Solar coronal stability theory is a powerful tool for understanding the complex behaviour of the Sun's atmosphere. It enables one to discover the driving forces behind some intriguing phenomena and to gauge the soundness of theoretical models for observed structures. In this thesis, the linear stability analysis of line-tied symmetric magnetohydrostatic equilibria is studied within the framework of ideal MHD, aimed at its application to the solar corona. Firstly, a tractable stability procedure based on a variational method is devised. It provides a necessary condition for stability to disturbances localised about a particular flux surface, and a sufficient condition for stability to all accessible perturbations that vanish at the photosphere. The tests require the minimisation of a line integral along the magnetic field lines. For 1-D equilibria, this can be performed analytically, and simple stability criteria are obtained. The necessary condition then serves as an extended Suydam criterion, incorporating the stabilising effect of line-tying. For 2-D equilibria, the minimisation requires the integration of a system of ordinary differential equations along the field lines. This stability technique is applied to arcade, loop, and prominence models, yielding tight bounds on the equilibrium parameters. Secondly, global modes in 1-D coronal loops are investigated using a normal mode method, in order to clarify their link with localised interchange modes. For nearly force-free fields it is shown that instability to localised modes implies the existence of a fast growing global kink mode driven in the neighbourhood of the radius predicted by the local analysis. This confers a new significance on the study of localised interchange modes and the associated extended Suydam criterion.

The nonuniform structure observed in the solar atmosphere, and in particular the corona, is thought to arise due to the interaction between the magnetic field and plasma. Using a linear theory, the nature of these interactions is investigated, and it is shown how coronal structure may be modelled in a simple way by extended standing disturbances. The effect of inertial forces in considered in both a Cartesian and cylindrical geometries, and a first correction due to gravity is calculated. The restrictions of a linear theory may be overcome by finding exact solutions. Solutions are presented which may model plasma flows in closed, partially open and open magnetic field line structures. A new method for finding particular classes of exact steady solutions in a gravitationally stratified, isothermal atmosphere is presented, along with some examples of possible solutions.

In this thesis the nature of magnetoacoustic surface waves at a single magnetic interface is examined for the case of parallel propagation. Above the interface is an isothermal medium permeated by a horizontal magnetic field. The lower region is a field-free medium of different density to the magnetic atmosphere. We consider both the incompressible and compressible situations and the effect of including gravity. In each case a transcendental dispersion relation is solved numerically for a range of parameters and the resulting dispersion curves plotted. In the first chapter we provide a general introduction to the work, reviewing previous work in this area and considering applications of surface waves. In the second chapter we consider the existence of surface waves for the case when the media are incompressible either side of the interface. We consider the cases of both uniform and non-uniform distributions of densities and the effect of including gravity. We show that the f-mode exists in a restricted band of horizontal wavenumber. In the subsequent chapters we consider the effect of compressibility on surface waves. The media either side of the interface are taken to be isothermal. In the absence of gravity the interface may support one or two surface modes determined by the relative temperatures and magnetism of the two media. This case is studied in Chapter 3 where phase-speeds and penetration depths of the waves and the associated pressure perturbations are investigated for a variety of field strengths and sound speeds. In Chapters 4 and 5 we consider the effect of gravity on the compressible modes described in Chapter 3. In Chapter 4 an exact dispersion relation is obtained for the case of a constant Alfven speed, whilst in Chapter 5 the case of a uniform magnetic field is considered. In the absence of the magnetic field the transcendental dispersion relation may be reduced to a polynomial. This polynomial possesses two acceptable solutions, only one of which may exist at any given circumstance depending on the densities either side of the interface. If the gas density within the field exceeds that in the field-free medium, then the f-mode may propagate; otherwise, a magnetic surface gravity mode propagates. As in the incompressible case, the f-mode exists in a restricted band of horizontal wavenumber. An analytical form for the wave speed of the f-mode is obtained for small values of the Alfven speed. It is shown that the f-mode is related to the fast magnetoacoustic surface wave, merging into that mode at short wavelengths.

Using the Magnetohydrodynamic model, two problems in the behaviour of magnetic field structures are investigated. Firstly, the stability of tokamak equilibria to coupled tearing modes is calculated. Secondly, the equilibrium structure of a solar coronal loop is examined. The flux co-ordinate method is used to construct toroidal equilibria of the type found in large aspect ratio tokamaks. In such a field configuration, the analysis of tearing modes is complicated by the coupling of different poloidal fourier modes. The effect of coupling through elliptic shaping of plasma surfaces is calculated. For certain current profiles, this effect may cause instability. The response of coronal loops to twisting at their photospheric footpoints is investigated. Long loops are shown to have an essentially 1-D nature. This observation is used to develop a 1-D, line-tied model for such loops. This model is used to conduct an extensive survey of the non-linear twist regime, including the effects of enhanced fluid pressure. The possibility of non-equilibrium, which would provide energy for coronal heating and compact flares, is examined. When the physical variable of footpoint displacement is specified, no loss of equilibrium is found by twisting. Loss of equilibrium is found for high pressures, which we do not, however, expect to find in the corona.

