The possibility of a collisionless shock in a plasma not subject to electrostatic instabilities nor immersed in a permanent magnetic field is investigated. The equations governing a steady shock in an electron-ion plasma are deduced and their limitations are discussed. Attention is given to the fact that the velocity distribution in a collisionless plasma may not be Maxwellian. This yields a parameter which is a measure of high-energy particle density. It is found that almost always no steady shock exists. The inverse of the electron plasma wave-number k0=ωp/c provides a scale length for the shock, though the dampening effect of the ions indicates a shock thickness much in excess of . It is concluded that the mechanism for such shocks may be important in a gas such as the solar wind.

The influence of the lack of neutrality on the drift instabilities (collisionless and collisional) of a non-uniform, isothermal plasma is considered. It is shown that the local dispersion relation depends on the derivatives of the electric field present in the equilibrium of the nonneutral plasma. In particular the influence of the first two derivatives is discussed.

A magnetically confiuied, slightly non-uniform and non-neutral plasma column is considered. The existence is shown, in plane, infinite geometry, of a mode whose characteristic properties agree with those of an unstable wave detected in a recent experiment.

The stability of a stationary non-linear wave set up by two counter-streaming cold beams of electrons immersed in a background of stationary ions is examined without recourse to an expansion in terms of the amplitude of this wave. It is found that this wave is unstable to small perturbations and hence one would not expect such waves to exist in practice.

Non-linear transverse waves in a classical non-relativistic collisionless, Maxwellian electron gas with external magnetic field B0 are considered. There is assumed a small, sinusoidal variation in the initial electric and magnetic fields, corresponding to excitation of a discrete wave-number mode. The non-linear Vlasov equation is solved to second order in the long time limit via the Montgomery—Gorman perturbation expansion, and the time-independent, spatially homogeneous part of the second-order distribution function is used to modify the linear dispersion relation. For frequencies near the electron cyclotron frequency a non-linear damping decrement results such that, for many values of the parameters, the damping is less than the linear rate. Thus at sufficiently long times, the rate of damping of transverse electron cyclotron waves should decrease, a result similar to that for non-linear damping of longitudinal electron plasma waves.

The high frequency conductivity tensor of an isotropic plasma is derived taking into account particle correlations at the lowest consistent order in the parameter ωp/ω these correlations describe a weakly Langmuir turbulent plasma. Two special cases are investigated in which the two-particle correlation function is related to the turbulent electrostatic field spectrum. Particular distribution functions and spectra are considered and approximate dispersion relations are derived in both cases in ‘the cold plasma limit’. The importance of the corrective term is discussed in terms of three dimensionless parameters measuring the strength of the turbulence, the shape of the spectrum, and the frequency. The effect could be important for frequencies not too far from the plasma frequency.

We point out in this paper that when one increases the intensity of the second applied pulse or the time interval between the two pulses, the amplitude of the first echo saturates and multiple echoes of comparable amplitude appear. Moreover, we predict a damping of the echoes which is due to a phase mixing effect; that is collisionless.

We discuss the propagation of wave packets of the form

in an infinite uniform plasma, where G(z, t) is a slowly varying function of space z and time t. One can very simply derive the equation of change of G(z, t) for the stable or unstable case. The terms in the equation are of physical interest and clearly define the limitations of linear theory. We then investigate the problem of whistler mode wave propagation in a collisionless Vlasov plasma in a given non-uniform magnetic field. We choose the electric field to be of a W.K.B. form and the particle distribution to be isotropic. We can express the perturbation in the particle distribution in terms of an integration along the zero-order par tide orbits (an integration overtime). These orbits can be found correct to a term linear in a smallness parameter ε (when ε equals zero we arrive back at a uniform magnetic field). The charge and current density due to the perturbation are related through Maxwell's equations to the electric and magnetic field of the wave in the usual self consistent Boltzmann—Vlasov description. We show that the contribution to the current arises from recent events in the history of a given particle because of the finite temperature of the plasma. This result leads to an expansion of slowly varying parameters which in turn gives rise to the equation governing the motion of the wave packet.

The effect of Hall current and finite electrical resistivity has been studied on the Rayleigh—Taylor instability of superposed incompressible fluids in the presence of a uniform horizontal magnetic field. It is found that Hall current has a negligible stabilizing influence in the high resistivity limit whereas it has a destabilizing influence in the low resistivity limit.

In this note the problem of stability of two hot collisionless streams of charged particles is considered. The masses, charges, densities, and temperatures are arbitrary and the distribution functions are modelled by one ‘resonance’ function for each stream. The problem of stability is resolved by Nyquist diagrams, and, for the case of equal plasma frequencies, also by solving the dispersion relation in ω. A comparison with two Maxwellians on the one hand, and a two step function model on the other, is given. Step functions appear to be too crude for this problem.

It is shown that complex variable transformations, suitable for obtaining the solution for the field boundary of a system of line currents confined in one cavity by a perfectly conducting uniform plasma, can be used for obtaining the solution to the inverse problem where a perfectly conducting uniform plasma is confined in one cavity by a system of line currents. It is deduced that the minimum number of line currents for confining (not stably) a plasma is two. The equilibrium configurations for several special but simple cases are investigated and discussed.

Non-linear periodic waves propagating across a magnetic field are investigated, taking relativistic effects into account. The governing equations of the phenomena are taken to be the relativistic two-fluid plasma equations for a cold plasma together with Maxwell's equations. The analytical solution in the non-relativistic case was obtained by Adlam and Allen and others, a case where the quasi- neutral approximation is a good one and the inertia of the electrons plays an essential role in the dispersion effect of the plasma.

In contrast, the analysis of the relativistic waves shows that the quasi- neutral approximation becomes invalid and that the charge separation plays an essential role in the dispersion effect while the inertia of the electrons plays minor role only. Therefore the phenomena to be investigated have a length scale of the appropriately defined Debye distance (the ion Larmor radius), while in the non-relativistic case the width of the wave is of the order of the geometric mean of the Larmor radii of the electrons and of the ions. In the present case where the electron mass may be neglected, there exists an exact solution which is a good approximation for a real plasma. The solitary wave solution is found as the limiting case of infinite periodicity.

Small amplitude waves and collisionless shock waves are investigated within the framework of the first-order Chew—Goldberger—Low equations. For linearized oscillations, two modes are present for propagation along an applied magnetic field. One is an acoustic type which contains no finite Larmor radius effects. The other which contains the ‘fire hose’ instability in its lowest order terms, does possess finite Larmor radius corrections. These corrections, however, do not produce instabilities or dissipation. There are no finite Larmor radius corrections to the single mode present for propagation normal to the applied magnetic field. Normal shock structure is investigated, but it is shown that jump solutions do not exist. An analytic solitary pulse solution is found and is compared with the Adlam—Allen pulse solution.

The low frequency, linear oscifiation of an unbounded, homogeneous, low density plasma in the presence of a strong, uniform magnetic field is examined using a set of three-dimensional hydromagnetic equations derived from the higher-order Chew—Goldberger—Low theory recently obtained by Frieman, Davidson & Langdon (1966). It is found from the higher-order corrections to the dispersion relation that the waves are generally dispersed and the mirror and firehose instabilities are stabilized.