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.

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 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.

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.

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.

The nonlinear interaction between two Alfvén waves and a sound wave is studied, using the normal-mode approach. This leads, in a simple way to a set of coupled equations, and consequently to a dispersion relation for the waves under consideration. It is shown that a large-amplitude Alfvén wave can give rise to two distinct types of parametric instabilities, namely the oscillating and the purely growing waves. In each case, the expressions for the threshold pump intensity, the frequency shift and the growth rate of the excited waves are obtained. In particular, the results for a propagating pump under perfect frequency matching conditions are compared with those of Sagdeev & Galeev.

The weak-coupling approximation takes stochastic perturbations of orbits into account, but neglects a part of mode coupling. This approximation, which is non-trivial, is discussed here. Its domain of validity appears to be T E ≪Tt provided the turbulence is strong enough so that γNL- γL ñ γL. If γL ≪ a stronger condition is required. It is shown that the growth rate γNL can be calculated either from the nonlinear dispersion equation εNL =0, or from the energy equation, provided thepropagator conserves particles. Dupree's propagator U does not do this; but it gives a correct approximation of γNL, provided Δv≫vth.

With electrostatic probes, the electric field component E∥ along the magnetic field B was comprehensively investigated in a collisionless plasma, the density of which was of the order of 1010 cm-3. The plasma in the experiment has several properties in common with the plasma of the ionosphere/magnetosphere scaled to laboratory dimensions. It is produced by means of electron cyclotron resonance in a microwave cavity located in the magnetic field gradient in one half of a magnetic mirror field. The magnetic field strength is 3600G in the resonance zone and 1800G in the middle of the mirror field. The measurements show that a stationary E∥ exists everywhere in the plasma, where the magnetic field gradient grad11 B in the direction of the field is different from zero. The direction of E‖ is opposite to that of grad‖B. The total potential drop along B between the resonance zone and the midplane of the mirror field is of the order of kilovolts. E‖ accelerates ions along B to energies of the order of kilo electron volts. Experimental parameters of importance for the production of E‖ are the neutral gas pressure p (normally a few times 10− Torr), the microwave power (usually about 2kW), and the mirror ratio γ in the mirror region opposite to the cavity side, γ was normally <2. For γ>2·3, an instability develops and no stationary E‖ remains. As p is increased, E‖ decreases successively. In terms of the mean free path λ, it is found that λ>5−10L is a necessary condition for the existence of E‖. L is twice the distance between the cavity and the midplane of the mirror field. In the experiment, the ion and electron pitch angle distributions are forced to be different; the ion velocity is mainly parallel to B, and the electron velocity essentially perpendicular to JB, and as consequence E‖ is created. In this way an experimental demonstration is presented of the theoretically predicted relation between E‖ and the pitch angle distributions. When imposing sufficiently strong radial electric fields Er (fields perpendicular to B), the distribution of the potential along B is deformed, probably due to changes in the particle distributions caused by E‖. We think that our results strongly support the idea that Et is produced in the magnetosphere, and is at least sometimes an important mechanism for the acceleration and precipitation of auroral particles.

We present a procedure to solve the Vlasov–Poisson equations for electrostatic wave packets. We obtain a Schrödinger type of wave equation, taking the wave– particle interaction into account. We use this equation to study the propagation of one finite-amplitude wave packet. We find a change in amplitude caused by interaction between the packet and particles propagating near to the group velocity. Also, we find a modulation of the plasma in the front of the packet, caused by trapping effects.

It is argued that the use of quasi-normality or ‘random phase’ approximations in the kinetic theory of the two-dimensional guiding-centre plasma gives rise to unphysical and inconsistent results.