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 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.

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.

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 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.

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.

The enhanced effective collision frequency in a turbulent plasma produced by a strong electric field is studied using a small toroidal device. The collision frequency is presented as a function of the electric field E and the average electron drift velocity. Two collision frequency regimes are identified. In the regime I where v ∝ E½, the current driven ion-acoustic turbulence is considered responsible.

The resistive evolution of a magnetic field through force-free states is discussed, and some unusual features described.

The definition of the generalized antipotential four-vector makes it possible to give a relativistically covariant variational formulation in the dynamics of ideal charged fluids. A special relativistically covariant form of Maxwell's equation is given. The antipotential four-vector does not explicitly appear in the Lagrangian density. The derivation of the equation of motion of a single charged particle is given, to illustrate the theory.

Expressions for the conductivity tensors of all orders are obtained, for both relativistic and unrelativistic magnetized Vlasov–Maxwell plasmas, closely related to the well-known formula for the linear conductivity tensor. Thus the unperturbed orbit integrations and the ø-integration in velocity space have been performed and each conductivity tensor is expressed as an infinite series with terms involving ordinary Bessel functions. Jα(K⊥V⊥/ωc) and denominators of type ω–KzVz–αωc for integers.α a. The symmetries related to the Manley-Rowe relations are explicitly seen in the formulae presented. These conductivity tensor formulae provide a much better starting point than the basic equations for many nonlinear investigations.

For a high β screw pinch it is shown explicitly how the perturbation theory distorts the spectrum of ideal magnetohydrodynamics and guiding centre plasmas. In particular it is seen why the Cauchy integrals (present in the nonperturbative linearized guiding centre plasma model, and which are necessary for the isotropic pressure guiding centre plasma eigenvalue problem to differ from that for magnetohydrodynamics) are absent in the perturbation theory. This result holds also for axially and helically symmetric systems. The distortions in the m = 1 growth rates are also presented.

The perturbation solution of the ordinary differential equations that describe exact nonlinear travelling plane waves leads to asymptotic expansions in powers of the (small) wave amplitude for both the proffle and the frequency of the waves. This paper shows how the Padé approximant method can be used to extend the validity of those expansions to larger amplitudes. The method is applied to the Duffing equation and to two types of nonlinear waves in a cold electron plasma: longitudinal oscillations and coupled transverse–longitudinal relativistic waves.