects a tube confined by an inhomogeneous pressure distribution. It is shown that, in general, a tube expands as it is twisted; this is illustrated both by extensions to slender tube theory and by some exact analytical solutions. A family of linear solutions is used to model the evolution of a finite tube confined by a falling external pressure. It is shown that, if the confining pressure falls too low, the tube may burst, with some dynamic process ensuing.
The equilibrium properties of a flux tube with a curved axis are then investigated, with the main aim of modelling coronal loops. Previous theory for the equilibrium of a curved slender flux tube in a gravitationally stratified atmosphere, with a balance between magnetic buoyancy and tension forces, is extended to take into account an external field and the effects of twist. Increasing the magnitude of the external field tends to lower the summit height of the tube. It is found that non-equilibrium sets in if the footpoints are separated more than a certain critical width, which does not depend on the magnitude of the external field. It is found that two possible equilibrium heights can exist for a twisted tube; however, if the tube is twisted too far, or if the footpoints are moved apart, non-equilibrium can set in. The critical width at which non-equilibrium occurs is lower for a twisted tube than for an untwisted one. This is suggested as an explanation for the rise of a filament prior to a two ribbon flare, and as a mechanism for coronal transients.
An alternative description of the coronal magnetic field is given, using a perturbation expansion for an almost potential field, with small pressure gradients. The field is assumed to be line-tied at the photospheric base.
Then the equilibrium properties of the global magnetic field of a star are investigated. A linear and non-linear family of solutions to the magnetostatic equilibrium equation are found. The linear solutions are used to investigate the twisting up of force-free dipolar and quadrupolar fields, including in a simple manner the effects of a stellar wind. In both cases, it was found that the field becomes physically unreasonable if it is twisted too far, with field lines detached from the star being formed, which would be pulled out by the stellar wind. Thus, if the field is twisted more than a critical amount, non-equilibrium sets in and some catastrophic behaviour takes place. This is suggested as a possible mechanism for stellar flares. Similar results are found in a study of the effects of increasing the pressure gradients at the stellar surface of a magnetostatic dipole-like field. The linear solutions are also used to study the equilibrium of a finite magnetosphere, and multiple equilibria are found.
Finally, one aspect of the propagation of waves in an inhomogeneous magnetic field is studied, with particular reference to the problem of heating the solar corona. The mechanism of phase-mixing, which provides a means of dissipating shear Alfven waves that propagate in an inhomogeneous magnetic field, is investigated. The onset of Kelvin-Helmholtz instability, which could disrupt the wave and thus enhance the dissipation, is studied. First, the dispersion relation of the instability is calculated for the case of fully developed phase-mixing. Then, the onset of the instability is investigated, to find out whether the instability can grow before the phase-mixing is fully developed. It is found that instability can set in after only a very few wave periods. It is suggested that the instability triggers off a turbulent cascade which dissipates the wave energy. The heating rates that could be produced by such a process are calculated, and are found to be more than adequate for coronal heating.
Sun, 01 Jan 1984 00:00:00 GMThttp://hdl.handle.net/10023/38301984-01-01T00:00:00ZBrowning, PhilippaThe magnetic field in the solar atmosphere is highly inhomogeneous. In the photosphere, the field is concentrated into intense flux tubes and the coronal magnetic field consists of many loops and regions of open field. This thesis investigates some of the basic properties of inhomogeneous solar magnetic fields.
First of all, the equilibrium properties of untwisted flux tubes, confined by a spatially varying external pressure distribution, are investigated. The behaviour of thick flux tubes, including the effects of a transverse field component and a variation in the field across the tube, is compared with slender flux tube theory. It is shown that slender tube theory is accurate for tubes which are approximately slender, but that completely misleading results can be obtained by applying slender tube theory if the pressure distribution is not slowly varying.
Twisted flux tubes are then studied, with the aim of finding how twisting affects a tube confined by an inhomogeneous pressure distribution. It is shown that, in general, a tube expands as it is twisted; this is illustrated both by extensions to slender tube theory and by some exact analytical solutions. A family of linear solutions is used to model the evolution of a finite tube confined by a falling external pressure. It is shown that, if the confining pressure falls too low, the tube may burst, with some dynamic process ensuing.
The equilibrium properties of a flux tube with a curved axis are then investigated, with the main aim of modelling coronal loops. Previous theory for the equilibrium of a curved slender flux tube in a gravitationally stratified atmosphere, with a balance between magnetic buoyancy and tension forces, is extended to take into account an external field and the effects of twist. Increasing the magnitude of the external field tends to lower the summit height of the tube. It is found that non-equilibrium sets in if the footpoints are separated more than a certain critical width, which does not depend on the magnitude of the external field. It is found that two possible equilibrium heights can exist for a twisted tube; however, if the tube is twisted too far, or if the footpoints are moved apart, non-equilibrium can set in. The critical width at which non-equilibrium occurs is lower for a twisted tube than for an untwisted one. This is suggested as an explanation for the rise of a filament prior to a two ribbon flare, and as a mechanism for coronal transients.
An alternative description of the coronal magnetic field is given, using a perturbation expansion for an almost potential field, with small pressure gradients. The field is assumed to be line-tied at the photospheric base.
Then the equilibrium properties of the global magnetic field of a star are investigated. A linear and non-linear family of solutions to the magnetostatic equilibrium equation are found. The linear solutions are used to investigate the twisting up of force-free dipolar and quadrupolar fields, including in a simple manner the effects of a stellar wind. In both cases, it was found that the field becomes physically unreasonable if it is twisted too far, with field lines detached from the star being formed, which would be pulled out by the stellar wind. Thus, if the field is twisted more than a critical amount, non-equilibrium sets in and some catastrophic behaviour takes place. This is suggested as a possible mechanism for stellar flares. Similar results are found in a study of the effects of increasing the pressure gradients at the stellar surface of a magnetostatic dipole-like field. The linear solutions are also used to study the equilibrium of a finite magnetosphere, and multiple equilibria are found.
Finally, one aspect of the propagation of waves in an inhomogeneous magnetic field is studied, with particular reference to the problem of heating the solar corona. The mechanism of phase-mixing, which provides a means of dissipating shear Alfven waves that propagate in an inhomogeneous magnetic field, is investigated. The onset of Kelvin-Helmholtz instability, which could disrupt the wave and thus enhance the dissipation, is studied. First, the dispersion relation of the instability is calculated for the case of fully developed phase-mixing. Then, the onset of the instability is investigated, to find out whether the instability can grow before the phase-mixing is fully developed. It is found that instability can set in after only a very few wave periods. It is suggested that the instability triggers off a turbulent cascade which dissipates the wave energy. The heating rates that could be produced by such a process are calculated, and are found to be more than adequate for coronal heating.The structure, stability and interaction of geophysical vortices
http://hdl.handle.net/10023/3729
This thesis examines the structure, stability and interaction of geophysical vortices. We do so by restricting our attention to relative vortex equilibria, or states which appear stationary in a co-rotating frame of reference. We approach the problem from three different perspectives, namely by first studying the single-vortex, quasi-geostrophic shallow-water problem, next by generalising it to an (asymmetric) two-vortex problem, and finally by re-visiting the single-vortex problem, making use of the more realistic, although more complicated, shallow-water model.
We find that in all of the systems studied, small vortices (compared to the Rossby deformation length) are more likely to be unstable than large ones. For the single-vortex problem, this means that large vortices can sustain much greater deformations before destabilising than small vortices, and for the two-vortex problem this means that vortices are able to come closer together before destabilising. Additionally, we find that for large vortices, the degree of asymmetry of a vortex pair does not affect its stability, although it does affect the underlying steady state into which an unstable state transitions. Lastly, by carefully defining the "equivalence" between cyclones and anticyclones which appear in the shallow-water system, we find that cyclones are more stable than anticyclones. This is contrary to what is generally reported in the literature.
Fri, 28 Jun 2013 00:00:00 GMThttp://hdl.handle.net/10023/37292013-06-28T00:00:00ZPłotka, HannaThis thesis examines the structure, stability and interaction of geophysical vortices. We do so by restricting our attention to relative vortex equilibria, or states which appear stationary in a co-rotating frame of reference. We approach the problem from three different perspectives, namely by first studying the single-vortex, quasi-geostrophic shallow-water problem, next by generalising it to an (asymmetric) two-vortex problem, and finally by re-visiting the single-vortex problem, making use of the more realistic, although more complicated, shallow-water model.
We find that in all of the systems studied, small vortices (compared to the Rossby deformation length) are more likely to be unstable than large ones. For the single-vortex problem, this means that large vortices can sustain much greater deformations before destabilising than small vortices, and for the two-vortex problem this means that vortices are able to come closer together before destabilising. Additionally, we find that for large vortices, the degree of asymmetry of a vortex pair does not affect its stability, although it does affect the underlying steady state into which an unstable state transitions. Lastly, by carefully defining the "equivalence" between cyclones and anticyclones which appear in the shallow-water system, we find that cyclones are more stable than anticyclones. This is contrary to what is generally reported in the literature.Equilibrium and stability properties of collisionless current sheet models
http://hdl.handle.net/10023/3548
The work in this thesis focuses primarily on equilibrium and stability properties of collisionless current sheet models, in particular of the force-free Harris sheet model.
A detailed investigation is carried out into the properties of the distribution function found by Harrison and Neukirch (Physical Review Letters 102, 135003, 2009) for the force-free Harris sheet, which is so far the only known nonlinear force-free Vlasov-Maxwell equilibrium. Exact conditions on the parameters of the distribution function are found, which show when it can be single or multi-peaked in two of the velocity space directions. This is important because it may have implications for the stability of the equilibrium.
One major aim of this thesis is to find new force-free equilibrium distribution functions. By using a new method which is different from that of Harrison and Neukirch, it is possible to find a complete family of distribution functions for the force-free Harris sheet, which includes the Harrison and Neukirch distribution function (Physical Review Letters 102, 135003, 2009). Each member of this family has a different dependence on the particle energy, although the dependence on the canonical momenta remains the same. Three detailed analytical examples are presented. Other possibilities for finding further collisionless force-free equilibrium distribution functions have been explored, but were unsuccessful.
The first linear stability analysis of the Harrison and Neukirch equilibrium distribution function is then carried out, concentrating on macroscopic instabilities, and considering two-dimensional perturbations only. The analysis is based on the technique of integration over unperturbed orbits. Similarly to the Harris sheet case (Nuovo Cimento, 23:115, 1962), this is only possible by using approximations to the exact orbits, which are unknown. Furthermore, the approximations for the Harris sheet case cannot be used for the force-free Harris sheet, and so new techniques have to be developed in order to make analytical progress. Full analytical expressions for the perturbed current density are derived but, for the sake of simplicity, only the long wavelength limit is investigated. The dependence of the stability on various equilibrium parameters is investigated.
Fri, 28 Jun 2013 00:00:00 GMThttp://hdl.handle.net/10023/35482013-06-28T00:00:00ZWilson, FionaThe work in this thesis focuses primarily on equilibrium and stability properties of collisionless current sheet models, in particular of the force-free Harris sheet model.
A detailed investigation is carried out into the properties of the distribution function found by Harrison and Neukirch (Physical Review Letters 102, 135003, 2009) for the force-free Harris sheet, which is so far the only known nonlinear force-free Vlasov-Maxwell equilibrium. Exact conditions on the parameters of the distribution function are found, which show when it can be single or multi-peaked in two of the velocity space directions. This is important because it may have implications for the stability of the equilibrium.
One major aim of this thesis is to find new force-free equilibrium distribution functions. By using a new method which is different from that of Harrison and Neukirch, it is possible to find a complete family of distribution functions for the force-free Harris sheet, which includes the Harrison and Neukirch distribution function (Physical Review Letters 102, 135003, 2009). Each member of this family has a different dependence on the particle energy, although the dependence on the canonical momenta remains the same. Three detailed analytical examples are presented. Other possibilities for finding further collisionless force-free equilibrium distribution functions have been explored, but were unsuccessful.
The first linear stability analysis of the Harrison and Neukirch equilibrium distribution function is then carried out, concentrating on macroscopic instabilities, and considering two-dimensional perturbations only. The analysis is based on the technique of integration over unperturbed orbits. Similarly to the Harris sheet case (Nuovo Cimento, 23:115, 1962), this is only possible by using approximations to the exact orbits, which are unknown. Furthermore, the approximations for the Harris sheet case cannot be used for the force-free Harris sheet, and so new techniques have to be developed in order to make analytical progress. Full analytical expressions for the perturbed current density are derived but, for the sake of simplicity, only the long wavelength limit is investigated. The dependence of the stability on various equilibrium parameters is investigated.Finite and infinite ergodic theory for linear and conformal dynamical systems
http://hdl.handle.net/10023/3220
The first main topic of this thesis is the thorough analysis of two families of piecewise linear
maps on the unit interval, the α-Lüroth and α-Farey maps. Here, α denotes a countably infinite
partition of the unit interval whose atoms only accumulate at the origin. The basic properties
of these maps will be developed, including that each α-Lüroth map (denoted Lα) gives rise to a
series expansion of real numbers in [0,1], a certain type of Generalised Lüroth Series. The first
example of such an expansion was given by Lüroth. The map Lα is the jump transformation
of the corresponding α-Farey map Fα. The maps Lα and Fα share the same relationship as the
classical Farey and Gauss maps which give rise to the continued fraction expansion of a real
number. We also consider the topological properties of Fα and some Diophantine-type sets of
numbers expressed in terms of the α-Lüroth expansion.
Next we investigate certain ergodic-theoretic properties of the maps Lα and Fα. It will turn
out that the Lebesgue measure λ is invariant for every map Lα and that there exists a unique
Lebesgue-absolutely continuous invariant measure for Fα. We will give a precise expression for
the density of this measure. Our main result is that both Lα and Fα are exact, and thus ergodic.
The interest in the invariant measure for Fα lies in the fact that under a particular condition on
the underlying partition α, the invariant measure associated to the map Fα is infinite.
Then we proceed to introduce and examine the sequence of α-sum-level sets arising from
the α-Lüroth map, for an arbitrary given partition α. These sets can be written dynamically in
terms of Fα. The main result concerning the α-sum-level sets is to establish weak and strong
renewal laws. Note that for the Farey map and the Gauss map, the analogue of this result has
been obtained by Kesseböhmer and Stratmann. There the results were derived by using advanced
infinite ergodic theory, rather than the strong renewal theorems employed here. This underlines
the fact that one of the main ingredients of infinite ergodic theory is provided by some delicate
estimates in renewal theory.
Our final main result concerning the α-Lüroth and α-Farey systems is to provide a fractal-geometric
description of the Lyapunov spectra associated with each of the maps Lα and Fα.
The Lyapunov spectra for the Farey map and the Gauss map have been investigated in detail by
Kesseböhmer and Stratmann. The Farey map and the Gauss map are non-linear, whereas the
systems we consider are always piecewise linear. However, since our analysis is based on a large
family of different partitions of U , the class of maps which we consider in this paper allows us
to detect a variety of interesting new phenomena, including that of phase transitions.
Finally, we come to the conformal systems of the title. These are the limit sets of discrete
subgroups of the group of isometries of the hyperbolic plane. For these so-called Fuchsian
groups, our first main result is to establish the Hausdorff dimension of some Diophantine-type
sets contained in the limit set that are similar to those considered for the maps Lα. These sets
are then used in our second main result to analyse the more geometrically defined strict-Jarník
limit set of a Fuchsian group. Finally, we obtain a “weak multifractal spectrum” for the Patterson
measure associated to the Fuchsian group.
Wed, 30 Nov 2011 00:00:00 GMThttp://hdl.handle.net/10023/32202011-11-30T00:00:00ZMunday, SaraThe first main topic of this thesis is the thorough analysis of two families of piecewise linear
maps on the unit interval, the α-Lüroth and α-Farey maps. Here, α denotes a countably infinite
partition of the unit interval whose atoms only accumulate at the origin. The basic properties
of these maps will be developed, including that each α-Lüroth map (denoted Lα) gives rise to a
series expansion of real numbers in [0,1], a certain type of Generalised Lüroth Series. The first
example of such an expansion was given by Lüroth. The map Lα is the jump transformation
of the corresponding α-Farey map Fα. The maps Lα and Fα share the same relationship as the
classical Farey and Gauss maps which give rise to the continued fraction expansion of a real
number. We also consider the topological properties of Fα and some Diophantine-type sets of
numbers expressed in terms of the α-Lüroth expansion.
Next we investigate certain ergodic-theoretic properties of the maps Lα and Fα. It will turn
out that the Lebesgue measure λ is invariant for every map Lα and that there exists a unique
Lebesgue-absolutely continuous invariant measure for Fα. We will give a precise expression for
the density of this measure. Our main result is that both Lα and Fα are exact, and thus ergodic.
The interest in the invariant measure for Fα lies in the fact that under a particular condition on
the underlying partition α, the invariant measure associated to the map Fα is infinite.
Then we proceed to introduce and examine the sequence of α-sum-level sets arising from
the α-Lüroth map, for an arbitrary given partition α. These sets can be written dynamically in
terms of Fα. The main result concerning the α-sum-level sets is to establish weak and strong
renewal laws. Note that for the Farey map and the Gauss map, the analogue of this result has
been obtained by Kesseböhmer and Stratmann. There the results were derived by using advanced
infinite ergodic theory, rather than the strong renewal theorems employed here. This underlines
the fact that one of the main ingredients of infinite ergodic theory is provided by some delicate
estimates in renewal theory.
Our final main result concerning the α-Lüroth and α-Farey systems is to provide a fractal-geometric
description of the Lyapunov spectra associated with each of the maps Lα and Fα.
The Lyapunov spectra for the Farey map and the Gauss map have been investigated in detail by
Kesseböhmer and Stratmann. The Farey map and the Gauss map are non-linear, whereas the
systems we consider are always piecewise linear. However, since our analysis is based on a large
family of different partitions of U , the class of maps which we consider in this paper allows us
to detect a variety of interesting new phenomena, including that of phase transitions.
Finally, we come to the conformal systems of the title. These are the limit sets of discrete
subgroups of the group of isometries of the hyperbolic plane. For these so-called Fuchsian
groups, our first main result is to establish the Hausdorff dimension of some Diophantine-type
sets contained in the limit set that are similar to those considered for the maps Lα. These sets
are then used in our second main result to analyse the more geometrically defined strict-Jarník
limit set of a Fuchsian group. Finally, we obtain a “weak multifractal spectrum” for the Patterson
measure associated to the Fuchsian group.The propagation and damping of slow magnetoacoustic waves in the solar atmosphere
http://hdl.handle.net/10023/3215
The propagation and damping of slow magnetoacoustic waves in the solar atmosphere is investigated, with
particular emphasis placed on waves with periodicities of five minutes. The basic model of a uniform
temperature loop is extended by the addition of an equilibrium temperature gradient allowing study of
wave propagation from the transition region to the corona. The inclusion of thermal conduction produces
a phase shift between the perturbations in velocity, density and temperature, which for a non-uniform
equilibrium temperature varies along the loop and may be observable as a phase shift between intensity and
Doppler shift observations. Forward modelling of the simulation results, for both constant and non-constant
equilibrium temperature profiles, is undertaken in order to establish the observational consequences for
TRACE, SoHO/CDS and Hinode/EIS. Slow waves propagating in a non-uniform equilibrium temperature
loop are seen to damp rapidly in the corona, however, as a result of the ionisation balance, the inclusion of
damping can actually increase the amplitude of some parts of the oscillation.
The ability of several data analysis techniques to identify oscillation signatures are examined. In particular,
empirical mode decomposition was found to be a very useful technique for extracting oscillations from a
wide range of data sets and is capable of intrinsically determining background trends. Co-spatial and cotemporal
TRACE 171 A, CDS and EIS data are analysed for evidence of propagating slow waves. Slow
waves with periods of 210 s to 370 s are found with amplitudes of 1.2% to 3.4% in the corona and 2.3% to
6.0% in the transition region.
