The propagation of nonlinear electromagnetic waves in a plasma is investigated by seeking travelling wave solutions to the macroscopic plasma equations. The model considered is that of a warm, anisotropic electron plasma in a magnetic field, in which the waves are allowed to propagate at any angle with respect to the magnetic field. A nonlinear dispersion relation is obtained which is valid for frequencies sufficiently close to the resonant frequencies. Travelling wave solutions are found for all amplitudes for which the local density remains positive. Also in sufficiently anisotropic plasmas the nature (i.e. propagating or cut-off) of one of the resonant waves can depend upon the amplitude.

This paper examines analytically the transition domain between collisionless and collision dominated flows in the environment of a plasma sheath. The onset of collisions also brings into focus the concurrent effects due to ionization, which are retained in the theory. An integral method of iteration is established, and the dominant correction terms due to collision and ionization effects are obtained. Various limiting cases are examined.

A modification of the Penrose criterion is applied to the local dispersion relation for low-frequency, electrostatic instabilities in a low-density, slightly inhomogeneous magnetoplasma. The Nyquist diagrams obtained can be classified according to their behaviour near the origin; this is sufficient in some cases to conclude that the equilibrium is unstable. With this method, in the limit of negligible ion Larmor radius, we find that the Langmuir waves propagating in a direction orthogonal to the external magnetic field are driven unstable by a temperature gradient.

A non-linear wave propagation mechanism is found for ionization fronts. The fronts propagate due to ionizing collisions between electrons and neutrals. A parameter is obtained which specifies the amount of electrical interaction. For small electrical interaction, charge separation may occur and the front can propagate at any speed greater than a critical value. For large electrical interaction, the number densities of ions and electrons are nearly equal and a lower bound on the propagation speed is also found.

The collisionless relativistic Boltzmann equation is used to investigate the stability of a large-scale two-component plasma containing numerically small anisotropic relativistic populations. Although most permitted waves are stable, Alfvén type waves are found to be unstable whenever the anisotropy exceeds O(v A/c). Growth times are estimated as days for supernova remnants and millennia for cosmic ray protons.

The problem of transition from ambipolar to free diffusion due to charged particle density gradients in a decaying plasma of spherical symmetry is investigated in this paper. This investigation is carried out within the framework of the approximate procedure set forth by Allis & Rose (1954). A further simplification is introduced by using only the time-dependent ‘imperfect’ ambipolar diffusion coefficients for the electrons and ions evaluated at the centre of the plasma for carrying out calculations. The results for the time dependence of these ‘imperfect’ ambipolar diffusion coefficients and spatial distributions for the charge density, electric field, and potential which are obtained by this procedure are valid for all values of time in the neighbourhood of the centre of the plasma and for all other positions of space only during the time while the plasma is diffusing in an almost ‘perfect’ ambipolar manner.

Starting with the Chew, Goldberger & Low equations, an analysis is made of instability arising due to a tangential velocity discontinuity in a dilute plasma. The velocities on either side are parallel but oppositely directed. Two cases are considered: (i) the magnetic field is uniform and everywhere transverse to the motion, and (ii) the magnetic field vectors on either side are orthogonal, being parallel to the motion on one side and perpendicular on the other. The conditions for instability are obtained and it is found that the effect of magnetic field is destabilizing in both cases. The effect of orthogonality of magnetic fields on the conventional fire-hose instability for a uniform, static plasma is also discussed as special case.

Time-dependent motions of the cavity formed when a uniform corpuscular flux is incident on the magnetic field of n line currents are considered. The general problem is formulated and solved in the case of an initially flat-faced cloud of flux approaching and passing the magnetic field of n line currents. The special case of a line dipole is analysed in some detail.

The electrostatic modes for which k is perpendicular to the impressed field B0 (the ‘Bernstein modes’) have been examined, and dispersion curves (w vs. k) have been calculated numerically for hybrid frequencies in the vicinity of the thirtieth to fortieth harmonic of the gyrofrequency. Although the calculations were performed for the lower hybrid case (ion branch), they also apply to the upper hybrid cases in which the plasma parameter The structure of the dispersion curves for these large values of plasma parameter differs from that for smaller values investigated by others, in that broad bands in wave-number space have very small group speeds in narrow frequency bands close to many of the gyrofrequency harmonics. All of the Bernstein modes examined have group speeds less than 37 % of the ion thermal speed, and can be classified as very slow waves. Furthermore, the spectrum of propagating Bernstein modes is broadened as becomes large, with an effective total transmission region from the gyrofrequency up to several harmonics of gyrofrequency above the hybrid frequency.

