The elements of magnetohydrodynamics (MHD) with the Hall current are presented. The governing partial differential equations are, like those of ideal MHD, quasi-linear, but unlike ideal MHD, are not hyperbolic. However, they do possess real characteristics which are shown to be associated with the sound wave and the surfaces ø(r, t) = const. determined by B. ∇ø = 0. Properties of weak and strong discontinuities are discussed, and comparisons are made with ideal MHD. It is shown that generalizations of dipole layers appear as a result of discontinuities in the magnetic field.

The kinetics of ions in a tokamak is analyzed on the basis of the Balescu – Lenard equation. We show that the injection of a neutral beam (acting through charge exchange with the plasma) produces a population of high energy ions. The global energy increase is evaluated.

Electrostatic instabilities excited by an energetic ion beam are examined as a possible plasma heating mechanism. The dielectric properties of electrostatic waves in a uniform magnetized plasma are studied for a warm ion beam propagating at an arbitrary angle with respect to the magnetic field. The instability threshold and growth rate are calculated for the resonant ion-beam and ion- acoustic modes. The quasi-linear moment equations are used to follow the self-consistent evolution of the macroscopic plasma properties. A comparison is made with the collisional slowing-down rate of the beam. For the fastest growing modes it is found that the quasi-linear slowing-down time is significantly shorter than the coffisional slowing-down time.

We observe radial expansion of an ion beam propagating along a homogeneous guide magnetic field. The beam spreading appears to arise from interactions with beam-excited resonant ion cyclotron waves. Magnitude and scaling of the expansion agree with a theory based on a resonant diffusion process.

Three-dimensional plots of dispersion in a cold anisotropic plasma are presented. The ω(κ, θ) surfaces provide a clear picture of the behaviour of cold plasma waves as the direction of propagation is varied. The group velocity (dω/dκ) has a simple geometrical interpretation on the surfaces.

The first two equations of the BBGKY hierarchy are discussed and solved in order to derive a kinetic equation for an electron gas (non-neutral plasma) where strong electric and magnetic fields as well as inhomogeneities are taken into account on scales relevant for collisions between particles. The gyrotropic assumption is not made. The magnetic field and the inhomogeneities are shown to have special effects on the collision terms. A strong magnetic field approximation is then made in order to simplify the collision term, and a new, proper collision term has been found when a strong magnetic field is present.

Kinetic theory for the nonlinear propagation of quasi-monochromatic extraordinary waves is presented. It reveals that propagation of an envelope of the extraordinary carriers is described by the nonlinear Schrödinger equation. In the cold plasma limit, a detailed analysis is carried out of the behaviour of the envelope of the upper- and the lower-hybrid waves at their respective resonant frequency ranges.

For a non-spherical ellipsoidal or conical implosion with an eccentricity of ε≃10 the minimum implosion velocity is computed to be less than 100km sec−1.

We demonstrate that self-consistent large-amplitude longitudinal waves in a cold plasma are unstable to at least transverse perturbations, with an instability growth rate which is roughly of the order of the plasma frequency of the system. The instability statements derived here are conservative, since modes not discussed may be more unstable.

We discuss the possible magnetohydrodynamic configurations that can be realized as ‘most probable’ states compatible with the existence of certain constraints. These constraints can be either experimentally imposed constraints such as constant total electric current or magnetic flux, or constants of the motion, or both.

We demonstrate that a self-consistent large-amplitude circularly polarized wave, propagating in a cold plasma in the presence of a large-scale magnetic field, is unstable if the constant bulk streaming speed of the plasma is zero in the frame in which the wave depends oniy on time. The growth rate is of the order of the plasma frequency or the gyrofrequency at short perturbation wavelengths, and is proportional to the perturbation wave vector at long wavelengths. For nonzero but small streaming the instability rate increases for one streaming direction and decreases for the other. We conclude that instability is the rule rather than the exception for large-amplitude waves in a cold plasma.

A phenomenological theory of drift type turbulence in inhomogeneous magnetized plasma is developed in the spirit of the Kolmogorov spectrum of hydrodynamic turbulence. We model the spectrum in terms of subspectra: production, equilibrium and dissipation. Different nonlinearities and either weak or strong turbulence characterize different subspectra.