Fri, 30 Nov 2012 00:00:00 GMThttp://hdl.handle.net/10023/32152012-11-30T00:00:00ZOwen, Nicholas RobertThe propagation and damping of slow magnetoacoustic waves in the solar atmosphere is investigated, with
particular emphasis placed on waves with periodicities of five minutes. The basic model of a uniform
temperature loop is extended by the addition of an equilibrium temperature gradient allowing study of
wave propagation from the transition region to the corona. The inclusion of thermal conduction produces
a phase shift between the perturbations in velocity, density and temperature, which for a non-uniform
equilibrium temperature varies along the loop and may be observable as a phase shift between intensity and
Doppler shift observations. Forward modelling of the simulation results, for both constant and non-constant
equilibrium temperature profiles, is undertaken in order to establish the observational consequences for
TRACE, SoHO/CDS and Hinode/EIS. Slow waves propagating in a non-uniform equilibrium temperature
loop are seen to damp rapidly in the corona, however, as a result of the ionisation balance, the inclusion of
damping can actually increase the amplitude of some parts of the oscillation.
The ability of several data analysis techniques to identify oscillation signatures are examined. In particular,
empirical mode decomposition was found to be a very useful technique for extracting oscillations from a
wide range of data sets and is capable of intrinsically determining background trends. Co-spatial and cotemporal
TRACE 171 A, CDS and EIS data are analysed for evidence of propagating slow waves. Slow
waves with periods of 210 s to 370 s are found with amplitudes of 1.2% to 3.4% in the corona and 2.3% to
6.0% in the transition region.Wave propagation, phase mixing and dissipation in Hall MHD
http://hdl.handle.net/10023/3182
In this thesis the effect of the Hall term in the generalised Ohm’s law on Alfvén (shear) and fast wave propagation and dissipation in the ion cyclotron frequency range is investigated.
The damping of an initially Gaussian field perturbation in a uniform Hall MHD plasma is treated analytically. Subsequently a 2D Lagrangian remap code (Lare2d) is used to study the damping and phase mixing of initially Gaussian field perturbations and a harmonic series of boundary-driven perturbations in a uniform field (in the presence of a transverse equilibrium density gradient). The same code is then used to study a range of initially shear and fast-wave perturbations in the vicinity of a magnetic X-type null point.
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 kδ where k is wavenumber and δ is ion skin depth. A similar decay law applies to whistler perturbations in the limit kδ>>>1.
We demonstrate that in both geometries considered, the inclusion of the Hall term reduces the effectiveness of phase-mixing in plasma heating. The reduction in the damping rate in the uniform ﬁeld (non-uniform density) cases, arising from dispersive effects, tends to zero in both the weak and strong phase mixing limits. In the Hall MHD X-point case, minimal reductions are seen for initially shear wave pulses, suggesting that little or no phase-mixing takes place. Nonlinear fast wave pulses which interact with the initial X-point destabilise the local field sufficiently to generate multiple null pairs; subsequent oscillatory current sheet behaviour appears unaffected by earlier differences between the MHD and Hall MHD cases.
Fri, 22 Jun 2012 00:00:00 GMThttp://hdl.handle.net/10023/31822012-06-22T00:00:00ZThrelfall, James W.In this thesis the effect of the Hall term in the generalised Ohm’s law on Alfvén (shear) and fast wave propagation and dissipation in the ion cyclotron frequency range is investigated.
The damping of an initially Gaussian field perturbation in a uniform Hall MHD plasma is treated analytically. Subsequently a 2D Lagrangian remap code (Lare2d) is used to study the damping and phase mixing of initially Gaussian field perturbations and a harmonic series of boundary-driven perturbations in a uniform field (in the presence of a transverse equilibrium density gradient). The same code is then used to study a range of initially shear and fast-wave perturbations in the vicinity of a magnetic X-type null point.
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 kδ where k is wavenumber and δ is ion skin depth. A similar decay law applies to whistler perturbations in the limit kδ>>>1.
We demonstrate that in both geometries considered, the inclusion of the Hall term reduces the effectiveness of phase-mixing in plasma heating. The reduction in the damping rate in the uniform ﬁeld (non-uniform density) cases, arising from dispersive effects, tends to zero in both the weak and strong phase mixing limits. In the Hall MHD X-point case, minimal reductions are seen for initially shear wave pulses, suggesting that little or no phase-mixing takes place. Nonlinear fast wave pulses which interact with the initial X-point destabilise the local field sufficiently to generate multiple null pairs; subsequent oscillatory current sheet behaviour appears unaffected by earlier differences between the MHD and Hall MHD cases.A non-linear force-free field model for the solar magnetic carpet
http://hdl.handle.net/10023/3114
The magnetic carpet is defined to be the small-scale photospheric magnetic field of the quiet Sun. Observations of the magnetic carpet show it to be highly dynamic, where the time taken for all flux within the magnetic carpet to be replaced is on the order of just a few hours. The magnetic carpet is continually evolving due to the Sun's underlying convection and the interaction of small-scale magnetic features with one another. Due to this, the small-scale coronal field of the magnetic carpet is also expected to be highly dynamic and complex. Previous modelling has shown that much of the flux from the magnetic carpet is stored along low-lying closed connections between magnetic features. This indicates that significant coronal heating could occur low down in the small-scale corona. In this thesis, a new two-component magnetic field model is developed for the evolution of the magnetic carpet. A 2D model is constructed to realistically simulate the evolution of the photospheric field of the magnetic carpet, where many of the parameters for the model are taken from observational studies. The photospheric model contains a granular and supergranular flow profile to describe the motion of the small-scale magnetic features, and includes the processes of flux emergence, cancellation, coalescence and fragmentation. This 2D model then couples to a 3D model as the lower boundary condition, which drives the evolution of the coronal field through a series of non-linear force-free states, via a magnetofrictional relaxation technique. We first apply the magnetofrictional technique to consider the coronal evolution of three basic small-scale photospheric processes: emergence, cancellation and flyby. We consider the interaction of the magnetic features with an overlying coronal magnetic field, and quantify magnetic energy build-up, storage and dissipation. The magnetofrictional technique is then applied to synthetic magnetograms produced from the 2D model, to simulate the evolution of the coronal field in a situation involving many hundreds of magnetic features. We conduct a preliminary analysis of the resultant 3D simulations, considering the magnetic energy stored and dissipated, as well as regions of enhanced velocity and electric current density within the coronal volume. The simulations show that the so-called 'quiet Sun' is not quiet and a significant amount of complex interactions take place.
Fri, 22 Jun 2012 00:00:00 GMThttp://hdl.handle.net/10023/31142012-06-22T00:00:00ZMeyer, Karen AlisonThe magnetic carpet is defined to be the small-scale photospheric magnetic field of the quiet Sun. Observations of the magnetic carpet show it to be highly dynamic, where the time taken for all flux within the magnetic carpet to be replaced is on the order of just a few hours. The magnetic carpet is continually evolving due to the Sun's underlying convection and the interaction of small-scale magnetic features with one another. Due to this, the small-scale coronal field of the magnetic carpet is also expected to be highly dynamic and complex. Previous modelling has shown that much of the flux from the magnetic carpet is stored along low-lying closed connections between magnetic features. This indicates that significant coronal heating could occur low down in the small-scale corona. In this thesis, a new two-component magnetic field model is developed for the evolution of the magnetic carpet. A 2D model is constructed to realistically simulate the evolution of the photospheric field of the magnetic carpet, where many of the parameters for the model are taken from observational studies. The photospheric model contains a granular and supergranular flow profile to describe the motion of the small-scale magnetic features, and includes the processes of flux emergence, cancellation, coalescence and fragmentation. This 2D model then couples to a 3D model as the lower boundary condition, which drives the evolution of the coronal field through a series of non-linear force-free states, via a magnetofrictional relaxation technique. We first apply the magnetofrictional technique to consider the coronal evolution of three basic small-scale photospheric processes: emergence, cancellation and flyby. We consider the interaction of the magnetic features with an overlying coronal magnetic field, and quantify magnetic energy build-up, storage and dissipation. The magnetofrictional technique is then applied to synthetic magnetograms produced from the 2D model, to simulate the evolution of the coronal field in a situation involving many hundreds of magnetic features. We conduct a preliminary analysis of the resultant 3D simulations, considering the magnetic energy stored and dissipated, as well as regions of enhanced velocity and electric current density within the coronal volume. The simulations show that the so-called 'quiet Sun' is not quiet and a significant amount of complex interactions take place.A highly adaptive three dimensional hybrid vortex method for inviscid flows and helically symmetric vortex equilibria
http://hdl.handle.net/10023/3091
This thesis is concerned with three-dimensional vortex dynamics, in particular the modelling of vortex structures in an inviscid context. We are motivated by the open problem of regularity of the inviscid equations, i.e. whether or not these equations possess solutions. This problem is manifest in small scales, where vortex filaments are stretched and intensify as they are drawn into increasingly thin tendrils. This creates great difficulty in the investigation of such flows. Our only means of experimentation is to perform numerical simulations, which require exceptionally high resolution to capture the small scale vortex structures.
A new numerical method to solve the inviscid Euler equations for three-dimensional, incompressible fluids is presented, with special emphasis on spatial adaptivity to resolve as broad a range of scales as possible in a completely self-similar fashion. We present a hybrid vortex method whereby we discretise the vorticity in Lagrangian filaments and perform and inversion to compute velocity on an arbitrary unstructured finite-volume grid. This allows for a two-fold adaptivity strategy. First, although naturally spatially adaptive by definition, the vorticity filaments undergo ‘renoding’. We redistribute nodes along the filament to concentrate their density in regions of high curvature. Secondly the Eulerian mesh is adapted to follow high strain by increasing resolution based on local filament dimensions. These features allow vortex stretching and folding to be resolved in a completely automatic and self-similar way. The method is validated via well known vortex rings and newly discovered helical vortex equilibria are also used to test the method.
We begin by presenting this new class of three-dimensional vortex equilibria which possess helical symmetry. Such vortices are observed in propeller and wind turbine wakes, and their equilibria shapes have until now been unknown. These vortices are described by contours bounding regions of uniform axial vorticity. Material conservation of axial vorticity enables equilibria to be calculated simply by a restriction on the helical stream function. The states are parameterised by their mean radius and centroid position. In the case of a single vortex, the parameter space cannot be fully filled by our numerical approach. We conjecture that multiply connected contours will characterise equilibria where the algorithm fails. We also consider multiple vortices, evenly azimuthally spaced about the origin. In such cases instabilities often lead to a single helical vortex.
Fri, 22 Jun 2012 00:00:00 GMThttp://hdl.handle.net/10023/30912012-06-22T00:00:00ZLucas, DanielThis thesis is concerned with three-dimensional vortex dynamics, in particular the modelling of vortex structures in an inviscid context. We are motivated by the open problem of regularity of the inviscid equations, i.e. whether or not these equations possess solutions. This problem is manifest in small scales, where vortex filaments are stretched and intensify as they are drawn into increasingly thin tendrils. This creates great difficulty in the investigation of such flows. Our only means of experimentation is to perform numerical simulations, which require exceptionally high resolution to capture the small scale vortex structures.
A new numerical method to solve the inviscid Euler equations for three-dimensional, incompressible fluids is presented, with special emphasis on spatial adaptivity to resolve as broad a range of scales as possible in a completely self-similar fashion. We present a hybrid vortex method whereby we discretise the vorticity in Lagrangian filaments and perform and inversion to compute velocity on an arbitrary unstructured finite-volume grid. This allows for a two-fold adaptivity strategy. First, although naturally spatially adaptive by definition, the vorticity filaments undergo ‘renoding’. We redistribute nodes along the filament to concentrate their density in regions of high curvature. Secondly the Eulerian mesh is adapted to follow high strain by increasing resolution based on local filament dimensions. These features allow vortex stretching and folding to be resolved in a completely automatic and self-similar way. The method is validated via well known vortex rings and newly discovered helical vortex equilibria are also used to test the method.
We begin by presenting this new class of three-dimensional vortex equilibria which possess helical symmetry. Such vortices are observed in propeller and wind turbine wakes, and their equilibria shapes have until now been unknown. These vortices are described by contours bounding regions of uniform axial vorticity. Material conservation of axial vorticity enables equilibria to be calculated simply by a restriction on the helical stream function. The states are parameterised by their mean radius and centroid position. In the case of a single vortex, the parameter space cannot be fully filled by our numerical approach. We conjecture that multiply connected contours will characterise equilibria where the algorithm fails. We also consider multiple vortices, evenly azimuthally spaced about the origin. In such cases instabilities often lead to a single helical vortex.Continued fractions which correspond to two series expansions and the strong Hamburger moment problem
http://hdl.handle.net/10023/2977
Just as the denominator polynomials of a J-fraction are
orthogonal polynomials with respect to some moment functional, the
denominator polynomials of an M-fraction are shown to satisfy a skew
orthogonality relation with respect to a stronger moment functional.
Many of the properties of the numerators and denominators of an M-
fraction are also studied using this pseudo orthogonality relation
of the denominator polynomials. Properties of the zeros of the
denominator polynomials when the associated moment functional is
positive definite are also considered.
A type of continued fraction, referred to as a J-fraction, is
shown to correspond to a power series about the origin and to another
power series about infinity such that the successive convergents of
this fraction include two more additional terms of anyone of the
power series. Given the power series expansions, a method of
obtaining such a J-fraction, whenever it exists, is also looked at.
The first complete proof of the so called strong Hamburger moment
problem using a continued fraction is given. In this case the
continued fraction is a J-fraction.
Finally a special class of J-fraction, referred to as positive
definite J-fractions, is studied in detail.
The four chapters of this thesis are divided into sections.
Each section is given a section number which is made up of the
chapter number followed by the number of the section within the
chapter. The equations in the thesis have an equation number
consisting of the section number followed by the number of the
equation within that section.
In Chapter One, in addition to looking at some of the
historical and recent developments of corresponding continued
fractions and their applications, we also present some preliminaries.
Chapter Two deals with a different approach of understanding
the properties of the numerators and denominators of corresponding
(two point) rational functions and, continued fractions. This
approach, which is based on a pseudo orthogonality relation of the
denominator polynomials of the corresponding rational functions,
provides an insight into understanding the moment problems. In
particular, results are established which suggest a possible type
of continued fraction for solving the strong Hamburger moment
problem.
In the third chapter we study in detail the existence
conditions and corresponding properties of this new type of continued
fraction, which we call J-fractions. A method of derivation of one
of these 3-fractions is also considered. In the same chapter we also
look at the all important application of solving the strong Hamburger
moment problem, using these 3-fractions.
The fourth and final chapter is devoted entirely to the study
of the convergence behaviour of a certain class of J-fractions,
namely positive definite J-fractions. This study also provides some
interesting convergence criteria for a real and regular 3-fraction.
Finally a word concerning the literature on continued fractions
and moment problems. The more recent and up-to-date exposition on
the analytic theory of continued fractions and their applications is
the text of Jones and Thron [1980]. The two volumes of Baker and
Graves-Morris [1981] provide a very good treatment on one of the
computational aspects of the continued fractions, namely Pade
approximants. There are also the earlier texts of Wall [1948] and
Khovanskii [1963], in which the former gives an extensive insight
into the analytic theory of continued fractions while the latter,
being simpler, remains the ideal book for the beginner. In his
treatise on Applied and Computational Complex Analysis, Henrici
[1977] has also included an excellent chapter on continued fractions.
Wall [1948] also includes a few chapters on moment problems and
related areas. A much wider treatment of the classical moment
problems is provided in the excellent texts of Shohat and Tamarkin
[1943] and Akhieser [1965].
Sun, 01 Jan 1984 00:00:00 GMThttp://hdl.handle.net/10023/29771984-01-01T00:00:00ZSri Ranga, A.Just as the denominator polynomials of a J-fraction are
orthogonal polynomials with respect to some moment functional, the
denominator polynomials of an M-fraction are shown to satisfy a skew
orthogonality relation with respect to a stronger moment functional.
Many of the properties of the numerators and denominators of an M-
fraction are also studied using this pseudo orthogonality relation
of the denominator polynomials. Properties of the zeros of the
denominator polynomials when the associated moment functional is
positive definite are also considered.
A type of continued fraction, referred to as a J-fraction, is
shown to correspond to a power series about the origin and to another
power series about infinity such that the successive convergents of
this fraction include two more additional terms of anyone of the
power series. Given the power series expansions, a method of
obtaining such a J-fraction, whenever it exists, is also looked at.
The first complete proof of the so called strong Hamburger moment
problem using a continued fraction is given. In this case the
continued fraction is a J-fraction.
Finally a special class of J-fraction, referred to as positive
definite J-fractions, is studied in detail.
The four chapters of this thesis are divided into sections.
Each section is given a section number which is made up of the
chapter number followed by the number of the section within the
chapter. The equations in the thesis have an equation number
consisting of the section number followed by the number of the
equation within that section.
In Chapter One, in addition to looking at some of the
historical and recent developments of corresponding continued
fractions and their applications, we also present some preliminaries.
Chapter Two deals with a different approach of understanding
the properties of the numerators and denominators of corresponding
(two point) rational functions and, continued fractions. This
approach, which is based on a pseudo orthogonality relation of the
denominator polynomials of the corresponding rational functions,
provides an insight into understanding the moment problems. In
particular, results are established which suggest a possible type
of continued fraction for solving the strong Hamburger moment
problem.
In the third chapter we study in detail the existence
conditions and corresponding properties of this new type of continued
fraction, which we call J-fractions. A method of derivation of one
of these 3-fractions is also considered. In the same chapter we also
look at the all important application of solving the strong Hamburger
moment problem, using these 3-fractions.
The fourth and final chapter is devoted entirely to the study
of the convergence behaviour of a certain class of J-fractions,
namely positive definite J-fractions. This study also provides some
interesting convergence criteria for a real and regular 3-fraction.
Finally a word concerning the literature on continued fractions
and moment problems. The more recent and up-to-date exposition on
the analytic theory of continued fractions and their applications is
the text of Jones and Thron [1980]. The two volumes of Baker and
Graves-Morris [1981] provide a very good treatment on one of the
computational aspects of the continued fractions, namely Pade
approximants. There are also the earlier texts of Wall [1948] and
Khovanskii [1963], in which the former gives an extensive insight
into the analytic theory of continued fractions while the latter,
being simpler, remains the ideal book for the beginner. In his
treatise on Applied and Computational Complex Analysis, Henrici
[1977] has also included an excellent chapter on continued fractions.
Wall [1948] also includes a few chapters on moment problems and
related areas. A much wider treatment of the classical moment
problems is provided in the excellent texts of Shohat and Tamarkin
[1943] and Akhieser [1965].Solar flare particle acceleration in collapsing magnetic traps
http://hdl.handle.net/10023/2839
The topic of this thesis is a detailed investigation of different aspects of the particle acceleration mechanisms operating in Collapsing Magnetic Traps (CMTs), which have been suggested as one possible mechanism for particle acceleration during solar flares.
The acceleration processes in CMTs are investigated using guiding centre test particle calculations.
Results including terms of different orders in the guiding centre approximation are compared to help identify which of the terms are important for the acceleration of particles. For a basic 2D CMT model the effects of different initial conditions (position, kinetic energy and pitch angle) of particles are investigated in detail. The main result is that the particles that gain most energy are those with initial pitch angles close to 90° and start in weak field regions in the centre of the CMT. The dominant acceleration mechanism for these particles is betatron acceleration, but other
particles also show signatures of Fermi acceleration.
The basic CMT model is then extended by (a) including a magnetic field component in the invariant direction and (b) by making it asymmetric. It is found that the addition of a guide field does not change the characteristics of particle acceleration very much, but for the asymmetric models the associated energy gain is found to be much smaller than in symmetric models, because the
particles can no longer remain very close to the trap centre throughout their orbit.
