An electron thermal instability is proposed, which could explain the occurrence of the break-up of a plasma shell into separated current elements during the early history of a discharge. Inclusion of electron thermal conduction and Faraday's law leads to an optimum wavelength for growth, in good agreement with recent results on a θ pinch at NRL, Washington. Finally, the addition of ion motion leads to a modification of the stability criterion. The non-convective instability exists only at low temperature and high density; but a convective instability now appears, which persists at high temperature, and could be relevant to fusion conditions.

We give a necessary criterion of validity for the Dupree–Weinstock theory of turbulence. In this connexion, we analyze two specific cases of high-frequency (HF) and low-frequency (LF) turbulence. In the HF case, we show that an appreciable modification of the growth rate of unstable modes cannot occur before trapping.

This paper discusses the procedure for constructing intense hollow relativistic electron beam equilibria within the framework of the steady-state (∂/∂t = 0) Vlasov–Maxwell equations. It is assumed that the electron beam propagates parallel to a uniform axial guide field Bext0 = B0 êz inside a grounded cylindrical conductor, and that the positive ions provide a partially neutralizing background with density n0i(r) = fn0e(r), where f = const. = fractional neutralization. The equilibrium properties are calculated for the specific choice of electron distribution function , where H, Pθ, and Pz are the energy, canonical angular momentum, and axialcanonical momentum, respectively, for an electron moving in the equilibrium fields, and , and P0 are constants. For this choice of distribution function, the mean axial velocity of the electron beam is equal to , and the beam density profile is hollow with inner and outer radii, R0 and R1, determined self-consistently from nonlinear boundary conditions that involve the equilibrium parameters Po, γb, γ0, etc., and the equilibrium self fields. Closed expressions for Ro and B1 are obtained for the case where the collisionless skin depth c/ωpe is large in comparison with the characteristic radius of the electron beam, and the beam density is sufficiently low that . The two cases, Po>0 and Po<0, are considered.

Using a simple model of particles in a discrete spectrum of waves, we investigate the influence of stochastic instability of particle motion on the mechanism of nonlinear Landau damping. We show that, as long as particle trajectories in the spectrum are stochastically unstable (at the beat resonances under consideration), diffusion of particles takes place. We discuss the influence of this effect on the normal nonlinear Landau mechanism, and consider briefly the possibility of consequent heating in beam–plasma experiments.

We investigate radiation of waves at double the frequency from an arbitrary inhomogeneous, isotropic plasma layer, when 8-polarized electromagnetic waves are obliquely incident on it and their lengths are considerably greater than the width of the layer. We calculate the amplitudes of the radiated waves. We show that, in general, the second harmonics are radiated in the direction of propagation of both the reflected and the transmitted wave being at the basic frequency. At small plasma densities, the second harmonics are emitted practically only in the direction of propagation of the reflected wave at the basic frequency. For normal incidence, there is no wave radiation at double the frequency of the incident wave.

The oblique echo model for the resonance near the local plasma frequency fN observed by topside sounders involves the propagation of slow waves away from the sounder, which later return as echoes after reflexion due to an electron density gradient. The model is investigated using the WKB technique introduced by Fejer & Yu. The resulting set of saddle-point equations is solved by a method which can, unlike previous work, be applied to both the ‘strong’ resonance (for whichfN > fH wherefH is the electron gyrofrequency) and to the ‘weak’ (fN < fH) resonance. Throughout the calculations, the validity of the approximations made is checked. It is found that the weak resonance solution is not valid for typical topside parameters. For the strong resonance, the electric field of a small pulsed dipole is calculated. Expressions are found for the frequency of the two echoes that are received by the sounder and the frequency of the beating between them. Although these are only strictly valid for times longer than typical observation times, they yield results that agree well with the corresponding results from ray trajectory computations.

The oblique echo model for the resonance near the local upper hybrid frequency fT observed by topside sounders involves the propagation of slow waves away from the sounder, which later return as echoes after refiexion due to an electron density gradient. The model is investigated using the WKB technique introduced by Fejer & Yu. The resulting set of saddle-point equations is solved by a method which can, unlike previous work, be applied to both the ‘strong’ resonance (for which fT < 2fH where fH is the electron gyrofrequency) and to the ‘weak’ (fT 2fH) resonance. Throughout the calculations the validity of the approximations made is checked. It is found that the weak resonance solution is not valid for typical topside parameters. A condition is derived for the existence of intercepting ray paths and for the maximum time delay which occurs when they do exist. For the strong resonance the electric field of a small pulsed dipole is calculated. Expressions are found for the frequency of the two echoes which are received by the sounder and the frequency of the beating between them. Although they are only strictly valid for times longer than typical observation times, they yield results that agree well with the corresponding results from ray- trajectory computations.

