This thesis will deal with certain problems of parametric instabilities in the inhomogeneous plasma. A large amplitude, 'pump' wave can deposit some of its energy into the plasma through resonance with two lower frequency waves (which may be damped). This type of process is a parametric decay of the pump wave and has applications in many fields. We consider, predominantly, that of laser fusion, in which the pump wave is electromagnetic and incident on the plasma. The objective is to deposit as much energy as possible within the plasma. Instabilities reducing this energy input are therefore of importance and it is, mostly, to these that this thesis will turn. They are mostly scattering processes in which one of the decay modes is electromagnetic. We examine the stimulated Brillouin backscattering process (the other decay mode being an ion accoustic wave) from a reference frame in which the plasma is streaming outwards. It is found that, if this velocity is near the sound velocity, the ion acoustic wave has a frequency Doppler-shifted to zero, the electromagnetic waves then having equal frequencies. In such a situation, any reflection of the pump wave at the critical surface will enhance the initial level of the backscattered wave. We find that, allowing for this, there is considerable enhancement of backscatter from the plasma, with consequent energy loss to the pump. Since the effect is noticeably unaffected by 'off- resonance' situations, it is felt that this process could mount a barrier to possible applications. We next consider the stimulated Compton scattering process, where the pump is scattered off the 'bare' or thermal electrons in the plasma. It is found that this rather weak instability occurs predominantly only when electron plasma waves are heavily dampled. Substantial reflection only occurs for high pump powers. Whilst there is little loss to the pump energy, there is substantial perturbation to the background distribution function. However, at the high powers involved filamentation and modulation of the pump can occur with a resulting enhancement of the scattering. Finally, we consider the effect on the decay instability (photon → plasmon + phonon) of the presence of substantial filamentation of the critical surface. It is found that the growth rate is substantially reduced.

The work of this thesis is concerned with the investigation of the propagation of waves in a magnetized plasma containing various parameter gradients, and with the stability of ion acoustic waves in a weakly collisional plasma with a strong temperature gradient. The thesis is divided into three sections. In the first section the intention is to derive in a compact and unambiguous tensor form the dispersion relation describing the propagation of waves in a magnetized plasma containing three-dimensional density and temperature gradients, an E̲⏜ B̲ drift, and differing temperatures parallel and perpendicular to the magnetic field. This is achieved by introducing and extending the polarized co-ordinate system first proposed by Buneman in 1961, and then carrying through the standard procedure of integration along unperturbed trajectories. The "local" approximation of Krall and Rosenbluth is used in order that an analytic result may be derived. The dispersion relation obtained includes certain moment tensors whose elements may be evaluated independently of the gradients involved in the problem. These elements may then be listed and the list referred to in order to obtain the elements required for a specific problem. The second section is concerned with the use of the theory and results of J.P. Dougherty to show that in the high-frequency regime the introduction of a small amount of collisions into a plasma is sufficient to disrupt the gyro-resonances which allow the existence of Bernstein waves at multiples of the gyro-frequencies perpendicular and near- perpendicular to the magnetic field. It is shown that a collision frequency v such that (k ρ) ⁻² ≲ v/Ω < (k ρ) ⁻¹ where k ρ >> 1 is sufficient to do this; k is the wave-number, ρ the Larmor radius, and the gyro-frequency. It is also shown that in this case the ion-acoustic dispersion relation is valid even for propagation perpendicular to the magnetic field. In the final section the result of the second section is used to derive a dispersion relation for high-frequency wave propagation in a weakly-collisional plasma containing an electron temperature gradient. The dispersion relation is solved numerically for various electron-ion temperature ratios and electron temperature gradient drift velocities. Earlier predictions, based on analytic calculations for small temperature ratios and drift velocities, are confirmed and some new results presented. In particular, it is shown that a temperature gradient is a more effective destabilizing agent then a simple drift between ions and electrons. Dispersion plots are given, along with analytic and physical explanations of their form; finally neutral stability curves are presented. The thesis concludes with a summary of the results obtained.

The question of topology in the coronal magnetic field is addressed in this thesis. Magnetic reconnection, which plays a major role in many of the fascinating phenomena seen in the solar atmosphere, is likely to occur at the boundaries between different topological regions of the magnetic field. By modelling the coronal field using discrete sources of flux, to represent the concentrations seen at the photospheric surface, we study the varying topological structures present in the field. We generate a criterion for determining the presence of null points above the photospheric surface and establish that any separatrix surfaces present in the field are due to the presence of either null points, or regions where the field tangentially grazes the surface. We follow the evolution of these separatrix surfaces and, in particular, determine the existence of a well-defined separator field line in the absence of coronal null points. Finally, we look locally at the configuration of the magnetic field in the region surrounding a straight current sheet. We derive an analytical expression to describe the topology of both potential and constant-current force-free fields in the neighbourhood of a sheet, and in so doing generalise the previously known expressions.

