The non-linear one-dimensional steady-state equations which govern the flow of an ion—electron beam emitted from a plane are solved in the phase plane, and it is shown that a perfectly neutralized beam follows for a large range of injection velocities of the electrons. When the velocity of the ions is less than the electron sound speed the transition region for the neutralization has a length of the order of a Debye length λD = (kT)½ (4πNe2)–½, which is a typical plasma sheath. The maximum velocity of injection of the electrons for which neutralization is predicted is, in this case, the sound speed of the electrons. If the electrons are injected with a supersonic speed, they cannot be decelerated continuously to the subsonic speed corresponding to the velocity of the ions. No bound is set on the electron injection velocity from below. When the velocity of the ion beam is greater than the electron sound speed, oscillations with an amplitude which depends on the velocity of injection of the electrons, and a wavelength which depends on the ratio of the ion velocity to the electron speed of sound, are found. In this case the injection speed of the electrons needed to obtain the steady-state oscillatory solution is bounded both from above and from below. Subsonic electrons cannot be accelerated continuously to the supersonic velocity required to match the velocity of the ions, and within the supersonic range there is shown to be a limit (depending on the ratio of the ion velocity to the electron speed of sound, so that faster electrons cannot be decelerated continuously to match the (supersonic) ion velocity.

Equations are derived which may be used to describe the propagation of electromagnetic waves in non-uniform magnetized plasma when the wave frequency is near the second electron cyclotron harmonic. The method used is to expand the linearized Vlasov equation in powers of the electron Larmor radius divided by a typical scale length. The general equations are then specialized to the problem of the coupling of transverse waves to the longitudinal modes (Bernstein modes) which exist when all quantities vary only in a plane perpendicular to a straight magnetic field. The form of these equations for two simple models of the equilibrium plasma is given. Comments are made about the equations for the higher harmonics, and the question of boundary conditions is discussed. Finally, the general equations are examined in the limit Ω→0 in order to provide equations suitable for the description of high frequency waves in non-magnetized plasmas.

We present here a simple and direct derivation of the equations governing the motion of a charged particle in space- and time-dependent fields in the guiding centre approximation in lowest order. The method of attack is based initially on the expansion scheme of Bogolyubov & Zubarev (1955) combined with Northrop's averaging process over one gyration period. Thus, we first resolve the instantaneous velocity of the particle into three components, r = ê1u + UE + Tw, where u is the parallel component, UE is the familiar E × B drift, and Tw is the instantaneous perpendicular component as measured in a frame moving with the UE drift. Substituting this expression into the particle's (non-relativistic) vector equation of motion, resolving it into parallel and perpendicular components, and applying Northrop's averaging process, we obtain the averaged equations of motion, in lowest order, for the components u and w. From the latter equation we prove immediately that the magnetic moment of the particle is an adiabatic invariant to lowest order. Next, we resolve the instantaneous velocity of the guiding centre into two components, R = r — p, where p is a rapidly rotating vector directed from the guiding centre to the instantaneous position of the particle. Upon applying Northrop's averaging process we obtain at once the averaged velocity of the guiding centre including the complete set of first-order drifts. From this equation we readily deduce the energy gain equation to lowest order. It is then shown that the simple averaging prescription applied here is mathematically rigorous and yields correctly all lowest order results in an expansion in reciprocal powers of the cyclotron frequency.

A model kinetic equation for a plasma containing many species is set up. It is an approximation to the correct kinetic equation but is considerably simpler in form. The collision operator has the property that particles, momentum and energy are conserved, and it gives to good approximation the correct rates of transfer of momentum and energy between all pairs of species. It is also shown that the equation leads to an H-theorem.

The Chapman—Enskog expansion is applied to the model Fokker—Planck equation for a plasma, derived in part 2. It is shown that the complete set of transport coefficients can be calculated without further approximations. Results are derived first in the absence of any external magnetic field. The transport coefficients are also derived when there is a strong magnetic field, in which case they become anisotropic.

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