A theory is presented for the rate of thermal relaxation of a two-dimensional plasma in a strong uniform d.c. magnetic field. The Vahala—Montgomery kinetic description is completed by providing a cut-off time for the time of interaction of two particles which contribute to the collision term. The kinetic equation preclicts that thermal relaxation occurs as a function of the dimensionless time (ωpt) (ωp/Ω) (n0λ2D)−½, where ωp, is the plasma frequency, Ω is the gyrofrequency, and n0 λ2D is the number of particles per Debye square. By contrast, in the absence of an external magnetic field, a two-dimensional plasma relaxes as a function of (ωpt) (n0λ2D)−1.

The energy transfer equation in wavenumber space of ion–acoustic wave turbulence is obtained through investigation of the effects of turbulent fluctuations on the propagator of a test wave in the plasma. Those turbulent fluctuations whose wavelengths are shorter than that of the test wave give rise to collisional effects, while those with longer wavelengths act to deform continuously the shape of the test wave, thereby producing a diffusive flow of the turbulence energy in k space. Dimensional analysis, coupled with the Kolmogorov hypothesis, is then carried out on the resulting equation; a universal spectrum proportional to in the large k subdivision and to k−4 in the small k subdivision of the inertial domain is thus obtained.

The problem of the two-stream instability in warm, collisionless, field-free plasmas is investigated for arbitrary direction of propagation vector, using the moment equations. It is found that the configurations characterized by (i) a beam of electrons with ions immobile, and (ii) all electrons moving through mobile ions, are stable for subsonic interpenetration speeds. For supersonic speeds, however, the configurations are unstable both for electrostatic and electromagnetic perturbations. The former instability is restricted only to a cone of propagation about the streaming direction, while the latter exists for all angles, and has growth rates that are even larger than the electrostatic growth rates in the limit of long-wavelength perturbations.

The longitudinal behaviour of the electron ion ring in a stationary electron ring accelerator (ERA) is studied by means of a filament model which neglects the radial thickness of the ring. There are two widely different time scales (short or STS of the order of a bounce time in the ring, and long or LTS) that characterize the system. The Vlasov equations for a system which is stationary on the STS can be solved in the ion-pickup region by an approximation in which each species of particle essentially sees only a potential well formed by the other species. This gives a generalization of the classical Bernstein–Greene–Kruskal (BGK) problem, in which both species move in the same potential. The conditions under which this generalized BGK problem is well defined are given, and broad classes of quasi-equilibria (i.e. equilibria on the STS) are obtained. The time dependence on the LTS of some of these quasi-equilibria is then obtained by invoking charge conservation, momentum conservation and the adtained invariance of longitudinal action integrals. The stability of these quasi-equilibria (i.e. their behaviour under time-dependent perturbations on the STS) is deferred to a subsequent paper.

The thermal relaxation process for a spatially uniform two-dimensional plasma in a uniform d.c. magnetic field is simulated numerically. Thermal relaxation times are defined in terms of the time necessary for the numerically computed Boltzmann H function to decrease through a given part of the distance to its minimum value. Dependence of relaxation time on two parameters is studied: number of particles per Debye square n0 λ2D and ratio of gyrofrequency to plasma frequency Ω/ωp. When Ω2/ω2p becomes ≫[ln (L/2πλD)]−½, where L is the linear dimension of the system, it is found that the relaxation time varies to a good approximation as (n0 λ2D)½ and Ω/ωp.

The Boltzmann transfer equation for electrons in an r.f. discharge plasma immersed in a stationary magnetic field is solved for the various coefficients of the expansion of the distribution function, expanded in a Taylor series in velocity space. Assuming a spatial variation of static electron number density as the mechanism of harmonic generation, explicit expressions for the inner field and the second harmonic current density are derived. The r.f. electric field is assumed to be spatially uniform. The geometry of the plasma considered is that of a rectangular waveguide, but with the parallel plates of one of the two pairs of the metal plate boundaries of the plasma much more closely spaced than those of the other pair. Cyclotron resonance is studied in a situation where the frequency of electron-neutral particle collisions v is much less than the frequency of the r.f. field ω/2π. The resonance effect is obtained, and is found to be more pronounced at the cyclotron frequency ω = ωc than at 2ω = ωc. For v≫ω/2π, the result is not sensitive to the value of the stationary magnetic field, and the resonance effects are absent. The effect of collisions on the process of harmonic generation is also studied.