The test particle method is then also applied to a CMT model from the literature which contains a magnetic X-line and open and closed field lines and the results are compared with the previous results and the findings in the literature.
Finally, the theoretical framework of CMT models is extended to 2.5D models with shear flow and to fully 3D models, allowing the construction of more realistic CMT models in the future.
Fri, 22 Jun 2012 00:00:00 GMThttp://hdl.handle.net/10023/28392012-06-22T00:00:00ZGrady, Keith J.The topic of this thesis is a detailed investigation of different aspects of the particle acceleration mechanisms operating in Collapsing Magnetic Traps (CMTs), which have been suggested as one possible mechanism for particle acceleration during solar flares.
The acceleration processes in CMTs are investigated using guiding centre test particle calculations.
Results including terms of different orders in the guiding centre approximation are compared to help identify which of the terms are important for the acceleration of particles. For a basic 2D CMT model the effects of different initial conditions (position, kinetic energy and pitch angle) of particles are investigated in detail. The main result is that the particles that gain most energy are those with initial pitch angles close to 90° and start in weak field regions in the centre of the CMT. The dominant acceleration mechanism for these particles is betatron acceleration, but other
particles also show signatures of Fermi acceleration.
The basic CMT model is then extended by (a) including a magnetic field component in the invariant direction and (b) by making it asymmetric. It is found that the addition of a guide field does not change the characteristics of particle acceleration very much, but for the asymmetric models the associated energy gain is found to be much smaller than in symmetric models, because the
particles can no longer remain very close to the trap centre throughout their orbit.
The test particle method is then also applied to a CMT model from the literature which contains a magnetic X-line and open and closed field lines and the results are compared with the previous results and the findings in the literature.
Finally, the theoretical framework of CMT models is extended to 2.5D models with shear flow and to fully 3D models, allowing the construction of more realistic CMT models in the future.Applications of statistics in flood frequency analysis
http://hdl.handle.net/10023/2666
Estimation of the probability of occurrence of future flood events at one
or more locations across a river system is frequently required for the design of
bridges, culverts, spillways, dams and other engineering works. This study
investigates some of the statistical aspects for estimating the flood frequency
distribution at a single site and on regional basis.
It is demonstrated that generalized logistic (GL) distribution has many
properties well suited for the modelling of flood frequency data. The GL
distribution performs better than the other commonly recommended flood frequency
distributions in terms of several key properties. Specifically, it is capable of
reproducing almost the same degree of skewness typically present in observed
flood data. It appears to be more robust to the presence of extreme outliers in the
upper tail of the distribution. It has a relatively simpler mathematical form. Thus all
the well known methods of parameter estimation can be easily implemented.
It is shown that the method of probability weighted moments (PWM)
using the conventionally recommended plotting position substantially effects the
estimation of the shape parameter of the generalized extreme value (GEV)
distribution by relocating the annual maximum flood series. A location invariant
plotting position is introduced to use in estimating, by the method of PWM, the
parameters of the GEV and the GL distributions.
Tests based on empirical distribution function (EDF) statistics are
proposed to assess the goodness of fit of the flood frequency distributions. A
modified EDF test is derived that gives greater emphasis to the upper tail of a
distribution which is more important for flood frequency prediction. Significance
points are derived for the GEV and GL distributions when the parameters are to be
estimated from the sample data by the method of PWMs. The critical points are
considerably smaller than for the case where the parameters of a distribution are
assumed to be specified. Approximate formulae over the whole range of the
distribution for these tests are also developed which can be used for regional
assessment of GEV and GL models based on all the annual maximum series
simultaneously in a hydrological region.
In order to pool at-site flood data across a region into a single series for
regional analysis, the effect of standardization by at-site mean on the estimation of
the regional shape parameter of the GEV distribution is examined. Our simulation
study based on various synthetic regions reveals that the standardization by the at-site
mean underestimates the shape parameter of the GEV by about 30% of its true
value and also contributes to the separation of skewness of observed and simulated
floods. A two parameter standardization by the at-site estimates of location and
scale parameters is proposed. It does not distort the shape of the flood frequency
data in the pooling process. Therefore, it offers significantly improved estimate of
the shape parameter, allows pooling data with heterogeneous coefficients of
variation and helps to explain the separation of skewness effect.
Regions on the basis of flood statistics L-CV and USKEW are derived
for Scotland and North England. Only about 50% of the basins could be correctly
identified as belonging to these regions by a set of seven catchment characteristics.
The alternative approach of grouping basins solely on the basis of physical
properties is preferable. Six physically homogeneous groups of basins are
identified by WARD's multivariate clustering algorithm using the same seven
characteristics. These regions have hydrological homogeneity in addition to their
physical homogeneity. Dimensionless regional flood frequency curves are produced
by fitting GEV and GL distributions for each region. The GEV regional growth
curves imply a larger return period for a given magnitude flood. When floods are
described by GL model the respective return periods are considerably smaller.
Sun, 01 Jan 1989 00:00:00 GMThttp://hdl.handle.net/10023/26661989-01-01T00:00:00ZAhmad, Muhammad IdreesEstimation of the probability of occurrence of future flood events at one
or more locations across a river system is frequently required for the design of
bridges, culverts, spillways, dams and other engineering works. This study
investigates some of the statistical aspects for estimating the flood frequency
distribution at a single site and on regional basis.
It is demonstrated that generalized logistic (GL) distribution has many
properties well suited for the modelling of flood frequency data. The GL
distribution performs better than the other commonly recommended flood frequency
distributions in terms of several key properties. Specifically, it is capable of
reproducing almost the same degree of skewness typically present in observed
flood data. It appears to be more robust to the presence of extreme outliers in the
upper tail of the distribution. It has a relatively simpler mathematical form. Thus all
the well known methods of parameter estimation can be easily implemented.
It is shown that the method of probability weighted moments (PWM)
using the conventionally recommended plotting position substantially effects the
estimation of the shape parameter of the generalized extreme value (GEV)
distribution by relocating the annual maximum flood series. A location invariant
plotting position is introduced to use in estimating, by the method of PWM, the
parameters of the GEV and the GL distributions.
Tests based on empirical distribution function (EDF) statistics are
proposed to assess the goodness of fit of the flood frequency distributions. A
modified EDF test is derived that gives greater emphasis to the upper tail of a
distribution which is more important for flood frequency prediction. Significance
points are derived for the GEV and GL distributions when the parameters are to be
estimated from the sample data by the method of PWMs. The critical points are
considerably smaller than for the case where the parameters of a distribution are
assumed to be specified. Approximate formulae over the whole range of the
distribution for these tests are also developed which can be used for regional
assessment of GEV and GL models based on all the annual maximum series
simultaneously in a hydrological region.
In order to pool at-site flood data across a region into a single series for
regional analysis, the effect of standardization by at-site mean on the estimation of
the regional shape parameter of the GEV distribution is examined. Our simulation
study based on various synthetic regions reveals that the standardization by the at-site
mean underestimates the shape parameter of the GEV by about 30% of its true
value and also contributes to the separation of skewness of observed and simulated
floods. A two parameter standardization by the at-site estimates of location and
scale parameters is proposed. It does not distort the shape of the flood frequency
data in the pooling process. Therefore, it offers significantly improved estimate of
the shape parameter, allows pooling data with heterogeneous coefficients of
variation and helps to explain the separation of skewness effect.
Regions on the basis of flood statistics L-CV and USKEW are derived
for Scotland and North England. Only about 50% of the basins could be correctly
identified as belonging to these regions by a set of seven catchment characteristics.
The alternative approach of grouping basins solely on the basis of physical
properties is preferable. Six physically homogeneous groups of basins are
identified by WARD's multivariate clustering algorithm using the same seven
characteristics. These regions have hydrological homogeneity in addition to their
physical homogeneity. Dimensionless regional flood frequency curves are produced
by fitting GEV and GL distributions for each region. The GEV regional growth
curves imply a larger return period for a given magnitude flood. When floods are
described by GL model the respective return periods are considerably smaller.Coupling of the solar wind, magnetosphere and ionosphere by MHD waves
http://hdl.handle.net/10023/2571
The solar wind, magnetosphere and ionosphere are coupled by magnetohydrodynamic waves, and this gives rise to new and often unexpected behaviours that cannot be produced by a single, isolated part of the system. This thesis examines two broad instances of coupling: field-line resonance (FLR) which couples fast and Alfvén waves, and magnetosphere-ionosphere (MI-) coupling via Alfvén waves.
The first part of this thesis investigates field-line resonance for equilibria that vary in two dimensions perpendicular to the background magnetic field. This research confirms that our intuitive understanding of FLR from 1D is a good guide to events in 2D, and places 2D FLR onto a firm mathematical basis by systematic solution of the governing equations. It also reveals the new concept of ‘imprinting’ of spatial forms: spatial variations of the resonant Alfvén wave correlate strongly with the spatial form of the fast wave that drives the resonance.
MI-coupling gives rise to ionosphere-magnetosphere (IM-) waves, and we have made a detailed analysis of these waves for a 1D sheet E-region. IM-waves are characterised by two quantities: a speed v_{IM} and an angular frequency ω_{IM} , for which we have obtained analytic expressions. For an ideal magnetosphere, IM-waves are advective and move in the direction of the electric field with speed v_{IM}. The advection speed is a non-linear expression that decreases with height-integrated E-region plasma-density, hence, wavepackets steepen on their trailing edge, rapidly accessing small length-scales through wavebreaking. Inclusion of electron inertial effects in the magnetosphere introduces dispersion to IM-waves. In the strongly inertial limit (wavelength λ << λ_{e} , where λ_{e} is the electron inertial length at the base of the magnetosphere), the group velocity of linear waves goes to zero, and the waves oscillate at ω_{IM} which is an upper limit on the angular frequency of IM-waves for any wavelength. Estimates of v_{IM} show that this speed can be a significant fraction (perhaps half) of the E_{⊥} × B_{0} drift in the E-region, producing speeds of up to several hundred metres per second. The upper limit on angular frequency, ωIM , is estimated to give periods from a few hundredths of a second to several minutes. IM-waves are damped by recombination and background ionisation, giving an e-folding decay time that can vary from tens of seconds to tens of minutes.
We have also investigated the dynamics and steady-states that occur when the magnetosphere-ionosphere system is driven by large-scale Alfvénic field-aligned currents. Steady-states are dominated by two approximate solutions: an ‘upper’ solution that is valid in places where the E-region is a near perfect conductor, and a ‘lower’ solution that is valid where E-region depletion makes recombination negligible. These analytic solutions are extremely useful tools and the global steady-state can be constructed by matching these solutions across suitable boundary-layers. Furthermore, the upper solution reveals that E-region density cavities form and widen (with associated broadening of the magnetospheric downward current channel) if the downward current density exceeds the maximum current density that can be supplied by background E-region ionisation. We also supply expressions for the minimum E-region plasma-density and shortest length-scale in the steady-state.
IM-waves and steady-states are extremely powerful tools for interpreting MI-dynamics. When an E-region density cavity widens through coupling to an ideal, single-fluid MHD magnetosphere, it does so by forming a discontinuity that steps between the upper and lower steady-states. This discontinuity acts as part of an ideal IM-wave and moves in the direction of the electric field at a speed U = \sqrt{v_{IM}^{+} v_{IM}^{-}}, which is the geometric mean of v_{IM} evaluated immediately to the left and right of the discontinuity. This widening speed is typically several hundreds of metres per second. If electron inertial effects are included in the magnetosphere, then the discontinuity is smoothed, and a series of undershoots and overshoots develops behind it. These undershoots and overshoots evolve as inertial IM-waves. Initially they are weakly inertial, with a wavelength of about λ_{e}, however, strong gradients of ω_{IM} cause IM-waves to phase-mix, making their wavelength inversely proportional to time. Therefore, the waves rapidly become strongly inertial and oscillate at ω_{IM}. The inertial IM-waves drive upgoing Alfvén waves in the magnetosphere, which populate a region over the downward current channel, close to its edge. In this manner, the E-region depletion mechanism, that we have detailed, creates small-scale Alfvén waves in large-scale current systems, with properties determined by MI-coupling.
Tue, 30 Nov 2010 00:00:00 GMThttp://hdl.handle.net/10023/25712010-11-30T00:00:00ZRussell, Alexander J. B.The solar wind, magnetosphere and ionosphere are coupled by magnetohydrodynamic waves, and this gives rise to new and often unexpected behaviours that cannot be produced by a single, isolated part of the system. This thesis examines two broad instances of coupling: field-line resonance (FLR) which couples fast and Alfvén waves, and magnetosphere-ionosphere (MI-) coupling via Alfvén waves.
The first part of this thesis investigates field-line resonance for equilibria that vary in two dimensions perpendicular to the background magnetic field. This research confirms that our intuitive understanding of FLR from 1D is a good guide to events in 2D, and places 2D FLR onto a firm mathematical basis by systematic solution of the governing equations. It also reveals the new concept of ‘imprinting’ of spatial forms: spatial variations of the resonant Alfvén wave correlate strongly with the spatial form of the fast wave that drives the resonance.
MI-coupling gives rise to ionosphere-magnetosphere (IM-) waves, and we have made a detailed analysis of these waves for a 1D sheet E-region. IM-waves are characterised by two quantities: a speed v_{IM} and an angular frequency ω_{IM} , for which we have obtained analytic expressions. For an ideal magnetosphere, IM-waves are advective and move in the direction of the electric field with speed v_{IM}. The advection speed is a non-linear expression that decreases with height-integrated E-region plasma-density, hence, wavepackets steepen on their trailing edge, rapidly accessing small length-scales through wavebreaking. Inclusion of electron inertial effects in the magnetosphere introduces dispersion to IM-waves. In the strongly inertial limit (wavelength λ << λ_{e} , where λ_{e} is the electron inertial length at the base of the magnetosphere), the group velocity of linear waves goes to zero, and the waves oscillate at ω_{IM} which is an upper limit on the angular frequency of IM-waves for any wavelength. Estimates of v_{IM} show that this speed can be a significant fraction (perhaps half) of the E_{⊥} × B_{0} drift in the E-region, producing speeds of up to several hundred metres per second. The upper limit on angular frequency, ωIM , is estimated to give periods from a few hundredths of a second to several minutes. IM-waves are damped by recombination and background ionisation, giving an e-folding decay time that can vary from tens of seconds to tens of minutes.
We have also investigated the dynamics and steady-states that occur when the magnetosphere-ionosphere system is driven by large-scale Alfvénic field-aligned currents. Steady-states are dominated by two approximate solutions: an ‘upper’ solution that is valid in places where the E-region is a near perfect conductor, and a ‘lower’ solution that is valid where E-region depletion makes recombination negligible. These analytic solutions are extremely useful tools and the global steady-state can be constructed by matching these solutions across suitable boundary-layers. Furthermore, the upper solution reveals that E-region density cavities form and widen (with associated broadening of the magnetospheric downward current channel) if the downward current density exceeds the maximum current density that can be supplied by background E-region ionisation. We also supply expressions for the minimum E-region plasma-density and shortest length-scale in the steady-state.
IM-waves and steady-states are extremely powerful tools for interpreting MI-dynamics. When an E-region density cavity widens through coupling to an ideal, single-fluid MHD magnetosphere, it does so by forming a discontinuity that steps between the upper and lower steady-states. This discontinuity acts as part of an ideal IM-wave and moves in the direction of the electric field at a speed U = \sqrt{v_{IM}^{+} v_{IM}^{-}}, which is the geometric mean of v_{IM} evaluated immediately to the left and right of the discontinuity. This widening speed is typically several hundreds of metres per second. If electron inertial effects are included in the magnetosphere, then the discontinuity is smoothed, and a series of undershoots and overshoots develops behind it. These undershoots and overshoots evolve as inertial IM-waves. Initially they are weakly inertial, with a wavelength of about λ_{e}, however, strong gradients of ω_{IM} cause IM-waves to phase-mix, making their wavelength inversely proportional to time. Therefore, the waves rapidly become strongly inertial and oscillate at ω_{IM}. The inertial IM-waves drive upgoing Alfvén waves in the magnetosphere, which populate a region over the downward current channel, close to its edge. In this manner, the E-region depletion mechanism, that we have detailed, creates small-scale Alfvén waves in large-scale current systems, with properties determined by MI-coupling.Atmospheric transport and critical layer mixing in the troposphere and stratosphere
http://hdl.handle.net/10023/2538
This thesis aims to improve the understanding of transport and critical layer mixing in the troposphere and stratosphere. A dynamical approach is taken based on potential vorticity which has long been recognised as the essential field inducing the flow and thermodynamic structure of the atmosphere. Within the dynamical framework of critical layer mixing of potential vorticity, three main topics are addressed.
First, an idealised model of critical layer mixing in the stratospheric surf zone is examined. The effect of the shear across the critical layer on the critical layer evolution itself is investigated. In particular it is found that at small shear barotropic instability occurs and the mixing efficiency of the critical layer increases due to the instability. The effect of finite deformation length is also considered which extends previous work.
Secondly, the dynamical coupling between the stratosphere and troposphere is examined by considering the effect of direct perturbations to stratospheric potential vorticity on the evolution of midlatitude baroclinic instability. Both zonally symmetric and asymmetric perturbations to the stratospheric potential vorticity are considered, the former representative of a strong polar vortex, the latter representative of the stratospheric state following a major sudden warming. A comparison of these perturbations gives some insight into the possible influence of pre or post-sudden warming conditions on the tropospheric evolution.
Finally, the influence of the stratospheric potential vorticity distribution on lateral mixing and transport into and out of the tropical pipe, the low latitude ascending branch of the Brewer-Dobson circulation, is investigated. The stratospheric potential vorticity distribution in the tropical stratosphere is found to have a clear pattern according to the phase of the quasi-biennial oscillation (QBO). The extent of the QBO influence is quantified, by analysing trajectories of Lagrangian particles using an online trajectory code recently implemented in the Met Office's Unified Model.
Fri, 22 Jun 2012 00:00:00 GMThttp://hdl.handle.net/10023/25382012-06-22T00:00:00ZSmy, Louise AnnThis thesis aims to improve the understanding of transport and critical layer mixing in the troposphere and stratosphere. A dynamical approach is taken based on potential vorticity which has long been recognised as the essential field inducing the flow and thermodynamic structure of the atmosphere. Within the dynamical framework of critical layer mixing of potential vorticity, three main topics are addressed.
First, an idealised model of critical layer mixing in the stratospheric surf zone is examined. The effect of the shear across the critical layer on the critical layer evolution itself is investigated. In particular it is found that at small shear barotropic instability occurs and the mixing efficiency of the critical layer increases due to the instability. The effect of finite deformation length is also considered which extends previous work.
Secondly, the dynamical coupling between the stratosphere and troposphere is examined by considering the effect of direct perturbations to stratospheric potential vorticity on the evolution of midlatitude baroclinic instability. Both zonally symmetric and asymmetric perturbations to the stratospheric potential vorticity are considered, the former representative of a strong polar vortex, the latter representative of the stratospheric state following a major sudden warming. A comparison of these perturbations gives some insight into the possible influence of pre or post-sudden warming conditions on the tropospheric evolution.
Finally, the influence of the stratospheric potential vorticity distribution on lateral mixing and transport into and out of the tropical pipe, the low latitude ascending branch of the Brewer-Dobson circulation, is investigated. The stratospheric potential vorticity distribution in the tropical stratosphere is found to have a clear pattern according to the phase of the quasi-biennial oscillation (QBO). The extent of the QBO influence is quantified, by analysing trajectories of Lagrangian particles using an online trajectory code recently implemented in the Met Office's Unified Model.The investigation of quasi-separatrix layers in solar magnetic fields
http://hdl.handle.net/10023/2106
The structure of the magnetic field is often an important factor in
many energetic processes in the solar corona.
To determine the topology of the magnetic field features such as null
points, separatrix surfaces, and separators must be found.
It has been found that these features may be preferred sites for the formation of current sheets associated with the
accumulation of free magnetic energy.
