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On global regularity of 2D generalized magnetohydrodynamic equations

Tran, Chuong Van — Wed, 15 May 2013 00:00:00 GMT

Abstract: In this article we study the global regularity of 2D generalized magnetohydrodynamic equations (2D GMHD), in which the dissipation terms are $- \nu \left( - \triangle \right)^{\alpha} u$ and $- \kappa \left( - \triangle \right)^{\beta} b$. We show that smooth solutions are global in the following three cases: $\alpha \geqslant 1 / 2, \beta \geqslant 1$; $0 \leqslant \alpha < 1 / 2, 2 \alpha + \beta > 2$; $\alpha \geqslant 2, \beta = 0$. We also show that in the inviscid case $\nu = 0$, if $\beta > 1$, then smooth solutions are global as long as the direction of the magnetic field remains smooth enough.

Sharp global nonlinear stability for a fluid overlying a highly porous material

Hill, Antony A. — Fri, 08 Jan 2010 00:00:00 GMT

Abstract: The stability of convection in a two-layer system in which a layer of fluid with a temperature-dependent viscosity overlies and saturates a highly porous material is studied. Owing to the difficulties associated with incorporating the nonlinear advection term in the Navier-Stokes equations into a stability analysis, previous literature on fluid/porous thermal convection has modelled the fluid using the linear Stokes equations. This paper derives global stability for the full nonlinear system, by utilizing a model proposed by Ladyzhenskaya. The nonlinear stability boundaries are shown to be sharp when compared with the linear instability thresholds.

Nonlinear stability of the one-domain approach to modelling convection in superposed fluid and porous layers

Hill, A A — Wed, 01 Sep 2010 00:00:00 GMT

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

Instability in internal solitary waves with trapped cores

Carr, Magda — Sun, 01 Jan 2012 00:00:00 GMT

Abstract: A numerical method that employs a combination of contour advection and pseudo-spectral techniques is used to investigate instability in internal solitary waves with trapped cores. A three-layer configuration for the background stratification in which the top two layers are linearly stratified and the lower layer is homogeneous is considered throughout. The strength of the stratification in the very top layer is chosen to be sufficient so that waves of depression with trapped cores can be generated. The flow is assumed to satisfy the Dubriel-Jacotin-Long equation both inside and outside of the core region. The Brunt-Vaisala frequency is modelled such that it varies from a constant value outside of the core to zero inside the core over a sharp but continuous transition length. This results in a stagnant core in which the vorticity is zero and the density is homogeneous and approximately equal to that at the core boundary. The time dependent simulations show that instability occurs on the boundary of the core. The instability takes the form of Kelvin-Helmholtz billows. If the instability in the vorticity field is energetic enough, disturbance in the buoyancy field is also seen and fluid exchange takes place across the core boundary. Occurrence of the Kelvin-Helmholtz billows is attributed to the sharp change in the vorticity field at the boundary between the core and the pycnocline. The numerical scheme is not limited by small Richardson number unlike the other alternatives currently available in the literature which appear to be.

Shear induced breaking of large amplitude internal solitary waves

Fructus, D — Sun, 01 Feb 2009 00:00:00 GMT

Convectively induced shear instability in large amplitude internal solitary waves

Carr, Magda — Mon, 01 Dec 2008 00:00:00 GMT

Abstract: Laboratory study has been carried out to investigate the instability of an internal solitary wave of depression in a shallow stratified fluid system. The experimental campaign has been supported by theoretical computations and has focused on a two layered stratification consisting of a homogeneous dense layer below a linearly stratified top layer. The initial background stratification has been varied and it is found that the onset and intensity of breaking are affected dramatically by changes in the background stratification. Manifestations of a combination of shear and convective instability are seen on the leading face of the wave. It is shown that there is an interplay between the two instability types and convective instability induces shear by enhancing isopycnal compression. Variation in the upper boundary condition is also found to have an effect on stability. In particular, the implications for convective instability are shown to be profound and a dramatic increase in wave amplitude is seen for a fixed (as opposed to free) upper boundary condition.

