To produce nuclear fusion in a Tokamak reactor requires the heating of a plasma to a temperature of the order of 10 keV. Electron cyclotron resonant heating (ECRH), in which the plasma is heated by radio waves in resonance with the Larmor frequency of the plasma's electrons, is one scheme under consideration for achieving this. A description of such a heating scheme requires a theory to explain the propagation and absorption of high frequency waves in a plasma in the presence of a magnetic field gradient. A WKB analysis can describe some of the processes involved but a complete explanation requires the use of full wave equations. In this thesis we shall develop a technique for deriving such equations which will be shown to be simpler and more general than calculations performed by earlier workers. The technique relies on including the effect of the magnetic gradient across the Larmor orbit of the electrons in the resonance condition of the wave, the so called Gyrokinetic correction, which has been ignored in calculations by previous workers. Once derived, the equations are solved numerically and the results applied to a number of experiments currently being performed on Tokamak fusion. In addition, we shall also look at the energy loss processes of runaway electrons, which have been shown experimentally to be shorter than would be expected.

In this thesis, several problems relating to thermal instabilities in the solar corona are examined. Chapter 1 gives a brief description of the Sun and corresponding events with particular attention focused on prominences, their formation and eruption. Various problems concerning thermal instabilities are then tackled in the later Chapters. In Chapter 2, the basic MHD equations are introduced and a physical description of the thermal instability mechanism given. The MHD equations are linearised in a uniform, infinite medium and the basic instability criteria obtained. Chapter 3 investigates the normal mode spectrum for the linearised MHD equations for a cylindrical equilibrium. This spectrum is examined for zero perpendicular thermal conduction, with both zero and non-zero scalar resistivity. Particular attention is paid to the continuous branches of this spectrum, or continuous spectra. For zero resistivity there are three types of continuous spectra present, namely the Alfven, slow and thermal continua. It is shown that when dissipation due to resistivity is included, the slow and Alfven continua are removed and the thermal continuum is shifted to a different position (where the shift is independent of the exact value of resistivity). The 'old' location of the thermal continuum is covered by a dense set of nearly singular discrete modes called a quasi-continuum, for equilibria with the thermal time scale much smaller than the Alfven time scale. This quasi-continuum is investigated numerically and the eigenfunctions are shown to have rapid spatial oscillating behaviour. These oscillations are confined to the most unstable part of the equilibrium based on the Field criterion and may be the cause of fine structure in prominences. In Chapter 4, the normal mode spectrum for the linearised MHD equations is examined for a plasma in a cylindrical equilibrium. The equations describing these normal modes are solved numerically using a finite element code. In the ideal case the Hain-Lust equation is expanded and a WKB solution obtained for large axial wave numbers. This is compared to the numerical solutions. In the non-ideal case, the ballooning equations describing localised modes are manipulated in an arcade geometry and a dispersion relation derived. It is illustrated that as the axial wave number k is increased, the fundamental thermal and Alfven modes can coalesce to form overstable magnetothermal modes. The ratio between the magnetic and thermal terms is varied and the existence of the magnetothermal modes examined. The corresponding growth rates are predicted by a WKB solution to the ballooning equations. The interaction of thermal and magnetic instabilities and the existence of these magnetothermal modes may be significant in the eruption of prominences into solar flares. Chapter 5 extends the work presented in Chapter 4 to include the effects of line-tying in a coronal arcade. The ballooning equations which were introduced in Chapter 4 are manipulated to give a dispersion relation. This relation is a quadratic in the square of the azimuthal wave number m if parallel thermal conduction is neglected and a cubic in m2 if parallel conduction is included. Rigid wall boundary conditions are applied to this dispersion relation. This dispersion relation is then solved numerically subject to these boundary conditions and the solutions plotted. Unfortunately the expression for the thermal continuum in line-tied arcades is required since the thermal continuum must play an important role in the proceedings. This calculation is left for future work. From the results obtained, it can be seen that the thermal instability can play a major part in prominence formation and destruction. The thermal instability may help create the prominence. Resistivity and perpendicular thermal conduction can cause of the observed fine scale structure. Finally, a neighbouring thermal instability may trigger a magnetic instability that causes the prominence to erupt.

Magnetic reconnection is a fundamental physical process by which stored magnetic energy may be released. It is already known that different reconnection regimes result from changes in the nature of the plasma inflow towards the reconnection site. In this thesis, we examine both how the outflow region responds to changes both in the inflow and outflow boundary conditions and also how introducing compressibility affects the results. We find that if the inflow is converging, the outflow velocity is least, the width of the outflow region is greatest and the ratio of outflowing thermal to kinetic energy is greatest. Also, there is one free outflow parameter which would naturally be specified by the velocity of plasma leaving the reconnection site. We suggest that reverse currents seen in numerical simulations may result from the specification of an extra boundary condition. In addition, we find that the main effects of including compressibility are: to enhance convergence or divergence of the inflow; to increase the maximum reconnection rate where the inflow is converging; to increase the flow speed near the reconnection site where the inflow is diverging; to give faster, narrower outflow jets; to increase variations between regimes in the energy conversion and to increase the ratio of thermal to kinetic energy in the outflow jet.