Over the last decade, it also became clear that the geometrical
analogs of the separatrices, the so-called quasi separatrix
layers, have similar properties.
This thesis has the aim of investigating these properties and to find correlations between these quantities.
Our goal is to determine the relation between the geometrical features associated with the QSLs and with current structures, sites of reconnection and topological features.
With these aims
we conduct three different studies.
First, we investigate a non linear force free magnetic field extrapolation from observed magnetogram data taken during a solar flare eruption concentrating our attention on two snapshots, one before the event and one after.
We determine the QSLs and related structures and by considering carefully how these change between the two snapshots we are able to propose a possible scenario for how the flare occurred.
In our second project we consider potential source distributions. We take different potential point source models: two four sources models already presented in the literature and a random distribution of fifteen sources.
From these potential models we conduct a detailed analysis of the relationship between topological features and QSLs.
It is found that the maxima of the Q-factor in the photosphere are located near and above the position of the subphotospheric null points (extending part way along their spines) and that their narrow QSLs are associated with the curves defined by the photospheric endpoints of all fan field lines that start from subphotospheric sources.
Our last study investigates two different flux rope emergence simulations. In particular, we take one case with and one without an overlying magnetic field.
Here, we can identify the QSLs, current, and sites of reconnection and determine the relation between them.
From this work we found that not all high-Q regions are associated with current and/or reconnection and vice-versa.
We also investigated the geometry of the field lines associated with high-Q regions to determine which geometrical behaviour of the magnetic field they are associated with. Those that are associated with reconnection also coincide with topological features such as separators.
Mon, 20 Jun 2011 00:00:00 GMThttp://hdl.handle.net/10023/21062011-06-20T00:00:00ZRestante, Anna LisaThe structure of the magnetic field is often an important factor in
many energetic processes in the solar corona.
To determine the topology of the magnetic field features such as null
points, separatrix surfaces, and separators must be found.
It has been found that these features may be preferred sites for the formation of current sheets associated with the
accumulation of free magnetic energy.
Over the last decade, it also became clear that the geometrical
analogs of the separatrices, the so-called quasi separatrix
layers, have similar properties.
This thesis has the aim of investigating these properties and to find correlations between these quantities.
Our goal is to determine the relation between the geometrical features associated with the QSLs and with current structures, sites of reconnection and topological features.
With these aims
we conduct three different studies.
First, we investigate a non linear force free magnetic field extrapolation from observed magnetogram data taken during a solar flare eruption concentrating our attention on two snapshots, one before the event and one after.
We determine the QSLs and related structures and by considering carefully how these change between the two snapshots we are able to propose a possible scenario for how the flare occurred.
In our second project we consider potential source distributions. We take different potential point source models: two four sources models already presented in the literature and a random distribution of fifteen sources.
From these potential models we conduct a detailed analysis of the relationship between topological features and QSLs.
It is found that the maxima of the Q-factor in the photosphere are located near and above the position of the subphotospheric null points (extending part way along their spines) and that their narrow QSLs are associated with the curves defined by the photospheric endpoints of all fan field lines that start from subphotospheric sources.
Our last study investigates two different flux rope emergence simulations. In particular, we take one case with and one without an overlying magnetic field.
Here, we can identify the QSLs, current, and sites of reconnection and determine the relation between them.
From this work we found that not all high-Q regions are associated with current and/or reconnection and vice-versa.
We also investigated the geometry of the field lines associated with high-Q regions to determine which geometrical behaviour of the magnetic field they are associated with. Those that are associated with reconnection also coincide with topological features such as separators.The period ratio P₁/2P₂ in coronal waves
http://hdl.handle.net/10023/2101
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.
In this thesis we consider the period ratio P₁/2P₂ of coronal loops from a theoretical standpoint. Previous
theory and observations indicate that the period ratio is likely to be less than unity for oscillations of
coronal loops. We consider the role of damping and density structuring on the period ratio.
In Chapter 2 we consider analytically the one-dimensional wave equation with the inclusion of a generic
damping term for both uniform and non-uniform media. Results suggest that the period ratio is dominated
by longitudinal structuring rather than damping.
In Chapter 3 we consider analytically the effects of thermal conduction and compressive viscosity on the
period ratio for a longitudinally propagating sound wave. We find that damping by either thermal conduction
or compressive viscosity typically has a small effect on the period ratio. 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.
In Chapter 4 we extend the analysis of Chapter 3 to include radiative cooling and find that it too has a
negligible effect on the period ratio for typical coronal values. As an extension to the investigation, damping
rates are considered for thermal conduction, compressive viscosity and radiative cooling. The damping
time is found to be optimal for each mechanism in a different temperature range, namely below 1 MK for
radiative cooling, 2 − 6 MK for thermal conduction and above 6 MK for compressive viscosity.
In Chapter 5 we consider analytically the period ratio for the fast kink, sausage and n = N modes of a
magnetic slab, discussing both an Epstein density profile and a simple step function profile. We find that
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 slab-like structures. The similarity in the behaviour of the period
ratio for both profiles means either can be used as a robust model. We consider also other profiles numerically
for the kink mode, which are found to be either slab-like or Epstein-like suggesting again that it is not
necessary to distinguish the nature of the density profile when considering the period ratio.
Wed, 30 Nov 2011 00:00:00 GMThttp://hdl.handle.net/10023/21012011-11-30T00:00:00ZMacnamara, Cicely K.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.
In this thesis we consider the period ratio P₁/2P₂ of coronal loops from a theoretical standpoint. Previous
theory and observations indicate that the period ratio is likely to be less than unity for oscillations of
coronal loops. We consider the role of damping and density structuring on the period ratio.
In Chapter 2 we consider analytically the one-dimensional wave equation with the inclusion of a generic
damping term for both uniform and non-uniform media. Results suggest that the period ratio is dominated
by longitudinal structuring rather than damping.
In Chapter 3 we consider analytically the effects of thermal conduction and compressive viscosity on the
period ratio for a longitudinally propagating sound wave. We find that damping by either thermal conduction
or compressive viscosity typically has a small effect on the period ratio. 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.
In Chapter 4 we extend the analysis of Chapter 3 to include radiative cooling and find that it too has a
negligible effect on the period ratio for typical coronal values. As an extension to the investigation, damping
rates are considered for thermal conduction, compressive viscosity and radiative cooling. The damping
time is found to be optimal for each mechanism in a different temperature range, namely below 1 MK for
radiative cooling, 2 − 6 MK for thermal conduction and above 6 MK for compressive viscosity.
In Chapter 5 we consider analytically the period ratio for the fast kink, sausage and n = N modes of a
magnetic slab, discussing both an Epstein density profile and a simple step function profile. We find that
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 slab-like structures. The similarity in the behaviour of the period
ratio for both profiles means either can be used as a robust model. We consider also other profiles numerically
for the kink mode, which are found to be either slab-like or Epstein-like suggesting again that it is not
necessary to distinguish the nature of the density profile when considering the period ratio.Small-scale magnetic feature evolution as observed by Hinode/NFI and SOHO/MDI
http://hdl.handle.net/10023/2083
The surface (photosphere) of the Sun is threaded throughout by magnetic fields. Groups of magnetic fields form magnetic features (of a wide range of sizes in flux and area) on the surface where the fields are directed into or out of the Sun. The aim of this thesis is to examine in detail the four key processes, emergence, cancellation, fragmentation and coalescence, which determine the behaviour of small-scale magnetic features, in the Sun’s photosphere. I identify features in both Hinode/NFI and SOHO/MDI full-disk to enable these processes to be examined at the currently smallest observable scales and over an entire solar cycle.
The emerging event frequency versus flux distribution, for intranetwork emerging regions to active regions, is found to follow a power-law distribution with index -2.50, which spans nearly 7 orders of magnitude in flux (10¹⁶ - 10²³ Mx) and 18 orders of magnitude in frequency. The global rate of flux emergence is found to be 3.9 x 10²⁴ Mx day⁻¹. Since the slope of all emerged fluxes is less than -2 this implies that most of the new flux that is fed into the solar atmosphere is from small-scale emerging events. This single power-law distribution over all emerged fluxes suggest a scale-free dynamo, therefore indicating that in addition to dynamo actions in the tachocline producing sunspots, a turbulent dynamo may act throughout the convection zone. Similarly for cancellations I find a power-law relationship between the frequency of cancellation and the peak flux lost per cancelling event (for events detected in both Hinode/NFI and SOHO/MDI full-disk), with slope -2.10. Again, the process of cancellation appears to be scale free and the slope is less than -2 indicating that numerous small-scale features are cancelling the majority of flux on the Sun. I also estimate the frequency of all surface processes at solar maximum and find, 1.3 x 10⁸, 4.5 x 10⁷, 4.0 x 10⁷ and 3.6 x 10⁶ events per day over the whole surface for emergence, cancellation, fragmentation and coalescence events, respectively. All the surface processes are found to behave in a similar manner over all flux scales. The majority of events for all processes occur in features with flux below 10²º Mx, which highlights the dynamic nature of the magnetic carpet. Using SOHO/MDI full-disk data I investigate the cyclic variation of the 4 key processes throughout cycle 23. It is found that the rate of emerging events, cancellations, fragmentations and coalescences varied in anti-phase with the solar cycle by factors of 3.4, 3.1, 2.4 and 2.2, respectively over the cycle. Not surprisingly, therefore, the number of network features detected throughout the cycle also exhibits an anti-phase variation over the solar cycle by a factor of 1.9. The mean peak flux of tracked small-scale network, fragmenting, coalescing and cancelling features showed in-phase relationships with the solar cycle by factors of 1.4, 1.7, 2.4 and 1.2, respectively. The total flux which is emerged and cancelled by small-scale events, varied in anti-phase with the solar cycle, by factors of 1.9 and 3.2. This is clearly due to the variation in the number of emerging and cancelling events and the fact that the flux of individual emerging events showed no cyclic variation. The results in this thesis show that the large-scale solar cycle plays a complex role in the surface processes features undergo. The fact that the number of ephemeral regions emerging has an anti-phase variation to the solar cycle has a knock-on effect in the number of features which are available to undergo surface processes. Also decaying active regions, during more active periods, contribute more small-scale features, with high flux density, into the network which has an effect on the surface processes. This work has revealed the significant importance of small-scale features in the flux budget through continual emergence and cancellation, plus highlighted how through dynamic surface motions, small-scale features form the fundamental components with which the network is developed.
Wed, 30 Nov 2011 00:00:00 GMThttp://hdl.handle.net/10023/20832011-11-30T00:00:00ZThornton, L. M.The surface (photosphere) of the Sun is threaded throughout by magnetic fields. Groups of magnetic fields form magnetic features (of a wide range of sizes in flux and area) on the surface where the fields are directed into or out of the Sun. The aim of this thesis is to examine in detail the four key processes, emergence, cancellation, fragmentation and coalescence, which determine the behaviour of small-scale magnetic features, in the Sun’s photosphere. I identify features in both Hinode/NFI and SOHO/MDI full-disk to enable these processes to be examined at the currently smallest observable scales and over an entire solar cycle.
The emerging event frequency versus flux distribution, for intranetwork emerging regions to active regions, is found to follow a power-law distribution with index -2.50, which spans nearly 7 orders of magnitude in flux (10¹⁶ - 10²³ Mx) and 18 orders of magnitude in frequency. The global rate of flux emergence is found to be 3.9 x 10²⁴ Mx day⁻¹. Since the slope of all emerged fluxes is less than -2 this implies that most of the new flux that is fed into the solar atmosphere is from small-scale emerging events. This single power-law distribution over all emerged fluxes suggest a scale-free dynamo, therefore indicating that in addition to dynamo actions in the tachocline producing sunspots, a turbulent dynamo may act throughout the convection zone. Similarly for cancellations I find a power-law relationship between the frequency of cancellation and the peak flux lost per cancelling event (for events detected in both Hinode/NFI and SOHO/MDI full-disk), with slope -2.10. Again, the process of cancellation appears to be scale free and the slope is less than -2 indicating that numerous small-scale features are cancelling the majority of flux on the Sun. I also estimate the frequency of all surface processes at solar maximum and find, 1.3 x 10⁸, 4.5 x 10⁷, 4.0 x 10⁷ and 3.6 x 10⁶ events per day over the whole surface for emergence, cancellation, fragmentation and coalescence events, respectively. All the surface processes are found to behave in a similar manner over all flux scales. The majority of events for all processes occur in features with flux below 10²º Mx, which highlights the dynamic nature of the magnetic carpet. Using SOHO/MDI full-disk data I investigate the cyclic variation of the 4 key processes throughout cycle 23. It is found that the rate of emerging events, cancellations, fragmentations and coalescences varied in anti-phase with the solar cycle by factors of 3.4, 3.1, 2.4 and 2.2, respectively over the cycle. Not surprisingly, therefore, the number of network features detected throughout the cycle also exhibits an anti-phase variation over the solar cycle by a factor of 1.9. The mean peak flux of tracked small-scale network, fragmenting, coalescing and cancelling features showed in-phase relationships with the solar cycle by factors of 1.4, 1.7, 2.4 and 1.2, respectively. The total flux which is emerged and cancelled by small-scale events, varied in anti-phase with the solar cycle, by factors of 1.9 and 3.2. This is clearly due to the variation in the number of emerging and cancelling events and the fact that the flux of individual emerging events showed no cyclic variation. The results in this thesis show that the large-scale solar cycle plays a complex role in the surface processes features undergo. The fact that the number of ephemeral regions emerging has an anti-phase variation to the solar cycle has a knock-on effect in the number of features which are available to undergo surface processes. Also decaying active regions, during more active periods, contribute more small-scale features, with high flux density, into the network which has an effect on the surface processes. This work has revealed the significant importance of small-scale features in the flux budget through continual emergence and cancellation, plus highlighted how through dynamic surface motions, small-scale features form the fundamental components with which the network is developed.Current sheets in the solar corona : formation, fragmentation and heating
http://hdl.handle.net/10023/2081
In this thesis we investigate current sheets in the solar corona. The well known 1D model for the tearing mode instability is presented, before progressing to 2D where we introduce a non-uniform resistivity. The effect this has on growth rates is investigated and we find that the inclusion of the non-uniform term in η cause a decrease in the growth rate of the dominant mode. Analytical approximations and numerical simulations are then used to model current sheet formation by considering two distinct experiments. First, a magnetic field is sheared in two directions, perpendicular to each other. A twisted current layer is formed and we find that as we increase grid resolution, the maximum current increases, the width of the current layer decreases and the total current in the layer is approximately constant. This, together with the residual Lorentz force calculated, suggests that a current sheet is trying to form. The current layer then starts to fragment. By considering the parallel electric field and calculating the perpendicular vorticity, we find evidence of reconnection. The resulting temperatures easily reach the required coronal values. The second set of simulations carried out model an initially straight magnetic field which is stressed by elliptical boundary motions. A highly twisted current layer is formed and analysis of the energetics, current structures, magnetic field and the resulting temperatures is carried out. Results are similar in nature to that of the shearing experiment.
Wed, 30 Nov 2011 00:00:00 GMThttp://hdl.handle.net/10023/20812011-11-30T00:00:00ZBowness, RuthIn this thesis we investigate current sheets in the solar corona. The well known 1D model for the tearing mode instability is presented, before progressing to 2D where we introduce a non-uniform resistivity. The effect this has on growth rates is investigated and we find that the inclusion of the non-uniform term in η cause a decrease in the growth rate of the dominant mode. Analytical approximations and numerical simulations are then used to model current sheet formation by considering two distinct experiments. First, a magnetic field is sheared in two directions, perpendicular to each other. A twisted current layer is formed and we find that as we increase grid resolution, the maximum current increases, the width of the current layer decreases and the total current in the layer is approximately constant. This, together with the residual Lorentz force calculated, suggests that a current sheet is trying to form. The current layer then starts to fragment. By considering the parallel electric field and calculating the perpendicular vorticity, we find evidence of reconnection. The resulting temperatures easily reach the required coronal values. The second set of simulations carried out model an initially straight magnetic field which is stressed by elliptical boundary motions. A highly twisted current layer is formed and analysis of the energetics, current structures, magnetic field and the resulting temperatures is carried out. Results are similar in nature to that of the shearing experiment.Magnetic flux transport simulations : applications to solar and stellar magnetic fields
http://hdl.handle.net/10023/2072
Magnetic fields play a key role in a wide variety of phenomena found on the Sun. One such phenomena
is the Coronal Mass Ejection (CME) where a large amount of material is ejected from the
Sun. CME’s may directly affect the earth, therefore understanding their origin is of key importance
for space weather and the near-Earth environment.
In this thesis, the nature and evolution of solar magnetic fields is considered through a combination
of Magnetic Flux Transport Simulations and Potential Field Source Surface Models. The Magnetic
Flux Transport Simulations produce a realistic description of the evolution and distribution of the
radial magnetic field at the level of the solar photosphere. This is then applied as a lower boundary
condition for the Potential Field Source Surface Models which prescribe a coronal magnetic field.
Using these two techniques, the location and variation of coronal null points, a key element in the
Magnetic Breakout Model of CMEs, are determined. Results show that the number of coronal null
points follow a cyclic variation in phase with the solar cycle. In addition, they preferentially form
at lower latitudes as a result of the complex active latitude field. Although a significant number of
coronal nulls may exist at any one time (≈ 17), it is shown that only half may satisfy the necessary
condition for breakout. From this it is concluded that while the Magnetic Breakout Model of CMEs
is an important model in understanding the origin of the CMEs, other processes must occur in order
to explain the observed number of CMEs.
Finally, the Magnetic Flux Transport Simulations are applied to stellar magnetic fields and in particular
to the fast rotating star HD171488. From this speculative study it is shown that the Magnetic Flux Transport Simulations constructed for the Sun may be applied in very different stellar circumstances
and that for HD171488 a significantly higher rate of meridional flow (1200-1400 ms⁻¹) is required to
match observed magnetic field distributions.
Tue, 01 Nov 2011 00:00:00 GMThttp://hdl.handle.net/10023/20722011-11-01T00:00:00ZCook, Graeme RobertMagnetic fields play a key role in a wide variety of phenomena found on the Sun. One such phenomena
is the Coronal Mass Ejection (CME) where a large amount of material is ejected from the
Sun. CME’s may directly affect the earth, therefore understanding their origin is of key importance
for space weather and the near-Earth environment.
In this thesis, the nature and evolution of solar magnetic fields is considered through a combination
of Magnetic Flux Transport Simulations and Potential Field Source Surface Models. The Magnetic
Flux Transport Simulations produce a realistic description of the evolution and distribution of the
radial magnetic field at the level of the solar photosphere. This is then applied as a lower boundary
condition for the Potential Field Source Surface Models which prescribe a coronal magnetic field.
Using these two techniques, the location and variation of coronal null points, a key element in the
Magnetic Breakout Model of CMEs, are determined. Results show that the number of coronal null
points follow a cyclic variation in phase with the solar cycle. In addition, they preferentially form
at lower latitudes as a result of the complex active latitude field. Although a significant number of
coronal nulls may exist at any one time (≈ 17), it is shown that only half may satisfy the necessary
condition for breakout. From this it is concluded that while the Magnetic Breakout Model of CMEs
is an important model in understanding the origin of the CMEs, other processes must occur in order
to explain the observed number of CMEs.
Finally, the Magnetic Flux Transport Simulations are applied to stellar magnetic fields and in particular
to the fast rotating star HD171488. From this speculative study it is shown that the Magnetic Flux Transport Simulations constructed for the Sun may be applied in very different stellar circumstances
and that for HD171488 a significantly higher rate of meridional flow (1200-1400 ms⁻¹) is required to
match observed magnetic field distributions.An investigation into the use of balance in operational numerical weather prediction
http://hdl.handle.net/10023/1903
Presented in this study is a wide-ranging investigation into
the use of properties of balance in an operational numerical
weather prediction context.
