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This dissertation reports research about the phase space perspective for solving wave problems, with particular emphasis on the phenomenon of mode conversion in multicomponent wave systems, and the mathematics which underlie the phase space perspective. Part I of this dissertation gives a review of the phase space theory of resonant mode conversion. We describe how the WKB approximation is related to geometrical structures in phase space, and how in particular ray-tracing algorithms can be used to construct the WKB solution. We further review how to analyze the phenomena of mode conversion from the phase space perspective. By making an expansion of the dispersion matrix about the mode conversion point in phase space, a local coupled wave equation is obtained. The solution of this local problem then provides a way to asymptotically match the WKB solutions on either side of the mode conversion region. We describe this theory in the context of a pedagogical example; that of a pair of coupled harmonic oscillators undergoing resonant conversion. Lastly, we present new higher order corrections to the local solution for the mode conversion problem which allow better asymptotic matching to the WKB solutions. The phase space tools used in Part I rely on the Weyl symbol calculus, which gives a way to relate operators to functions on phase space. In Part II of this dissertation, we explore the mathematical foundations of the theory of symbols. We first review the theory of representations of groups, and the concept of a group Fourier transform. The Fourier transform for commutative groups gives the ordinary transform, while the Fourier transform for non-commutative groups relates operators to functions on the group. We go on to present the group theoretical formulation of symbols, as developed recently by Zobin. This defines the symbol of an operator in terms of a double Fourier transform on a non-commutative group. We give examples of this new type of symbol, using the discrete Beisenberg-Wey1 group to construct the symbol of a matrix. We then go on to show how the path integral arises when calculating the symbol of a function of an operator. We also show how the phase space and configuration space path integrals arise when considering reductions of the regular representation of the Heisenberg-Wey1 group to the primary representations and irreducible representations, respectively. We also show how the path integral can be interpreted as a Fourier transform on the space of measures, opening up the possibility of using tools from statistical mechanics (such as maximum entropy techniques) to analyze the path integral. We conclude with a survey of ideas for future research and describe several potential applications of this group theoretical perspective to problems in mode conversion.

Computer simulations of complex phenomena have become an invaluable tool for scientists in all disciplines. These simulations serve as a tool both for theorists attempting to test the validity of new theories and for experimentalists wishing to obtain a framework for the design of new experiments. Lattice Boltzmann Methods (LBM) provide a kinetic simulation technique for solving systems governed by non-linear conservation equations. Direct LBMs use the linearized single time relaxation form of the Boltzmann equation to temporally evolve particle distribution functions on a discrete spatial lattice. We will begin with a development of LBMs from basic kinetic theory and will then show how one can construct LBMs to model incompressible resistive magnetohydrodynamic (MHD) conservation laws. We will then present our work in extending existing models to the octagonal lattice, showing that the increased isotropy of the octagonal lattice produces better numerical stability and higher Reynolds numbers in MHD simulations. Finally, we will develop LBMs that use non-uniform grids and apply them to one dimensional MHD systems.

In this dissertation, we study the control of near-integrable systems. A near-integrable system is one whose phase space has a similar structure to an integrable system during short time periods and for some parameter regime. We begin by studying the control of integrable Hamiltonian systems. The controller targets an exact solution to the integrable system using dissipative and conservative terms. We find that a Takens-Bogdanov bifurcation occurs in the limit of no dissipative control. The presence of a Takens-Bogdanov bifurcation implies that the control is highly susceptible to noise. We illustrate our results using a two- and four-dimensional integrable systems generated by low order truncations of solutions to the nonlinear Schrodinger equation (NLS). We then extend our results to a near-integrable system related to the NLS; the Ginzburg-Landau equation. We attempt to control the Ginzburg-Landau equation to a plane wave solution of the NLS. We show that for a certain parameter regime; a Takens-Bogdanov bifurcation occurs in the limit of no dissipative control. Through this example, we show that solutions of integrable systems can be viable control targets for related near-integrable systems.

The Lattice Boltzmann Method (LBM) is a simple and highly efficient method for computing nearly incompressible fluid flow. However, it is well known to suffer from numerical instabilities for low values of the transport coefficients. This dissertation examines a number of methods for increasing the stability of the LBM over a wide range of parameters. First, we consider a simple transformation that renders the standard LB equation implicit. It is found that the stability is largely unchanged. Next, we consider a stabilization method based on introducing a Lyapunov function which is essentially a discrete-time H-function. The uniqueness of an H-function that appears in the literature is proven, and the method is extended to stabilize some of the more popular LB models. We also introduce a new method for implementing boundary conditions in the LBM. The hydrodynamic fields are imposed in a transformed moment space, whereas The non-hydrodynamic fields are shifted over from neighboring nodes. By minimizing population gradients, this method exhibits superior numerical stability over other widely employed schemes when tested on the widely-used benchmark of incompressible flow over a backwards-facing step.

