A statistical equilibrium theory is developed which characterizes the large-scale coherent structures that emerge during the course of the evolution of an ideal two-dimensional magnetofluid. Macrostates are defined to be local joint probability distributions, or Young measures, on the values of the fluctuating magnetic field and velocity field at each point in the spatial domain. The most probable macrostate is found by maximizing a Kullback-Liebler entropy functional subject to constraints dictated by the conserved integrals of the ideal dynamics. This maximum entropy macrostate is, for each point in the spatial domain, a Gaussian probability distribution, whose local mean is an exact stationary solution of the evolution equations of the magnetohydrodynamic system. The predictions of the statistical equilibrium model are found to be in excellent qualitative and quantitative agreement with recent high resolution numerical simulations of turbulence in slightly dissipative two-dimensional magnetofluids.

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The research work presented herein addresses the problem of the mathematical modelling of the mould filling processes as encountered in the foundry industry. The quality of castings, especially aerospace components, is primarily pre-determined at the stage of mould filling within the entire casting process. The entrapment of oxide films, air voids and other impurities into the cast, caused by waves and the breaking of the molten metal surface during filling must be avoided. Otherwise, substandard casting products will result which cost the foundry industry millions of pounds in lost revenue. A three-dimensional control-volume, free surface flow technique known as the Scalar Equation Algorithm (SEA) has been developed as an attachment to the PHOENICS and Harwell-FLOW3D CFD codes for this study. The SEA technique uses a conserved scalar variable to represent the liquid, with an adaptation of the van Leer TVD scheme to define the instantaneous position of the interface. It is similar to the approach used by the well known Volume Of Fluid (VOF) method. However, the SEA technique deals with both air and liquid explicitly, whereas the VOF method does not. A technique has also been developed to allow the liquid temperature to be determined from a conserved `mixture' enthalpy. The liquid temperature is subsequently used in a solidification algorithm to simulate the effect of phase change. The filling model without heat transfer and solidification has been validated against experimental data in both water experiments and actual mould filling experiments. The capability of the SEA method in capturing convoluted waves and air voids has been successfully demonstrated in an example of filling part of a mould running system. It has also been compared against the predictions from the SOLution Algorithm-Volume Of Fluid (SOLA-VOF) and Marker And Cell (MAC) methods. Examples of the developed filling model coupled with heat transfer and solidification are also given. (DX182,921)

The main aim of this research project is to investigate techniques to improve the resolution of flow variables on unstructured skewed meshes whilst working within a Finite Volume (FV) context. A three-dimensional vertex-based FV algorithm for the solution of thermo- fluid flow problems has been developed and integrated within a multi-physics FV framework PHYSICA. Currently PHYSICA employs a cell-centred discretisation technique for fluid mechanics problems and a vertex-based discretisation technique for solid mechanics problems. The vertex-based discretisation approach is validated for a variety of heat transfer problems and comparisons are made with cell-centred solutions. A coupled thermo-mechanical problem, including solidification and radiation, is simulated using vertex-based and cell-centred techniques. Results, run-time and memory requirements are compared. Hybrid vertex-based/cell-centred discretisation of the hydrodynamic variables is also investigated. The components of velocity are solved vertex-based with pressure cell-centred or conversely pressure is solved vertex-based with velocity cell-centred. The methods are applied to flow in a lid-driven cavity and solutions are obtained on a number of distorted meshes. Comparisons are made with the benchmark solutions. The hybrid discretisation enables solutions on distorted meshes where purely cell-centred techniques fail. The hybrid methods produce final solutions containing errors due to mesh distortion. The co-located vertex-based flow solutions obtained on the distorted meshes are comparable to solutions obtained on a uniform Cartesian mesh. Having a good resolution of the flow field on distorted meshes enables the solution of other transported variables using cell-centred techniques. Finally, this hybrid vertex-based/cell-centred technique is applied to thermally driven flow, turbulent flow, and three-dimensional flow over an aircraft wing.