This paper analyses the dispersion relation for a collisionless moving electron plasma, when the direction of motion is normal to the magnetic field and that of the wave propagation along the magnetic field. It is shown that, in strong magnetic fields, the one continuous allowed band of the left-handed wave (of the stationary plasma) splits into two, and the right-handed wave shows a second resonance besides the cyclotron resonance. In weak magnetic fields, the lefthanded wave develops a backward wave band, which shows resonance at its low frequency edge, and the right-handed wave also develops an extra band of propagation. The effect of the motion of the plasma, on waves of frequency much lower than the plasma frequency, is identical to a doppler shift, but, on those of frequency much higher than that, is negligible.

The modified Chew—Goldberger—Low (CGL) equations are applied to the effect of Hall current on the instability of an incompressible plasma jet surrounded by non-conducting, compressible matter. The dispersion relation is obtained and discussed. The following is found. (i) When λ (the ratio of plasma density to the density of surrounding medium) is much greater than unity, the plasma jet is unstable for all wavenumbers for which k* = Küα < [(4R22 – 1)ü(1 + V2α)], where R2 = p∥/p⊥, V2α = H2/4αρ, K = (l2 + α2)½. Also, the jet is unstable for R2 > 1 + V2α. (ii) When λ ≪ 1, the critical wavenumber ratio for the instability to set in is k* < [(V2α + 3R2)ü(1 + V2α)½. Also, the jet becomes unstable for R2 < ⅓. (iii) When either l = 0 or α= 0, we must have R2 > 1 + V2α for instability. It is established that the Hall current has a destabilizing effect for certain wave- numbers. The dispersion relation for the incompressible plasma jet in cylindrical geometry is solved numerically on a computer.

This paper derives an expression for the velocity space diffusion tensor (time asymptotic) for resonant particles in the weak-coupling limit of strong plasma turbulence theory. The analysis is based on an improved approximation scheme for the determination of the actual particle trajectories, rather than the determination of &ensemble averaged particle orbits’ with the help of the &average Vlasov propagator’. Except for numerical factors of order unity, the results agree with those of Birmingham & Bornatici, both when Tc → 0 (Tc is the correlation width of the fluctuating field as seen by the resonant particles), and when Tc is finite but small. The paper indicates the limits of this method, as well as those of Birmingham & Bornatici.

General relations for quasi-linear diffusion coefficients in pitch angle and energy are applied to resonant particle interactions with ion–cyclotron and whistler waves. Expressions for the diffusion coefficients, valid for any distribution of wave energy with frequency and wave normal angle, are derived and normalized to be independent of the ambient magnetic field intensity and the electron density. The results illustrate how resonant particle diffusion rates vary with pitch angle and energy, and how the diffusion rates depend upon the distribution of wave energy.

By applying quasi-linear diffusion theory, general expressions are obtained for particle diffusion coefficients in both pitch angle and energy. The results are valid for resonant interactions with any mode of weak plasma turbulence, with wave energy distributed over any range of frequencies and wave normal angles. Simple relations are found between the diffusion coefficients, these relations reflecting the geometry of the diffusion surfaces for resonant particles.

The magnetoplasma dispersion function is defined as

It is found in all the dispersion relations for wave propagation perpendicular to the magnetic field in hot plasmas. We review the properties of fω(z), and present several new expansions, which may be of interest in the study of properties of waves in plasmas, or may suggest computational methods for calculating fω(z).

The excitation characteristics of the current-driven ion acoustic instability are studied, using a linearized kinetic model for a weakly ionized, unmagnetized plasma. Convective instability is predicted for typical low-pressure positive column conditions. The calculated spatial growth rates show a variation with frequency, which is similar to that of the amplitude variation with frequency of the self-excited instability measured in our positive column experiments in helium and argon. The comparison between theory and experiment indicates that ion Landau damping is significant for typical experimental conditions.