Number of degrees of freedom and energy spectrum of surface quasi-geostrophic turbulence

Tran, Chuong V. — Sat, 01 Oct 2011 00:00:00 GMT

Abstract: We study both theoretically and numerically surface quasi-geostrophic turbulence regularized by the usual molecular viscosity, with an emphasis on a number of classical predictions. It is found that the system's number of degrees of freedom N, which is defined in terms of local Lyapunov exponents, scales as Re-3/2, where R e is the Reynolds number expressible in terms of the viscosity, energy dissipation rate and system's integral scale. For general power-law energy spectra k(-alpha), a comparison of N with the number of dynamically active Fourier modes, i.e. the modes within the energy inertial range, yields alpha = 5/3. This comparison further renders the scaling Re-1/2 for the exponential dissipation rate at the dissipation wavenumber. These results have been predicted on the basis of Kolmogorov's theory. Our approach thus recovers these classical predictions and is an analytic alternative to the traditional phenomenological method. The implications of the present findings are discussed in conjunction with related results in the literature. Support for the analytic results is provided through a series of direct numerical simulations. Description: L.A.K.B. was supported by an EPSRC post-graduate studentship.

Coronal heating by the partial relaxation of twisted loops

Bareford, Michael — Fri, 01 Feb 2013 00:00:00 GMT

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

Damping of kink waves by mode coupling : I. Analytical treatment

Hood, Alan William — Tue, 01 Jan 2013 00:00:00 GMT

Abstract: Aims. To investigate the spatial damping of propagating kink waves in an inhomogeneous plasma. In the limit of a thin tube surrounded by a thin transition layer, an analytical formulation for kink waves driven in from the bottom boundary of the corona is presented. Methods. The spatial form for the damping of the kink mode was investigated using various analytical approximations. When the density ratio between the internal density and the external density is not too large, a simple di.erential-integral equation was used. Approximate analytical solutions to this equation are presented. Results. For the first time, the form of the spatial damping of the kink mode is shown analytically to be Gaussian in nature near the driven boundary. For several wavelengths, the amplitude of the kink mode is proportional to (1 + exp(-z2 /L2 g))/2, where L2g = 16/ǫκ2 k2 . Although the actual value of 16 in Lg depends on the particular form of the driver, this form is very general and its dependence on the other parameters does not change. For large distances, the damping profile appears to be roughly linear exponential decay. This is shown analytically by a series expansion when the inhomogeneous layer width is small enough.

The influence of a fluid-porous interface on solar pond stability

Hill, A. A — Fri, 01 Feb 2013 00:00:00 GMT

Abstract: The linear instability of the gradient zone of a solar pond containing a fluidporous interface is investigated. It is found that the gradient zone can retain the same stability for lower values of the solute Rayleigh number with the introduction of a porous material compared with a purely fluid layer, whilst maintaining the same lower convective zone temperature. Interestingly, it is also shown that for certain parameter values the penetration of a porous medium into the gradient zone can cause the temperature of the lower convective zone to rise. However, for certain parameter ranges, when the fluid-porous interface is towards the top of the gradient zone, the solar pond can become highly unstable.

A Bayesian approach to fitting Gibbs processes with temporal random effects

King, Ruth — Sat, 01 Dec 2012 00:00:00 GMT

Abstract: We consider spatial point pattern data that have been observed repeatedly over a period of time in an inhomogeneous environment. Each spatial point pattern can be regarded as a “snapshot” of the underlying point process at a series of times. Thus, the number of points and corresponding locations of points differ for each snapshot. Each snapshot can be analyzed independently, but in many cases there may be little information in the data relating to model parameters, particularly parameters relating to the interaction between points. Thus, we develop an integrated approach, simultaneously analyzing all snapshots within a single robust and consistent analysis. We assume that sufficient time has passed between observation dates so that the spatial point patterns can be regarded as independent replicates, given spatial covariates. We develop a joint mixed effects Gibbs point process model for the replicates of spatial point patterns by considering environmental covariates in the analysis as fixed effects, to model the heterogeneous environment, with a random effects (or hierarchical) component to account for the different observation days for the intensity function. We demonstrate how the model can be fitted within a Bayesian framework using an auxiliary variable approach to deal with the issue of the random effects component. We apply the methods to a data set of musk oxen herds and demonstrate the increased precision of the parameter estimates when considering all available data within a single integrated analysis.

Finite and infinite ergodic theory for linear and conformal dynamical systems

Munday, Sara Ann — Wed, 30 Nov 2011 00:00:00 GMT

Abstract: 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.

