Cross-modulation in plasma is an electromagnetic wave interaction in which the modulation of one ‘disturbing’ wave is imposed nonlinearly on the transport properties of the medium, and thence onto a second, ‘wanted’ wave propagating linearly through it. This analysis is restricted to weakly ionized plasma with allowance for ambient magnetic field, as in the lower ionosphere. A kinetic description is used, based on the Boltzmann equation for the electrons, with electron-molecule collisions described by Boltzmann's collision integral. Because of the small mass ratio this simplifies to a differential form. The perturbation of the electron velocity distribution function f(v, t) due to the disturbing wave is calculated up to terms quadratic in wave amplitude, which are the lowest order to show the effect. The part of the term quadratic in wave amplitude at zero times the fundamental frequency, and isotropic in velocity space, which represents the perturbation in electron energy distributions, is selectively enhanced by an inverse factor of mass ratio, since the excess energy imparted by the wave to the electrons is transferred collisionally to the molecules at a rate inversely proportional to mass ratio. Modulation of the wave induces modulation of the electron energy distribution. A more general expansion scheme, in velocity-space spherical harmonics, is also presented. To calculate the dispersion relation for the second, ‘wanted’ wave, the linear part of the disturbing wave analysis is adapted, and the amplitude of the wanted wave is given in the WKB approximation as a phase integral of the refractive index along the ray path; this contains moments of the electron energy distribution and is modulated. The predictions of older semi-empirical theories, that the effect is enhanced when the fundamental frequency of the disturbing wave is close to the electron gyrofrequency, and that the second harmonic of the modulation is also imposed on the wanted wave, are confirmed. The wanted wave is predominantly amplitude-modulated, and only amplitude modulation of the disturbing wave is picked up; phase modulation is not transferred. There is no cross-modulation if the collision frequency is independent of collision speed, when contributions from all parts of velocity space cancel.

Using kinetic theory, the electron-acoustic instability is investigated in a three-component plasma consisting of a hot electron beam and stationary cool electrons and ions. In the model considered here both the electrons and ions are magnetized, with the beam drift along the external magnetic field. The dependence of the growth rate on plasma parameters, such as electron-beam density, electron-beam speed, magnetic field strength and propagation angle, is studied. In addition, the effects of anisotropies in the velocity distributions of the hot electron beam and the cool electrons on the instability growth rate are examined.

The stability of straight field-aligned electron beams, immersed in an external magnetic field of finite magnitude, with respect to the excitation in them of circularly polarized (spiral) electromagnetic waves is a problem calling for detailed investigation, particularly in the context of the study and development of free-electron lasers. Traditionally the problem is treated using the theory of electromagnetic waves scattering off electron-beam density oscillations. This is done, however, without considering the inverse influence of the beam on the dispersion properties of the electromagnetic waves. On the other hand, it is well known that the presence of the beam introduces substantial changes in the characteristics of the electromagnetic waves interacting with the beam, and, moreover, this results in the appearance of radically new types of waves that are entirely absent in free space. The paper is dedicated to the study of the nonlinear dynamics of the interaction of such radically changed electromagnetic waves with the beam density oscillations.

An expression is obtained for the collision integral describing the simultaneous interaction of plasma particles with resonant and non-resonant waves. It is shown that this collision integral is determined by two processes: a ‘direct’ nonlinear interaction of particles and waves, and the influence of the non-stationarity of the system. The expression for the nonlinear collision integral is found to be quite different from the expression for a quasi-linear collision integral; in particular, the nonlinear integral contains higher-order derivatives of the distribution function with respect to momentum than the quasi-linear one.

The stability of a viscous and heat-conducting plasma is considered, with temperature and density gradients arranged so that the resulting pressure is homogeneous, as suggested by some experiments on laser-plasma interactions. Gravity and magnetic field effects are ignored throughout. A non-dissipative fluid is considered first. Under the assumption of an Epstein profile for the equilibrium temperature, the dispersion curves for the propagation of sound waves are determined using a relaxation technique applied to a second-order spheroidal boundary-value problem for mode numbers 1,…, 5: it is found that these modes may be destablized by inhomogeneity. The dynamics of vorticity is analysed in the framework of the WKB approximation, which allows reduction to a fourth-order boundary-value problem: this is tackled in the ‘quasi-classical’ limit by solution of the relevant Bohr-Sommerfeld eigenvalue problem. Convective flows arise in the classically accessible layers and evanesce outside. For the same Epstein temperature profile as above, two such layers are formed, in agreement with observation. Given the growth rate of the unstable flow, the modified Rayleigh number, the width of the convective layer and the temperature drop across it are calculated as functions of the transverse wavenumber and for mode numbers 1–5. Conversely, given the modified Rayleigh number, the growth rate is calculated. The modified Rayleigh number and the growth rate thus found have a minimum and a maximum respectively as functions of the wavenumber of the perturbation. These can be taken as the most favourable values for convection to develop. The value of the wavenumber at which these extrema are attained can be used to determine the aspect ratio of the convective cells.

