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.

An improved equation is derived for the interchange mode in the limit k‖ = 0. The possibilities are discussed of explosive instabilities and spectrum cascade processes by mode coupling in interchange mode turbulence in two limits.

The method of generating steady azimuthal currents in a cylindrical plasma column by means of a rotating magnetic field is investigated numerically and analytically for the case where an external steady azimuthal magnetic field is applied to the plasma. For a plasma of low resistivity, the effect of imposing a strong azimuthal field is a substantial reduction in the amount of current that can be driven. For a resistive plasma the effect is more complicated. In some cases the azimuthal field enhances current drive as has been observed in recent experiments.

This paper gives an extensive analytical and numerical characterization of the growth-rate curves (imaginary frequency versus wavenumber) derived from the free electron laser dispersion relation for a warm relativistic electron beam propagating through a constant-amplitude helical magnetic wiggler field. The electron beam is treated as infinite in transverse extent. A detailed mathematical analysis is given of the exact dispersion relation and its Compton approximation for the case of a water-bag equilibrium distribution function (uniform distribution in axial momentum pz). Applicability of the water-bag results to the case of a Gaussian equilibrium distribution in pz is tested numerically. One result of the water-bag analysis is a set of validity conditions for the Compton approximation. Numerical and analytical results indicate that these conditions are applicable to the Gaussian case far outside the parameter range where the individual water-bag and corresponding Gaussian growth rate curves agree.

Ordinary solutions of the weakly relativistic, electromagnetic dispersion relation are investigated for waves propagating perpendicular to a uniform magnetic field in a Maxwellian plasma. Weakly relativistic resonance broadening, frequency downshift and damping are found to alter dramatically the dispersion predicted by the corresponding strictly non-relativistic (‘classical’) theory in the neighbourhood of harmonics of the cyclotron frequency Ωe. All classical resonances and cut-offs are removed except the cut-off at the plasma frequency ωp. At frequencies above ωp the infinite family of classically predicted modes is replaced by a single weakly damped mode whose dispersion differs only slightly from that predicted by cold plasma theory. No weakly damped modes exist in the range of harmonics s satisfying (ωp/Ωe)⅔/8 ≲ S < Ωp/Ωe, however, one such mode is located immediately below each harmonic for s ≲ (ωp/Ωe)⅔/8. A companion paper investigates extraordinary solutions of the dispersion relation.

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.

Extraordinary solutions of the weakly relativistic, electromagnetic dispersion relation are investigated for waves propagating perpendicular to a uniform magnetic field in a Maxwellian plasma. As in a companion paper, which treated ordinary modes, weakly relativistic effects are found to modify dramatically the dispersion predicted by strictly non-relativistic ‘classical’ theory in the neighbourhood of harmonics of the cyclotron frequency Ωe. The infinite families of classical Gross–Bernstein and Dnestrovskii–Kostomarov modes are truncated to include only harmonics s satisfying s ≲ (ω2p mc2/4kB TΩ2e)⅓ and s ≲(ωp/Ωe)⅔/8 respectively where ωp is the plasma frequency and T the temperature. All classical cut-offs and resonances are removed apart from the x– and z– mode cut-offs. The only coupling between large- and small-wave-vector modes is between the z mode and a Gross–Bernstein mode near the upper-hybrid frequency and between the x mode and the second Gross–Bernstein mode near 2Ωe. Dispersion of the weakly relativistic counterpart of the x mode departs only slightly from that predicted by cold plasma theory except near Ωe and 2Ωe.

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.

Three-dimensional, pseudo-spectral computation is used to follow the evolution of a resistive, incompressible magnetofluid. The magnetofluid is confined by rigid, free-slip, perfectly-conducting square boundaries in the x, y directions (‘poloidal’ boundaries), and periodic boundary conditions are assumed in the z direction (‘toroidal’ direction). A constant, uniform d.c. magnetic field B0 is assumed in the z direction and a non-uniform current density j flows along it initially. Starting from a non-equilibrium hollow current profile, the evolution is followed for several tens of Alfvén transit times. Considerable small-scale turbulence develops, which causes energy to decay more rapidly than magnetic helicity. The average toroidal magnetic field at the (x, y) boundary reverses sign spontaneously. The near spatial constancy of the ratio jB/(jB) ≡ cos θ, in the relaxed state at late times, suggests that the state is nearly force-free. However, the ratio j. B/B2 ≡ α is considerably less uniform than is cos θ suggesting more residual disorder than a pure minimum-energy state would display.

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.

In the above mentioned paper, table 1 is incorrect and the correct version is given below. Thus the result which was there obtained, namely that the coefficient of the nonlinear term of the modified KdV equation becomes negative, is not correct and, in fact, the coefficient is always positive.

The theory of the slow reaction of charged particles in the presence of a high-frequency electromagnetic field (oscillation-centre motion) is developed by using a Hamiltonian formalism with non-canonical variables and pseudo-canonical transformations. The flexibility introduced by the latter features allows us to construct a theory which is manifestly gauge-invariant and involves only physical concepts (electromagnetic fields and particle velocities instead of potentials and canonical momenta). A complete description of the oscillation-centre dynamics is derived. The known expressions of the ponderomotive force are derived as special cases of our theory.

The effect of the ion temperature on ion acoustic solitary waves in a collisionless relativistic plasma is discussed using the Korteweg–de Vries equation. The phase velocity of the ion acoustic waves decreases as the relativistic effect increases, and increases as the ion temperature increases. Only a compressional soliton of the ion acoustic wave is formed in this system. Since its amplitude increases for the lower ion temperature as the relativistic effect increases, we deduce the formation of a precursor by the presence of the streaming ions. In contrast, for the higher ion temperature, the amplitude decreases slowly. Furthermore, it is shown that the oscillatory solution of the Korteweg–de Vries equation smoothly links with the nonlinear Schrödinger equation in a relativistic plasma.

The motion of a single particle under the influence of the ponderomotive force directed perpendicular to the external magnetostatic field is analysed. By solving the exact equation of motion for a specific applied electromagnetic field, the resultant ponderomotive drift is compared with the prediction of a single-particle theory using the oscillation-centre approximation. The regime of validity of this theory is discussed. It is shown that, for certain values of the amplitude and frequency of the electromagnetic field, the particle motion is unstable and therefore the concept of a single-particle ponderomotive force is meaningless.