In this investigation, we study the effects of non-linear interaction of plasma waves in the weak turbulence approximation on a stream-plasma system which is subject to the two-stream instability. The momentum distribution of the stream is assumed to be a Gaussian function and the background plasma is assumed to be a Maxwellian with a temperature T. The non-linear wave-particle interaction of induced scattering on the polarization clouds of ions may lead to stabilization of the stream—plasma waves in the resonant cone are scattered into the nonresonant region. The non-linear wave-particle interaction of induced scattering on the polarization clouds of ions may lead to stabilization of the stream; plasma waves in the resonant cone are scattered into the non-resonant region and vice versa, leading to a well-defined average energy density in plasma waves which is much less than the energy density of the stream. Non-linear scattering also causes transformation of plasma waves into electromagnetic radiation near the fundamental of the electron plasma frequency of the background plasma, and combination of two plasma waves leads to electromagnetic radiation near the second harmonic of this frequency. Expressions are given for the latter process for emission and absorption for arbitrary values of the ratio of the wave number of plasma waves to the wave number of the electromagnetic wave. The major features of the whole phenomenon are illustrated with the help of a numerical example, with conditions typical for the solar corona in which case an ion stream would be stabilized, but an electron or neutral stream (with the chosen density) would not.

The time asymptotic method of non-linear mechanics (Bogoljubov & Mitropoiski 1961) is used to solve the hierarchy equations of low-amplitude wave correlations. We start with a formal description of individual dispersive and undamped waves and derive the usual kinetic equation for the energy spectrum and the three-wave correlation in the lowest order (three-wave processes). We show that the equations are automatically closed (‘Quasi-Gaussian’) even in the case where the correlations have the same order of magnitude as the corresponding moments. This approach parallels the multiple-time formalism of Sandri and Frieman, but it represents an alternate systematic and unique method (without ‘extension’ in the sense of Sandri).

In electrostatic turbulence under appropriate conditions we show that we can have an equation for ϕk (time variation of electrostatic potential amplitude) which is essentially instantaneous in time. The coefficients in this equation are uniquely determined by the Poisson equation of the corresponding order. By applying time asymptotic methods of Bogoljubov and Mitropoiski to the corresponding hierarchy equations we obtain a kinetic equation for the spectrum 〈|ϕk|2〉. The resonant mode coupling terms agree with the usual ones. But the non-linear Landau damping terms, as given, e.g. by Frieman & Rutherford (1964), Kadomtsev (1965), Tsytovich (1968), and Rogister & Oberman (1969) are complemented by additional terms which are, at least formally, non-zero. For simplicity the calculations are only done for the weakly unstable (and weakly damped) modes.

The interaction of three electrostatic waves in a collisionless plasma is treated to fourth order neglecting the wave—particle interaction (damping). Using the principles of energy conservation, and invariance under time reversal, conditions on the coupling constants are derived, enabling the solution to be expressed as a function of only two coupling constants. Phase plane diagrams of the solutions are sketched showing that the only singular points are stable equili bria. It is suggested how the theory may be applied to the explanation of the ‘floating spike’ resonance observable when one-half the hybrid frequency is the difference between the plasma and gyro frequencies (Hagg & Muldrew 1968).

The stability of the screw pinch is investigated taking into account the presence of Hall current in Ohm's law. The dispersion relation is obtained as the exact solution of the characteristic value equations governing the general perturbations. It is found that the Hall current has no effect on the hydromagnetic stability of the screw pinch.

Using a reformulation of ordinary perturbation theory to first and second order, explicit solutions for the penetration of electrostatic waves into a semi-infinite Vlasov plasma are presented. Two types of electron velocity distribution are used, the uniform step (‘water-bag’ model) and the Maxwellian. The existence of propagating modes below the plasma frequency, behaviour of the second harmonic field and the formation of a narrow sheet of electrons near the bounding wall are displayed. Replacement of boundary fields with equivalent charge and current sources is discussed.