The propagation and damping of slow magnetoacoustic waves in the solar atmosphere

Owen, Nicholas Robert — Fri, 30 Nov 2012 00:00:00 GMT

Abstract: 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.

Wave propagation, phase mixing and dissipation in Hall MHD

Threlfall, James W. — Fri, 22 Jun 2012 00:00:00 GMT

Abstract: 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.

Collisionless distribution function for the relativistic force-free Harris sheet

Stark, C. R. — Sun, 01 Jan 2012 00:00:00 GMT

Abstract: A self-consistent collisionless distribution function for the relativistic analogue of the force-free Harris sheet is presented. This distribution function is the relativistic generalization of the distribution function for the non-relativistic collisionless force-free Harris sheet recently found by Harrison and Neukirch [Phys. Rev. Lett. 102, 135003 (2009)], as it has the same dependence on the particle energy and canonical momenta. We present a detailed calculation which shows that the proposed distribution function generates the required current density profile (and thus magnetic field profile) in a frame of reference in which the electric potential vanishes identically. The connection between the parameters of the distribution function and the macroscopic parameters such as the current sheet thickness is discussed. (C) 2012 American Institute of Physics. [doi: 10.1063/1.3677268]

A non-linear force-free field model for the solar magnetic carpet

Meyer, Karen A. — Fri, 22 Jun 2012 00:00:00 GMT

Abstract: 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.

A highly adaptive three dimensional hybrid vortex method for inviscid flows and helically symmetric vortex equilibria

Lucas, Daniel — Fri, 22 Jun 2012 00:00:00 GMT

Abstract: 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.

Numerical simulation of shear-induced instabilities in internal solitary waves

Carr, Magda — Sun, 25 Sep 2011 00:00:00 GMT

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

Continued fractions which correspond to two series expansions and the strong Hamburger moment problem

Sri Ranga, A. — Sun, 01 Jan 1984 00:00:00 GMT

Abstract: 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

Grady, Keith J. — Fri, 22 Jun 2012 00:00:00 GMT

Abstract: 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

Ahmad, Muhammad Idrees — Sun, 01 Jan 1989 00:00:00 GMT

Abstract: 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.

Coupling of the solar wind, magnetosphere and ionosphere by MHD waves

Russell, Alexander J. B. — Tue, 30 Nov 2010 00:00:00 GMT

Abstract: 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

Smy, Louise Ann — Fri, 22 Jun 2012 00:00:00 GMT

Abstract: 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.

Behind and beyond a theorem on groups related to trivalent graphs

Havas, George — Mon, 01 Dec 2008 00:00:00 GMT

Abstract: In 2006 we completed the proof of a five-part conjecture that was made in 1977 about a family of groups related to trivalent graphs. This family covers all 2-generator, 2-relator groups where one relator specifies that a generator is an involution and the other relator has three syllables. Our proof relies upon detailed but general computations in the groups under question. The proof is theoretical, but based upon explicit proofs produced by machine for individual cases. Here we explain how we derived the general proofs from specific cases. The conjecture essentially addressed only the finite groups in the family. Here we extend the results to infinite groups, effectively determining when members of this family of finitely presented groups are simply isomorphic to a specific quotient.

Lower-hybrid waves generated by anomalous Doppler resonance in auroral plasmas

Bingham, R. — Sun, 01 Aug 2010 00:00:00 GMT

Abstract: This paper describes sonic aspects of lower-hybrid wave activity in space plasmas. Lower-hybrid waves are particularly important since they can transfer energy efficiently between electrons and ions in a collisionless magnetized plasma. We consider the 'fan' or anomalous Doppler resonance instability driven by energetic electron tails and show that it is responsible for the generation of lower-hybrid waves. We also demonstrate that observations of their intensity are sufficient to drive the modulational instability.

