A modified form of the Bogoliubov plasma cluster expansion is applied to the derivation of a divergence-free kinetic equation from the BBGKY hierarchy. Special attention is given to the conditions under which the Landau kinetic equation may be derived from this more general formulation.

An important class of overlooked MHD perturbations which are necessary to satisfy arbitrary initial conditions is discussed.

The test charge problem for a semi-infinite plasma is formulated. Expressions for the far field potential are derived for stationary and slowly moving test charges. By applying the superposition principle for uncorrelated dressed test particles we calculate the dynamic form factor as well as the energy spectrum of the spontaneous radiation field.

This paper is concerned with the propagation of one-dimensional nonlinear waves in the relatively simple physical situation of a cold, collisionless, field-free, slightly non-uniform plasma comprising a stream of oscillating electrons and a stream of ions whose oscillatory motion is neglected. The waves are considered as a small perturbation from the corresponding uniform periodic solution. The equations which govern the slow variation of certain gross quantities associated with the waves are obtained by applying the ‘two-timing’ technique to the basic plasma equations. The properties of the resulting modulation equations are discussed and an exact solution of an initial value problem for them is given.

The acceleration of trapped particles by a Langmuir wave in an inhomogeneous plasma is investigated for different density profiles. In particular the important case of a wave moving towards its refiexion point is considered both analytically and numerically. To obtain estimates of the energy gain of the trapped particles, their stability is investigated. Our findings differ from those of previous authors. The effect of wave damping, and a slowly varying frequency (whistler-like phenomena), on the stability of trapping is estimated.

This paper reports on theoretical and experimental refinements in the use of forced radial magneto-acoustic oscillations of a magnetized plasma as a diagnostic technique. The theory of such oscillations is extended to include radial variations in electron number density, neutral atom density and plasma temperature. In experiments carried out in an argon afterglow plasma, observations of the radial variation of the oscillation amplitude, and of the total flux in the plasma, are combined with theory to yield radial density and temperature profiles. These profiles are in good agreement with independent measurements made using laser interferometry and spectroscopic techniques.

This paper studies the hydromagnetic stability of a cylindrical jet of a perfectly conducting and inviscid fluid of very high compressibility. The fluid velocities and magnetic fields, inside and outside the jet, are uniform and in the axial direction, with possible cliscontinuities in their values across the jet surface. In the limit of infinite compressibifity, the jet is stable against axisymmetric disturbances, but instability is present for asymmetric disturbances when the magnetic field is sufficiently small. For these disturbances the infinitely compressible jet behaves like an incompressible one when the wavenumber of disturbances is small. The effect of (infinite) compressibility becomes more evident as the wavenumber of disturbances increases and the jet becomes stable when the wavelength tends to vanish.

It is shown that solitary electrostatic ion cyclotron waves exist. An explicit solution is presented for the case of propagation near the ion-acoustic speed.

General equations are derived for the interaction of four waves to second order (i.e. two triplets with two waves common to both). The solutions are analysed under various initial conditions and it is shown that the weaker triplet (as measured by the coupling constraints) is stabilized by the stronger one against explosive instabilities, whereas the converse does not happen. It is further found that under certain circumstances one of the common waves may act as a ‘catalyst’, remaining fixed in amplitude, while the other waves oscillate or even grow exponentially. This work extends the treatment of Karplyuk, Oraevskii & Pavlenko to include the possibility of negative energy waves.

Relativistic electron-mass variations due to the presence of intense electromagnetic radiation in the plasma cause a nonlinear refractive index. Using a variational principle the latter is obtained up to fourth order in the electric field amplitude and it is shown that nonlinear effects of the second order lead to self-focusing of a beam of radiation. By nonlinear optics considerations, the self- focusing length of an axially symmetric beam is obtained. Including higher- order dispersive effects it is shown that within the thin-beam approximation thecomplex electric field envelope obeys a cubic nonlinear Schrödinger equation with an attractive self-consistent potential. The cylindrically symmetric nonlinear Schrödinger equation predicts collapse of the radiation at the self-focusing distance. The nature of the self-focusing singularity is analysed and it is shown that higher-order nonlinearities saturate the amplitude. Then oscillations of beam radius along the axial direction occur.