This thesis is about the construction of low dimensional diffusion models of climate variables. It assesses the predictive skill of models derived from a principled averaging procedure and a purely empirical approach. The averaging procedure starts from the equations for the original system then approximates the \weather" variables by a stochastic process. They are then averaged with respect to their invariant measure. This assumes that they equilibriate much faster than the climate variables. The empirical approach argues for a very general model form, then parameters are estimated using likelihood based inference for Stochastic Differential Equations. This is computationally demanding and relies upon Markov Chain Monte Carlo methods. A large part of this thesis is focused upon techniques to improve the efficiency of these algorithms. The empirical approach works well on simple one dimensional models but performs poorly on multivariate problems due to the rapid increase in unknown parameters. The averaging procedure is skillful in multivariate problems but is sensitive to lack of complete time scale separation in the system. In conclusion, the averaging procedure is better and can be improved by estimating parameters in a principled way based on the likelihood function and by including a latent noise process in the model.

In tokamaks, turbulence is a key contributor to cross field transport. However, it is also responsible for the spontaneous generation of large scale structures such as zonal ows. These are of relevance to fusion plasmas as they can create transport barriers which aid plasma confinement. The interaction between drift waves and zonal ows can be investigated using reduced models such as the Hasegawa- Mima and Hasegawa-Wakatani equations. A four-wave truncated model is developed for the Extended-Hasegawa-Mima (EHM) equation. This produces a set of four ordinary differential equations (ODEs) that are used to investigate the modulational instability (MI), a mechanism by which drift waves can produce a zonal ow. These equations are linearised to produce a dispersion relation for the MI which is used to produce a set of maps of the linear growth rate of the MI. These show how additional modes become unstable as the gyroradius is increased. The truncated model and dispersion relation are then compared to measurements taken from simulations of the full EHM partial differential equation (PDE) which has been seeded with an appropriate initial condition. Good agreement is found when the pump wave has no component in the direction of the density gradient. A similar truncated model is derived for the Extended-Hasegawa-Wakatani (EHW) equations. As the EHW system has separate equations for density and potential this leads to a set of eight ODEs. The linearisation technique used for the EHM system cannot be applied here. Instead, approximations based on the built in EHW instability are made to calculate a linear growth rate for the zonal ow using the ODEs describing it. These analytical predictions are then compared to a full PDE simulation of the system, which is initialised using random noise. It is found that for particular sets of waves the ODEs provide a good prediction of the linear growth rate. A driving term is added to the EHM equation to reproduce the effect of the built in instability of the EHW equations. This causes a drift wave spectrum to grow when full EHW PDE simulations are seeded with random noise. The four-wave ODE model is updated to include this driving. The ODE model again produces good predictions for the growth rate of the zonal flow.

We consider the evolution of sharp fronts and almost-sharp fronts for the ↵-equation, where for an active scalar q the corresponding velocity is defined by u = r?(−#)−(2 − ↵)/2q for 0 < ↵ < 1. This system is introduced as a model interpolating between the two-dimensional Euler equation (↵ = 0) and the surface quasi-geostrophic (SQG) equation (↵ = 1). The study of such fronts for the SQG equation was introduced as a natural extension when searching for potential singularities for the three-dimensional Euler equation due to similarities between these two systems, with sharp-fronts corresponding to vortex-lines in the Euler case (Constantin et al., 1994b). Almost-sharp fronts were introduced in C´ordoba et al. (2004) as a regularisation of a sharp front with thickness $, with interest in the study of such solutions as $ ! 0, in particular those that maintain their structure up to a time independent of $. The construction of almost-sharp front solutions to the SQG equation is the subject of current work (Fe↵erman and Rodrigo, 2012). The existence of exact solutions remains an open problem. For the ↵-equation we prove analogues of several known theorems for the SQG equations and extend these to investigate the construction of almost-sharp front solutions. Using a version of the Abstract Cauchy Kovalevskaya theorem (Safonov, 1995) we show for fixed 0 < ↵ < 1, under analytic assumptions, the existence and uniqueness of approximate solutions and exact solutions for short-time independent of $; such solutions take a form asymptotic to almost-sharp fronts. Finally, we obtain the existence and uniqueness of analytic almost-sharp front solutions.