Solar coronal observations have shown that the corona has a highly complex structure which presumably owes its existence to the magnetic field. Models in thermal and hydrostatic equilibrium are here calculated in order to try and explain many of these observations. Coronal holes occur where open field lines reach out into space. The model of McWhirter, et al. (1975) for the inner corona in such a configuration is generalised to allow different types and magnitudes of heating as well as different area divergences and flows. It is found that hot, fast upflows cannot always exist in thermal equilibrium. The choice of boundary conditions can appreciably alter the results, and so different choices are compared. Most of the corona, especially in active regions, appears to consist of coronal loops. Subtle relations for energy balance models of such loops are found to exist between the physical parameters of a loop's length, base density, and heat input. No solution exists at coronal temperatures in certain cases, which may explain the observations of very cool loops. The effect of a loop's geometry and field line divergence on the structure is found. Results predicted from scaling laws are compared, and the uniqueness of the solution for a loop with a fixed mass is studied. The error in the predicted emission measure through assuming uniform pressure is shown to be considerable. The life-time of a loop can often be many days, suggesting the existence of a thermally stable state. A global stability analysis is performed, and it is found that a loop's stability may depend critically upon its length. Thermally isolated loops, which are the most unstable type, can be thermally stable, provided their pressure falls off sufficiently rapidly with height (due to hydrostatic equilibrium).

The nature of motions in intense magnetic fields is investigated. For a flux tube in a uniform atmosphere a dispersion relation is derived for the modes of vibration and analytic approximations are obtained for a slender tube. In a stratified atmosphere an expansion procedure is used to derive an equation for the vertical velocity perturbation. The behaviour of motions within the flux tube is shown to depend upon a transition frequency ω_v such that vertically propagating waves are possible only for frequencies greater than ω_v. Also, the nature of convective instability in a slender magnetic flux tube is explored. A sufficient condition for stability is derived for the case of an arbitrary temperature profile in the external medium. For a tube of infinite depth, with a uniform-temperature gradient inside the tube equal to that in the exterior, a necessary and sufficient condition for convective stability to occur inside the tube is derived. Under the assumptions of the model, intense flux tubes are convectively stable if sufficiently shallow (with depths 1 - 2 x 10³ km or less). Tubes that extend deeper into the convection zone are potentially (convectively) unstable, but may be stabilised for sufficiently strong magnetic fields. Radiative damping of waves is important in the upper photosphere and the effect of radiative relaxation on the propagation of waves in an intense flux tube is examined both for a uniform and stratified atmosphere. The cut-off frequency is generalized to include the effects of radiative relaxation. The phase-shift between velocity oscillations at two different levels and the phase difference between temperature and velocity perturbations are derived and compared with the available observations. Finally, the consequences of the observed steady downflow are discussed.

Current sheets are widely believed to play an important role in astrophysics when regions of magnetic flux are in motion. Several models based on the formation of current sheets have been proposed to explain such phenomena as geomagnetic storms, solar flares and prominences. In this thesis three aspects of current sheet theory are studied with particular reference to the solar flare problem. Firstly the development of two-dimensional current sheets is investigated for several simple configurations. These include converging line current sources, converging and diverging line dipole sources and a dipole of increasing moment situated in either a uniform magnetic field or a constant dipole field. These last two may be thought of as modelling the emergence of bipolar flux from beneath the photosphere, a phenomena frequently observed prior to solar flares. The length, position and shape of the current sheet is determined from the requirement that the magnetic field be frozen-into the plasma. The sheet is found to be curved, except in the symmetrical case of converging line sources. In addition, the extra energy due to the presence of the current sheet is determined. Comparison with estimates of the energy dissipated during a flare indicate that the formation of current sheets may store an adequate amount of preflare magnetic energy, provided no reconnection occurs during the formation process. A three-dimensional axi-symmetric model for current sheet formation is then considered. Two equal and co-directional dipoles approach along the axis of symmetry to form an annular current sheet between them. The equations determining the magnetic field for this configuration are reduced to a single integral equation for the current density in the sheet as a function of radial distance from the axis. A numerical method is used to solve this integral equation. The inner and outer radii of the sheet are then determined from the conditions of flux conservation as for the two-dimensional case. Finally the energetics of a current sheet that forms between newly emerging flux and an ambient field are considered. As more and more flux emerges, so the sheet rises in the solar atmosphere. The various contributions to the thermal energy balance in the sheet are approximated and the resulting equation is solved for the internal temperature of the sheet. It is found that, for certain choices of the ambient magnetic field strength and velocity, the internal temperature increases until, when the sheet reaches some critical height, no neighbouring stable state exists. The temperature then increases rapidly seeking a hotter branch of the solution curve. During this dynamic heating the threshold temperature for the onset of microinstabilities may be attained. It is suggested that this may be a suitable trigger mechanism for the recently proposed "emerging flux" model of a solar flare.