Quiescent solar prominences are amongst the most interesting and yet least understood of the phenomena observed on the Sun and provide both the theorist and the observer with equally demanding challenges. The theoretical study of prominences is an important branch of solar physics as it contributes significantly to the overall understanding of the Sun and its atmosphere. One only needs to be presented with the illuminating fact that there is more mass contained in these bodies than in the remainder of the entire corona to be convinced of their importance. Although many of the physical mechanisms associated with prominence theory are important in their own right, they are also of much wider relevance for various other astrophysical phenomena. For example, radiative and magnetic instabilities are explored in detail in the context of solar prominences; yet clearly these are important processes that relate to many other branches of astrophysics. Prominences are intimately associated with solar flares which occur when a prominence loses equilibrium. Also, prominence eruptions are very important as they are closely connected with coronal mass ejections. These account for a large fraction of the total mass lost from the Sun and so are extremely important events, particularly when one considers the consequences as this plasma interacts with the Earth's environment. It is the period of global equilibrium of quiescent prominences, though, that is the focus of this thesis. Various models are proposed to help understand both the topology and supporting mechanisms of the external, coronal magnetic field, and also the internal prominence structure and the way in which the two regimes fit together. In Chapter 3 we extend a model for the equilibrium of a prominence sheet in a twisted magnetic flux-tube, given by Ridgway, Priest and Amari (1991), to incorporate a current sheet of finite height. This removes the discontinuity at the edge of the tube and provides a shear-free outer boundary which enables the tube to be matched onto a background potential field. In addition, internal prominence solutions are found by expanding the sheet to a finite width and matching suitable magnetic profiles across this region. Next we consider a global model for the magnetic field structure surrounding a polar-crown prominence. We examine potential configurations generated from typical distributions of photospheric flux, and select solutions for which there is a location of dipped magnetic field where prominence material may collect and form. Once such a configuration is available, it is necessary to construct the ensuing prominence solution. We achieve this in Chapter 4 by considering a simplified form for the photospheric field. We show that the equilibrium contains a weighted, curved prominence sheet supported in the location of dipped magnetic field. The equilibrium requires an enhanced magnetic pressure below the sheet to support the component of weight in the normal direction. The internal equilibrium of curved or inclined prominence material has not been considered previously and so we formulate, in Chapter 6, a simple one-dimensional isothermal solution for a cut across the prominence. This is developed to allow for variations along the sheet and in this way an internal solution for the curved prominence of Chapter 4 is given, which matches onto the external potential polar-crown field. Finally, in Chapter 7, we rewrite this solution in terms of its constituent internal and external components and show how the composite solution switches between the two in a region of overlap, or transition region. From this, the internal plasma properties are deduced and realistic profiles for the pressure, density and temperature are obtained.

An outstanding problem in the field of nuclear fusion research is the precise mechanism by which a hot, magnetically inhomogeneous plasma is heated when illuminated by a constant beam of small amplitude radio waves matched in frequency to harmonics of the ion Larmor frequency. An accurate model must include microscopic dynamics and inevitably a kinetic theory is required. Highly energetic ions (> 1MeV) born from fusion reactions or powered by gyroresonance have large Larmor radii (> 10cm) which are comparable in size to the wavelength of the incident radiation. In particular we will focus on fast magnetosonic waves. Exact full wave equations describing a thermal plasma in a weakly inhomogeneous field are presently at least fourth order integro-differential equations (Sauter, 1992). These are computationally taxing. Recently a method was proposed to reduce the problem to a second order integro-differential equation at the expense of information related to the propagation of mode-converted waves (Holt, 1992). We present here a generalisation of the theory to allow for arbitrary velocity-dependent equilibria while at the same time retaining a general functional form for the field profile. We consider the specific case of a bi-Maxwellian plasma immersed in a linearly inhomogenous magnetic field. We find that thermal anisotropy produces resonance localisation when the perpendicular ion temperature is greater than that parallel to the ambient field. A study of the symmetry properties of the conductivity tensor reveals that the Onsager reciprocal relations are obeyed only for an isotropic plasma in an inhomogeneous field. This is a generalisation of the result obtained by Nambu (1995). We present a generalisation of the reduction method to include effects due to changes in wave amplitude. We find that we are able to include the odd-order field derivatives responsible for energy conservation. Our numerical study of fundamental Helium-3 gyroresonance in a majority Deuterium plasma reveals that we have > 99.9% energy conservation in all cases. We show that locally-uniform theory can be very inaccurate (≃ 70% in one case presented in our recent paper, Cairns et al., 1995) particularly for higher energy ions whose non-locality is more extreme. We present a representative sample of results for minority heating and mode conversion heating schemes. We report the appearance of an unexpected cut-off on the low field side of the minority gyroresonance which may have important consequences for antennae presently placed on the outside of Tokamaks.