We consider the nonlinear development of a long-wavelength finite-amplitude Langmuir wave. The wavenumber k0 of the initial Langmuir wave is chosen such that the three-wave decay is forbidden. We then describe the coupling of the initial Langmuir wave to Stokes and anti-Stokes Langmuir perturbations (with wavenumbers k0 ∓ ks) due to the presence of a low-frequency density perturbation of wavenumber ks. We then show that for a wide range of experimental conditions, the Stokes and anti-Stokes Langmuir waves are generated with wavenumbers well separated from k0. In order to describe the nonlinear evolution of these perturbations and the pump wave we make the static approximation for the ions and describe the high-frequency waves by three distinct wave envelopes. These coupled nonlinear differential equations are then solved exactly for a number of special cases. For the temporal evolution, we obtain periodic solutions and, when damping is included, we find a slow exponential decay of the amplitudes with a corresponding increase in the nonlinear period of oscillation. The stationary spatially varying solutions are shown to include four basic types of behaviour: periodic, solitary wave, phase jump and shock-like profiles. These latter solutions are of interest since they are obtained for zero dissipation and for a coherent wave interaction.

We examine the effect of finite ion Larmor radius on the hydromagnetic growth rates of the firehose and mirror instabilities. We perturb an infinite, collisionless, magnetized plasma which is initially uniform, and we do not restrict the direction of the wave vector, k.

The concept of merging tokamak plasmas is proposed. It has potential applications for solving the problems of pseudo steady-state tokamak operation, refuelling, impurity removal and plasma heating. The currents in the two merging tokamak plasmas are in the same direction so that the columns attract each other. A model for the merging process is given.

Incompressible magnetohydrodynamic turbulence is treated in the presence of cylindrical boundaries which are perfectly conducting and rigidly smooth. The model treated is non-dissipative and two-dimensional, the variation of all quantities in the axial direction beingignored. Equilibrium Gibbs ensemble predictions are explored assuming the constraint of constant axial current (appropriate to tokamak operation). No small-amplitude approximations are made. The expectation value of the turbulent kinetic energy is found to approach zero for the state of maximum mean-square vector potential to energy ratio. These are the only states for which large velocity fluctuations are not expected.

An inverstigation is made of the transverse instability of plane soliton solutions of the nonlinear Schrödinger equation, including the case of an exponential nonlinearity. The analysis is based on a variational method, where specific trial functions are applied for the perturbations. Explicit expressions for the growth rates are given, showing good agreement with numerical results.

Low ‘Ohmic’ voltages are adequate for establishing intense runaway electron beams in toroidal devices. These beams do not destroy conventional tokamak equilibrium and stability but still prove sufficient for turbulent heating by beam–plasma interaction.

The previous companion papers have shown the important effect of a local increase in resistivity upon the fast-reconnexion configuration. The present paper exaniines another significant situation where the injection rate of magnetic flux into the reconnexion region is strongly constrained by some external agency. It is shown that as long as a local resistivity enhancement is retained, stationary configurations can ultimately be established which depend on the imposed boundary values in each case. The rate of magnetic field line reconnexion in the stationary configuration is indeed closely controlled by the boundary values. However, the reconnexion rate (measured by the local Mach number at a distance from the neutral point as in the ordinary theoretical treatments) is scarcely influenced by them. Also, it is shown that without the local resistivity enhancement the large-scale X-type field configuration, which should be required for any fast reconnexion process, can no longer be sustained.

This paper examines the nature of plane, steady, incompressible MHD flow in a purely hyperbolic magnetic field. It is shown that in such a magnetic field the MHD equations can yield exact analytic solutions for the plasma flow. The flow has the following properties. In the far region where the conductivity is assumed to be sufficiently high so that the plasma is effectively ‘ frozen ’ to the magnetic field, the flow pattern is radial. The plasma motion is directed towards the neutral line in the incident ‘ sectors ’ and away from it in the outgoing ‘ sectors ’ with a consequent reversal in direction across the magnetic separatrices where the solution becomes singular. The plasma pressure and density in this flow are calculated and it is shown that the latter remains constant along a streamline. It is further shown that a uniform finite conductivity is not compatible with a stagnation point at the magnetic null point. However, for a parabolic increase of conductivity with increasing distance from that point, plasma flow with uniform density along hyperbolic streamlines is shown to be possible. The relevance of these flows to the magnetic field merging problem is discussed.