Initially, a joint numerical and observational study is undertaken. We used
the Unified Model (UM), the suite of atmospheric and oceanic prediction
software used at the UK Met Office (UKMO), to locate symmetric
instabilities (SIs), an indicator of imbalanced motion. These are
areas of negative Ertel potential vorticity (in the Northern
hemisphere) calculated on surfaces of constant potential temperature.
Once located, the SIs were compared with satellite and aircraft
observational data. As a full three-dimensional calculation of Ertel PV
proved outwith the scope of this study we calculated the
two-dimensional, vertical component of the absolute vorticity, to assess
the inertial stability criterion. We found that at the synoptic scale in
the atmosphere, if there existed a symmetric instability, it was dominated
by an inertial instability.
With the appropriate observational data, evidence of inertial instability
from the vertical component of the absolute vorticity, predicted by
the UM was found at 12km horizontal grid resolution. Varying the
horizontal grid resolution allowed the estimation of a grid length scale,
above which, the inertial instability was not captured by the observational
data, of approximately 20km. Independently, aircraft data was used to
estimate that horizontal grid
resolutions above 20-25km should not model any features
of imbalance providing a real world estimate of the
lower bound of the grid resolution that should be employed by a
balanced atmospheric prediction model. A further investigation of the UM
concluded that the data assimilation scheme and time of initialisation
had no effect on the generation of SIs.
An investigation was then made into the robustness of balanced
models in the shallow water context, employing the contour-advective
semi-Lagrangian (CASL) algorithm, Dritschel & Ambaum (1997), a novel
numerical algorithm that exploits the underlying balance observed
within a geophysical flow at leading order. Initially two algorithms
were considered, which differed by the prognostic variables employed.
Each algorithm had their three-time-level semi-implicit time integration
scheme de-centred to mirror the time integration scheme of the UM. We
found that the version with potential vorticity (PV), divergence and
acceleration divergence, CA[subscript(δ,γ)], as prognostic variables
preserved the Bolin-Charney balance to a much greater degree than the
model with PV, divergence and depth anomaly CA[subscript(tilde{h},δ)],
as prognostic variables. This demonstrated that CA[subscript(δ,γ)] was better equipped to benefit from de-centring, an essential property
of any operational numerical weather prediction (NWP) model.
We then investigate the robustness of CA[subscript(δ,γ)] by simulating flows with Rossby and Froude number O(1), to find the
operational limits of the algorithm. We also investigated increasing
the efficiency of CA[subscript(δ,γ)] by increasing the
time-step Δt employed while decreasing specific convergence
criteria of the algorithm while preserving accuracy. We find that
significant efficiency gains are possible for predominantly
mid-latitude flows, a necessary step for the use of
CA[subscript(δ,γ)] in an operational NWP context.
The study is concluded by employing CASL
in the non-hydrostatic context under the Boussinesq approximation,
which allows weak stratification to be considered,
a step closer to physical reality than the shallow water case. CASL is
compared to the primitive equation pseudospectral (PEPS) and
vorticity-based pseudospectral (VPS) algorithms, both as the names suggest,
spectral-based algorithms, which again
differ by the prognostic variables employed. This
comparison is drawn to highlight the computational advantages that
CASL has over common numerical methods used in many operational
forecast centres. We find that CASL requires
significantly less artificial numerical diffusion than its
pseudospectral counterparts in simulations of Rossby number ~O(1).
Consequently, CASL obtains a much less diffuse, more accurate solution,
at a lower resolution and therefore lower computational cost.
At low Rossby number, where the flow is strongly influence by the Earth's
rotation, it is found that CASL is the most cost-effective
method. In addition, CASL also preserves a much greater proportion
of balance, diagnosed with nonlinear quasigeostrophic balance (NQG), another significant advantage
over its pseudospectral counterparts.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/10023/19032011-01-01T00:00:00ZDevlin, David J.J.Presented in this study is a wide-ranging investigation into
the use of properties of balance in an operational numerical
weather prediction context.
Initially, a joint numerical and observational study is undertaken. We used
the Unified Model (UM), the suite of atmospheric and oceanic prediction
software used at the UK Met Office (UKMO), to locate symmetric
instabilities (SIs), an indicator of imbalanced motion. These are
areas of negative Ertel potential vorticity (in the Northern
hemisphere) calculated on surfaces of constant potential temperature.
Once located, the SIs were compared with satellite and aircraft
observational data. As a full three-dimensional calculation of Ertel PV
proved outwith the scope of this study we calculated the
two-dimensional, vertical component of the absolute vorticity, to assess
the inertial stability criterion. We found that at the synoptic scale in
the atmosphere, if there existed a symmetric instability, it was dominated
by an inertial instability.
With the appropriate observational data, evidence of inertial instability
from the vertical component of the absolute vorticity, predicted by
the UM was found at 12km horizontal grid resolution. Varying the
horizontal grid resolution allowed the estimation of a grid length scale,
above which, the inertial instability was not captured by the observational
data, of approximately 20km. Independently, aircraft data was used to
estimate that horizontal grid
resolutions above 20-25km should not model any features
of imbalance providing a real world estimate of the
lower bound of the grid resolution that should be employed by a
balanced atmospheric prediction model. A further investigation of the UM
concluded that the data assimilation scheme and time of initialisation
had no effect on the generation of SIs.
An investigation was then made into the robustness of balanced
models in the shallow water context, employing the contour-advective
semi-Lagrangian (CASL) algorithm, Dritschel & Ambaum (1997), a novel
numerical algorithm that exploits the underlying balance observed
within a geophysical flow at leading order. Initially two algorithms
were considered, which differed by the prognostic variables employed.
Each algorithm had their three-time-level semi-implicit time integration
scheme de-centred to mirror the time integration scheme of the UM. We
found that the version with potential vorticity (PV), divergence and
acceleration divergence, CA[subscript(δ,γ)], as prognostic variables
preserved the Bolin-Charney balance to a much greater degree than the
model with PV, divergence and depth anomaly CA[subscript(tilde{h},δ)],
as prognostic variables. This demonstrated that CA[subscript(δ,γ)] was better equipped to benefit from de-centring, an essential property
of any operational numerical weather prediction (NWP) model.
We then investigate the robustness of CA[subscript(δ,γ)] by simulating flows with Rossby and Froude number O(1), to find the
operational limits of the algorithm. We also investigated increasing
the efficiency of CA[subscript(δ,γ)] by increasing the
time-step Δt employed while decreasing specific convergence
criteria of the algorithm while preserving accuracy. We find that
significant efficiency gains are possible for predominantly
mid-latitude flows, a necessary step for the use of
CA[subscript(δ,γ)] in an operational NWP context.
The study is concluded by employing CASL
in the non-hydrostatic context under the Boussinesq approximation,
which allows weak stratification to be considered,
a step closer to physical reality than the shallow water case. CASL is
compared to the primitive equation pseudospectral (PEPS) and
vorticity-based pseudospectral (VPS) algorithms, both as the names suggest,
spectral-based algorithms, which again
differ by the prognostic variables employed. This
comparison is drawn to highlight the computational advantages that
CASL has over common numerical methods used in many operational
forecast centres. We find that CASL requires
significantly less artificial numerical diffusion than its
pseudospectral counterparts in simulations of Rossby number ~O(1).
Consequently, CASL obtains a much less diffuse, more accurate solution,
at a lower resolution and therefore lower computational cost.
At low Rossby number, where the flow is strongly influence by the Earth's
rotation, it is found that CASL is the most cost-effective
method. In addition, CASL also preserves a much greater proportion
of balance, diagnosed with nonlinear quasigeostrophic balance (NQG), another significant advantage
over its pseudospectral counterparts.MHD evolution of magnetic null points to static equilibria
http://hdl.handle.net/10023/1897
In magnetised plasmas, magnetic reconnection is the process of magnetic field merging and recombination through which considerable amounts of magnetic energy may be converted into other forms of energy. Reconnection is a key mechanism for solar flares and coronal mass ejections in the solar atmosphere, it is believed to be an important source of heating of the solar corona, and it plays a major role in the acceleration of particles in the Earth's magnetotail. For reconnection to occur, the magnetic field must, in localised regions, be able to diffuse through the plasma. Ideal locations for diffusion to occur are electric current layers formed from rapidly changing magnetic fields in short space scales. In this thesis we consider the formation and nature of these current layers in magnetised plasmas.
The study of current sheets and current layers in two, and more recently, three dimensions, has been a key field of research in the last decades. However, many of these studies do not take plasma pressure effects into consideration, and rather they consider models of current sheets where the magnetic forces sum to zero. More recently, others have started to consider models in which the plasma beta is non-zero, but they simply focus on the actual equilibrium state involving a current layer and do not consider how such an equilibrium may be achieved physically. In particular, they do not allow energy conversion between magnetic and internal energy of the plasma on their way to approaching the final equilibrium.
In this thesis, we aim to describe the formation of equilibrium states involving current layers at both two and three dimensional magnetic null points, which are specific locations where the magnetic field vanishes. The different equilibria are obtained through the non-resistive dynamical evolution of perturbed hydromagnetic systems. The dynamic evolution relaxes via viscous damping, resulting in viscous heating.
We have run a series of numerical experiments using LARE, a Lagrangian-remap code, that solves the full magnetohydrodynamic (MHD) equations with user controlled viscosity and resistivity. To allow strong current accumulations to be created in a static equilibrium, we set the resistivity to be zero and hence simply reach our equilibria by solving the ideal MHD equations.
We first consider the relaxation of simple homogeneous straight magnetic fields embedded in a plasma, and determine the role of the coupling between magnetic and plasma forces, both analytically and numerically. Then, we study the formation of current accumulations at 2D magnetic X-points and at 3D magnetic nulls with spine-aligned and fan-aligned current. At both 2D X-points and 3D nulls with fan-aligned current, the current density becomes singular at the location of the null. It is impossible to be precisely achieve an exact singularity, and instead, we find a gradual continuous increase of the peak current over time, and small, highly localised forces acting to form the singularity. In the 2D case, we give a qualitative description of the field around the magnetic null using a singular function, which is found to vary within the different topological regions of the field. Also, the final equilibrium depends exponentially on the initial plasma pressure. In the 3D spine-aligned experiments, in contrast, the current density is mainly accumulated along and about the spine, but not at the null. In this case, we find that the plasma pressure does not play an important role in the final equilibrium.
Our results show that current sheet formation (and presumably reconnection) around magnetic nulls is held back by non-zero plasma betas, although the value of the plasma pressure appears to be much less important for torsional reconnection. In future studies, we may consider a broader family of 3D nulls, comparing the results with the analytical calculations in 2D, and the relaxation of more complex scenarios such as 3D magnetic separators.
Wed, 18 May 2011 00:00:00 GMThttp://hdl.handle.net/10023/18972011-05-18T00:00:00ZFuentes Fernández, JorgeIn magnetised plasmas, magnetic reconnection is the process of magnetic field merging and recombination through which considerable amounts of magnetic energy may be converted into other forms of energy. Reconnection is a key mechanism for solar flares and coronal mass ejections in the solar atmosphere, it is believed to be an important source of heating of the solar corona, and it plays a major role in the acceleration of particles in the Earth's magnetotail. For reconnection to occur, the magnetic field must, in localised regions, be able to diffuse through the plasma. Ideal locations for diffusion to occur are electric current layers formed from rapidly changing magnetic fields in short space scales. In this thesis we consider the formation and nature of these current layers in magnetised plasmas.
The study of current sheets and current layers in two, and more recently, three dimensions, has been a key field of research in the last decades. However, many of these studies do not take plasma pressure effects into consideration, and rather they consider models of current sheets where the magnetic forces sum to zero. More recently, others have started to consider models in which the plasma beta is non-zero, but they simply focus on the actual equilibrium state involving a current layer and do not consider how such an equilibrium may be achieved physically. In particular, they do not allow energy conversion between magnetic and internal energy of the plasma on their way to approaching the final equilibrium.
In this thesis, we aim to describe the formation of equilibrium states involving current layers at both two and three dimensional magnetic null points, which are specific locations where the magnetic field vanishes. The different equilibria are obtained through the non-resistive dynamical evolution of perturbed hydromagnetic systems. The dynamic evolution relaxes via viscous damping, resulting in viscous heating.
We have run a series of numerical experiments using LARE, a Lagrangian-remap code, that solves the full magnetohydrodynamic (MHD) equations with user controlled viscosity and resistivity. To allow strong current accumulations to be created in a static equilibrium, we set the resistivity to be zero and hence simply reach our equilibria by solving the ideal MHD equations.
We first consider the relaxation of simple homogeneous straight magnetic fields embedded in a plasma, and determine the role of the coupling between magnetic and plasma forces, both analytically and numerically. Then, we study the formation of current accumulations at 2D magnetic X-points and at 3D magnetic nulls with spine-aligned and fan-aligned current. At both 2D X-points and 3D nulls with fan-aligned current, the current density becomes singular at the location of the null. It is impossible to be precisely achieve an exact singularity, and instead, we find a gradual continuous increase of the peak current over time, and small, highly localised forces acting to form the singularity. In the 2D case, we give a qualitative description of the field around the magnetic null using a singular function, which is found to vary within the different topological regions of the field. Also, the final equilibrium depends exponentially on the initial plasma pressure. In the 3D spine-aligned experiments, in contrast, the current density is mainly accumulated along and about the spine, but not at the null. In this case, we find that the plasma pressure does not play an important role in the final equilibrium.
Our results show that current sheet formation (and presumably reconnection) around magnetic nulls is held back by non-zero plasma betas, although the value of the plasma pressure appears to be much less important for torsional reconnection. In future studies, we may consider a broader family of 3D nulls, comparing the results with the analytical calculations in 2D, and the relaxation of more complex scenarios such as 3D magnetic separators.Numerical modelling of two HMX-based plastic-bonded explosives at the mesoscale
http://hdl.handle.net/10023/1709
Mesoscale models are needed to predict the effect of changes to the microstructure of plastic-bonded explosives on their shock initiation and detonation behaviour. This thesis describes the considerable progress that has been made towards a mesoscale model for two HMX-based explosives PBX9501 and EDC37. In common with previous work in the literature, the model is implemented in hydrocodes that have been designed for shock physics and detonation modelling. Two relevant physics effects, heat conduction and Arrhenius chemistry, are added to a one-dimensional Lagrangian hydrocode and correction factors are identified to improve total energy conservation. Material models are constructed for the HMX crystals and polymer binders in the explosives, and are validated by comparison to Hugoniot data, Pop-plot data and detonation wave profiles. One and two-dimensional simulations of PBX9501 and EDC37 microstructures are used to investigate the response of the bulk explosive to shock loading. The sensitivity of calculated temperature distributions to uncertainties in the material properties data is determined, and a thermodynamic explanation is given for time-independent features in temperature profiles. Hotspots are widely accepted as being responsible for shock initiation in plastic-bonded explosives. It is demonstrated that, although shock heating of crystals and binder is responsible for temperature localisation, it is not a feasible hotspot mechanism in PBX9501 and EDC37 because the temperatures generated are too low to cause significant chemical reaction in the required timescales. Critical hotspot criteria derived for HMX and the binders compare favourably to earlier studies. The speed of reaction propagation from hotspots into the surrounding explosive is validated by comparison to flame propagation data, and the temperature of the gaseous reaction products is identified as being responsible for negative pressure dependence. Hotspot size, separation and temperature requirements are identified which can be used to eliminate candidate mechanisms in future.
Wed, 22 Jun 2011 00:00:00 GMThttp://hdl.handle.net/10023/17092011-06-22T00:00:00ZHandley, Caroline A.Mesoscale models are needed to predict the effect of changes to the microstructure of plastic-bonded explosives on their shock initiation and detonation behaviour. This thesis describes the considerable progress that has been made towards a mesoscale model for two HMX-based explosives PBX9501 and EDC37. In common with previous work in the literature, the model is implemented in hydrocodes that have been designed for shock physics and detonation modelling. Two relevant physics effects, heat conduction and Arrhenius chemistry, are added to a one-dimensional Lagrangian hydrocode and correction factors are identified to improve total energy conservation. Material models are constructed for the HMX crystals and polymer binders in the explosives, and are validated by comparison to Hugoniot data, Pop-plot data and detonation wave profiles. One and two-dimensional simulations of PBX9501 and EDC37 microstructures are used to investigate the response of the bulk explosive to shock loading. The sensitivity of calculated temperature distributions to uncertainties in the material properties data is determined, and a thermodynamic explanation is given for time-independent features in temperature profiles. Hotspots are widely accepted as being responsible for shock initiation in plastic-bonded explosives. It is demonstrated that, although shock heating of crystals and binder is responsible for temperature localisation, it is not a feasible hotspot mechanism in PBX9501 and EDC37 because the temperatures generated are too low to cause significant chemical reaction in the required timescales. Critical hotspot criteria derived for HMX and the binders compare favourably to earlier studies. The speed of reaction propagation from hotspots into the surrounding explosive is validated by comparison to flame propagation data, and the temperature of the gaseous reaction products is identified as being responsible for negative pressure dependence. Hotspot size, separation and temperature requirements are identified which can be used to eliminate candidate mechanisms in future.Theoretical magnetic flux emergence
http://hdl.handle.net/10023/1692
Magnetic flux emergence is the subject of how magnetic fields from
the solar interior can rise and expand into the atmosphere to produce
active regions. It is the link that joins dynamics in the convection
zone with dynamics in the atmosphere. In this thesis, we study many
aspects of magnetic flux emergence through mathematical modelling
and computer simulations. Our primary aim is to understand the key
physical processes that lie behind emergence.
The first chapter introduces flux emergence and the theoretical framework,
magnetohydrodynamics (MHD), that describes it. In the second
chapter, we discuss the numerical techniques used to solve the
highly non-linear problems that arise from flux emergence. The third
chapter summarizes the current literature. In the fourth chapter, we
consider how changing the geometry and parameter values of the initial
magnetic field can affect the dynamic evolution of the emerging
magnetic field. For an initial toroidal magnetic field, it is found that
its axis can emerge to the corona if the tube’s initial field strength is
large enough. The fifth chapter describes how flux emergence models
can produce large-scale solar eruptions. A 2.5D model of the breakout
model, using only dynamic flux emergence, fails to produce any large scale
eruptions. A 3D model of toroidal emergence with an overlying
magnetic field does, however, produce multiple large-scale eruptions
and the form of these is related to the breakout model. The sixth
chapter is concerned with signatures of flux emergence and how to
identify emerging twisted magnetic structures correctly. Here, a flux
emergence model produces signatures found in observations. The signatures
from the model, however, have different underlying physical
mechanisms to the original interpretations of the observations. The
thesis concludes with some final thoughts on current trends in theoretical
magnetic flux emergence and possible future directions.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/10023/16922011-01-01T00:00:00ZMacTaggart, DavidMagnetic flux emergence is the subject of how magnetic fields from
the solar interior can rise and expand into the atmosphere to produce
active regions. It is the link that joins dynamics in the convection
zone with dynamics in the atmosphere. In this thesis, we study many
aspects of magnetic flux emergence through mathematical modelling
and computer simulations. Our primary aim is to understand the key
physical processes that lie behind emergence.
The first chapter introduces flux emergence and the theoretical framework,
magnetohydrodynamics (MHD), that describes it. In the second
chapter, we discuss the numerical techniques used to solve the
highly non-linear problems that arise from flux emergence. The third
chapter summarizes the current literature. In the fourth chapter, we
consider how changing the geometry and parameter values of the initial
magnetic field can affect the dynamic evolution of the emerging
magnetic field. For an initial toroidal magnetic field, it is found that
its axis can emerge to the corona if the tube’s initial field strength is
large enough. The fifth chapter describes how flux emergence models
can produce large-scale solar eruptions. A 2.5D model of the breakout
model, using only dynamic flux emergence, fails to produce any large scale
eruptions. A 3D model of toroidal emergence with an overlying
magnetic field does, however, produce multiple large-scale eruptions
and the form of these is related to the breakout model. The sixth
chapter is concerned with signatures of flux emergence and how to
identify emerging twisted magnetic structures correctly. Here, a flux
emergence model produces signatures found in observations. The signatures
from the model, however, have different underlying physical
mechanisms to the original interpretations of the observations. The
thesis concludes with some final thoughts on current trends in theoretical
magnetic flux emergence and possible future directions.Application of stochastic differential equations and real option theory in investment decision problems
http://hdl.handle.net/10023/1691
This thesis contains a discussion of four problems arising from the application of
stochastic differential equations and real option theory to investment decision problems in a continuous-time framework. It is based on four papers written jointly with
the author’s supervisor.
