A re-normalized expansion is presented for the solution of the fluctuating Vlasov equation for turbulent plasmas. Its first term can be identified with the Weinstock result in the weak-coupling approximation. The use of time-ordering operators allows one to solve the Weinstock UA evolution operator explicitly. Detailed results are given for the weak-coupling approximation, presented here as a simple re-normalization of the quasi-linear theory. Results of Weinstock and Birmingham & Bornatici, concerning the frequency shift, resonance broadening and Dupree damping appear to be deeply modified by similar corrective terms. These arise from a consistent handling of expressions involving non-commuting operators.

In part 1 (Misguich & Balescu 1975) the Dupree description of plasma turbulence (i.e. the weak-coupling approximation of Weinstock) was derived and completed in the framework of the re-normalized quasi-linear approximation. The time behaviour of the frequency shift and of the Dupree damping was proved to be modified by time-ordering effects, usually neglected. In part 2 we show that a Markovian truncation must be applied to the basic average propagator Ū, to recover the short-time expansion of Birmingham & Bornatici. Such an expansion is shown to be generally non-convergent. The corresponding short-time results are invalid.

Collective binary correlations are deduced from the re-normalized quasi-linear (RQL) equation describing the evolution of the singlet average function. For a constant growth rate, a crude calculation suggests that generalized Dupree damping could seriously reduce the trapping of resonant particles.

The solutions of Maxwell's equations and of their adjoint equations, being essentially different in absorbing media, are written in terms of matrizants. A relation between these is established, utilizing the constancy of the bi-linear concomitant. With this, and with the concept of the conjugate field, introduced in Suchy & Altmann (1975), a theorem, connecting the transfer matrices of the field and the conjugate field, is derived. This, in turn, leads to a corresponding theorem connecting the scattering matrices, which are composed of the refiexion and transmission matrices.

The Kelvin-Helmholtz problem is analyzed by a set of general hydromagnetic equations, which includes ideal magnetohydrodynamic and Chew-Goldberger-Low models as particular cases. A formalism is given that facilitates comparison between results from different models. A sheared flow is one in which the velocity has no component in the y direction, and such that the x and z components of the velocity depend on the y co-ordinate. A sheared field is defined similarly. The differential equations for linear modes of oscillation of a sheared flow in a sheared magnetic field is obtained; and the energy of these modes is studied. As a particular case of oscillations of a sheared flow, the properties of the modes excited by arbitrary modulation of a tangential discontinuity are studied. The relationship between radiation of waves from such a discontinuity and instability of the system is brought out by considering the system energy. Domains of absolute stability are given; and the different hydromagnetic models are compared by examining the predicted domains. It is found that anisotropy plays an important role in the conditions of stability.

Transport laws in collisionless systems must be derived from the one-particle Liouville equation. The simplest cases are those of the CGL invariants along the magnetic field lines, together with the resulting equations of continuity and motion, in circumstances where a supersonic particle flux is parallel to a diminishing magnetic field. We give functional expressions for the two contributions to the parallel heat flux, with integrated forms of kinetic theory. The general expressions, corresponding to the moments of greater order, coincide with those obtained by series development of the differential equation of moments. Moreover, we illustrate a case of both parallel and convergent flux, for which the equation of continuity gives considerable acceleration to the precipitation flux average velocity in the absence of any important electric field, because that acceleration is important in understanding the relationship between the plasmasheet and the ionosphere.

The self-modulation parameter of the ordinary and the extraordinary waves propagating in a cold magnetoplasma is given explicitly, by the familiar reductive perturbation method, in the framework of the hydrodynamic approach. Both quantitative and qualitative order-of-magnitude analysis of this parameter are carried out for the lower-hybrid and the upper-hybrid branches of the extraordinary wave. The usefulness of the stability criterion, which is linear with respect to an envelope of these linear waves, is also discussed.

A model for the kinetic formulation of a weakly-ionized monatomic gas, which interacts with thermal radiation, is proposed. The collision terms for particle-initiated ionization, developed in terms of a non-equilibrium degree of ionization, and the collision terms for photon-initiated ionization, developed by assuming particles remain in local equilibrium during photon relaxation, all satisfy the usual conservation principles. It is also shown that the proposed model leads to an H theorem.

This paper considers an example of the propagation of a finite wave-train in a plasma of relatively simple dispersive properties. In this case, the wave field can be evaluated exactly. The concept of a wave-packet moving at the group speed can therefore be tested rigorously. Various approximations to the field, physically more transparent than the exact solution, are presented and compared with the latter. One of these, relevant in the neighbourhood of the ideal group front, is in fact applicable to a wide range of dispersion laws. The paper demonstrates broadening with time, and eventual disappearance, of the pulse.

The paper describes a model, in which ions are accelerated against the direction of the electric field at a density gradient. A one-dimensional, inhomogeneous, steady-state is constructed, in which cold electrons stream relative to cold ions. The wavenumber and amplitude of the fluctuating potential are calculated, then used to compute the trajectories of representative ions trapped in the potential troughs. Ion acceleration to energies an order of magnitude greater than the applied potential is demonstrated.

The instability of the ordinary mode propagating perpendicular to the magnetic field is discussed for two ion beams counter-streaming along the magnetic field through a plasma of stationary electrons and ions. The instability criterion gives the necessary and sufficient condition for a purely growing mode. Although the instability threshold is high (u/Voi ≳ (2βoiNoi/Noi)-½ if the streaming velocity is 10% larger than the threshold value), large growth rates of the order of the ion cyclotron frequency result.