The Yang-Mills-Higgs theory has its origins in Physics. It describes particles with masses via the Higgs mechanism and predicts magnetic monopoles. We study here the mathematical aspects of the theory following an analytical and geometric approach. Our motivation comes from physics and we work all the time with the full Lagrangian of the theory. At the same time, we are interested in it from the variational point of view, as a functional on an infinite dimensional space and as a system of non-linear equations on a non-compact manifold with finite energy as the only constraint. We are concerned mainly with the configuration space of the theory, the existence of solutions and their behaviour at infinity.

In this thesis we construct solitons of geometric flows with applications in three different settings. The first setting is related to nonuniqueness for geometric heat flows. We show that certain double cones in Euclidean space have several self-expanding evolutions under mean curvature flow. The construction of the associated self-expanding solitons leads to an application in fluid dynamics. We present a new model for the behaviour of oppositely charged droplets of fluid, based on the mean curvature flow of double cones. If two oppositely charged droplets of fluid are close to each other, they start attracting each other and touch eventually. Surprisingly, experiments have shown, that if the strength of the charges is high enough, then the droplets are repelled from each other, after making short contact. The constructed self-expanders can be used to correctly predict the experimental results, using our theoretical model. Secondly we employ space-time solitons of the mean curvature flow to give a geometric proof of Hamilton's Harnack estimate for the mean curvature flow. This proof is based on the observation that the associated Harnack quantity is the second fundamental form of a space-time self-expander. Moreover the self-expander is asymptotic to a cone over the convex initial hypersurface. Hence the self-expander can be seen as the mean curvature evolution of a convex cone, which we exploit to show that preservation of convexity directly implies the Harnack estimate. In the last chapter we study solutions of the mean curvature flow in a Ricci flow backgound. We show that the space-time track of such a solution can be seen as a soliton. Moreover the second fundamental form of this soliton matches the evolution of a functional, which is the analogue of G. Perelman's F-functional for the Ricci flow on a manifold with boundary and which also has relations to quantum gravity. Furthermore our construction provides a link between the Harnack estimate for the mean curvature flow and the Harnack estimate for the Ricci flow.

The particle-in-cell code EPOCH was extended to include field and collisional ionisation for use in simulating initially neutral or partially-ionised targets in laser-plasma inter- actions. The means by which particles ionise in the the field of an intense laser was described and physical models were included to determine the instantaneous ionisa- tion rate at particles within the simulation domain for multiphoton, tunnelling, barrier- suppression and electron-impact ionisation. The algorithms used to implement these models were presented and demonstrated to produce the correct ionisation statistics. A scheme allowing for modelling small amounts of ionisation for an arbitrarily low number of superparticles was also presented for comparison and it was shown that for sufficient simulation time the two schemes converge. The three major mechanisms of ionisation in laser-plasma interactions were described as being ionisation-induced defocussing, fast shuttering and ionisation injection. Simulations for these three effects were presented and shown to be in good agreement with theory and experiment. For fast-shuttering, plasma mirrors were simulated using the pulse profile for the Astra Gemini laser at the Central Laser Facility. Rapid switch-on and the theoretical maximum for contrast ratio was observed. For ionisation injection, simulations for laser wakefield acceleration in a helium gas were performed and the accelerated electron population was shown to be greatly increased through use of a 1% nitrogen dopant consistent with the experimental results of McGuffey et al. A study of the laser filamentation instability due to SRS backscatter at the relativistically corrected quarter critical surface (RCQCS) was per- formed in collaboration with C.S. Brady and T.D. Arber at the University of Warwick [1]. It was found that for hydrogen and plastic the instability was unaffected by the in- clusion of ionisation. Further study with argon revealed a attening of the RCQCS and it was demonstrated that for a material with multiple ionisation levels ionising strongly near the self-focussed intensities at the RCQCS, rapid ionisation caused an inversion of the RCQCS that suppressed the filamentation instability.