Solar flares and solar prominences are amongst the best known features of solar activity. Despite this familiarity, however, there are still significant gaps in our knowledge of these phenomena. In this thesis some theoretical aspects of these events are considered. We first consider solar prominences. We propose a model for the static equilibrium of quiescent prominences which will simultaneously explain the support mechanism for the dense prominence material and take account roughly of the required energy balance. This model contains two parameters, namely the coronal plasma beta and the horizontal shear angle Φ, that the magnetic fieldlines make with the prominence normal. We obtain limits on both these parameters which, when exceeded, imply that no equilibrium state is possible. The results obtained provide a possible explanation for several prominence features. For the remainder of the thesis we consider one aspect of the solar flare problem, namely the possibility of a trigger mechanism for the rapid release of energy in a flare. One candidate for this mechanism is the sudden release of energy stored in excess of potential by a force-free magnetic field which becomes unstable as a result of photospheric motions. For this reason we seek simple analytic solutions to the force-free field equations which may exhibit such an instability. An alternative trigger mechanism, which requires the presence of a current sheet, is given by the emerging flux model for solar flares. We thus develop a one-dimensional model for current sheets in general, where the conditions within the current sheet are given in terms of several non-dimensional parameters which describe the external conditions. These results are then applied to the emerging flux model.

In this thesis, a study is made of the global solar oscillations known as p-modes, modelled by a plane-parallel stratified plasma, within which is embedded a horizontal layered magnetic field. A magnetohydrodynamic formalism is used to investigate the models. The main aim of the thesis is to model the turnover effect in the frequency shifts of the p-modes observed over the course of the solar cycle. Radial oscillations (modes of degree zero) of the Sun are studied for several atmospheric temperature and magnetic field profiles. It is found that the turnover in frequency shifts may be obtained by an increase in the strength of the atmospheric horizontal magnetic field (assumed to be uniform), coupled with a simultaneous increase in atmospheric temperature. The effect of a thin superadiabatic layer in the upper convection zone on p-mode frequencies is also considered. For this model we study modes of general degree, and find that the observed rise and subsequent downturn in the frequency shifts can be duplicated, in the absence of a magnetic field, by simultaneously steepening the temperature gradient of the superadiabatic layer and increasing the atmospheric temperature. In the presence of a magnetic field, where the atmosphere is permeated by a uniform horizontal magnetic field, turnover is reproduced by a combination of an increase in magnetic field strength, a steepening of the temperature gradient in the superadiabatic region, and an increase in atmospheric temperature. The unstable superadiabatic layer also gives rise to convective modes, which are considered briefly. Finally, a model incorporating a magnetic layer residing at the base of the convection zone is constructed and its influence on the frequencies of p-modes assessed. By simply changing the magnetic field strength of this layer, we are unable to reproduce the observed solar cycle variations in p-mode frequencies. The buried magnetic layer supports surface and body magnetoacoustic waves, and a brief study is made of their properties.

The theme of this thesis is ideal linear MHD wave propagation in structured media, using models relevant to structures in the solar atmosphere. We derive dispersion relations governing the ideal linear MHD modes for stationary states which are discretely structured in velocity and other plasma properties, in a direction transverse to the magnetic field, with field-aligned steady flow; the discrete structures considered are the single interface, uniform slab and uniform cylinder. This represents an extension of earlier models for the static case (Edwin 1984), by the inclusion of structured flows. The basic effects of flow are described, drawing on a discussion of the dispersion relations. The dispersion relations for the case of incompressible surface modes are examined in detail. We identify the qualitative effects of flow, including the onset of instability, by tracing the evolution of stable solutions and their propagation windows, as the relative flow is increased. Our analysis is presented in terms of a general formulation applicable to all three geometries (interface, slab and cylinder), revealing the combined role of dispersion and the ratio of densities in the two media. We go on to consider the relevance of the incompressible approximation to compressible surface modes, with particular reference to the static case of a single interface one side of which is field-free. We present and investigate analytical solutions for several special cases. The properties of the solutions obtained are compared with those for the equivalent incompressible case. Finally, we turn to the topic of global oscillations of quiescent prominences. A uniform slab model (Joarder 1993) yields, under conditions appropriate to the prominence-coronal inhomogeneity with the magnetic field threading the prominence being line-tied in the photosphere, modes which are analogous to the oscillations of a uniform string loaded with a point mass, and a formula approximating the period is given. We investigate the robustness of this formula for various plasma density profiles, assessing the applicability of the results from the uniform slab calculation to more realistic density profiles of the prominence-coronal inhomogeneity.