Falling towards forgetfulness : synaptic decay prevents spontaneous recovery of memory

Stone, James V. — Fri, 22 Aug 2008 00:00:00 GMT

Abstract: Long after a new language has been learned and forgotten, relearning a few words seems to trigger the recall of other words. This "free-lunch learning'' (FLL) effect has been demonstrated both in humans and in neural network models. Specifically, previous work proved that linear networks that learn a set of associations, then partially forget them all, and finally relearn some of the associations, show improved performance on the remaining (i.e., nonrelearned) associations. Here, we prove that relearning forgotten associations decreases performance on nonrelearned associations; an effect we call negative free-lunch learning. The difference between free-lunch learning and the negative free-lunch learning presented here is due to the particular method used to induce forgetting. Specifically, if forgetting is induced by isotropic drifting of weight vectors (i.e., by adding isotropic noise), then free-lunch learning is observed. However, as proved here, if forgetting is induced by weight values that simply decay or fall towards zero, then negative free-lunch learning is observed. From a biological perspective, and assuming that nervous systems are analogous to the networks used here, this suggests that evolution may have selected physiological mechanisms that involve forgetting using a form of synaptic drift rather than synaptic decay, because synaptic drift, but not synaptic decay, yields free-lunch learning.

On the relationship between equilibrium bifurcations and ideal MHD instabilities for line-tied coronal loops

Neukirch, T. — Fri, 01 Jan 2010 00:00:00 GMT

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

Mathematics for history's sake : a new approach to Ptolemy's Geography

Mintz, Daniel V. — Wed, 22 Jun 2011 00:00:00 GMT

Abstract: Almost two thousand years ago, Claudius Ptolemy created a guide to drawing maps of the world, identifying the names and coordinates of over 8,000 settlements and geographical features. Using the coordinates of those cities and landmarks which have been identified with modern locations, a series of best-fit transformations has been applied to several of Ptolemy’s regional maps, those of Britain, Spain, and Italy. The transformations relate Ptolemy’s coordinates to their modern equivalents by rotation and skewed scaling. These reflect the types of error that appear in Ptolemy’s data, namely those of distance and orientation. The mathematical techniques involved in this process are all modern. However, these techniques have been altered in order to deal with the historical difficulties of Ptolemy’s maps. To think of Ptolemy’s data as similar to that collected from a modern random sampling of a population and to apply unbiased statistical methods to it would be erroneous. Ptolemy’s data is biased, and the nature of that bias is going to be informed by the history of the data. Using such methods as cluster analysis, Procrustes analysis, and multidimensional scaling, we aimed to assess numerically the accuracy of Ptolemy’s maps. We also investigated the nature of the errors in the data and whether or not these could be linked to historical developments in the areas mapped.

Automatic presentations and semigroup constructions

Cain, Alan J. — Sun, 01 Aug 2010 00:00:00 GMT

Abstract: An automatic presentation for a relational structure is, informally, an abstract representation of the elements of that structure by means of a regular language such that the relations can all be recognized by finite automata. A structure admitting an automatic presentation is said to be FA-presentable. This paper studies the interaction of automatic presentations and certain semigroup constructions, namely: direct products, free products, finite Rees index extensions and subsemigroups, strong semilattices of semigroups, Rees matrix semigroups, Bruck-Reilly extensions, zero-direct unions, semidirect products, wreath products, ideals, and quotient semigroups. For each case, the closure of the class of FA-presentable semigroups under that construction is considered, as is the question of whether the FA-presentability of the semigroup obtained from such a construction implies the FA-presentability of the original semigroup[s]. Classifications are also given of the FA-presentable finitely generated Clifford semigroups, completely simple semigroups, and completely 0-simple semigroups.

Automatic presentations for semigroups

Cain, Alan J. — Sun, 01 Nov 2009 00:00:00 GMT

Abstract: This paper applies the concept of FA-presentable structures to semigroups. We give a complete classification of the finitely generated FA-presentable cancellative semigroups: namely, a finitely generated cancellative semigroup is FA-presentable if and only if it is a subsemigroup of a virtually abelian group. We prove that all finitely generated commutative semigroups are FA-presentable. We give a complete list of FA-presentable one-relation semigroups and compare the classes of FA-presentable semigroups and automatic semigroups. (C) 2009 Elsevier Inc. All rights reserved. Description: Special Issue: 2nd International Conference on Language and Automata Theory and Applications (LATA 2008)

Cancellative and Malcev presentations for finite Rees index subsemigroups and extensions

Cain, Alan James — Fri, 01 Feb 2008 00:00:00 GMT

Abstract: It is known that, for semigroups, the property of admitting a finite presentation is preserved on passing to subsemigroups and extensions of finite Rees index. The present paper shows that the same holds true for Malcev, cancellative, left-cancellative and right-cancellative presentations. (A Malcev (respectively, cancellative, left-cancellative, right-cancellative) presentation is a presentation of a special type that can be used to define any group-embeddable (respectively, cancellative, left-cancellative, right-cancellative) semigroup.).