The voltage refiexion coefficient from a vacuum-plasma boundary in a co-axial transmission line with an axial magnetic field B0 applied has been measured. The results agree well with a previously published theory for conditions where the microwave-, plasma-, electron collision-, and electron cyclotron-frequencies are of the same order. A 9 GHz co-axial microwave probe is mounted along the axis of a 44mm diameter, hydrogen driven, dry air filled, shock tube in an axial d.c. magnetic field. Shock ionized air (Ms = 9–14, T 4000 °K, electron density nc = 1017 to 3 x 1019 m−3, initial pressure p0 = 1–10 Torr, electron collision frequency v = 1010 to 1011/S) fills the coaxial line and partially reflects a microwave signal. Initially this probe, and a similar rectangular waveguide probe, were used with B0 = 0 to calibrate the plasma (ne, v) in terms of the shock tube parameters (p0, Ms). Measurement of the saturated-ion current to electrostatic probes inset into a fiat plate in the shock tube flow showed that the sheath-edge ion density is close to the predicted free-stream equilibrium ion density. The apparent ionization potential derived from electrostatic probe results decreased as p0 was reduced from 10 to 1 Torr.

An infinite, inhomogeneous electron plasma driven by a spatially uniform oscillating electric field is investigated. The multi-time perturbation method is used to analyze possible parametric excitations of transverse waves and to evaluate their growth rates. It is shown that there exist subharmonic excitations of: (1) a pair of transverse waves in an unmagnetized plasma and (2) a pair of one right and one left circularly polarized wave in a magnetoplasma. Additionally, parametric excitation of two right or two left circularly polarized waves with different frequencies can exist in a magnetoplasma. The subharmonic excitations are impossible whenever the density gradient and the applied electric field are perpendicular. However, parametric excitation is possible with all configurations.

It is shown that the reported absence of a contribution of electronic excitation on to the total specific heats of helium and nitrogen (10,000–35,000°K, 10−1−10 atm) should be attributed to the presence of compensation effects among the various terms into which the total specific heat can be separated and that these terms are influenced by the electronic contribution.

We first discuss wave propagation perpendicular to a magnetic field. Variables depend on x only. The medium is collision-free, described by the Vlasov theory. We can prove that there exist two, and only two, unique constants of the motion, that are independent of y, z and t, and that at least one of them depends on x. Thus a necessary condition for the existence of stationary shocks is fulfilled. Examples show that the existence proof also suggests a practical tool for solving problems. The oblique case which is also briefly discussed turns out to be qualitatively different in that, owing to the presence of reflected particles, a uniform upstream state cannot be prescribed a priori.

A recent paper on quasiinear theory by Klozenberg & Bernstein is commented upon. It is argued that improper treatment of the perturbed electron distribution for damped waves has led to a diffusion-type equation with a negative diffusion coefficient.

The criticisms of Montgomery & Bodner (preceding paper) seem to stem from a misunderstanding of the claims in the paper of Klozenberg & Bernstein (1970) in part perhaps semantic. The derivation criticized was intended to apply only to those particle velocities v corresponding to the range in phase velocities βк/к for which there is an appreciable wave excitation. The term ‘resonant phase velocity’ was intended to mean those phase velocities in a small interval Δ such that ν − ½Δ < βк/к < ν + ½Δ, which interval in the integral determining the diffusion coefficients leads to the familiar expression involving δ(βк кν). When this contribution is non-vanishing it determines the lowest significant order contribution, in the small parameters of the problem, of ∂ƒ0/∂t, assuming that the wave excitation at the time in question is so large that effects associated with initial conditions are indeed negligible.

We wish to point out that the revised quasilinear equations found recently by Klozenberg & Bernstein (1970) for continuous wave-number spectra, which are a simplified version of those derived by us for discrete wave spectra (Abraham-Shrauner 1970), can be derived easily from the conventional quasilinear equations (Drummond & Pines 1962; Vedenov, Veliklov & Sagdeev 1962) by analytic continuation. The analytic continuation is the same type used in discussions of the plasma dispersion relation for the linearized Vlasov equation and of the weakly unstable kinetic equation (Balescu 1963; Rogister & Obermann 1969).

By solving for the changes 〈Δν〉 and 〈ΔνΔν〉 for a particle, it is shown that there is no diffusion equation for non-resonant particles when the electric field wave is damping. By indirect means it is then shown for this case that the appropriate quasi-linear solution for ƒ1(к, ν, t) is proportional to ∂ƒ0(ν, 0)/∂ν rather than ∂ƒ(ν, t)/∂ν.

Resonance frequencies of a plane plasma condenser in crossed electric and magnetic fields are determined, using the two-component magnetic hydrodynamical equations. The perturbed plasma parameters (ion and electron densities and velocities and the electric field in the plasma), due to an external a.c. electric field are found. The calculated resonance is not exact since dissipative processes are neglected.