This paper considers the steady two-dimensional ‘magnetic bottle’ in which a moving, compressible and electrically conducting plasma is confined by a horizontally aligned magnetic field. It is assumed that dissipation by viscosity and resistivity is negligible and that the plasma-magnetic field interface is free from instabilities of all kinds.

We consider semi-quantitatively the cyclotron resonance instability of ion cyclotron and magnetosonic waves propagating at an angle to the magnetic field in an infinite uniform plasma. The velocity distributions of electrons and ions consist of a dense cold component and a diffuse high-energy tail. If the high-energy protons are sufficiently intense and their pitch angle distributions sufficiently anisotropic, instability occurs for those waves propagating parallel to the magnetic field. If the spectrum of resonant protons is sufficiently hard, a reasonably large cone of propagating angles about the magnetic field can be unstable. Observed fluxes of trapped protons in the magnetosphere should destabilise the ion cyclotron wave at a lower intensity threshold than for at least one class of electrostatic waves.

The stability of oblique magnetohycirodynamic shock waves is studied with respect to a disturbance that excites magneto-acoustic waves. The problem is solved numerically by the normal mode analysis and it is shown that slow shocks are unstable in the sense that the disturbance grows exponentially with time. Growth rates are calculated for a particular Mach number and for different values of the magnetic field and obliqueness. The fast shock appears to be stable.

A formalism is developed describing the time evolution of wave correlations in a uniformly turbulent ensemble of weakly non-linear systems. The statistics are built into the formalism a priori. With closure (of the hierarchy of equations for wave correlations) appropriate to the inclusion of resonant three-wave interactions, a (non-linear) kinetic equation for the two-wave correlations is derived and various properties of this equation are discussed. The effects of both bilinear and trilinear non-linearities are considered. Modifications of the formalism in situations where there is a weak (linear) instability γк, (γк/ωк ≪ 1) are also considered.

The problem concerning the interaction of electromagnetic fields with a warm, slightly ionized magneto-plasma is analysed by means of the multiple time scales asymptotic technique. Only initial value problems in an infinite plasma are here considered in detail. It is shown how the full set of Vlasov–Maxwell equations, modified through a standard Boltzmann collision integral to take account of the elastic electron-neutral encounters, can be reduced, in the basic limit assumed, to a set of two coupled non-linear integro-difl�erential equations for the dominant terms of the longitudinal fields and of the electron distribution function. Once the initial conditions are specified, the behaviour of the transverse initial field is also described. A particular example has been given for the so-called progressing-wave initial conditions.

The loss-cone instability of plasma confined in a mirror-type trap is considered. Relaxation of the particle distribution in the trap with a length larger than the mean free path between ‘turbulent’ collisions is described by conventional quasilinear theory. A quasi-linear equation for the ion distribution is solved analytically for the case of a small loss-cone volume of particles in velocity space (it takes place, for instance, in the case of a trap with a larger mirror ratio).

The propagation of small amplitude hydromagnetic waves of a collisionless, anisotropic plasma in a strong magnetic field is treated using the Chew—Goldberger—Low (CGL) fluid equations. The Alfvén, fast and slow magnetoacoustic, and entropy modes are analysed by phase speed and wave front diagrams and compared to their counterparts in magnetohydrodynamics (MHD) where collisions maintain an isotropic pressure. Two main classes of CGL waves are identified, pseudo-MHD and reverse-MHD where the latter has the slow wave speed greater than the Alfvén wave speed. Significant differences of the CGL hydromagnetic waves as compared to MHD waves are found which have special relevance for a study of infinitesimal and finite discontinuities in space plasmas.