Weyl's well-known theorem describing the behaviour of the entropy on the Hugoniot for gasdynamics is extended to magnetohydrodynamics. In so doing, it is pointed out why similar thermodynamical arguments are not easily established for more complicated reacting fluids such as the gasdynamical equations with mass-loading.

The evolution of the distribution function due to the simultaneous nonlinear interaction of plasma particles with resonant and non-resonant waves is studied. A stationary particle distribution resulting from a balance of the quasi-linear interaction and the nonlinear one is found. The temporal evolution of an initial δ-function-shaped distribution (like a ‘beam’) is examined in the one-dimensional case. General formulae are obtained for stochastic particle acceleration (taking account of the nonlinear interaction studied here).

An analysis is presented of linear stability against tearing modes of a current sheet formed between two oppositely magnetized plasmas forced towards each other in two-dimensional steady stagnation-point flow. The velocity vector in this flow is confined to planes perpendicular to the reversing component of the magnetic field. The unperturbed state is an exact resistive and viscous equilibrium in which the resistive diffusion outwards from the current sheet is exactly balanced by the inward motion associated with the stagnation-point flow. Thus the behaviour of the tearing mode can be examined even when the resistive diffusion time is comparable to or smaller than the growth time of the instability. The linear ordinary differential equation describing the mode structure is integrated numerically. For large Lundquist number S and viscous Reynolds number Re the Furth-Killeen-Rosenbluth scaling of the growth rate is recovered with excellent accuracy. The influence of the stagnation-point flow on the tearing mode is as follows: (i) long-wavelength perturbations are stabilized so that the unstable regime falls between a short-wavelength and a long-wavelength marginal state; (ii) for sufficiently low Lundquist number (S < 12.25) the current sheet is completely stable to tearing-mode perturbations; (iii) the presence of high viscosity reduces the growth rate of the tearing instability. This effect is more important at small wavelength. Finally, application of the results from this study to the problem of solar-wind plasma flow past the earth's magnetosphere is briefly discussed.

The problem of nonlinear surface Alfvén waves propagating on an interface between a plasma and a vacuum is discussed, with dispersion provided by the finite-frequency effect, i.e. the finite ratio of the frequency to the ion-cyclotron frequency. A set of simplified nonlinear wave equations is derived using the method of stretched co-ordinates, and another approach uses the generation of a second-harmonic wave and its interaction with the first harmonic to obtain a nonlinear dispersion relation. A nonlinear Schrödinger equation is then derived, and soliton solutions found that propagate as solitary pulses in directions close to parallel and antiparallel to the background magnetic field.

Some properties of the frequency spectrum of magnetic fluctuations in a tokamak two-species plasma are investigated. We start from the nonlinear resistive ballooning kinetic equation. A method based on the fluctuation-dissipation theorem of thermodynamics is employed. The frequency range |ωDe|/ω>|ω*e| is considered. The dependence of magnetic fluctuations on frequency and various macroscopic parameters is discussed. The influence of magnetic fluctuations on plasma thermodynamics is investigated.

A study is made of the induction of a direct field-aligned current due to Alfvén waves excited through an initial–boundary condition for an incompressible fluid with high conductivity in a non-uniform magnetic field. A simple but exact case is considered in which the direction of inhomogeneity of the magnetic field is perpendicular to that of the shear fluid motion at the boundary, and the fluid moves across the magnetic field. The boundary does not satisfy the frozen-flux condition, but has a large Lundquist number. The conductivity of the fluid held between the two boundaries is high, and the frozen-flux condition is approximately satisfied there. A preflare stage of the sun is investigated, and a reasonable magnetic field intensity and d.c. field-aligned current in the magnetic arcade are obtained for a given horizontal shear motion that is antisymmetric at both boundaries of the arcade foot points, taking account of the observed horizontal fluid motion.

Starting from a three-wave interaction coefficient for a homogeneous Vlasov plasma, we show that the parametric decay to half-harmonics has zero growth rate if the wavenumber of the pump wave vanishes. Earlier proofs of this fact for electrostatic decay modes are thus extended for a general case. As a specific example, we study the decay of a fast magnetosonic wave to two slow waves.

Using the magnetohydrodynamic model, the evolution of a resistive plasma can be represented as a relaxation through a sequence of force-free equilibrium states. We show, by extending existing work, that this process is equivalent to magnetic field diffusion in a strongly anisotropie static conductor. The latter evolution is easier to simulate numerically, and is carried out for laboratory based plasmas confined in cylinders and toroids. We obtain a variety of universal equilibrium profiles that are consistent with experiment and relaxation theory and that predict the existence of states arising in reversed-field pinches. The existence of a critical axial flux is predicted about which there exist stable modes of operation corresponding to high and low current. We also show the existence of a critical aspect ratio at which it is most desirable to build toroidal devices. This corresponds to the value at which maximum current, for a fixed driving field, can be generated.