In the first problem, we study an evolutionary stock market model in a continuous-time framework where uncertainty in dividends is produced by a single Wiener process.
The model is an adaptation to a continuous-time framework of a discrete evolutionary
stock market model developed by Evstigneev, Hens and Schenk-Hoppé (2006). We
consider the case of fix-mix strategies and derive the stochastic differential equations
which determine the evolution of the wealth processes of the various market players.
The wealth dynamics for various initial set-ups of the market are simulated.
In the second problem, we apply an entry-exit model in real option theory to study
concessionary agreements between a private company and a state government to
run a privatised business or project. The private company can choose the time to
enter into the agreement and can also choose the time to exit the agreement if the
project becomes unprofitable. An early termination of the agreement by the company
might mean that it has to pay a penalty fee to the government. Optimal times for
the company to enter and exit the agreement are calculated. The dynamics of the
project are assumed to follow either a geometric mean reversion process or geometric
Brownian motion. A comparative analysis is provided. Particular emphasis is given
to the role of uncertainty and how uncertainty affects the average time that the
concessionary agreement is active. The effect of uncertainty is studied by using Monte
Carlo simulation.
In the third problem, we study numerical methods for solving stochastic optimal
control problems which are linear in the control. In particular, we investigate methods
based on spline functions for solving the two-point boundary value problems that
arise from the method of dynamic programming. In the general case, where only
the value function and its first derivative are guaranteed to be continuous, piecewise
quadratic polynomials are used in the solution. However, under certain conditions,
the continuity of the second derivative is also guaranteed. In this case, piecewise
cubic polynomials are used in the solution. We show how the computational time
and memory requirements of the solution algorithm can be improved by effectively
reducing the dimension of the problem. Numerical examples which demonstrate the
effectiveness of our method are provided.
Lastly, we study the situation where, by partial privatisation, a government gives
a private company the opportunity to invest in a government-owned business. After
payment of an initial instalment cost, the private company’s investments are assumed
to be flexible within a range [0, k] while the investment in the business continues. We
model the problem in a real option framework and use a geometric mean reversion
process to describe the dynamics of the business. We use the method of dynamic
programming to determine the optimal time for the private company to enter and
pay the initial instalment cost as well as the optimal dynamic investment strategy
that it follows afterwards. Since an analytic solution cannot be obtained for the
dynamic programming equations, we use quadratic splines to obtain a numerical
solution. Finally we determine the optimal degree of privatisation in our model from
the perspective of the government.
Tue, 30 Nov 2010 00:00:00 GMThttp://hdl.handle.net/10023/16912010-11-30T00:00:00ZChavanasporn, WalailuckThis thesis contains a discussion of four problems arising from the application of
stochastic differential equations and real option theory to investment decision problems in a continuous-time framework. It is based on four papers written jointly with
the author’s supervisor.
In the first problem, we study an evolutionary stock market model in a continuous-time framework where uncertainty in dividends is produced by a single Wiener process.
The model is an adaptation to a continuous-time framework of a discrete evolutionary
stock market model developed by Evstigneev, Hens and Schenk-Hoppé (2006). We
consider the case of fix-mix strategies and derive the stochastic differential equations
which determine the evolution of the wealth processes of the various market players.
The wealth dynamics for various initial set-ups of the market are simulated.
In the second problem, we apply an entry-exit model in real option theory to study
concessionary agreements between a private company and a state government to
run a privatised business or project. The private company can choose the time to
enter into the agreement and can also choose the time to exit the agreement if the
project becomes unprofitable. An early termination of the agreement by the company
might mean that it has to pay a penalty fee to the government. Optimal times for
the company to enter and exit the agreement are calculated. The dynamics of the
project are assumed to follow either a geometric mean reversion process or geometric
Brownian motion. A comparative analysis is provided. Particular emphasis is given
to the role of uncertainty and how uncertainty affects the average time that the
concessionary agreement is active. The effect of uncertainty is studied by using Monte
Carlo simulation.
In the third problem, we study numerical methods for solving stochastic optimal
control problems which are linear in the control. In particular, we investigate methods
based on spline functions for solving the two-point boundary value problems that
arise from the method of dynamic programming. In the general case, where only
the value function and its first derivative are guaranteed to be continuous, piecewise
quadratic polynomials are used in the solution. However, under certain conditions,
the continuity of the second derivative is also guaranteed. In this case, piecewise
cubic polynomials are used in the solution. We show how the computational time
and memory requirements of the solution algorithm can be improved by effectively
reducing the dimension of the problem. Numerical examples which demonstrate the
effectiveness of our method are provided.
Lastly, we study the situation where, by partial privatisation, a government gives
a private company the opportunity to invest in a government-owned business. After
payment of an initial instalment cost, the private company’s investments are assumed
to be flexible within a range [0, k] while the investment in the business continues. We
model the problem in a real option framework and use a geometric mean reversion
process to describe the dynamics of the business. We use the method of dynamic
programming to determine the optimal time for the private company to enter and
pay the initial instalment cost as well as the optimal dynamic investment strategy
that it follows afterwards. Since an analytic solution cannot be obtained for the
dynamic programming equations, we use quadratic splines to obtain a numerical
solution. Finally we determine the optimal degree of privatisation in our model from
the perspective of the government.MHD mode conversion of fast and slow magnetoacoustic waves in the solar corona
http://hdl.handle.net/10023/1361
There are three main wave types present in the Sun’s atmosphere: Alfvén waves and fast and slow magnetoacoustic waves. Alfvén waves are purely magnetic and would not exist if it was not for the Sun’s magnetic field. The fast and slow magnetoacoustic waves are so named due to their relative phase speeds. As the magnetic field tends to zero, the slow wave goes to zero as the fast wave becomes the sound wave. When a resonance occurs energy may be transferred between the different modes, causing one to increase in amplitude whilst the other decreases. This is known as mode conversion. Mode conversion of fast and slow magnetoacoustic waves takes place when the characteristic wave speeds, the sound and Alfvén speeds, are equal. This occurs in regions where the ratio of the gas pressure to the magnetic pressure, known as the plasma β, is approximately unity.
In this thesis we investigate the conversion of fast and slow magnetoacoustic waves as they propagate from low- to high-β plasma. This investigation uses a combination of analytical and numerical techniques to gain a full understanding of the process. The MacCormack finite-difference method is used to model a wave as it undergoes mode conversion. Complementing this analytical techniques are employed to find the wave behaviour at, and distant from, the mode-conversion region. These methods are described in Chapter 2.
The simple, one-dimensional model of an isothermal atmosphere permeated by a uniform magnetic field is studied in Chapter 3. Gravitational acceleration is included to ensure that mode conversion takes place. Driving a slow magnetoacoustic wave on the upper boundary conversion takes place as the wave passes from low- to high-β plasma. This is expanded upon in Chapter 4 where the effects of a non-isothermal temperature profile are examined. A tanh profile is selected to mimic the steep temperature gradient found in the transition region. In Chapter 5 the complexity is increased by allowing for a two-dimensional model. For this purpose we choose a radially-expanding magnetic field which is representative of a coronal hole. In this instance the slow magnetoacoustic wave is driven upwards from the surface, again travelling from low to high β. Finally, in Chapter 6 we investigate mode conversion near a two-dimensional, magnetic null point. At the null the plasma β becomes infinitely large and a wave propagating towards the null point will experience mode conversion.
The methods used allow conversion of fast and slow waves to be described in the various model atmospheres. The amount of transmission and conversion are calculated and matched across the mode-conversion layer giving a full description of the wave behaviour.
Tue, 30 Nov 2010 00:00:00 GMThttp://hdl.handle.net/10023/13612010-11-30T00:00:00ZMcDougall-Bagnall, A. M. DeeThere are three main wave types present in the Sun’s atmosphere: Alfvén waves and fast and slow magnetoacoustic waves. Alfvén waves are purely magnetic and would not exist if it was not for the Sun’s magnetic field. The fast and slow magnetoacoustic waves are so named due to their relative phase speeds. As the magnetic field tends to zero, the slow wave goes to zero as the fast wave becomes the sound wave. When a resonance occurs energy may be transferred between the different modes, causing one to increase in amplitude whilst the other decreases. This is known as mode conversion. Mode conversion of fast and slow magnetoacoustic waves takes place when the characteristic wave speeds, the sound and Alfvén speeds, are equal. This occurs in regions where the ratio of the gas pressure to the magnetic pressure, known as the plasma β, is approximately unity.
In this thesis we investigate the conversion of fast and slow magnetoacoustic waves as they propagate from low- to high-β plasma. This investigation uses a combination of analytical and numerical techniques to gain a full understanding of the process. The MacCormack finite-difference method is used to model a wave as it undergoes mode conversion. Complementing this analytical techniques are employed to find the wave behaviour at, and distant from, the mode-conversion region. These methods are described in Chapter 2.
The simple, one-dimensional model of an isothermal atmosphere permeated by a uniform magnetic field is studied in Chapter 3. Gravitational acceleration is included to ensure that mode conversion takes place. Driving a slow magnetoacoustic wave on the upper boundary conversion takes place as the wave passes from low- to high-β plasma. This is expanded upon in Chapter 4 where the effects of a non-isothermal temperature profile are examined. A tanh profile is selected to mimic the steep temperature gradient found in the transition region. In Chapter 5 the complexity is increased by allowing for a two-dimensional model. For this purpose we choose a radially-expanding magnetic field which is representative of a coronal hole. In this instance the slow magnetoacoustic wave is driven upwards from the surface, again travelling from low to high β. Finally, in Chapter 6 we investigate mode conversion near a two-dimensional, magnetic null point. At the null the plasma β becomes infinitely large and a wave propagating towards the null point will experience mode conversion.
The methods used allow conversion of fast and slow waves to be described in the various model atmospheres. The amount of transmission and conversion are calculated and matched across the mode-conversion layer giving a full description of the wave behaviour.Aspects of three-dimensional MHD : magnetic reconnection and rotating coronae
http://hdl.handle.net/10023/947
Solutions of the magnetohydrodynamic (MHD) equations are very important for modelling laboratory, space and astrophysical plasmas, for example the solar and stellar coronae, as well as for modelling many of the dynamic processes that occur in these different plasma environments such as the fundamental process of magnetic reconnection. Our previous understanding of the behavior of plasmas and their associated dynamic processes has been developed through two-dimensional (2D) models. However, a more realistic model should be three-dimensional (3D), but finding 3D solutions of the MHD equations is, in general, a formidable task. Only very few analytical solutions are known and even calculating solutions with numerical methods is usually far from easy.
In this thesis, 3D solutions which model magnetic reconnection and rigidly rotating magnetized coronae are presented. For magnetic reconnection, a 3D stationary MHD model is used. However, the complexity of the problem meant that so far no generic analytic solutions for reconnection in 3D exist and most work consists of numerical simulations. This has so far hampered progress in our understanding of magnetic reconnection. The model used here allows for analytic solutions at least up to a certain order of approximation and therefore gives some better insight in the significant differences between 2D and 3D reconnection. Three-dimensional numerical solutions are also obtained for this model.
Rigidly rotating magnetized coronae, on the other hand, are modeled using a set of magnetohydrostatic (MHS) equations. A general theoretical framework for calculating 3D MHS solutions outside massive rigidly rotating central bodies is presented. Under certain assumptions, the MHS equations are reduced to a single linear partial differential equation referred to as the fundamental equation of the theory. As a first step, an illustrative case of a massive rigidly rotating magnetized cylinder is considered, which somehow allows for analytic solutions in a certain domain of validity. In general, the fundamental equation of the theory can only be solved numerically and hence numerical example solutions are presented. The theory is then extended to include a more realistic case of massive rigidly rotating spherical bodies. The resulting fundamental equation of the theory in this case is too complicated to allow for analytic solutions and hence only numerical solutions are obtained using similar numerical methods to the ones used in the cylindrical case.
Wed, 23 Jun 2010 00:00:00 GMThttp://hdl.handle.net/10023/9472010-06-23T00:00:00ZAl-Salti, Nasser S.Solutions of the magnetohydrodynamic (MHD) equations are very important for modelling laboratory, space and astrophysical plasmas, for example the solar and stellar coronae, as well as for modelling many of the dynamic processes that occur in these different plasma environments such as the fundamental process of magnetic reconnection. Our previous understanding of the behavior of plasmas and their associated dynamic processes has been developed through two-dimensional (2D) models. However, a more realistic model should be three-dimensional (3D), but finding 3D solutions of the MHD equations is, in general, a formidable task. Only very few analytical solutions are known and even calculating solutions with numerical methods is usually far from easy.
In this thesis, 3D solutions which model magnetic reconnection and rigidly rotating magnetized coronae are presented. For magnetic reconnection, a 3D stationary MHD model is used. However, the complexity of the problem meant that so far no generic analytic solutions for reconnection in 3D exist and most work consists of numerical simulations. This has so far hampered progress in our understanding of magnetic reconnection. The model used here allows for analytic solutions at least up to a certain order of approximation and therefore gives some better insight in the significant differences between 2D and 3D reconnection. Three-dimensional numerical solutions are also obtained for this model.
Rigidly rotating magnetized coronae, on the other hand, are modeled using a set of magnetohydrostatic (MHS) equations. A general theoretical framework for calculating 3D MHS solutions outside massive rigidly rotating central bodies is presented. Under certain assumptions, the MHS equations are reduced to a single linear partial differential equation referred to as the fundamental equation of the theory. As a first step, an illustrative case of a massive rigidly rotating magnetized cylinder is considered, which somehow allows for analytic solutions in a certain domain of validity. In general, the fundamental equation of the theory can only be solved numerically and hence numerical example solutions are presented. The theory is then extended to include a more realistic case of massive rigidly rotating spherical bodies. The resulting fundamental equation of the theory in this case is too complicated to allow for analytic solutions and hence only numerical solutions are obtained using similar numerical methods to the ones used in the cylindrical case.Development and application of a global magnetic field evolution model for the solar corona
http://hdl.handle.net/10023/734
Magnetic ﬁelds are fundamental to the structure and dynamics of the Sun’s corona. Observations show them to be locally complex, with highly sheared and twisted ﬁelds visible in solar ﬁlaments/prominences. The free magnetic energy contained in such ﬁelds is the primary source of energy for coronal mass ejections, which are important—but still poorly understood drivers of space weather in the near-Earth environment.
In this thesis, a new model is developed for the evolution of the large-scale magnetic ﬁeld in the global solar corona. The model is based on observations of the radial magnetic ﬁeld on the solar photosphere (visible surface). New active regions emerge, and their transport and dispersal by surface motions are simulated accurately with a surface ﬂux transport model. The 3D coronal magnetic ﬁeld is evolved in response to these photospheric motions using a magneto-frictional technique. The resulting sequence of nonlinear force-free equilibria traces the build-up of magnetic helicity and free energy over many months.
The global model is applied to study two phenomena: ﬁlaments and coronal mass ejections. The magnetic ﬁeld directions in a large sample of observed ﬁlaments are compared with a 6-month simulation. Depending on the twist of newly-emerging active regions, the correct chirality
is simulated for up to 96% of ﬁlaments tested. On the basis of these simulations, an explanation for the observed hemispheric pattern of ﬁlament chirality is put forward, including why exceptions occur for ﬁlaments in certain locations. Twisted magnetic ﬂux ropes develop in the simulations, often losing equilibrium and lifting off, removing helicity. The physical basis for such losses of equilibrium is demonstrated through 2D analytical models. In the 3D global simulations, the twist of emerging regions is a key parameter controlling the number of lift-offs, which may explain around a third of observed coronal mass ejections.
Wed, 24 Jun 2009 00:00:00 GMThttp://hdl.handle.net/10023/7342009-06-24T00:00:00ZYeates, Anthony RobinsonMagnetic ﬁelds are fundamental to the structure and dynamics of the Sun’s corona. Observations show them to be locally complex, with highly sheared and twisted ﬁelds visible in solar ﬁlaments/prominences. The free magnetic energy contained in such ﬁelds is the primary source of energy for coronal mass ejections, which are important—but still poorly understood drivers of space weather in the near-Earth environment.
In this thesis, a new model is developed for the evolution of the large-scale magnetic ﬁeld in the global solar corona. The model is based on observations of the radial magnetic ﬁeld on the solar photosphere (visible surface). New active regions emerge, and their transport and dispersal by surface motions are simulated accurately with a surface ﬂux transport model. The 3D coronal magnetic ﬁeld is evolved in response to these photospheric motions using a magneto-frictional technique. The resulting sequence of nonlinear force-free equilibria traces the build-up of magnetic helicity and free energy over many months.
The global model is applied to study two phenomena: ﬁlaments and coronal mass ejections. The magnetic ﬁeld directions in a large sample of observed ﬁlaments are compared with a 6-month simulation. Depending on the twist of newly-emerging active regions, the correct chirality
is simulated for up to 96% of ﬁlaments tested. On the basis of these simulations, an explanation for the observed hemispheric pattern of ﬁlament chirality is put forward, including why exceptions occur for ﬁlaments in certain locations. Twisted magnetic ﬂux ropes develop in the simulations, often losing equilibrium and lifting off, removing helicity. The physical basis for such losses of equilibrium is demonstrated through 2D analytical models. In the 3D global simulations, the twist of emerging regions is a key parameter controlling the number of lift-offs, which may explain around a third of observed coronal mass ejections.The contour-advective semi-Lagrangian hybrid algorithm approach to weather forecasting and freely propagating inertia-gravity waves in the shallow-water system
http://hdl.handle.net/10023/716
This thesis is aimed at extending the spherical barotropic contour-advective semi-Lagrangian (CASL) Algorithm, written in 1996 by David Dritschel and Maarten Ambaum, to more complex test cases within the shallow-water context. This is an integral part for development of any numerical model and the accuracy obtained depends on many factors, including knowledge of the initial state of the atmosphere or ocean, the numerical methods applied, and the resolutions used.
The work undertaken throughout this thesis is highly varied and produces important steps towards creating a versatile suite of programs to model all types of flow, quickly and accurately. This, as will be explained in later chapters, impacts both public safety and the world economy, since much depends on accurate medium range forecasting. There shall be an investigation of a series of tests which demonstrate certain aspects of a dynamical system and its progression into more unstable situations - including the generation and feedback of freely propagating inertia-gravity waves (hereafter “gravity waves"), which transmit throughout the system. The implications for increasing forecast accuracy will be discussed.
Within this thesis two main CASL algorithms are outlined and tested, with the accuracy of the results compared with previous results. In addition, other dynamical fields (besides geopotential height and potential vorticity) are analysed in order to assess how well the models deal with gravity waves. We shall see that such waves are sensitive to the presence, or not, of sharp potential vorticity gradients, as well as to numerical parameter settings. In particular, large time-steps (convenient for semi-Lagrangian schemes) may not only seriously affect gravity waves, but may also have an adverse impact on the primary fields of height and velocity. These problems are exacerbated by a poor resolution of potential vorticity gradients, which we shall attempt to improve.