The beam-plasma interaction which has been proposed by Stix is examined both by approximate analysis and by integration of the non-linear equations of motion which describe the interaction of an individual electron with a monochromatic large amplitude electrostatic plasma wave. The quasi-stochastic model of the large amplitude plasma waves, introduced by Stix (1964), has been used in the calculations by programming a random phase function into the argument of the periodic plasma wave function. It is found that electrons are subject to a quasi- cyclotron acceleration which proceeds to a higher energy in the stochastic case than the limit found in the non-stochastic case (a phase-coherent plasma wave). This behaviour is interpreted in terms of a limit cycle phenomenon, and the stochastic phase shift appears to push electrons across these limit cycles. It is also found that favoured groups of electrons in a 2eV plasma excited by a 5 keV beam in a2000 Gauss field can achieve energies in the range 85 to 170 keY in less than a nanosecond by a single acceleration occurring in one plasma wave coherence length L‖ as defined by Stix.

The jump conditions for finite discontinuities in a collisionless plasma are derived in the Chew—Goldberger—Low approximation.

The theory of Allis and Rose based on the hypotheses of congruence and proportionality is used to formulate the problem of ambipolar diffusion and drift of a plasma of added carriers in a semiconductor in the presence of a magnetic field. These hypotheses together with the linearized equations for the particle flow densities of the electrons and holes permit an expression for the particle flow density of the added carriers to be derived. In its final form this expression is composed of a drift and diffusion term in which the ambipolar group mobility and diffusion coefficient appear as tensorial quantities. These antisymmetric tensors will reduce to the scalar quantities of van Roosbroeck (1953) when the magnetic field is set equal to zero and to the proper tensorial quantities for the added minority carriers in strong extrinsic n or p type semiconductors. Finally, as an illustrative example, the ambipolar continuity equation for the particle density of added carriers is solved as an initial-value problem in an unbounded domain in order to analyze the space and time behaviour of a pulse of added carriers whose initial distribution is a Dirac delta function.

Numerically exact solutions of the problems of heat transfer between concentric circular cylinders and spheres have been obtained.

The non-linear behaviour of one-dimensional electrostatic oscillations in a homogeneous, unbounded, collisionless and fully ionized plasma is considered for the case in which a single wave of small, but finite amplitude is excited initially. The Vlasov–Poisson equations are solved using the method of strained co-ordinates in which the independent variable t, the electric field and the distribution function are expanded in the form of asymptotic series, the terms of which are founded by an iterative procedure. An ordering parameter e is introduced, which is proportional to the initial amplitude of the electric field given by linear theory. Differential equations are derived which can be solved sequentially to obtain uniformly valid solutions to all orders in ε. Solutions are given to second order and applied to the case in which the background distribution function is Maxwellian. It is found that the changes in the real and imaginary part of the frequency are small in comparison to the values obtained in the linear theory; that the free-streaming terms decay exponentially in time with a damping rate proportional to ε2, in contrast with the linear theory where they are Un- damped; and that the analysis allows us to calculate the changes in the background distribution function for large time, resulting from particle-wave interactions.

A study is made of finite amplitude, oblique, hydromagnetic waves in a cold collision-free plasma. It is shown that the equations admit steady solutions which describe flow through a front. Ahead of the front, the flow is uniform, but behind the front the magnetic field and flow variables oscillate irregularly in a random manner. The fronts are called ‘quasi-shocks’, although they have more in common with the laminar-turbulent transition of fluid mechanics than with the classical shock wave. The structure of the quasi-shocks is examined, and estimates are made for the mean values behind the front. The relevance of the quasishocks to the so-called ‘collision-free shock’ is considered, and comparison is made with the ‘bow shock’ on the solar wind near the earth. It is shown that the quasi-shock is consistent with some of the observed data.

The problem of combined Rayleigh—Taylor and Kelvin—Helmholtz instability for incompressible plasmas carrying a uniform magnetic field is investigated taking account of the Hall current. The resistivity and the finite ion Larmor radius effects are left out. It is found that the finite Larmor frequency is destabilizing in nature. The growth rate of an otherwise (without the Hall current) unstable configuration is increased, and unstable modes may be produced in otherwise stable situations for reasonably large values of the Hall current. The Hall effect also results in overstable modes.