This dissertation reports on the first direct comparison between the results of ray-based and full-wave calculations for mode conversion in plasma. This study was motivated by the modular method originally developed by Ye and Kaufman to treat a magnetosonic wave crossing a cold minority-ion gyroresonance layer. We start with the cold plasma fluid model and introduce a system of evolution equations for electrons and two ion species: deuterium and hydrogen. We first study this system of equations for uniform plasma by Fourier methods, which gives the dispersion relations. We discuss how the traditional approach---which eliminates all other dynamical variables in terms of the electric field---leads to singular denominators at the resonances. We then introduce the Kaufman & Ye approach, which retains the ion velocities as dynamical variables. In this formulation, the ion resonances appear as 'avoided crossings' between the familiar 'fast wave' and a zero-group-velocity ion 'mode' associated with the particle velocities. We then extend our problem to nonuniform plasma where the resonance is localized in space. Away from the resonance, WKB methods apply, but they break down in the vicinity of the resonance. In this region, we introduce the notion of 'uncoupled modes' and discuss how to use them to systematically carry out a simplification of the problem. This leads directly to the modular method of Kaufman & Ye in the mode conversion region, and provides the connection coefficients for the WKB solutions across the resonance layer. We specialize to an incoming wave packet and use the full-wave equations and the reduced 2x2 form to numerically study the wave packet conversion. This allows us to observe the emission of the reflected wave packet after a time delay (the linear 'ion-cyclotron echo'). We calculate the incoming, transmitted and reflected wave packet energies. We compare them to the transmission and reflection coefficients predicted by the S matrix approach of Kaufman and Ye for a wide range of ion density ratios and find good agreement.

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Experimentally it is observed that the plasma in Elmo Bumpy Torus (EBT) shows discontinuous changes in the electron line density, electron and ion temperatures and fluctuation levels as the ambient gas pressure or electron cyclotron heating is varied continuously. We use the Point Model of Hedrick et al. for the toroidal core plasma in EBT. The Point Model is not a gradient dynamic system. Hence the Elementary Catastrophe Theory is not directly applicable to the Point Model. Nonetheless, the Point Model equilibria will be shown to exhibit properties which are quite akin to the canonical cusp catastrophe. The ambipolar electric field is taken as a control parameter. When electrons are nonresonant, the equilibrium surfaces show only one fold; but when ions are nonresonant, equilibrium surfaces may show single or multiple folds. In the former (electron) case, qualitative agreement with experiments is quite good. For the case of nonresonant ions, predictions are made as to the possible plasma behavior which, when ICRH heating is sufficiently intense, may be checked by experiments. The nonlinear time evolution of the Point Model equations show that the plasma follows the Delay Convention. This simple model of the EBT plasma exhibits rich structure, i.e., equilibrium surfaces with single or multiple folds, attractors, repellors, appearance and disappearance of folds, reversal of direction on equilibrium trajectories, catastrophes, hysterisis, competition between multiple point attractors, and basins of attractors. This leads us to conclude that a simple model of physical systems governed by external parameters can unravel general, complicated qualitative behavior of the system.

It is shown numerically, both for the two-dimensional Navier-Stokes equations and for two-dimensional magnetohydrodynamics, that the long-time asymptotic state in a forced inverse-cascade situation is one in which the spectrum is completely dominated by its own fundamental. The growth continues until the fundamental is dissipatively limited by its own dissipation rate. An algebraic model is proposed for the dynamics of such a final state.

The subject of this thesis is an analysis of results from pseudospectral simulation of the Strauss equations of reduced three-dimensional magnetohydrodynamics. We have solved these equations in a rigid cylinder of square cross section, a cylinder with perfectly conducting side walls, and periodic ends. We assume that the uniform-density magnetofluid which fills the cylinder is resistive, but inviscid. Situations which we are considering are in several essential ways similar to a tokamak-like plasma; an external magnetic field is imposed, and the plasma carries a net current which produces a poloidal magnetic field of sufficient strength to induce current disruptions. These disruptions are characterized by helical "m = 1, n = 1" current filaments which wrap themselves around the magnetic axis. An ordered, helical velocity field grows out of the broad-band, low amplitude noise with which we initialize the velocity field. Kinetic energy peaks near the time the helical current filament disappears, and the current column broadens and is flattens itself out. We find that this is a nonlinear, turbulent phenomenon, in which many Fourier modes participate. By raising the Lundquist number used in the simulation, we are able to generate situations in which multiple disruptions are induced. When an external electric field is imposed on the plasma, the initial disruption, from a quiescent, state, is found to be very similar to those observed in the undriven runs. After the lobed "m = 1, n = 1" stream function pattern develops, however, a quasi-steady state with flow is maintained for tens of Alfven transit times. If viscous damping is included in the driven problem, the steady state may be avoided, and additional disruptions produced in a time less than a large-scale resistive decay time.