The existence of a non-trivial free boundary solution of the nonlinear Grad-Shafranov equation is studied. In the case of axisymmetric toroids, the existence of the solution is proved by the standard variational approach using assumptions which are not physically restrictive. Also, the existence of a cylindrically symmetric solution is proved in the case of straight cylinders with circular cross-section.

In order to study the absorption and emission properties of a magnetized plasma in the electron cyclotron range of frequencies, the weakly relativistic (Shkarofsky) plasma dispersion functions are simply and exactly expressed in terms of the Z function. This gives a useful working form to the dielectric tensor, for any wave vector and harmonic number, covering also the case of electron Maxwellian distributions drifting along the magnetic field.

The structure of driven electrostatic oscillations is determined for a cold plasma with a zero-order density gradient along the magnetic field. A set of well-behaved structures, bounded at plasma resonance, exists for discrete values of the perpendicular wavenumber. A Green's function suitable for sources located in the overdense region of the plasma is constructed. The driven medium responds as a loss free guide.

A mathematical description of the weakly nonlinear regime of the coherent cyclotron resonance instability in the magnetosphere which has hitherto been neglected in the literature has been developed. Using the concept that the fundamental role played by the inhomogeneity of the Earth's magnetic field is to limit the interaction time between the resonance electrons and whistler mode wave, perturbation expansions of the current equations to third order have been obtained. It is then found that the real component of resonance current can qualitatively account for excess growth found in some magnetospheric transmissions while the imaginary component gives a good quantitative estimate of frequency shifts observed in transmissions of the mains harmonics.

Ion-acoustic waves can be launched into plasmas by applying an oscillating voltage wave form to an electrode. The precise mechanism by which these waves are launched from such an electrode is not fully understood. The work described here examines the modelling of sheath edge motion in connexion with acoustic wave launching in both planar and spherical geometries. A novel demonstration of the applicability of the kinetic Bohm criterion is given by using the method of characteristics. A characteristics analysis is also used to show that an expanding, but decelerating, non-planar sheath can give rise to quasi-neutral compression-like features in a plasma. It is also shown that in planar geometry neither this kind of expansion, nor an oscillatory sheath motion, generates compression-like features.

We present the first detailed experimental study of the plasma response to movement of a plane ion-rich sheath. The results show clearly that ion rarefaction waves are associated with sheath expansion while ion enhancement (‘compressive’) waves are formed on sheath collapse. The data are compared successfully with a nonlinear theory which includes the presence of a presheath prior to the sheath motion.

Expressions are derived for transport by weak electrostatic microfluctuations driven unstable by a density gradient perpendicular to a uniform magnetic field in a Vlasov plasma. These expressions, which are evaluated for the universal and lower-hybrid density drift instabilities, provide improved physical interpretations for various wave-particle exchange frequencies previously reported. It is demonstrated that wave-particle effects generally reduce the free energy driving these instabilities, and that some forms of steady-state distribution functions are inappropriate for nonlinear theory.

The first-order CGL fluid equations for electrons including the first-order heat fluxes are applied to the propagation of whistler waves. The dispersion relation of whistler waves is derived for two types of equilibrium electron distribution functions with cold and hot components. The effect of electron temperature anisotropy and the existence of cold electrons on the field-aligned propagation of whistler waves is analysed. It is shown that the electron temperature anisotropy intensifies the tendency of whistler waves to follow the lines of force of static magnetic field, that the existence of cold electrons in an anisotropic plasma further intensifies this tendency, and that under certain conditions the waves propagate only along the static magnetic field.

For the solution of Petschek's problem of field-line reconnexion, a new method is elaborated which is based on the introduction of a special co-ordinate system in which the streamlines and the magnetic lines of force become co-ordinates simultaneously. We have constructed the zero-order and the first-order approximation (for small Alfvén Mach numbers) for the solution of Petschek's problem in the steady-state, compressible, two-dimensional symmetric case. It is shown that the density across the slow shock wave increases by a factor

and the pressure by

(β = 8πρ0/B20, γ being the adiabatic exponent), and the plasma accelerates up to the Alfvén velocity. On the bases of the results obtained and of the analysis of numerical experiments on the reconnexion problem we draw the conclusion that during the initial phase of the process there develops a current sheet as described by Syrovatskii and that simultaneously there is a development of the tearing mode instability whose nonlinear phase creates the condition for the reconnexion process in the sense of Petschek.

A dispersion relation is derived for axisymmetric perturbations of an infinitely extended circular incompressible Z pinch with a step-like volume current profile. This profile is characterized by constant but different volume currents in different regions of the plasma and at the step surface there is a sheet current. The stability boundaries are shifted compared with stability limits in ideal MHD theory. For equilibria with no current reversal there is a new stable range whereas for equilibria with current reversal there is a new unstable range. The number of solutions of the dispersion relation depends on the equilibrium. The behaviour of the eigenvalues near the stability boundaries is treated in accordance with bifurcation theory.

A detailed mathematical analysis of plane steady-state reconnexion is given for the case when the plasma parameters and the magnetic fields are not identical on both sides of the current sheet. Asymptotic solutions in the sense that the inflow velocity is much less than the local Alfvén velocity as well as the arrangement of shock waves are obtained. Rotational (Alfvén) waves, slow shock waves, rarefaction waves (expansion fans), and a contact discontinuity may occur. Four different types of solution, corresponding to different shock wave configurations, are possible. They depend on the parameters of the inflow regions in a unique way.