This thesis studies the time dependent evolution of MHD waves in cold, fully compressible non-uniform plasmas. We used a 1-D box model (e.g., Southwood (1974)) to study wave mode coupling, and concentrate upon developing an understanding of the underlying physics that governs waves in the Earth's magnetosphere. We begin by discussing the form of the (often singular) governing eigenmodes of the system, and subsequently use these eigenmodes as a basis with which to construct the solution to a variety of initial value problems. We consider a detailed analysis of both the widths and the internal length scales developed by cavity mode driven held line resonances (FLRs), and compare our results to observations presented in the literature.

Astrophysical outflows are common in a large variety of objects with very different length-scales. They can be almost spherical, as in the case of the solar wind, or show a high degree of anisotropy as in pre-main sequence stars, star-forming regions or even extragalactic objects. This work is aimed at finding exact solutions of the axisymmetric wind equations in which all variables depend not only on the distance to the central object but on latitude as well. The geometry of the stream/field-lines is taken as helicoidal and this seems to be a good approximation in some examples of collimated flows. From a simple hydrodynamic approach, a straightforward technique based on separation of the variables yields the most general solution of the wind equations under the above assumptions. The way the different variables depend on latitude is controlled by three anisotropy parameters which are related to typical ratios at the base of the atmosphere. The density needs to be higher at the equator than at the pole for the outflow to be able to accelerate. In these circumstances, the radial velocity always increases from equator to pole. Contrary to Parker's model of the solar wind, the solution does not pass through any critical point, since no polytropic law is assumed. However, the general behaviour is similar, with a high acceleration at the base and the velocity rapidly attaining an almost constant asymptotic value. The heating rate that sustains this rapid increase is mostly concentrated near the surface of the central object. The inclusion of the magnetic field in the analysis introduces two critical points: the Alfvenic point and an extra X -type point filtering the solution that gives a vanishing pressure at infinity. If the density anisotropy is too low the wind is unable to accelerate to large asymptotic values. The dependence of the angular velocity of the roots of the fieldlines with latitude reproduces well the observed rotation profile of photospheric magnetic features. The mass loss rate can be substantially increased if the structure of the outflow is highly anisotropic. Some applications to the solar wind are also discussed. In particular, recent results from ULYSSES (pointing out that solar speed increases with latitude while the density decreases from equator to the pole) are in good agreement with the general behaviour of the solutions presented in this work.

It has been supposed since the middle of this century that it is the global magnetic field surrounding a quiescent prominence that provides the force to prevent its collapse due to the sun's gravitational field. Many theoretical models, assuming that the prominence plasma is supported in a dip in the magnetic field lines associated by the magnetic tension force, have since been put forward. The aim of this thesis is to propose further models of quiescent prominences to widen our understanding and knowledge of these remarkable features. A short overview over the magnetohydrodynamic equations used to describe solar prominences, or most of the solar phenomena for that matter, are discussed in chapter 2, and a short summary of prominence observations and attempts to model them is given in chapter 3. A brief description of the numerical code used in chapters 5 and 7 is given in chapter 4. Observations of Kim (1990) and Leroy (1985) have found that most large quiescent prominences are of inverse polarity type for which the magnetic field passes through the prominence in the opposite direction to that expected from the photospheric magnetic field. Many theoretical models have been proposed, but failed. Hence, in chapter 5 we investigate first – without the inclusion of a prominence sheet – when an inverse polarity magnetic field must have the correct topology for an inverse polarity configuration before the formation of the prominence itself. Only very recently, the first basic successful model of an I-type polarity prominence was proposed by Low (1993). In chapter 6 we examine this model and investigate current sheets more complicated and realistic than the one used by Low. These analytical models deal with the force-free solution, which is matched onto an external, unsheared, potential coronal magnetic field. These solutions are mathematically interesting and allow an investigation of different profiles of the current intensity of the magnetic field vector and of the mass density in the sheet. The prominence properties predicted by these models have been examined and have been found to match the observational values. The mathematics of current sheets in general is also briefly discussed. Chapter 7 deals with numerical solutions of inverse polarity prominences embedded in a force-free magnetic flux tube, matched onto an unsheared potential coronal field. Unfortunately the solutions gained are quite sensitive to the boundary conditions imposed on them through the numerical box, showing a loss of convergence and a tendency for the solution to blow up. Finally, a short summary as well as possible future work is given in chapter 8.

