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.

Theoretical and numerical results concerning the nonlinear interaction of an initially cold beam with a single wave (ω ≈ k∥ vb) in either magnetized or field- free plasma are presented and discussed. By using the energy and momentum conservation laws, the equations describing the development of the wave amplitude and phase, both in the temporal and spatial problem, are derived and then solved numerically together with the equations of motion of the beam particles. The fundamental role of an absorption (or transformation) of the wave in its interaction with a beam modulated at a frequency belonging to the upper branch of the electron plasma oscillation is recognized in two model situations.

This paper is a theoretical and experimental study of the propagation of longitudinal electron waves in a collisionless, homogeneuos, isotropic plasma, composed of Maxwellian and supra-thermal electrons. Assuming the supra-thermal electrons are mono-energetic, the velocity distribution function is chosen as a Maxwellian plus water-bag distribution; then, using the method of residues (inverse Fourier–Laplace transform), the behaviour of the electric field is found to be in good agreement with experimental results obtained for a plasma of Maxwellian and quasi-mono-energetic electrons. The plasma parameters can be deduced from the theoretical dispersion curves matching the experimental points; they are found to agree well with those measured by electrostatic probes. Experimental values of damping and excitation coefficients are also compared with the theoretical. The results suggest that a longitudinal electron wave propagation experiment may be used as a powerful diagnostic tool to examine non-Maxwellian plasmas.

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.

The interaction between an ion wave and its second harmonic is discussed theoretically, on the basis of coupled-mode equations derived from the Korteweg–de Vries equation. Using an exact solution of the coupled-mode equations, we give a numerical analysis of the properties of the solutions; and we show that superposition of two waves can describe the formation of two solitons, the interaction between them, and the recurrence of an initial state. Our theory can explain completely recent experimental results on ion wave solitons excited by a continuous sine wave.

The propagation of a nonlinear wave in a dispersive medium has been extensively studied in the last decade. In a plasma, a finite-amplitude ion wave can form solitons in the course of its evolution, if wave damping is neglected.

A closed solution for the increment in the logarithmic derivative &dgr;(Q) over the inner (‘resistivity’) region for resistive tearing theory in cylindrical geometry including ‘parallel’ viscosity is compared with an earlier numerical solution. The singular point Qcrit about which δ(Q) exhibits branch-point behaviour in the presence of compressibility is identified; modification of the viscous theory, to include (for example) finite Larmor radius effects, is suggested. In the incompressible limit, a familiar (inviscid) stability criterion is recovered, in the presence of ‘parallel’ viscosity.

Wave propagation in a hot, collisionless electron plasma (without ambient magnetic field) is analyzed by coisidering the frame of reference in which the field has no space dependence. It is shown that the governing equations are of the same form as those for a cold plasma, and are likely to have corresponding exact (nonlinear, relativistic) solutions. In particular, it is shown that there exists a solution representing a purely transverse, circularly polarized, monochromatic wave. Three approximate forms of the dispersion relation of this wave are obtained explicitly, the first being valid when the temperature correction is small, the second applying to weak waves, and the third to strong waves. Purely longitudinal waves are also discussed.

The kinetic equation obtained in parts 1 and 2 is treated stochastically: the external field is stochastic, with an average and a fluctuating part. The turbulence of the system is described by the induced fluctuations in the plasma, and a general equation is derived for the average distribution function. As a particular case, the stochastic Vlasov equation is treated explicitly, and compared with the descriptions of Dupree, Weinstock and Benford & Thomson.

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.

A re-examination of the linear E x B instability problem for infinite stratified media gives results differing from those published. The main point is to warn against the application to inhomogeneous media of the usual instability criterion, as developed for homogeneous media. The fact that the dispersion relation gives complex values of ω for real k does not always indicate the existence of an instabifity. It could also describe propagation of a constant wave energy through an inhomogeneous medium. It is demonstrated that in certain situations it is possible to choose dependent variables in such a way that these two effects disentangle, allowing for a definite solution of the instability problem.

In this paper the linear stability' problem for an infinite inhomogeneous, partially ionized gas penetrated by crossed electric and magnetic fields, is re-examined. Several authors (e.g. Reid 1968; Whitehead 1967, 1971; Cunnold 1969; Chimonas 1969; Linson & Workman 1970; looper & Walker 1971; Sato 1971; Sato & Tsuda 1972; Kato 1972) have considered the possibility that the so-called cross-field instability exists in the ionosphere. Our reason for treating the problem again is to point out a deficiency in the literature.

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.

In close analogy with part 1, the dynamics of stochastic correlations is used to reformulate the preceding treatment of turbulent equations. The result is an exact and closed equation for an average function, involving only asymptotic stochastic correlations, and no initial fluctuation. This stochastic kinetic equation is valid for all initial conditions, and the corresponding function is a significant particular projection on an independent subspace R(t).

It is shown that the multiple water-bag model, with both analytic and numerical properties, is fairly well suited to the study of forced oscillations launched by grid systems, for different geometries with possible non-uniform charge distribution, and for any distance within the range 0–100 Debye lengths.

Freeston (1968) and Freeston & Malik (1971) compared measurements precisely with theoretical values of the electric field near a system of grids and antennas immersed in a plasma. Results provided a valuable diagnostic tool in space research; and radio-frequency probing was used successfully (Beghin & Debrie 1972; Chasseriaux, Debrie & Renard 1972).

We calculated relative and absolute maximum linear growth rates of the electromagnetic ion–cyclotron instability produced by the addition of cold plasma to an infinite uniform anisotropic plasma. In this, both accurate numerical solution of the linearized dispersion relation, and approximate analytic calculations, were carried out, and the results were compared critically.

Qualitatively, results similar to those obtained in the past for the right-hand polarized electron–cyclotron electromagnetic (whistler) waves (Cuperman & Landau 1974) are found also for the left-hand polarized waves considered in this paper. Specifically, the electromagnetic ion–cyclotron instability in a warm, anisotropic proton plasma, characterized by Pp ≡ βP,A2p(Ap + 1) ≲ 1, may be significantly enhanced by addition of cold plasma.

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 relationship between energy absorption and the continuous frequency spectrum of the linearized equations of ideal MHD is investigated. We limit ourselves to incompressible fluid perturbations for which the continuum stems from resonant surfaces where the oscillation frequency equals the local frequency of an Alfvén wave. Details of this absorption process are illustrated by obtaining the response of the diffuse sheet pinch to an external current source. Expressions are derived that give the rate at which energy is transferred to the plasma, and show the spatial distribution of the absorbed energy. The results indicate that the absorption is enhanced if the plasma profile has regions of large spatial gradients. This enhanced absorption is a consequence of the presence of surface waves.

This paper investigates the fluctuation spectrum of the surface waves in a semi-infinite plasma, and in a plasma cylinder, with an inhomogeneous boundary layer. It is shown that the correlations of the surface waves are dependent on temperature and density distributions in the boundary layer. The correlation function is determined by temperature values in the points of transition layer, where the longitudinal dielectric permeability becomes equal to zero. Dependence of the obtained correlation function on co-ordinates, in the electrostatic limit, is characterized by the complex correlation length, which is fully determined by the inhomogenity of the plasma density near the boundary.

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.

The paper studies the confluence of three shock waves meeting at a point, in an ideally conducting perfect gas, for transverse shocks and shocks in an aligned MHD field. The method used involves the hodograph mapping technique. A presentation of the shock polars in the total pressure P*, flow deflexion δ(δ, P*/P*0) plane is thus given. The algebraic complexity of the problem prohibits discussion of an alternative mathematical approach.