The motion of electrons and ions in the self-consistent fields of a compact toroidal equilibrium maintained by means of a rotating magnetic field is studied. It is found that the particles are confined although the lines of the instantaneous magnetic field are open. The results are compared with those obtained in an earlier study of the motion of charged particles in the self-consistent fields appropriate to cylindrical plasma equilibrium maintained by means of rotating magnetic fields.

The role of relativistic effects in the problem of drift instabilities in an inhomogeneous magnetized plasma is analysed. The kinetic approach is developed, taking into account a non-zero plasma pressure and the electromagnetic character of the perturbations. The general dispersion relation for the perturbations in an inhomogeneous relativistic plasma is obtained. As in the case of a non-relativistic plasma, the dispersion relation is written in terms of the modified dielectric permittivity tensor. This tensor is calculated by taking into account the non-Maxwellian character of the equilibrium momentum distributions of the particles. The relation between the modified dielectric permittivity tensor and the ordinary tensor of dielectric permittivity of a homogeneous plasma is elucidated.

The generalization of the kinetic theory of low-frequency long-wavelength drift instabilities (instabilities due to the number density and temperature gradients and an instability of the Kelvin–Helmholtz type) is given for the case of a relativistic plasma. The role of relativistic effects, taking into account the anisotropy of the particle momentum distribution function, is clarified. Particular attention is paid to the investigation of instabilities of these perturbations in a relativistic electron-positron plasma.

This paper considers the linear theory of ion-acoustic-like instabilities in a homogeneous Vlasov plasma with two ion components, a less dense beam and a more dense core, with a relative drift velocity. Numerical solutions of the full electrostatic dispersion equation are presented, and the properties of the ion–ion acoustic instability are studied in detail. The following properties are demonstrated: (i) At relatively cold beam temperatures, the instability is fluid-like, but it becomes a beam resonant kinetic instability as the beam temperature becomes of the order of the core temperature; (ii) if the mode is unstable, its threshold lies well below the threshold of the electron–ion acoustic instability; (iii) an electron temperature anisotropy T⊥/T‖e > 1 enhances the instability and (iv) at sufficiently large beam-core relative drift speeds, electron magnetization can either detract from or enhance the instability. If field-aligned ion beams are to drive the ion-acoustic-like enhanced fluctuations observed upstream of the Earth's bow shock, the effective beam temperature must be much smaller than values quoted in the literature.

The extraordinary-mode eigenvalue equation is used to investigate the local stability properties of relativistic, non-neutral electron flow in a planar diode. The local stability analysis assumes gentle equilibrium gradients and short perturbation wavelengths. The lowest-order local dispersion relation is derived assuming that localized solutions for the eigenfunction exist, and stability properties are investigated numerically over a wide range of System parameters for perturbations with frequency small in comparison with the electron cyclotron frequency. It is found that the local dispersion relation supports three solutions in this frequency regime. One of the solutions corresponds to a stable diocotron mode driven by the local density gradient. The other two branches are found to exhibit instability over a wide range of electron density. These modes are electromagnetic in nature and require relativistic electron flow with velocity shear in order for instability to exist. Moreover, the growth rate of the unstable electromagnetic mode can be substantial (a few per cent of the electron cyclotron frequency).

Two coupled equations describing the nonlinear evolution of modified convective cells with a finite but small wavenumber parallel to the external magnetic field B are derived. It is found that, for oblique propagation, the electrostatic and magnetostatic convective mode do not decouple as in the strictly perpen-dicular case. Various aspects of the governing equations are investigated. In the weakly nonlinear lirait, a three-wave interaction study shows the possibility of dual energy cascading, or of cascading to larger wavenumbers only, depending on the direction of the parallel wavenumber component of the pump wave. Furthermore, solitary wave solutions in the form of double vortices are considered.

Nonlinear waves and solitons satisfying the Zakharov-Kuznetsov equation for a dilute plasma immersed in a strong magnetic field are studied numerically. Growth rates of perpendicular instabilities, found theoretically in part 1, are confirmed and extended to arbitrary wavelengths of the perturbations (the calculations of part 1 were limited to long-wave perturbations). The effects of instabilities on nonlinear waves and solitons are illustrated graphically. Pre-vious, approximate results of other authors on the perpendicular growth rates for solitons are improved on. Similar results for perturbed nonlinear waves are presented. The effects of two-soliton collisions on instabilities are investigated. Rather surprisingly, we find that the growth of instabilities can be retarded by collisions. Instabilities can also be transferred from one soliton to another in a collision. This paper can be read independently of part 1.

The method of multiple scales is used to derive a nonlinear Schrödinger equation, which describes the nonlinear evolution of electron plasma ‘slow waves’ propagating along a hot cylindrical plasma column, surrounded by a dielectric medium and immersed in an essentially infinite axial magnetic field. The temperature is included as well as mobile ion effects for ail possible modes of propagation along the magnetic field. From this equation the condition for modulational instability for a uniform plasma wave train is determined.

Special relativistic magnetohydrodynamic shock waves in a perfect gas of infinite conductivity and constant adiabatic index are analysed. It is shown that the Rankine-Hugoniot equations for such shocks may be reduced to a seventh degree polynomial for the downstream dynamical volume ω, with the polynomial coefficients depending on the upstream state (ω equals the specific volume times the ratio of the energy density of the fluid (omitting electromagnetic terms) to the fluid rest mass energy density). In the non-relativistic limit, the polynomial equation reduces to a relation between the upstream and downstream Alfvénic Mach numbers, previously obtained by Cabannes. The equation for ω classifies in a natural way both shocks in which the electric field may be eliminated by transforming to the de Hoffman–Teller frame, and shocks for which this is not possible. The equation is used to determine the downstream state of relativistic shocks for a given upstream state as specified by the plasma beta, magnetic field obliquity, and flow speed.

We consider obliquely propagating (with respect to the ambient field) nonlinear magnetosonic waves in a hot plasma with arbitrary beta. It is shown that the two-dimensional propagation of both the fast and the slow modes is governed by the Kadomstev–Petiashvilli soliton equation. Explicit expressions are obtained for the various physical quantities involved via the soliton solution.

Thermal effects on the dispersion of right-handed (RH) electron cyclotron waves propagating parallel to a uniform, ambient magnetic field are investigated in the strictly non-relativistic (‘classical’) and weakly relativistic approximations for real frequency and complex wave vector. In each approximation, the two branches of the RH mode reconnect near the cyclotron frequency as the plasma temperature is increased or the density is lowered. This reconnection occurs in a manner different from that previously assumed at parallel propagation and from that at perpendicular propagation, giving rise to a new mode near the cold plasma cut-off frequency ωxC. For both parallel and perpendicular propagation, it is noted that reconnection occurs approximately when the cyclotron line-width equals the width of the stop-band in the cold plasma dispersion relation. Inclusion of weakly relativistic effects is found to be necessary for quantitative calculations and for an accurate treatment of the new mode near ωxC. Weakly relativistic effects also modify the analytic properties of the dispersion relation so as to introduce a new family of weakly damped and undamped solutions.