The structures of some laminar boundary layers in high-density, shock heated, 1 eV argon plasma flows have been investigated theoretically. The analysis is based upon a three-fluid continuum formulation. Boundary-layer equations have been solved numerically on a digital computer by a finite difference technique for the case of thermochemical equilibrium and no radiation and applied electromagnetic fields. The induced electric field has been considered and shown to be important. It strongly couples the diffusive motions of the electron and ion fluids, thus forming ambipolar motion except in a sheath region adjacent to the wall. Argon transport properties, calculated from simple kinetic theory, have been used in the analysis. Important parameters, such as the Prandtl number and the density-viscosity product have been found to vary one or two orders of magnitude in the argon plasma boundary layer, a finding in sharp contrast with results for classical, non-ionized boundary layers. Solutions have been developed for the simple Rayleigh's boundary layer (forming over an infinite flat plate with an impulsively started motion in its own plane) and for the shock-tube side-wall boundary layer (forming behind a plane, ionizing shock wave moving over an infinite, plane wall). Even in terms of appropriate similarity parameters, solutions (for e.g. velocity and temperature profiles) exhibit strong dependence upon free- stream conditions. Assumptions of chemical and temperature equilibria have been checked from the equilibrium solution. Results indicate equilibrium ionization would not be present in typical argon boundary layers, e.g. at temperatures below 9000 °K, at a pressure of 1 atm. Similarly, due to ineffective energy transfer rates between the electron and the heavy-particle fluids and the difference in electron and ion–atom thermal conductivities, the electron temperature would deviate from the heavy-particle temperature in the same temperature region. The electron temperature has been calculated in a linearized model and found to be larger than the ion–atom temperature.

The shock-tube side-wall boundary layer in a 1 eV, high-density argon plasma was studied experimentally using anew, simple, quantitative schlieren technique. The angular refraction of light which enters the shock-tube test section parallel to a side wall and passes through typically 1 mm thick boundary layers was determined in two separate wavelengths. This was done by measuring the displacements of two shadows formed by two thin wires placed in the point source light, which is reflected non-centrally by a concave spherical mirror. The experiments were of exploratory nature only, but clearly demonstrated the feasibility of the new technique in analysing plasma-boundary-layer flows. Measured electron density profiles in the high-temperature region of the sidewall boundary layers agreed within experimental errors with those calculated from the equilibrium-boundary-layer theory.

The mathematical formulation for the problem of non-linear oscillations in a self-consistent, non-uniform, collisionless plasma is considered. The fully nonlinear treatment illuminates the effect of the wave on the background distribution of the plasma through which it is passing. It is assumed that, although the overall non-uniformity may be large, significant changes occur only over time or length scales which are large compared with the plasma period or Debye length respectively. Exclusion of secular terms from the solution leads to a Liouvile type equation, which must be satisfied by the background distribution, and to propagation laws for the waves.

The theory is restricted to almost one-dimensional electrostatic waves and a general presentation is given from a relativistically-invariant point of view. Then the equations are derived in terms of physical variables for the special case in which: (i) the distribution functions and electrostatic potential depend on one space co-ordinate (that of propagation of the wave) and the former on the corresponding particle velocity component only, (ii) the wave is slowly-varying only with respect to this co-ordinate and time, and (iii) the magnetic field is zero. Finally, the non-relativistic limit of this case is considered in more detail. The boundary conditions satisfied by the distribution functions are discussed and this leads to the conclusion that in some circumstances thin sheets of probability fluid are formed in phase space and the background distribution cannot be strictly defined. This motivates a reformulation and subsequent re-solution of the problem (for this non-relativistic special case) in terms of weak functions, corresponding to the physical assumption of the presence of a small-scale mixing mechanism, which is excited by and smears the sheeted distribution but is otherwise dormant.

The results of the investigation are given as a system of differentio-integral equations which must be solved if necessary conditions for the absence of nonsecular solutions (of the Vlasov and Maxwell equations) are to be satisfied. No solution of this system is attempted here.

The dispersion equation for helicon waves in a plasma-filled wave-guide is derived. The relation between the solution and the quasi-static modes of Trivelpiece & Gould is discussed. An approximate dispersion relation, appropriate for cases where Ωe≫ ν ≫ ω, is obtained allowing the real and imaginary parts of the propagation constant to be expressed very simply in a normalized form. Curves are presented for this approximation.

‘On the a.c. electrical conductivity of a Lorentz gas’ by J. H. M. Fu and R. S. B.Ong, J. Plasma Phys. 2, 1968, P. 1.

Calculation of the electrical conductivity

The current density is expressed by

Using the dimensionless velocity c = (m/2θ;)½v and noting that the current density associated with the equilibrium distribution function f0 vanishes, we can write

g(k, c, w) is given formally by

where gns(k, c, w) is given by (2.15) in the original paper, and

with λ = (2θ/m)½/(w/k and δ the Dirac delta function.

‘Propagation along the magnetic field of a pulse current originated dlisturbance in a cold two-fluid plasma’, by G. J. Lewak, J. Plasma Physics, vol. 1, 1967 p. 435.

On page 449, the sixth of the list of references should read

Davis, L., Lust, R. & Schluter, A. 1958 Z. Naturforsch. 13a, 916.