A pair of modified Zakharov equations coupling the high-frequency Langmuir oscillations to the low-frequency ion-acoustic perturbations in a turbulent, collisional plasma with a neutral background is derived. The time evolution of the coupled ion-acoustic–Langmuir solitons show that the electron–ion and electron–neutral collisions damp the Langmuir oscillations and let the ion density perturbations radiate away. On the other hand, the ion–neutral collisions damp the ion density perturbations letting the Langmuir oscillations flow out. In the case of ion viscosity, however, both the ion density perturbations and the Langmuir oscillations radiate away one after the other. In these processes, electron–ion and electron–neutral collisions seem to dominate over the other two.

We present a general formulation for treating the linear stability of inhomogeneous plasmas for which at least one species is described by the Vlasov equation. Use of Poisson bracket notation and expansion of the perturbation distribution function in terms of eigenfunctions of the unperturbed Liouville operator leads to a concise representation of the stability problem in terms of a symmetric dispersion functional. A dispersion matrix is derived which characterizes the solutions of the linearized initial-value problem. The dispersion matrix is then expressed in terms of a dynamic spectral matrix which characterizes the properties of the unperturbed orbits, in so far as they are relevant to the linear stability of the system.

The solution of the linearized Vlasov equation is given for an arbitrary equilibrium Hamiltonian in which there is only one non-ignorable co-ordinate. The solution written in terms of integrals with respect to time which only extend over the bounce period of an equilibrium orbit in its equivalent one-dimensional potential. A closed-form solution and a solution based on a Fourier expansion are given. Explicit formulae are presented for Cartesian and cylindrical co-ordinates.

In this third of a series of papers on the stability of Vlasov equilibria, the dispersion functional technique developed in part 1 is applied to the multi-species Vlasov plasma and to the Vlasov-fluid model. An alternative form of the dispersion functional is derived for the Vlasov-fluid model in which magnetohydrodynamic and kinetic aspects of the problem are separated explicitly. A necessary and sufficient condition is derived for stability with the Vlasov-fluid model. Stability of the multi-species Vlasov plasma is discussed and a necessary and sufficient condition for stability near a marginal point is derived. Explicit formulae for quantities occurring in the dispersion functional are given in cylindrical co-ordinates for the Vlasov-fluid model when there is one nonignorable co-ordinate in the equilibrium. The dispersion functional for a multi-species Vlasov plasma is shown to be related to the ponderomotive Hamiltonian.

A systematic study is made of the dynamical properties of the charge density fluctuations in strongly coupled two-component plasmas. The time-correlation function of the partial density fluctuations is described with the use of the memory function formalism. A second-order memory function is approximately expressed in terms of its self-part by taking into account the total momentum conservation law. After assuming a Gaussian time-dependence for the self-part, the excitation spectrum of the charge density fluctuation is calculated for the semi-classical hydrogen plasma, and is in fairly good agreement with the molecular dynamics calculation due to Hansen & McDonald. It is found that the collisional effects play an important role in the strongly coupled system.

Wave excitation by an inhomogeneous suprathermal electron beam in a homogeneous magnetized plasma is investigated. Not only is the beam density non-uniform, but the beaming electrons also have a sheared bulk velocity. The general dispersion equation including both electrostatic and electromagnetic effects is derived. Particular attention is paid to the whistler mode. It is found that the density-gradient and velocity-shear effects are important for waves with frequencies close to the lower-hybrid resonance frequency.

Certain unfamiliar terms in renormalized plasma turbulence theory are interpreted in terms of the familiar physics of the Balescu–Lenard collision operator. Specifically, it is argued that the so-called polarization parts of the operator which renormalizes the particle propagator are related to fluctuations of the Fokker–Planck coefficient which describes polarization drag, and to fluctuations of the effective dielectric function of the medium.

A systematic and self-consistent analysis of the problem of Langmuir solitons in the entire range of Mach numbers (0 < M < 1) has been presented. A coupled set of nonlinear equations for the amplitude of the modulated, high-frequency Langmuir waves and the associated low-frequency ion waves is derived without using the charge neutrality condition or any a priori ordering schemes. A technique has been developed for obtaining analytic solutions of these equations where any arbitrary degree of ion nonlinearity consistent with the nonlinearity retained in the Langmuir field can be taken into account self-consistently. A class of solutions with non-zero Langmuir field intensity at the centre (ξ = 0) are found for intermediate values of the Mach number. Using these solutions, a smooth transition from single-hump solitons to the double-hump solitons with respect to the Mach number has been established through the definitions of critical and cut-off Mach numbers. Further, under appropriate limiting conditions, various solutions discussed by other authors are obtained. Sagdeev potential analyses of the solutions for the Langmuir field as well as the ion field are carried out. These analyses confirm the transition from single-hump solitons to the double-hump solitons with respect to the Mach number. The existence of many-hump solitons for higher-order nonlinearities in the low-frequency ion wave potential has been conjectured. The method of solution developed here can be applied to similar equations in other fields.