Oscillatory fluid flows arise naturally in many systems. Whether or not these systems are stable is an important question and external periodic forcing of the flow may result in rich and complicated behaviours. Here three distinct oscillatory fluid flows are examined in detail, with the stability of each being established using a range of analytical and computational methods. The first system comprises standing surface capillary-gravity waves in second-harmonic resonance subject to Faraday excitation. Using the perturbation technique of multiple scales, the amplitude equations for the system are derived. At exact resonance, and with the absence of damping, the only fixed point of the equations is found to be the origin. A computational approach reveals that the amplitudes of the two waves remain either bounded or grow to infinity depending on initial data. With the introduction of detuning and damping into the system families of fixed points now exist and some special cases are considered. The second class of flows are unbounded time-periodic flows with fixed ellipsoidal stream surfaces, and having spatially uniform but time-periodic strain rates. Using a recently developed method based on theoretical study of the Schrodinger equation with quasi-periodic potential, a computational approach is adopted which determines the stability of the flow to three-dimensional plane wave disturbances. Results for the growth rate and winding number of the disturbance clearly reveal the regions of instability. It is found that almost all these flows are highly unstable. The third class is another set of three-dimensional time-periodic flows with spatially uniform strain rates. These flows are non-axisymmetric and have sinusoidally-fluctuating rates of strain directed along the fixed coordinate axes. The same computational method is employed and it is found that instability increases along with the non-axisymmetric nature of the flow.

Small brightenings called x-ray bright points (Golub et al, 1974) occur in the solar corona. They are observed with the soft x-ray telescope on Skylab to be approximately 22 Mm in diameter with a brighter inner core of width 4-7 Mm although with the Normal Incidence X-ray Telescope their dimensions are observed to be typically 6 Mm x 9 Mm. By comparison with magnetograms of the photosphere it has been noticed recently that there is a high correlation between the occurrence of x-ray bright points and the mutual reduction of flux between two opposite polarity magnetic fragments. These fragments are originally unconnected magnetically, but move towards each other and simultaneously lose equal amounts of flux (cancel): they are called cancelling magnetic features (Martin et al, 1984). The observations relating to these features were reviewed by Priest et al. (1994) who suggested that they naturally evolve through three phases: the pre-interaction, interaction and cancellation phases. From this evidence qualitative pictures of the magnetic field structure for an x-ray bright point and associated cancelling magnetic feature were established. The aim of this thesis has been to build on the ideas of Priest et al. (1994) to produce a detailed theoretical model of an x-ray bright point and a cancelling magnetic feature. The magnetic field structures are estimated, and the position and lifetime of the bright point are calculated, as is the total amount of energy released during the bright point. This work is also extended to study more complex cancelling configurations representing the main basic types of cancelling magnetic feature. The results of these models determine the factors that affect the lifetime and position of a bright point and indicate which types of cancelling magnetic features are most likely to produce bright points that are long-lived, lie directly above the cancellation site and occur simultaneously with the cancellation phase. The complex structure of a bright point cannot be explained from the above two-dimensional models: thus two recently observed bright points were studied to see if the above model could be extended into three dimensions to explain the structure seen in soft x-ray images. The available observational data was used and leads to reasonable explanations for the complex shapes of both bright points. Finally, a more realistic model for the overlying field was set up involving a model of the field above a supergranule cell field with fragments of finite width. The interaction of an ephemeral region within this field was then studied and led to five different scenarios. The results obtained reaffirmed those found in the previous simpler models and suggest where bright points may appear in a cell relative to the cancelling magnetic feature and for how long the bright points might last. Predictions for the lifetimes of cancelling magnetic features are also made, indicating when the cancelling magnetic feature occurs relative to the bright point.