The investigation of quasi-separatrix layers in solar magnetic fields

Restante, Anna Lisa — Mon, 20 Jun 2011 00:00:00 GMT

Abstract: 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.

The period ratio P₁/2P₂ in coronal waves

Macnamara, Cicely K. — Wed, 30 Nov 2011 00:00:00 GMT

Abstract: 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.

The steady-state form of large-amplitude internal solitary waves

King, Stuart Edward — Mon, 10 Jan 2011 00:00:00 GMT

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

Small-scale magnetic feature evolution as observed by Hinode/NFI and SOHO/MDI

Thornton, L. M. — Wed, 30 Nov 2011 00:00:00 GMT

Abstract: 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

Bowness, Ruth — Wed, 30 Nov 2011 00:00:00 GMT

Abstract: 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.

Magnetic flux transport simulations : applications to solar and stellar magnetic fields

Cook, Graeme Robert — Tue, 01 Nov 2011 00:00:00 GMT

Abstract: 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.

An investigation into the use of balance in operational numerical weather prediction

Devlin, David J.J. — Sat, 01 Jan 2011 00:00:00 GMT

Abstract: 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

Fuentes Fernández, Jorge — Wed, 18 May 2011 00:00:00 GMT

Abstract: 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.

Numerical modelling of two HMX-based plastic-bonded explosives at the mesoscale

Handley, Caroline A. — Wed, 22 Jun 2011 00:00:00 GMT

Abstract: 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

MacTaggart, David — Sat, 01 Jan 2011 00:00:00 GMT

Abstract: 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.

Application of stochastic differential equations and real option theory in investment decision problems

Chavanasporn, Walailuck — Tue, 30 Nov 2010 00:00:00 GMT

Abstract: 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.

Impeded inverse energy transfer in the Charney--Hasegawa--Mima model of quasi-geostrophic flows

Tran, Chuong Van — Sat, 25 Mar 2006 00:00:00 GMT

Abstract: The behaviour of turbulent flows within the single-layer quasi-geostrophic (Charney-Hasegawa-Mima) model is shown to be strongly dependent on the Rossby deformation wavenumber lambda (or free-surface elasticity). Herein, we derive a bound oil the inverse energy transfer, specifically on the growth rate dl/dt of the characteristic length scale e representing the energy centroid. It is found that dl/dt <= 2 parallel to q parallel to(infinity)/(l(s)lambda(2)), where parallel to q parallel to(infinity) is the supremum of the potential vorticity and l(s) represents the potential enstrophy centroid of the reservoir, both invariant. This result implies that in the potential-energy-dominated regime (l >= l(s) >> lambda(-1)) the inverse energy transfer is strongly impeded, in the sense that under the usual time scale no significant transfer of energy to larger scales occurs. The physical implication is that the elasticity of the free surface impedes turbulent energy transfer in wavenumber space, effectively rendering large-scale vortices long-lived and inactive. Results from numerical simulations of forced-dissipative turbulence confirm this prediction.

Vanishing enstrophy dissipation in two-dimensional Navier--Stokes turbulence in the inviscid limit

Tran, Chuong Van — Tue, 25 Jul 2006 00:00:00 GMT

Abstract: Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) developed a theory of two-dimensional turbulence based on the assumption that the dissipation of enstrophy (mean-square vorticity) tends to a finite non-zero constant in the limit of infinite Reynolds number Re. Here, by assuming power-law spectra, including the one predicted by Batchelor's theory, we prove that the maximum dissipation of enstrophy is in fact zero in this limit. Specifically, as Re -> infinity, the dissipation approaches zero no slower than (ln Re)(-1/2). The physical reason behind this result is that the decrease of viscosity enhances the production of both palinstrophy (mean-square vorticity gradients) and its dissipation - but in such a way that the net growth of palinstrophy is less rapid than the decrease of viscosity, resulting in vanishing enstrophy dissipation. This result generalizes to a rich class of quasi-geostrophic models as well as to the case of a passive tracer in layerwise-two-dimensional turbulent flows having bounded enstrophy.