The three-dimensional magnetic structure of solar filament channels and filaments is considered. A simple analytical potential model of a filament channel is setup with line sources representing the overlying arcades and point sources the flux of the filament. A possible explanation of the distinct upper and lower bounds of a filament is given. A more detailed numerical force-free model with discrete flux sources is then developed and the effect of magnetic shear on the separatrix surface explored. Dextral channels are shown to exist for a wider range of negative values of the force-free alpha and sinistral channels for positive values of alpha. Potential models of a variety of coronal structures are then considered. The bending of a filament is modelled and a method of determining the horizontal component of a filament's magnetic field is proposed. Next, the observed opposite skew of arcades lying above switchbacks of polarity inversion lines is shown to be produced by a local flux imbalance at the corner of the switchback. Then, the magnetic structure of a particular filament in a filament channel is modelled using observations from a photospheric magnetogram. It is shown that dips in the filaments magnetic field could result from opposite polarity fragments lying below the filament. Finally, the formation of a specific filament channel and filament is modelled. The formation of the channel is shown to be due to the emergence of new flux in a sheared state. It is shown that convergence and reconnections between the new flux and old remnant flux is required for the filament to form. The field lines that represent the filament form a thin vertical sheet of flux. The changing angle of inclination of the sheet gives the appearance of twist. The method of formation is then generalised to other cases and it is shown that a hemispheric pattern consistent with the results of Martin et al. (1995) can be obtained.

This thesis aims to provide a basic theoretical explanation for the oscillatory motions observed in solar quiescent prominences. The prominence is treated as a simple plasma slab embedded in a hotter and rarer uniform coronal plasma. Both the slab and its environment are permeated by a uniform magnetic field. The field lines are anchored at rigid walls placed on either side of the plasma slab and representing the photospheric line-tying effect. The magnetohydrodynamic modes of oscillation of the plasma slab are then examined for different orientations of the magnetic field with respect to the long axis of the slab. Particularly interesting in this study is the appearance of the 'string MHD' modes that are analogous to the fundamental vibrations of a mass- loaded stretched elastic string. Such modes appear whenever the magnetic field vector is inclined to the long axis of the slab, thus producing a magnetic field component in the direction transverse to the axis of the slab. Observationally, this inclination of the field is generally small. For realistic values of the angle of inclination of the magnetic field lines, the 'string Alfven' mode and an 'internal slow' mode yield periods in the range 1/2-2 hr. These modes may correspond to the observed long period (40-90 minutes) oscillations in quiescent prominences. Intermediate periodicities, in the range 8-20 min, may be associated with an 'internal Alfven' mode and a 'fast string' mode of the prominence slab. The observed short periodicities, in the range 2-5 min, may correspond to an 'internal fast' mode in prominences. Having thus established a foundation for the theoretical modelling of prominence oscillations in terms of the magnetohydrodynamic modes of oscillation of a non-gravitating plasma slab, we discuss several factors, such as the effects of gravitational stratification, the curvature of the magnetic field lines, and the fine-structures in a prominence, that may complicate a description of its oscillatory modes. Some preliminary investigations of simple magnetohydrostatic equilibrium models suggest that gravity and the curvature of the magnetic field lines play only a secondary role in determining the periods of the oscillatory modes in prominences, the basic structure of the modes being similar to that present in simple slab models.

Equations are derived to describe the evolution of an electron distribution function under the action of electromagnetic instabilities in a non-uniform plasma using an extension of the quasilinear theory of Kennel and Engelmann. Variations in both the electron density and temperature and the background magnetic field are taken into account. These equations are simplified in the limit of small electron beta so that an electrostatic approximation is justified. Methods are then presented which allow the solution of these equations (or, in principle, the more complex electromagnetic equations). In particular, a method of solving the kinetic dispersion relation for an arbitrary background (first-order) distribution function with the minimum of additional assumptions and approximations is described in detail. The electrostatic equations are solved for a number of different cases in order to study the action of the modified two stream instability on the electron distribution function. Throughout, realistic values of the ratios of electron to ion mass and electron plasma to cyclotron frequency ratio are used. The applications to collisionless plasma shock waves are discussed, and it is found that the modified two stream instability can produce the (relatively small) amounts of electron heating observed at quasi-perpendicular terrestrial bow shocks, and the flat-topped electron distribution functions seen to evolve. Extensions to the model which would greatly improve its applicability and accuracy, as well as the amount of computational effort required, are discussed.

Electron cyclotron resonance heating has been successfully used in a number of experiments, firstly to raise the plasma temperature and secondly to drive currents noninductively. Recently the microwaves in tokamak experiment (MTX) has been proposed at the Lawrence Livermore Laboratory, which will involve pulsed heating at powers much higher than have previously been possible, using a Free Electron Laser (PEL). The physics of such an experiment differs greatly from the physics of experiments using less powerful but continuous operation gyrotron sources. An analytical model of the interaction between a wave and an electron is presented on the assumption that the wave amplitude experienced along the electron guiding centre changes slowly with time as it passes through the beam. This model is tested numerically by integrating the equations of motion governing the electron's motion as it interacts with the wave. Finally this model is used to predict the possible growth of instabilities in a plasma heated by a FEL. The growth rates of these waves may be large enough to act on the plasma in time scales much shorter than typical electron collision times.