By making use of the Feynman–Schwinger formalism of quantum electrodynamics, the classical dispersion relation for electrostatic and electromagnetic waves in a homogeneous plasma is derived. It is then argued that such techniques can be helpful in dealing with quasi-linear and nonlinear problems in plasmas.

The second order elliptic differential equation which describes the static equilibrium of an axisymmetric ideally conducting plasma toroid, in the magnetohydrodynamic approximation, may be transformed into an equivalent equation which admits at least one class of exact analytic solutions for which the original equation is essentially nonlinear. Necessary and sufficient conditions are given for the existence of systems in this category having closed toroidal flux surfaces and a result is proved which gives some insight into the mathematics by which the spatial variation of the hydrodynamic pressure can be represented by families of nested closed curves.

This paper makes a few remarks on the method of the averaged Lagrangian developed by Whitham to describe slow variations of nonlinear wave trains. The concept of multiple scales is incorporated into the variational formalism, and a consistent development to higher approximation is suggested as a formal perturbation based on the variational principle. The propagation of weakly dispersive long waves is also reconsidered in relation to the Lagrangian method. It is demonstrated that the nonlinear Schrodinger equation and the Korteweg–de Vries equation can be derived from the Euler–Lagrange equations of the perturbed Lagrangian.

We consider the nonlinear coupling of stationary whistler wave turbulence with low-frequency kinetic Alfvén and ion quasi-modes in a magnetized plasma. It is found that such couplings lead to instabilities. The influence of various plasma parameters on the linear growth rates of the instabilities is obtained. Our results may be useful to the understanding of numerous phenomena associated with electron whistler waves in space and laboratory plasmas.

This paper presents an investigation of the effect of self-focusing on the absolute growth rates of parametric back-scattering (e.g. stimulated Raman and Brillouin scatterings) and absorptive instabilities (e.g. two-plasmon decay and ion–plasmon decay) in a homogeneous and collisionless plasma. Gaussian intensity distribution of the pump laser beam causes the redistribution of carriers in the plane transverse to the beam propagation on account of the ponderomotive force, and leads the beam to propagate in an oscillatory wave guide mode. The modifications in plasma frequency and intensity of the beam cause the growth rates of the above parametric instabilities to show an oscillatory behaviour with the distance of propagation.

Finite wavelength guiding centre plasma stability of the bumpy θ-pinch is examined by a normal mode analysis. It is shown that previous bumpy θ-pinch calculations are recoverable as special cases of this analysis. The ideal magnetohydrodynamic and guiding centre plasma growth rates are compared for various pressure anisotropies and for various wavenumbers of the field line bumpiness. The well-posedness conditions on the guiding centre plasma equations are shown to give upper and lower bounds on the permissible pressure anisotropy which corresponds to the Aifvén continuum staying on the stable side of the spectrum and to the particle mirror force not having a singularity. It is also found that the higher azimuthal m ≥ 2 modes have growth rates larger than the m = 1 mode.

We consider the propagation of a large-amplitude lower-hybrid wave. Modulational instabilities arising from its interaction with low-frequency electrostatic perturbations are investigated. The growth lengths of the convective instabilities are obtained and compared with previous results for adiabatic perturbations.

The CMA diagram as a means for classifying electromagnetic waves propagating in cold plasmas is generalized to include boundaries on which the ray surface, rather than the refractive index or wave normal surface, undergoes topological changes. Since the ray surface has the shape of a wavefront for a wave originating from a point source, such boundaries are of importance in understanding wave behaviour. Formulae are presented applying to the case of a cold plasma including the effect of positive ions. Calculations are presented for the special casea hydrogen plasma.

Non-linear ideal MHD equilibria in axisymmetric systems are examined, both in diffuse and free boundary cases. Attention is restricted to the situation in present low-β tokamak experiments, in which there are no current reversals. Both general qualitative results on the uniqueness and bifurcation of solutions are provided and exact solutions of several problems in a circular cylinder are given, exhibiting bifurcation phenomena. In particular sufficient conditions for bifurcation behaviour are conjectured, for both diffuse and free boundary equilibria.