Quasi-geostrophic vortices in compressible atmospheres

Scott, Richard Kirkness — Tue, 10 May 2005 00:00:00 GMT

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

Subsemigroups of virtually free groups : finite Malcev presentations and testing for freeness

Cain, AJ — Sat, 01 Jul 2006 00:00:00 GMT

Abstract: This paper shows that, given a finite subset X of a finitely generated virtually free group F, the freeness of the subsemigroup of F generated by X can be tested algorithmically. (A group is virtually free if it contains a free subgroup of finite index.) It is then shown that every finitely generated subsemigroup, of F has a finite Malcev presentation (a type of semigroup presentation which can be used to define any semigroup that embeds in a group), and that such a presentation can be effectively found from any finite generating set.

The critical merger distance between two co-rotating quasi-geostrophic vortices

Reinaud, Jean Noel — Mon, 10 Jan 2005 00:00:00 GMT

Abstract: This paper examines the critical merger or strong interaction distance between two equal-potential-vorticity quasi-geostrophic vortices. The interaction between the two vortices depends on five parameters: their volume ratio, their height-to-width aspect ratios and their vertical and horizontal offsets. Due to the size of the parameter space, a direct investigation solving the full quasi-geostrophic equations is impossible. We instead determine the critical merger distance approximately using an asymptotic approach. We associate the merger distance with the margin of stability for a family of equilibrium states having prescribed aspect and volume ratios, and vertical offset. The equilibrium states are obtained using an asymptotic solution method which models vortices by ellipsoids. The margin itself is determined by a linear stability analysis. We focus on the interaction between oblate to moderately prolate vortices, the shapes most commonly found in turbulence. Here, a new unexpected instability is found and discussed for prolate vortices which is manifested by the tilting of vortices toward each other. It implies than tall vortices may merge starting from greater separation distances than previously thought.

The shape of vortices in quasi-geostrophic turbulence

Reinaud, Jean Noel — Fri, 10 Jan 2003 00:00:00 GMT

Abstract: The present work discusses the most commonly occurring shape of the coherent vortical structures in rapidly rotating stably stratified turbulence, under the quasi-geostrophic approximation. In decaying turbulence, these vortices-coherent regions of the materially-invariant potential vorticity-dominate the flow evolution, and indeed the flow evolution is governed by their interactions. An analysis of several exceptionally high-resolution simulations of quasi-geostrophic turbulence is performed. The results indicate that the population of vortices exhibits a mean height-to-width aspect ratio less than unity, in fact close to 0.8. This finding is justified here by a simple model, in which vortices are taken to be ellipsoids of uniform potential vorticity. The model focuses on steady ellipsoids within a uniform background strain flow. This background flow approximates the effects of surrounding vortices in a turbulent flow on a given vortex. It is argued that the vortices which are able to withstand the highest levels of strain are those most likely to be found in the actual turbulent flow. Our calculations confirm that the optimal height-to-width aspect ratio is close to 0.8 for a wide range of background straining flows.

The quasi-geostrophic ellipsoidal vortex model

Dritschel, David Gerard — Sun, 25 Apr 2004 00:00:00 GMT

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

The merger of vertically offset quasi-geostrophic vortices

Reinaud, Jean Noel — Fri, 25 Oct 2002 00:00:00 GMT

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

The persistence of balance in geophysical flows

Dritschel, David Gerard — Wed, 10 Jan 2007 00:00:00 GMT

Abstract: Rotating stably stratified geophysical flows can exhibit a near 'balanced' evolution controlled by the conservative advection of a single scalar quantity, the potential vorticity (PV). This occurs frequently in the Earth's atmosphere and oceans where motions tend to be weak compared with the background planetary rotation and where stratification greatly inhibits vertical motion. Under these circumstances, both high-frequency acoustic waves and lower-frequency inertia-gravity waves (IGWs) contribute little to the flow evolution compared with the even-lower-frequency advection of PV. Moreover, this 'slow' PV-controlled balanced evolution appears unable to excite these higher-frequency waves in any significant way-i.e. balance persists. The present work pushes the limits of balance by systematically exploring the evolution of a range of highly nonlinear flows in which motions are comparable with the background rotation. These flows do not possess a frequency separation between PV advection and IGWs. Nonetheless, the flows exhibit a remarkable persistence of balance. Even when flows are not initialized to minimize the amount of IGWs initially present, and indeed even when flows are deliberately seeded with significant IGW amplitudes, the flow evolution-over many inertial periods (days)-remains strongly controlled by PV advection. Description: This paper introduces a novel, powerful way to understand the why geophysical flows are largely under the control of a single scalar field, the potential vorticity, a materially conserved dynamical tracer in the absence of viscous and diabatic effects.