During propagation of a relativistic electron beam in hydrogen gas at sub-torr pressures using a foil-less diode, low-frequency oscillations in the megahertz range are observed in the net current wave forms, and continue even after passage of the beam. The novel feature of the experiment is that both beam generation and propagation take place in the same low-pressure regime, and hence the beam parameters are functions of gas pressures. Analysis of the experimental results shows that the low-frequency oscillations result from the resistive-hose instability.

The theory of particle aspect analysis is extended to a drift wave in the presence of an inhomogeneous magnetic field and an applied electric field parallel to the ambient magnetic field. The effect of a parallel electric field is included in the zeroth-order distribution function through modification of the particle thermal velocity in that direction. The plasma under consideration is assumed to be anisotropic and with low β. The dispersion relation and growth rate are evaluated and discussed for particular plasma parameters. The limitations of the theory are also pointed out.

It is well known that ion-acoustic double layers occur in plasmas containing two-temperature electron species and positive ions. In a recent study Verheest suggested that such structures can occur when there is only one electron species but both negative and positive ion species. There a stationary modified Korteweg de Vries equation was derived to support his deduction. This prediction, however, contradicts our own arbitrary-amplitude studies, which also cover the Verheest parameter regime. Here we resolve the discrepancy by examining both approaches in the pseudo-potential formulation in more relevant detail. The implications of this study extend beyond the realm of the present subject.

A plasma with a boundary transition layer of variable depth in the presence of a powerful electromagnetic field is considered. It is shown that a displacement of the boundary will grow, and will propagate as a nonlinear surface wave in the direction in which the depth of the transition layer decreases.

Solutions of the full nonlinear Vlasov–Poisson system for a one-dimensional unmagnetized plasma that correspond to undamped travelling waves near Maxwellian equilibria are analysed numerically using the splitting scheme algorithm. The numerical results are clearly in favour of the existence of such waves and confirm that there is a critical phase velocity below which they cannot be constructed.

For relativistic amplitudes (‘intense propagation’), modulations of electromagnetic (EM) waves in a cold plasma are strongly coupled to longitudinal modes via the ponderomotive force, even if the unmodulated wave is purely transverse. The effect of longitudinal motions is comparable to that of relativistic nonlinearities. Hence a correct and proper expansion procedure requires solution of all the field equations up the appropriate order, including the longitudinal equations. (The plasma is assumed to be fully ionized and cold, with no net charge. The positive component may consist of either positrons or singularly charged ions. Quasi-neutrality is not imposed, but deviations are shown to be small for slow modulations.) We begin by deriving envelope equations (i.e. a reduced set of partial differential equations) governing modulations of intensely propagating, circularly polarized, plane EM plasma waves. (Here the envelope equations are not merely accurate to the appropriate order, but in fact exact.) We then analyse their modulational instabilities. Since the envelope equations are exact, they include the weakly relativistic case (transverse velocities ≪ c) as a limit. We first verify that, in this limit, the envelope equations yield the nonlinear Schrödinger equation (NLS) complete with the correct coefficients and thus the correct modulational instability conditions. Next, although the NLS does not hold for the general case of fully relativistic systems (transverse velocities comparable to c), we still succeed in deriving the modulational instability properties of intensely propagating EM waves in the fully relativistic case directly from the envelope equations, again, reproducing the weakly relativistic case as a test of the perturbation method used. Finally, we apply the results in special cases. For example, in the ultrarelativistic limit (transverse velocities close to c) of either an electron—positron or ion—electron plasma, waves with frequencies below twice the relativistic plasma frequency ωp, where , are unstable. The time scale for growth is comparable to the time for a wave packet to move past a fixed point. (This time scale is much shorter than in the weakly relativistic case, where the time scale for instability is that of the spreading of the wave packet due to linear dispersion.) Our results have important consequences for observations of pulsar micropulses, possible technological applications, and general implications for applications of Whitham theory.

The range of ▿ ∧ (u ∧ B) is determined for an axisymmetric poloidal incompressible B and incompressible u, by solving ▿ ∧ (u ∧ B) = ▿ ∧ v for u, assumed to be confined to within a surface of revolution. Two constraints on V are shown to be necessary for the existence of solutions, viz that the integral of rç/|B| must vanish on each flux surface, and that the integral of v1 around any dosed field line must vanish. The construction of the general solution proves that the constraints are sufficient conditions, providing also that the second derivative, with respect to the poloidal flux function, of the volume contained by flux surfaces does not vanish. The general solution is stated for the homogeneous case, v = 0. For the particular case of poloidal v1 an integral property of the solution u is established.