Wed, 24 Jun 2009 00:00:00 GMThttp://hdl.handle.net/10023/7162009-06-24T00:00:00ZSmith, Robert K.This thesis is aimed at extending the spherical barotropic contour-advective semi-Lagrangian (CASL) Algorithm, written in 1996 by David Dritschel and Maarten Ambaum, to more complex test cases within the shallow-water context. This is an integral part for development of any numerical model and the accuracy obtained depends on many factors, including knowledge of the initial state of the atmosphere or ocean, the numerical methods applied, and the resolutions used.
The work undertaken throughout this thesis is highly varied and produces important steps towards creating a versatile suite of programs to model all types of flow, quickly and accurately. This, as will be explained in later chapters, impacts both public safety and the world economy, since much depends on accurate medium range forecasting. There shall be an investigation of a series of tests which demonstrate certain aspects of a dynamical system and its progression into more unstable situations - including the generation and feedback of freely propagating inertia-gravity waves (hereafter “gravity waves"), which transmit throughout the system. The implications for increasing forecast accuracy will be discussed.
Within this thesis two main CASL algorithms are outlined and tested, with the accuracy of the results compared with previous results. In addition, other dynamical fields (besides geopotential height and potential vorticity) are analysed in order to assess how well the models deal with gravity waves. We shall see that such waves are sensitive to the presence, or not, of sharp potential vorticity gradients, as well as to numerical parameter settings. In particular, large time-steps (convenient for semi-Lagrangian schemes) may not only seriously affect gravity waves, but may also have an adverse impact on the primary fields of height and velocity. These problems are exacerbated by a poor resolution of potential vorticity gradients, which we shall attempt to improve.Balance, gravity waves and jets in turbulent shallow water flows
http://hdl.handle.net/10023/708
This thesis contains a thorough investigation of the properties of freely decaying turbulence in a rotating shallow water layer on a sphere. A large number
of simulations, covering an extensive range of Froude and Rossby numbers, have
been carried out using a novel numerical algorithm that exploits the underly-
ing properties of the flow. In general these flows develop coherent structures;
vortices interact, merge and migrate polewards or equatorwards depending or
their sign, leaving behind regions of homogenized potential vorticity separated
by sharp zonal jets. In the first half of the thesis we investigate new ways of looking at these structures. In the second half of the thesis we examine the properties
of the potential vorticity (PV) induced, balanced component and the residual,
unbalanced component of the flows.
Cyclone-anticyclone asymmetry has long been observed in atmospheric and
oceanic data, laboratory experiments and numerical simulations. This asymmetry is usually seen to favour anticyclonic vorticity with the asymmetry becoming more pronounced at higher Froude numbers (e.g. Polvani et al. [1994a]). We find a similar result but note that the cyclones, although fewer, are significantly
more intense and coherent. We present several ways of quantifying this across
the parameter space.
Potential vorticity homogenization is an important geophysical mechanism
responsible for sharpening jets through the expulsion of PV gradients to the edge of flow structures or domains. Sharp gradients of PV are obvious in contour plots
of this field as areas where the contours are bunched together. This suggests that
we can estimate the number of zonal jets by performing a cluster analysis on
the mean latitude of PV contours (this diagnostic is also examined by Dritschel
and McIntyre [2007]). This provides an estimate rather than an exact count of
the number of jets because the jets meander signficantly. We investigate the
accuracy of the estimates provided by different clustering techniques. We find
that the properties of the jets defy such simple classification and instead demand
a more local examination. We achieve this by examining the palinstrophy field.
This field, calculated by taking the gradient of the PV, highlights the regions
where PV contours come closer together, exactly what we would expect in regions
of strong jets. Plots of the palinstrophy field reveal the complex structure of these
features.
The potential vorticity field is even more central to the flow evolution than
the strong link with jets suggests. From a knowledge of the spatial distribution
of PV, it is possible to diagnose the balanced components of all other fields.
These components will not contain inertia-gravity waves but will contain the
dominant, large scale features of the flow. This inversion, or decomposition into
balanced (vortical) and unbalanced (wave) components, is not unique and can be
defined to varying orders of accuracy. We examine the results of four dfferent
definitions of this decomposition, two based on truncations of the full equations
and two based on an iterative procedure applied to the full equations. We find the
iterative procedure to be more accurate in that it attributes more of the flow to
the PV controlled, balanced motion. However, the truncated equations perform
surprisingly well and do not appear to suffer in accuracy at the equator, despite
the fact that the scaling on which they are based has been thought to break down
there.
We round off this study by considering the impact of the unbalanced motion on the flow. This is accomplished by splitting the integration time of the model into
intervals τ < t < τ+dτ and comparing, at the end of each interval, the balanced
components of the flow obtained by a) integrating the model from t = 0 and b)
integrating the full equations, initialised at t = τ with the balanced components
from a) at t = τ. We find that any impact of the unbalanced component of the
flow is less than the numerical noise of the model.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10023/7082009-01-01T00:00:00ZShipton, JemmaThis thesis contains a thorough investigation of the properties of freely decaying turbulence in a rotating shallow water layer on a sphere. A large number
of simulations, covering an extensive range of Froude and Rossby numbers, have
been carried out using a novel numerical algorithm that exploits the underly-
ing properties of the flow. In general these flows develop coherent structures;
vortices interact, merge and migrate polewards or equatorwards depending or
their sign, leaving behind regions of homogenized potential vorticity separated
by sharp zonal jets. In the first half of the thesis we investigate new ways of looking at these structures. In the second half of the thesis we examine the properties
of the potential vorticity (PV) induced, balanced component and the residual,
unbalanced component of the flows.
Cyclone-anticyclone asymmetry has long been observed in atmospheric and
oceanic data, laboratory experiments and numerical simulations. This asymmetry is usually seen to favour anticyclonic vorticity with the asymmetry becoming more pronounced at higher Froude numbers (e.g. Polvani et al. [1994a]). We find a similar result but note that the cyclones, although fewer, are significantly
more intense and coherent. We present several ways of quantifying this across
the parameter space.
Potential vorticity homogenization is an important geophysical mechanism
responsible for sharpening jets through the expulsion of PV gradients to the edge of flow structures or domains. Sharp gradients of PV are obvious in contour plots
of this field as areas where the contours are bunched together. This suggests that
we can estimate the number of zonal jets by performing a cluster analysis on
the mean latitude of PV contours (this diagnostic is also examined by Dritschel
and McIntyre [2007]). This provides an estimate rather than an exact count of
the number of jets because the jets meander signficantly. We investigate the
accuracy of the estimates provided by different clustering techniques. We find
that the properties of the jets defy such simple classification and instead demand
a more local examination. We achieve this by examining the palinstrophy field.
This field, calculated by taking the gradient of the PV, highlights the regions
where PV contours come closer together, exactly what we would expect in regions
of strong jets. Plots of the palinstrophy field reveal the complex structure of these
features.
The potential vorticity field is even more central to the flow evolution than
the strong link with jets suggests. From a knowledge of the spatial distribution
of PV, it is possible to diagnose the balanced components of all other fields.
These components will not contain inertia-gravity waves but will contain the
dominant, large scale features of the flow. This inversion, or decomposition into
balanced (vortical) and unbalanced (wave) components, is not unique and can be
defined to varying orders of accuracy. We examine the results of four dfferent
definitions of this decomposition, two based on truncations of the full equations
and two based on an iterative procedure applied to the full equations. We find the
iterative procedure to be more accurate in that it attributes more of the flow to
the PV controlled, balanced motion. However, the truncated equations perform
surprisingly well and do not appear to suffer in accuracy at the equator, despite
the fact that the scaling on which they are based has been thought to break down
there.
We round off this study by considering the impact of the unbalanced motion on the flow. This is accomplished by splitting the integration time of the model into
intervals τ < t < τ+dτ and comparing, at the end of each interval, the balanced
components of the flow obtained by a) integrating the model from t = 0 and b)
integrating the full equations, initialised at t = τ with the balanced components
from a) at t = τ. We find that any impact of the unbalanced component of the
flow is less than the numerical noise of the model.Equilibrium and dynamics of collisionless current sheets
http://hdl.handle.net/10023/705
In this thesis examples of translationally invariant one-dimensional (1D) Vlasov-Maxwell (VM) equilibria are investigated. The 1D VM equilibrium equations are equivalent to the motion of
a pseudoparticle in a conservative pseudopotential, with the pseudopotential being proportional to one of the diagonal components of the plasma pressure tensor. A necessary condition on the pseudopotential (plasma pressure) to allow for force-free 1D VM equilibria is formulated. It is
shown that linear force-free 1D VM solutions correspond to the case where the pseudopotential is an attractive central potential. The pseudopotential for the force-free Harris sheet is found and a Fourier transform method is used to find the corresponding distribution function. The solution is extended to include a family of equilibria that describe the transition between the Harris sheet and the force-free Harris sheet. These equilibria are used in 2.5D particle-in-cell simulations of
magnetic reconnection. The structure of the diffusion region is compared for simulations starting from anti-parallel magnetic field configurations with different strengths of guide field and self-consistent linear and non-linear force-free magnetic fields. It is shown that gradients of off-diagonal
components of the electron pressure tensor are the dominant terms that give rise to the
reconnection electric field. The typical scale length of the electron pressure tensor components in the weak guide field case is of the order of the electron bounce widths in a field reversal. In the strong guide field case the scale length reduces to the electron Larmor radius in the guide magnetic field.
Wed, 24 Jun 2009 00:00:00 GMThttp://hdl.handle.net/10023/7052009-06-24T00:00:00ZHarrison, Michael GeorgeIn this thesis examples of translationally invariant one-dimensional (1D) Vlasov-Maxwell (VM) equilibria are investigated. The 1D VM equilibrium equations are equivalent to the motion of
a pseudoparticle in a conservative pseudopotential, with the pseudopotential being proportional to one of the diagonal components of the plasma pressure tensor. A necessary condition on the pseudopotential (plasma pressure) to allow for force-free 1D VM equilibria is formulated. It is
shown that linear force-free 1D VM solutions correspond to the case where the pseudopotential is an attractive central potential. The pseudopotential for the force-free Harris sheet is found and a Fourier transform method is used to find the corresponding distribution function. The solution is extended to include a family of equilibria that describe the transition between the Harris sheet and the force-free Harris sheet. These equilibria are used in 2.5D particle-in-cell simulations of
magnetic reconnection. The structure of the diffusion region is compared for simulations starting from anti-parallel magnetic field configurations with different strengths of guide field and self-consistent linear and non-linear force-free magnetic fields. It is shown that gradients of off-diagonal
components of the electron pressure tensor are the dominant terms that give rise to the
reconnection electric field. The typical scale length of the electron pressure tensor components in the weak guide field case is of the order of the electron bounce widths in a field reversal. In the strong guide field case the scale length reduces to the electron Larmor radius in the guide magnetic field.Magnetic skeletons and 3D magnetic reconnection
http://hdl.handle.net/10023/475
The upper atmosphere of the sun, the solar corona, is approximately 1,000,000K hotter than the surface of the Sun, a property which cannot be explained by the
normal processes of heat conduction and radiation. It is now commonly believed
that the magnetic fields which fill the solar atmosphere, and propagate down
into the interior of the Sun, are important for transferring and transforming
energy from the strong plasma flows inside the Sun into the corona as heat. I have
investigated an elementary flux interaction which forms a fundamental building
block of the coronal heating process. This interaction involves two opposite
polarity sources on the Sun's surface in the presence of an overlying magnetic
field. To fully understand how this interaction transfers heat into the solar
corona, the magnetic skeleton is required, which shows possible sites of
heating that are due to magnetic reconnection.
A magnetic field is best described by its magnetic skeleton. The most
important parts of the magnetic skeleton to find are the null points, from
which separatrix surfaces extend that divide magnetic flux of different
topology. Part of this thesis proposes a new method of finding null
points, for which the accuracy is shown and then compared with another commonly
used method (which gave false results).
Using these techniques for finding the magnetic skeleton in the magnetic
interaction above, the evolution of the skeleton was found to head through
seven distinct states, some of which were far more complicated than expected.
This included a high number of separators (the intersection of two separatrix
surfaces), which are a known location of magnetic reconnection. This
separator reconnection was shown to be the main heating mechanism in
this interaction, from which the total amount and rates of reconnection in the
experiment was calculated. This led to the discovery of recursive reconnection, a process where magnetic flux is reconnected before reconnecting
back to its original state, to allow for the process to repeat again. This
recursive reconnection was shown to allow far more reconnection than would have
been previously expected, all of which releases heat into the neighbouring
areas of the atmosphere.
Finally, the interaction was modelled with sources of different magnetic radii
but of equal flux. This showed that when the antisymmetric nature of the
previous interactions was removed, there was little change in the reconnection
rates, but when the strength of the overlying magnetic field was increased, the
reconnection rates were found to increase. This increase in the overlying
magnetic field strength also produced a new magnetic feature called a
bald-edge, which was found to replace some of the null points. These
bald-edges were found to be associated with surfaces similar to separatrix
surfaces that divide flux of different topology but do not extend from a null
point. Also features similar to separators extend from these bald-edges.
Electronic version does not contain additional mpeg files
Mon, 23 Jun 2008 00:00:00 GMThttp://hdl.handle.net/10023/4752008-06-23T00:00:00ZHaynes, Andrew L.The upper atmosphere of the sun, the solar corona, is approximately 1,000,000K hotter than the surface of the Sun, a property which cannot be explained by the
normal processes of heat conduction and radiation. It is now commonly believed
that the magnetic fields which fill the solar atmosphere, and propagate down
into the interior of the Sun, are important for transferring and transforming
energy from the strong plasma flows inside the Sun into the corona as heat. I have
investigated an elementary flux interaction which forms a fundamental building
block of the coronal heating process. This interaction involves two opposite
polarity sources on the Sun's surface in the presence of an overlying magnetic
field. To fully understand how this interaction transfers heat into the solar
corona, the magnetic skeleton is required, which shows possible sites of
heating that are due to magnetic reconnection.
A magnetic field is best described by its magnetic skeleton. The most
important parts of the magnetic skeleton to find are the null points, from
which separatrix surfaces extend that divide magnetic flux of different
topology. Part of this thesis proposes a new method of finding null
points, for which the accuracy is shown and then compared with another commonly
used method (which gave false results).
Using these techniques for finding the magnetic skeleton in the magnetic
interaction above, the evolution of the skeleton was found to head through
seven distinct states, some of which were far more complicated than expected.
This included a high number of separators (the intersection of two separatrix
surfaces), which are a known location of magnetic reconnection. This
separator reconnection was shown to be the main heating mechanism in
this interaction, from which the total amount and rates of reconnection in the
experiment was calculated. This led to the discovery of recursive reconnection, a process where magnetic flux is reconnected before reconnecting
back to its original state, to allow for the process to repeat again. This
recursive reconnection was shown to allow far more reconnection than would have
been previously expected, all of which releases heat into the neighbouring
areas of the atmosphere.
Finally, the interaction was modelled with sources of different magnetic radii
but of equal flux. This showed that when the antisymmetric nature of the
previous interactions was removed, there was little change in the reconnection
rates, but when the strength of the overlying magnetic field was increased, the
reconnection rates were found to increase. This increase in the overlying
magnetic field strength also produced a new magnetic feature called a
bald-edge, which was found to replace some of the null points. These
bald-edges were found to be associated with surfaces similar to separatrix
surfaces that divide flux of different topology but do not extend from a null
point. Also features similar to separators extend from these bald-edges.Solar flux emergence : a three-dimensional numerical study
http://hdl.handle.net/10023/441
Flux is continually emerging on the Sun, making its way from the solar interior up into the atmosphere. Emergence occurs on small-scales in the quiet Sun where magnetic fragments emerge, interact and cancel and on large-scales in active regions where magnetic fields emerge and concentrate to form sunspots. This thesis has been concerned with the large-scale emergence process and in particular the results from previous solar flux emergence modelling endeavours.
Modelling uses numerical methods to evolve a domain representing simplified layers of the Sun’s atmosphere, within which the subsurface layer contains magnetic flux. The flux is initialised such that it will rises towards the surface at the start of the simulation. Once the flux reaches the solar surface, it can only emerge into the atmosphere if a magnetic buoyancy instability occurs, after
which it expands rapidly both vertically and horizontally.
The aim of this thesis is to test the robustness of these general findings from simulations to date upon the seed magnetic field. More explicitly, we have used three-dimensional numerical simulations to investigate how variations in the subsurface magnetic field modify the emergence process
and the resulting atmospheric field. We initially consider a simple constant twist flux tube for the seed field and vary the tube’s magnetic field strength and degree of twist. Additionally, we have examined the effects of using non-constant twist flux tubes as the seed field by choosing two different profiles for the twist that are functions of the tube’s radius. Finally, we have investigated the effects of increasing the complexity of the seed field by positioning two flux tubes below the solar
surface and testing two different configurations for the tubes. In both cases, the magnetic fields of the two tubes are such that, once the tubes come into contact with each other, reconnection occurs and a combined flux system is formed.
From our investigations, we conclude that the general emergence results given by previous simulations are robust. However, for constant twist tubes with low field strength and twist, the buoyancy instability fails to be launched when the tubes reach the photosphere and they remain trapped in the low atmosphere. Similarly, when the non-constant twist profile results in a low tension force
throughout the tube, we find that the buoyancy instability is not initialised.
Sun, 01 Jun 2008 00:00:00 GMThttp://hdl.handle.net/10023/4412008-06-01T00:00:00ZMurray, Michelle J.Flux is continually emerging on the Sun, making its way from the solar interior up into the atmosphere. Emergence occurs on small-scales in the quiet Sun where magnetic fragments emerge, interact and cancel and on large-scales in active regions where magnetic fields emerge and concentrate to form sunspots. This thesis has been concerned with the large-scale emergence process and in particular the results from previous solar flux emergence modelling endeavours.
Modelling uses numerical methods to evolve a domain representing simplified layers of the Sun’s atmosphere, within which the subsurface layer contains magnetic flux. The flux is initialised such that it will rises towards the surface at the start of the simulation. Once the flux reaches the solar surface, it can only emerge into the atmosphere if a magnetic buoyancy instability occurs, after
which it expands rapidly both vertically and horizontally.
The aim of this thesis is to test the robustness of these general findings from simulations to date upon the seed magnetic field. More explicitly, we have used three-dimensional numerical simulations to investigate how variations in the subsurface magnetic field modify the emergence process
and the resulting atmospheric field. We initially consider a simple constant twist flux tube for the seed field and vary the tube’s magnetic field strength and degree of twist. Additionally, we have examined the effects of using non-constant twist flux tubes as the seed field by choosing two different profiles for the twist that are functions of the tube’s radius. Finally, we have investigated the effects of increasing the complexity of the seed field by positioning two flux tubes below the solar
surface and testing two different configurations for the tubes. In both cases, the magnetic fields of the two tubes are such that, once the tubes come into contact with each other, reconnection occurs and a combined flux system is formed.
From our investigations, we conclude that the general emergence results given by previous simulations are robust. However, for constant twist tubes with low field strength and twist, the buoyancy instability fails to be launched when the tubes reach the photosphere and they remain trapped in the low atmosphere. Similarly, when the non-constant twist profile results in a low tension force
throughout the tube, we find that the buoyancy instability is not initialised.The origin and dynamic interaction of solar magnetic fields
http://hdl.handle.net/10023/417
The dynamics of the solar corona are dominated by the magnetic field which creates its structure. The
magnetic field in most of the corona is ‘frozen’ to the plasma very effectively. The exception is in small
localised regions of intense current concentrations where the magnetic field can slip through the plasma
and a restructuring of the magnetic field can occur. This process is known as magnetic reconnection and is
believed to be responsible for a wide variety of phenomena in the corona, from the rapid energy release of solar flares to the heating of the high-temperature corona.
The coronal field itself is three-dimensional (3D), but much of our understanding of reconnection has
been developed through two-dimensional (2D) models. This thesis describes several models for fully 3D
reconnection, with both kinematic and fully dynamic models presented. The reconnective behaviour is
shown to be fundamentally different in many respects from the 2D case. In addition a numerical experiment
is described which examines the reconnection process in coronal magnetic flux tubes whose photospheric
footpoints are spun, one type of motion observed to occur on the Sun.