Bending and twisting instabilities of columnar elliptical vortices in a rotating strongly stratified fluid

Billant, Paul — Fri, 25 Aug 2006 00:00:00 GMT

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

Revisiting Batchelor's theory of two-dimensional turbulence

Dritschel, David Gerard — Sun, 25 Nov 2007 00:00:00 GMT

Abstract: Recent mathematical results have shown that a central assumption in the theory of two-dimensional turbulence proposed by Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) is false. That theory, which predicts a X-2/3 k(-1) enstrophy spectrum in the inertial range of freely-decaying turbulence, and which has evidently been successful in describing certain aspects of numerical simulations at high Reynolds numbers Re, assumes that there is a finite, non-zero enstrophy dissipation X in the limit of infinite Re. This, however, is not true for flows having finite vorticity. The enstrophy dissipation in fact vanishes. We revisit Batchelor's theory and propose a simple modification of it to ensure vanishing X in the limit Re -> infinity. Our proposal is supported by high Reynolds number simulations which confirm that X decays like 1/ln Re, and which, following the time of peak enstrophy dissipation, exhibit enstrophy spectra containing an increasing proportion of the total enstrophy (omega(2))/2 in the inertial range as Re increases. Together with the mathematical analysis of vanishing X, these observations motivate a straightforward and, indeed, alarmingly simple modification of Batchelor's theory: just replace Batchelor's enstrophy spectrum X(2/3)k(-1) with (omega(2))k(-1)(In Re)(-1).

A balanced approach to modelling rotating stably stratified geophysical flows

Dritschel, David Gerard — Sun, 10 Aug 2003 00:00:00 GMT

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

Penetrative convection in a superposed porous-medium–fluid layer via internal heating

Carr, Magda — Tue, 01 Jun 2004 00:00:00 GMT

MHD mode conversion of fast and slow magnetoacoustic waves in the solar corona

McDougall-Bagnall, A. M. Dee — Tue, 30 Nov 2010 00:00:00 GMT

Abstract: 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.

Aspects of three-dimensional MHD : magnetic reconnection and rotating coronae

Al-Salti, Nasser S. — Wed, 23 Jun 2010 00:00:00 GMT

Abstract: 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

Yeates, Anthony Robinson — Wed, 24 Jun 2009 00:00:00 GMT

Abstract: 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.

The contour-advective semi-Lagrangian hybrid algorithm approach to weather forecasting and freely propagating inertia-gravity waves in the shallow-water system

Smith, Robert K. — Wed, 24 Jun 2009 00:00:00 GMT

Abstract: 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

Shipton, Jemma — Thu, 01 Jan 2009 00:00:00 GMT

Abstract: 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.

Equilibrium and dynamics of collisionless current sheets

Harrison, Michael George — Wed, 24 Jun 2009 00:00:00 GMT

Abstract: 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.

Magnetic skeletons and 3D magnetic reconnection

Haynes, Andrew L. — Mon, 23 Jun 2008 00:00:00 GMT

Abstract: 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. Description: Electronic version does not contain additional mpeg files

Solar flux emergence : a three-dimensional numerical study

Murray, Michelle J. — Sun, 01 Jun 2008 00:00:00 GMT

Abstract: 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

Wilmot-Smith, A.L. — Sun, 01 Jun 2008 00:00:00 GMT

Abstract: 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.

Bambrey, Ross R. — Mon, 01 Jan 2007 00:00:00 GMT

Abstract: 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

McEwan, Michael P. — Fri, 01 Jun 2007 00:00:00 GMT

Abstract: 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

McGregor, Duncan Ekundayo — Mon, 01 Jan 2007 00:00:00 GMT

Abstract: 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.

Topological structure of the magnetic solar corona

Maclean, Rhona Claire — Mon, 01 Jan 2007 00:00:00 GMT

Abstract: 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.