The hydrodynamic approach to a relativistic gas is studied on the basis of methods used by Chew, Goldberger & Low and by Scargle. As the result of this study, the explicit form of the energy–momentum tensor is obtained by straight-forward application of Lorentz transformation. The term corresponding to non-zero momentum density in the plasma frame of reference is included in the energy–momentum tensor. For the limiting case of small bulk velocities compared with the speed of light in vacua, the energy flux as described by non- relativistic theory is immediately recovered. For the special case of scalar pressure, the energy–momentum tensor considered by Taub and by Harris follows directly from our expression. In a small-perturbation approximation, it is possible to close the system of MHD equations. As a result the solution describing all possible wave modes is derived. This solution coincides with the solution obtained by means of the kinetic theory.

Kaira & Kathuria used the method of multiple scales to develop nonlinear analysis of Rayleigh–Taylor instability of a plasma in a magnetic field. Their calculations remain valid only for wavenumbers k away from the linear cut-off value kc, and break down for wavenumbers near kc. The purpose of this paper is to treat the latter case. The solution uses the method of strained parameters. The results show the instability persists even at k = kc, despite the cut-off predicted by the linear theory.

This paper is concerned with the Boltzmann collision integral for the one-particle distribution function of a test species of particle undergoing elastic collisions with particles of a second species which is in thermal equilibrium. This expression is studied as a function of the ratio of the masses of the test and host particles for the case when the test particle distribution function is isotropic in velocity space. The analysis can also be considered as referring to the zeroth-order spherical harmonic in velocity space of a general velocity distribution function. The resulting collision term, due originally to Davydov, is of Fokker–Planck form and effectively describes a diffusion in energy. The method of derivation employed here is more systematic than hitherto, and is used to calculate the first correction to the Davydov term. Differences between classical and quantum cross-sections are considered; the correction to the Davydov term is checked by means of a comparison with the exact solution of the associated eigenvalue problem for the special case of Maxwell interactions treated classically.

Using similarity methods, an investigation is made of a generalized nonlinear diffusion equation arising in plasma physics in connexion with several recently proposed models for turbulent plasma energy transport and particle diffusion.

In almost every field of physics, diffusion or transport equations of the form Ψt = Δ. (DΔΨ) play an important role. The theory of linear diffusion equations, where the diffusion constant D is independent of Ψ, has been developed to a high degree of sophistication (e.g. Carsiaw & Jaeger 1959).

An important aspect of pre-flare current sheets in the solar atmosphere is the sudden enhancement of the effective electrical resistivity in the sheet due to the onset of a plasma micro-instability. Numerical and analytical solutions to the isothermal MHD equations are here presented that describe the evolution of a current sheet subsequent to such an enhancement in the resistivity. The solutions show that, if the initial width of the current sheet is less than the acoustic-diffusion length obtained by dividing the resistivity by the sound speed, then isomagnetic shocks are formed. These shocks propagate outward from the the centre of the current sheet and are transformed into fast-mode magneto-acoustic waves when they reach the edges of the current sheet. The fast-mode waves thus formed continue to propagate outward beyond the confines of the current sheet. In contrast to a previous study by Cheng, the present solutions demonstrate that flow speeds several times greater than the local fast-mode wave speed can be produced if the plasma beta parameter and the initial sheet width are sufficiently small. The results may be relevant to the triggering of a solar flare, as in the emerging flux model of flares.

A perturbation method is applied to the pair of second-order, coupled, nonlinear differential equations that describe the propagation, through a cold electron plasma, of plane waves of fixed profile, with direction of propagation and electric vector perpendicular to the ambient magnetic field. The equations are expressed in terms of polar variables π, φ, and solutions are sought as power series in the small parameter n, where c/n is the wave speed. When n = 0 periodic solutions are represented in the (π,φ) plane by circles π = constant, and when n is small it is found that there are corresponding periodic solutions represented to order n2 by ellipses. It is noted that further investigation is required to relate these finite-amplitude solutions to the conventional solutions of linear theory, and to determine their behaviour in the vicinity of certain resonances that arise in the perturbation treatment.

In a recent paper, Balescu (1980) criticizes the self-similar solution in the problem of anomalous plasma resistivity (Vekstein, Ryutov & Sagdeev 1970) and comes to the conclusion that this solution is not correct. The aim of this comment is to show why Balescu's arguments are erroneous.

We consider here the simplest case of the one-dimensional collisionless plasma in the presence of an external electric field.

I cannot agree with the reasoning of Vekstein, Ryutov & Sagdeev (1982) (VRS), the more so as I see no basic contradiction between their results and mine.

The basic equations of the quasi-linear theory are the kinetic equations determining the time evolution of the distribution functions of the electrons and ions, ∂tfe,i and of the spectral densities ∂tψσ(κ) (plus the definition of the diffusion coefficient and the dispersion relations). It is agreed that these equations must be satisfied in any event. The marginal stability is an added approximation (as VRS, of course, admit), and can only be accepted if it is compatible with the basic equations.