A study has been made of the global solar oscillations known as p-modes. The Sun is represented by a plane-parallel stratified plasma. Solutions are found to the magnetohydrodynamic equations of motion in such a plasma, and normal mode frequencies are calculated by applying realistic boundary conditions to these solutions. The normal modes model solar p-modes. For a model consisting of an isothermal chromosphere with a uniform horizontal magnetic field, it is demonstrated that modes may form at all frequencies. Consideration is also given to the related problem of vertical propagation of fast magnetoacoustic waves in a uniform magnetic field. An investigation is carried out into the observed solar cycle variations in the frequencies of p-modes in the classical, low frequency range (1-5 mHz). A possible mechanism for the observed "turnover" effect is discussed. Through the use of a modified Bohr- Sommerfeld condition, the effect of a non-isothermal chromosphere is also considered, and a physical description of chromospheric effects on p-mode frequencies is given. The formation of modes above the acoustic cut-off frequency is investigated. The theoretically calcidated forms of frequency shift curves in this high frequency range agree well with observations. The special case of modes of degree zero is also briefly examined. A mathematical formulation for such modes is constructed, and frequency shifts are determined for a simple chromospheric model atmosphere.

Magnetohydrodynamic stability theory provides a powerful tool for understanding and testing hypothesized mathematical and physical models of observed phenomena on the surface of the Sun. In this thesis, the problem of applying the 'correct' boundary conditions at the photospheric/coronal interface used in modelling coronal arcades is tackled. Then some aspects of the stability of coronal loops and arcades are investigated using a Fourier truncated series approximation for the equation of motion. The problem involving the boundary conditions has been the subject of a controversy for the past decade with two principal conditions suggested, the 'rigid-wall' conditions where all perturbations vanish at the interface, and 'flow-through' conditions where flows parallel to the equilibrium magnetic field take place. By modelling the photosphere and corona as two different density regions and then varying the ratio of the densities of the two regions, growth rates and eigen-functions of both ideal and resistive modes are investigated in order to follow the evolution of the modes as the density ratio is increased. In order to simplify the analysis, the 2-D equations are reduced to 1-D equations by taking a WKB approximation for the spatial variations across the field to give a localized ballooning approach with ordinary differential equations along the fieldlines. Stability of coronal loops to kink modes transformed to localized modes by increasing the poloidal wavenumber, m, is investigated. Two fields generated numerically from the Grad-Shafranov equation and three analytic fields are investigated in detail and the effect of pressure on the marginal loop length is found, both for near force-free conditions such as is found in the solar corona, and away from force-free conditions. It was found that for near force-free conditions, kink modes are the most unstable with localized modes the most stable. As pressure and pressure gradients become important, there is a reversal in the most unstable modes with localized modes the most unstable.

The magnetohydrodynamic (MHD) model is a simple mathematical model that treats a plasma as a perfectly conducting fluid acted upon by magnetic and pressure-driven forces. Many instabilities in plasmas can be predicted using this model. In this Thesis, aspects of the linear stability of solar and laboratory plasmas are studied using the MHD model. Firstly, we investigate the thermal instability of coronal plasmas with line-tied magnetic fields and with anisotropical heat conduction, using an analytical analysis which concentrates on isobaric perturbations, and a time-dependent numerical code. We find that including perpendicular thermal conduction means that condensations are restricted to a narrow layer around the region where the local isobaric growth rate is largest and that, while the growth rate of the thermal mode is largely unaffected by perpendicular thermal conduction, this may be an important factor in determining the lengthscale for the width of condensations. Secondly, the effect of a finitely conducting wall on the linear stability of Spheromak and Reversed Field Finch equilibria is investigated. We find growth rates for the modes that are present because of the finite resistivity of the wall, which grow proportionally to the "long" time constant of the wall. Finally, we apply a tractable method, derived by De Bruyne (1990), for investigating the stability of 2-D line-tied magnetic fields, to cylindrically symmetric spheromak equilibria. The method involves the solution of two sets of ordinary differential equations, integrated along the field lines, which give necessary and sufficient conditions for stability. The role of plasma pressure and of the width of the entrance region are investigated.