The large-scale coronal field itself is thought to be generated by a magnetohydrodynamic dynamo operating
in the solar interior. Although the dynamo effect itself is not usually associated with reconnection,
since the essential element of the problem is to account for the presence of large-scale fields, reconnection
is essential for the restructuring of the amplified small-scale flux. Here we examine some simple models of
the solar-dynamo process, taking advantage of their simplicity to make a full exploration of their behaviour
in a variety of parameter regimes. A wide variety of dynamic behaviour is found in each of the models,
including aperiodic modulation of cyclic solutions and intermittency that strongly resembles the historic
record of solar magnetic activity.
Sun, 01 Jun 2008 00:00:00 GMThttp://hdl.handle.net/10023/4172008-06-01T00:00:00ZWilmot-Smith, A.L.The dynamics of the solar corona are dominated by the magnetic field which creates its structure. The
magnetic field in most of the corona is ‘frozen’ to the plasma very effectively. The exception is in small
localised regions of intense current concentrations where the magnetic field can slip through the plasma
and a restructuring of the magnetic field can occur. This process is known as magnetic reconnection and is
believed to be responsible for a wide variety of phenomena in the corona, from the rapid energy release of solar flares to the heating of the high-temperature corona.
The coronal field itself is three-dimensional (3D), but much of our understanding of reconnection has
been developed through two-dimensional (2D) models. This thesis describes several models for fully 3D
reconnection, with both kinematic and fully dynamic models presented. The reconnective behaviour is
shown to be fundamentally different in many respects from the 2D case. In addition a numerical experiment
is described which examines the reconnection process in coronal magnetic flux tubes whose photospheric
footpoints are spun, one type of motion observed to occur on the Sun.
The large-scale coronal field itself is thought to be generated by a magnetohydrodynamic dynamo operating
in the solar interior. Although the dynamo effect itself is not usually associated with reconnection,
since the essential element of the problem is to account for the presence of large-scale fields, reconnection
is essential for the restructuring of the amplified small-scale flux. Here we examine some simple models of
the solar-dynamo process, taking advantage of their simplicity to make a full exploration of their behaviour
in a variety of parameter regimes. A wide variety of dynamic behaviour is found in each of the models,
including aperiodic modulation of cyclic solutions and intermittency that strongly resembles the historic
record of solar magnetic activity.Strong interaction between two co-rotating vortices in rotating and stratified flows.
http://hdl.handle.net/10023/341
In this study we investigate the interactions between two co-rotating vortices.
These vortices are subject to rapid rotation and stable stratification such as
are found in planetary atmospheres and oceans. By conducting a large number
of simulations of vortex interactions, we intend to provide an overview of the
interactions that could occur in geophysical turbulence.
We consider a wide parameter space covering the vortices height-to-width
aspect-ratios, their volume ratios and the vertical offset between them. The vortices
are initially separated in the horizontal so that they reside at an estimated
margin of stability. The vortices are then allowed to evolve for a period of approximately
20 vortex revolutions.
We find that the most commonly observed interaction under the quasi-geostrophic
(QG) regime is partial-merger, where only part of the smaller vortex is incorporated
into the larger, stronger vortex. On the other hand, a large number of filamentary
and small scale structures are generated during the interaction. We find
that, despite the proliferation of small-scale structures, the self-induced vortex energy
exhibits a mean `inverse-cascade' to larger scale structures. Interestingly we
observe a range of intermediate-scale structures that are preferentially sheared
out during the interactions, leaving two vortex populations, one of large-scale
vortices and one of small-scale vortices.
We take a subset of the parameter space used for the QG study and perform
simulations using a non-hydrostatic model. This system, free of the layer-wise
two-dimensional constraints and geostrophic balance of the QG model, allows for
the generation of inertia-gravity waves and ageostrophic advection. The study of
the interactions between two co-rotating, non-hydrostatic vortices is performed
over four different Rossby numbers, two positive and two negative, allowing for
the comparison of cyclonic and anti-cyclonic interactions. It is found that a
greater amount of wave-like activity is generated during the interactions in anticyclonic
situations. We also see distinct qualitative differences between the interactions
for cyclonic and anti-cyclonic regimes.
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10023/3412007-01-01T00:00:00ZBambrey, Ross R.In this study we investigate the interactions between two co-rotating vortices.
These vortices are subject to rapid rotation and stable stratification such as
are found in planetary atmospheres and oceans. By conducting a large number
of simulations of vortex interactions, we intend to provide an overview of the
interactions that could occur in geophysical turbulence.
We consider a wide parameter space covering the vortices height-to-width
aspect-ratios, their volume ratios and the vertical offset between them. The vortices
are initially separated in the horizontal so that they reside at an estimated
margin of stability. The vortices are then allowed to evolve for a period of approximately
20 vortex revolutions.
We find that the most commonly observed interaction under the quasi-geostrophic
(QG) regime is partial-merger, where only part of the smaller vortex is incorporated
into the larger, stronger vortex. On the other hand, a large number of filamentary
and small scale structures are generated during the interaction. We find
that, despite the proliferation of small-scale structures, the self-induced vortex energy
exhibits a mean `inverse-cascade' to larger scale structures. Interestingly we
observe a range of intermediate-scale structures that are preferentially sheared
out during the interactions, leaving two vortex populations, one of large-scale
vortices and one of small-scale vortices.
We take a subset of the parameter space used for the QG study and perform
simulations using a non-hydrostatic model. This system, free of the layer-wise
two-dimensional constraints and geostrophic balance of the QG model, allows for
the generation of inertia-gravity waves and ageostrophic advection. The study of
the interactions between two co-rotating, non-hydrostatic vortices is performed
over four different Rossby numbers, two positive and two negative, allowing for
the comparison of cyclonic and anti-cyclonic interactions. It is found that a
greater amount of wave-like activity is generated during the interactions in anticyclonic
situations. We also see distinct qualitative differences between the interactions
for cyclonic and anti-cyclonic regimes.Effect of structuring on coronal loop oscillations
http://hdl.handle.net/10023/265
In this Thesis the theoretical understanding of oscillations in coronal structures is developed. In particular, coronal loops are modelled as magnetic slabs of plasma. The effect of introducing inhomogeneities on the frequency of oscillation is studied. Current observations indicate the existence of magnetohydrodynamic (MHD) modes in the corona, so there is room for improved modelling of these modes to understand the physical processes more completely. One application of the oscillations, on which this Thesis concentrates, is coronal seismology. Here, the improved theoretical models are applied to observed instances of coronal MHD waves with the aim of determining information regarding the medium in which these waves propagate.
In Chapter two, the effect of gravity on the frequency of the longitudinal slow MHD mode is considered. A thin, vertical coronal slab of magnetised plasma, with gravity acting along the longitudinal axis of the slab is studied, and the effect on the frequency of oscillation for the uniform, stratified and structured cases is addressed. In particular, an isothermal plasma, a two-layer plasma and a plasma with a linear temperature profile are studied. Here, a thin coronal loop, with its footpoints embedded in the chromosphere-photosphere is modelled, and the effects introduced by both gravity and the structuring of density at the footpoint layers are studied. In this case, gravity increases the frequency of oscillation and causes amplification of the eigenfunctions by stratification. Furthermore, density enhancements at the footpoints cause a decrease in the oscillating frequency, and can inhibit wave propagation, depending on the parameter regime.
In Chapter three, the effects introduced to the transverse fast MHD mode when gravity acts across a thin coronal slab of magnetised plasma are considered. This study concentrates on the modification of the frequency due to the dynamical effect of gravity in the equation of motion, neglecting the effect of stratification. Here, gravity causes a reduction of the oscillating frequency of the fundamental fast mode, and increases the lower cutoff frequency. In effect, for this configuration, gravity allows the transition between body and surface modes, in a slab geometry.
It is found, in these two studies, that each harmonic is affected in a unique manner due to structuring or stratification of density. With this knowledge, in Chapter four, a new parameter is derived; P1/2P2, the ratio of the period of the fundamental harmonic of oscillation to twice the period of its first harmonic. This parameter is shown to be a measure of the longitudinal structuring of density along a coronal loop, and the departure of this ratio from unity can yield information regarding the lengthscales of the structure. This process is highlighted using the known observations, indicating that P1/2P2 may prove to be a useful diagnostic tool for coronal seismology.
Finally, in Chapter five, outwardly propagating coronal slow MHD modes are observed and are used to infer coronal parameters. The possibility of using these oscillations to infer near-resolution lengthscales in coronal loops -- fine-scale strands -- is also discussed. TRACE observations are used to determine the average period, phase speed, detection length, amplitude and energy flux for the propagating slow MHD mode. The indication is that the source of these oscillations appears very localised in space, and the driver only acts for a few periods, suggesting the perturbations are driven by leaky p-modes (solar surface modes).
Fri, 01 Jun 2007 00:00:00 GMThttp://hdl.handle.net/10023/2652007-06-01T00:00:00ZMcEwan, Michael P.In this Thesis the theoretical understanding of oscillations in coronal structures is developed. In particular, coronal loops are modelled as magnetic slabs of plasma. The effect of introducing inhomogeneities on the frequency of oscillation is studied. Current observations indicate the existence of magnetohydrodynamic (MHD) modes in the corona, so there is room for improved modelling of these modes to understand the physical processes more completely. One application of the oscillations, on which this Thesis concentrates, is coronal seismology. Here, the improved theoretical models are applied to observed instances of coronal MHD waves with the aim of determining information regarding the medium in which these waves propagate.
In Chapter two, the effect of gravity on the frequency of the longitudinal slow MHD mode is considered. A thin, vertical coronal slab of magnetised plasma, with gravity acting along the longitudinal axis of the slab is studied, and the effect on the frequency of oscillation for the uniform, stratified and structured cases is addressed. In particular, an isothermal plasma, a two-layer plasma and a plasma with a linear temperature profile are studied. Here, a thin coronal loop, with its footpoints embedded in the chromosphere-photosphere is modelled, and the effects introduced by both gravity and the structuring of density at the footpoint layers are studied. In this case, gravity increases the frequency of oscillation and causes amplification of the eigenfunctions by stratification. Furthermore, density enhancements at the footpoints cause a decrease in the oscillating frequency, and can inhibit wave propagation, depending on the parameter regime.
In Chapter three, the effects introduced to the transverse fast MHD mode when gravity acts across a thin coronal slab of magnetised plasma are considered. This study concentrates on the modification of the frequency due to the dynamical effect of gravity in the equation of motion, neglecting the effect of stratification. Here, gravity causes a reduction of the oscillating frequency of the fundamental fast mode, and increases the lower cutoff frequency. In effect, for this configuration, gravity allows the transition between body and surface modes, in a slab geometry.
It is found, in these two studies, that each harmonic is affected in a unique manner due to structuring or stratification of density. With this knowledge, in Chapter four, a new parameter is derived; P1/2P2, the ratio of the period of the fundamental harmonic of oscillation to twice the period of its first harmonic. This parameter is shown to be a measure of the longitudinal structuring of density along a coronal loop, and the departure of this ratio from unity can yield information regarding the lengthscales of the structure. This process is highlighted using the known observations, indicating that P1/2P2 may prove to be a useful diagnostic tool for coronal seismology.
Finally, in Chapter five, outwardly propagating coronal slow MHD modes are observed and are used to infer coronal parameters. The possibility of using these oscillations to infer near-resolution lengthscales in coronal loops -- fine-scale strands -- is also discussed. TRACE observations are used to determine the average period, phase speed, detection length, amplitude and energy flux for the propagating slow MHD mode. The indication is that the source of these oscillations appears very localised in space, and the driver only acts for a few periods, suggesting the perturbations are driven by leaky p-modes (solar surface modes).Electron cyclotron heating and current drive using the electron Bernstein modes
http://hdl.handle.net/10023/212
Electron Bernstein waves are a mode of oscillation in a plasma, thought a candidate for providing radiofrequency heating and non-inductive current drive in spherical tokamaks. Previous studies of these modes have relied on neglecting or simplifying the contribution made by relativistic effects.
This work presents fully relativistic numerical results that show the mode's dispersion relation for a wide range of parameters. Relativistic effects are shown to shift the location of the resonance as in previous studies, but the effects beyond this are shown to matter only in high temperature (10-20keV) plasmas. At these higher temperatures however, the fully relativistic model differs markedly. The assumption that the mode is electrostatic is looked at, and found to be inadequate for describing fully the electron Bernstein modes dispersion relation.
Simple estimates that neglect toroidal effects show current drive efficiency is expected to be an order of magnitude higher than that for conventional electron cyclotron current drive using the O or X modes. It is shown for a number of model tokamaks that heating the center of the plasma and driving current using EBWs is impossible launching from the outside due to strong damping of the wave at higher cyclotron harmonics.
Results from a code based on a more complicated semi-analytic model of current drive, that includes toroidal effects and calculates the average current drive over the magnetic surface, confirm the higher expected current drive efficiency, and the code is shown to give good agreement with a Fokker-Planck code. The higher values of perpendicular refractive index associated with the EBWs are shown to mitigate the deleterious effects of trapping on current drive efficiency to a small extent. The details of the magnetic field are found to be unimportant to the calculation beyond determing where the wave is absorbed.
The codes written to produce these results are outlined before each set of results. The last of these is considerably faster than conventional Fokker-Planck codes and a useful tool in studying electron cyclotron current drive in the future.
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10023/2122007-01-01T00:00:00ZMcGregor, Duncan EkundayoElectron Bernstein waves are a mode of oscillation in a plasma, thought a candidate for providing radiofrequency heating and non-inductive current drive in spherical tokamaks. Previous studies of these modes have relied on neglecting or simplifying the contribution made by relativistic effects.
This work presents fully relativistic numerical results that show the mode's dispersion relation for a wide range of parameters. Relativistic effects are shown to shift the location of the resonance as in previous studies, but the effects beyond this are shown to matter only in high temperature (10-20keV) plasmas. At these higher temperatures however, the fully relativistic model differs markedly. The assumption that the mode is electrostatic is looked at, and found to be inadequate for describing fully the electron Bernstein modes dispersion relation.
Simple estimates that neglect toroidal effects show current drive efficiency is expected to be an order of magnitude higher than that for conventional electron cyclotron current drive using the O or X modes. It is shown for a number of model tokamaks that heating the center of the plasma and driving current using EBWs is impossible launching from the outside due to strong damping of the wave at higher cyclotron harmonics.
Results from a code based on a more complicated semi-analytic model of current drive, that includes toroidal effects and calculates the average current drive over the magnetic surface, confirm the higher expected current drive efficiency, and the code is shown to give good agreement with a Fokker-Planck code. The higher values of perpendicular refractive index associated with the EBWs are shown to mitigate the deleterious effects of trapping on current drive efficiency to a small extent. The details of the magnetic field are found to be unimportant to the calculation beyond determing where the wave is absorbed.
The codes written to produce these results are outlined before each set of results. The last of these is considerably faster than conventional Fokker-Planck codes and a useful tool in studying electron cyclotron current drive in the future.Topological structure of the magnetic solar corona
http://hdl.handle.net/10023/151
The solar corona is a highly complex and active plasma environment, containing many exotic
phenomena such as solar flares, coronal mass ejections, prominences, coronal loops, and bright
points. The fundamental element giving coherence to all this apparent diversity is the strong
coronal magnetic field, the dominant force shaping the plasma there.
In this thesis, I model the 3D magnetic fields of various coronal features using the techniques
of magnetic charge topology (MCT) in a potential field. Often the real coronal field has departures
from its potential state, but these are so small that the potential field method is accurate enough to
pick out the essential information about the structure and evolution of the magnetic field.
First I perform a topological analysis of the magnetic breakout model for an eruptive solar
flare. Breakout is represented by a topological bifurcation that allows initially enclosed flux from
the newly emerging region in my MCT model of a delta sunspot to reconnect out to large distances.
I produce bifurcation diagrams showing how this behaviour can be caused by changing
the strength or position of the emerging flux source, or the force-free parameter α.
I also apply MCT techniques to observational data of a coronal bright point, and compare the
results to 3D numerical MHD simulations of the effects of rotating the sources that underlie the
bright point. The separatrix surfaces that surround each rotating source are found to correspond
to locations of high parallel electric field in the simulations, which is a signature of magnetic
reconnection. The large-scale topological structure of the magnetic field is robust to changes in
the method of deriving point magnetic sources from the magnetogram.
Next, I use a Green’s function expression for the magnetic field to relax the standard topological
assumption of a flat photosphere and extend the concept of MCT into a spherical geometry,
enabling it to be applied to the entire global coronal magnetic field. I perform a comprehensive
study of quadrupolar topologies in this new geometry, producing several detailed bifurcation
diagrams. These results are compared to the equivalent study for a flat photosphere. A new topological
state is found on the sphere which has no flat photosphere analogue; it is named the dual
intersecting state because of its twin separators joining a pair of magnetic null points.
The new spherical techniques are then applied to develop a simple six-source topological
model of global magnetic field reversal during the solar cycle. The evolution of the large-scale
global magnetic field is modelled through one complete eleven-year cycle, beginning at solar minimum.
Several distinct topological stages are exhibited: active region flux connecting across the
equator to produce transequatorial loops; the dominance of first the leading and then the following
polarities of the active regions; the magnetic isolation of the poles; the reversal of the polar field;
the new polar field connecting back to the active regions; the polar flux regaining its dominance;
and the disappearance of the transequatorial loops.
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10023/1512007-01-01T00:00:00ZMaclean, Rhona ClaireThe solar corona is a highly complex and active plasma environment, containing many exotic
phenomena such as solar flares, coronal mass ejections, prominences, coronal loops, and bright
points. The fundamental element giving coherence to all this apparent diversity is the strong
coronal magnetic field, the dominant force shaping the plasma there.
In this thesis, I model the 3D magnetic fields of various coronal features using the techniques
of magnetic charge topology (MCT) in a potential field. Often the real coronal field has departures
from its potential state, but these are so small that the potential field method is accurate enough to
pick out the essential information about the structure and evolution of the magnetic field.
First I perform a topological analysis of the magnetic breakout model for an eruptive solar
flare. Breakout is represented by a topological bifurcation that allows initially enclosed flux from
the newly emerging region in my MCT model of a delta sunspot to reconnect out to large distances.
I produce bifurcation diagrams showing how this behaviour can be caused by changing
the strength or position of the emerging flux source, or the force-free parameter α.
I also apply MCT techniques to observational data of a coronal bright point, and compare the
results to 3D numerical MHD simulations of the effects of rotating the sources that underlie the
bright point. The separatrix surfaces that surround each rotating source are found to correspond
to locations of high parallel electric field in the simulations, which is a signature of magnetic
reconnection. The large-scale topological structure of the magnetic field is robust to changes in
the method of deriving point magnetic sources from the magnetogram.
Next, I use a Green’s function expression for the magnetic field to relax the standard topological
assumption of a flat photosphere and extend the concept of MCT into a spherical geometry,
enabling it to be applied to the entire global coronal magnetic field. I perform a comprehensive
study of quadrupolar topologies in this new geometry, producing several detailed bifurcation
diagrams. These results are compared to the equivalent study for a flat photosphere. A new topological
state is found on the sphere which has no flat photosphere analogue; it is named the dual
intersecting state because of its twin separators joining a pair of magnetic null points.
The new spherical techniques are then applied to develop a simple six-source topological
model of global magnetic field reversal during the solar cycle. The evolution of the large-scale
global magnetic field is modelled through one complete eleven-year cycle, beginning at solar minimum.
Several distinct topological stages are exhibited: active region flux connecting across the
equator to produce transequatorial loops; the dominance of first the leading and then the following
polarities of the active regions; the magnetic isolation of the poles; the reversal of the polar field;
the new polar field connecting back to the active regions; the polar flux regaining its dominance;
and the disappearance of the transequatorial loops.