The magnetohydrodynamic properties near a two-dimensional X-type magnetic neutral line in a viscous and resistive plasma in the steady-state are investigated.

The time evolution of weak shocks in the classical theory of shock waves may be described by Burgers' equation, which is a small-amplitude, long-wavelength, but nonlinear wave equation. The present paper derives Burgers' equation for three different hydrodynamical models of cosmic ray shocks. The main development of the paper concerns the hydrodynamical model in which Alfvénic effects are neglected. For this model it is shown that the steady-state solution of Burgers' equation is equivalent to the weak, but smooth, transition solutions of the shock structure equation obtained previously. It is shown that Burgers' equation may be regarded as a wave energy equation, in which the interaction of the wave with the background medium is taken into account. Burgers' equation is also derived for Alfvénic models in which the Alfvén waves that scatter the cosmic rays are generated by the cosmic ray streaming instability, and propagate down the cosmic ray pressure gradient, at the Alfvén velocity relative to the background fluid.

In the Fokker-Planck approximation, a general solution of the kinetic equation for the distribution function of non-thermal α-particles in spatially uniform DT plasma is obtained. The previously published solutions are then derived as special cases. Transport equations describing diffusion of non-thermal α-particles are obtained and diffusion coefficients are calculated. To compare the hydro-dynamic and kinetic descriptions of non-thermal α-particle energy transport, the critical ignition length of the overheating instability in DT fusion plasma, determined by non-local plasma heating due to α-particle diffusion, is calculated using both approaches. The two results are in good agreement.

Ignition of a self-sustained fusion reaction in a strong magnetic field is studied. The critical ignition dimensions and the threshold ignition energy are calculated as functions of the fuel density and the magnitude of the transverse magnetic field. A new method for producing ultra-high magnetic fields is proposed which provides at once heating of the fuel by induction currents and localization of the micro-explosion by the magnetic field. Simple analytic estimates show that a certain combination of inertial, magnetic and wall methods of plasma confinement may decrease the ignition energy below 100 kJ.

If an external current pulse is applied to a diffuse plasma sheet pinch, surface wave modes are generated, which decay by collisionless damping, leaving only oscillations of the Alfvén continuum along the Alfvén resonance surface. The transverse perturbations within this surface phase-mix to zero. It is shown that perturbations induced by an initial pulse are modulated by a (later applied) second pulse of different wavelength, to yield non-vanishing second-order transverse perturbations, even though the first-order transverse perturbations have phase-mixed to zero. This analysis shows the importance of nonlinear effects in the evolution of inhomogeneous magnetohydrodynamic motions.

A method is presented for computing the class of axisymmetric current distributions flowing in a torus whose peripheral surface is a flux surface for the magnetic field produced by the current itself. The method allows the correct calculation of the ‘self-induced’ magnetic forces arising from the interaction between these currents and their own field. The general expression for the self-induced force is given and an approximate formula is presented in the large aspect-ratio limit.

It is shown that a finite-amplitude upper-hybrid wave can decay into a perpendicularly (to an external magnetic field B0zˇ) propagating electrostatic lower-hybrid/ion-cyclotron wave and a left-handed circularly polarized (LHCP) electromagnetic wave (along B0) or an ordinary mode radiation (across B0). Nonlinear dispersion relations and the growth rates are obtained. It is suggested that the present nonlinear instability can be responsible for the enhanced narrow band radiation which is frequently observed in the Earth's magnetosphere.

Following the method developed by Bernstein, Greene & Kruskal we obtain expressions for the distribution function of the trapped particles for a collisionless plasma, for cases when the plasma is unbounded and bounded (in a cylindrical waveguide). Figures are drawn showing the relationship between the width and amplitude of the solitary BGK wave for various cases. Analytic expressions depicting this relationship are also derived.

Detailed computational results are presented for the stability of linear tearing modes in low-q cylindrical tokamaks with various radial profiles. The stability properties of a widely used ‘classical’ profile are compared with those of a ‘piecewise’ profile. It is shown that the stability properties of low-q tokamak discharges show a sensitive dependence on small changes in the current density profile near the singular surface. While the ‘classical’ profile is stable for a large range of q0, where q0 is the value of the safety factor on the magnetic axis, a similar ‘piecewise’ profile is unstable for the same set of parameters.

Backscattering of Langmuir waves from low-frequency electrostatic waves in a plasma traversed by an electron beam is studied. The analysis is based on the use of beam electron, plasma electron, and ion susceptibilities provided by kinetic theory. For the case of a warm electron beam, formulae are derived for the growth rate and threshold associated with resonant backscattering from an ion-acoustic wave modified by the presence of the beam. For the case of a cold electron beam, formulae are derived from the growth rates associated with resonant back-scattering from a modified ion-acoustic wave and from a higher-frequency beam-plasma mode. A numerical study of the effects of an electron beam on these parametric instabilities is included.

In the presence of a low-frequency ion-acoustic turbulence and a high-frequency whistler-mode test wave, a new plasma instability occurs owing to a nonlinear force which originates from the resonant interaction between electrons and modulated nonlinear electric fields. The growth rate of the whistler mode is calculated and compared with observations.

Conditions for whistler-mode trapping in the magnetospheric ducts are considered. The plasma model includes electron temperature and anisotropy, and there are no restrictions on the value of the wave normal angle. It is pointed out that the range of the wave normal angles for which whistler-mode waves can be trapped in the ducts with enhanced electron density increases when the electron temperature and (or) anisotropy increases; the corresponding increase also takes place for the average wave normal angle when whistler-mode waves are trapped in the ducts that have a deficiency in electron density. The limits of applicability of a simplified formula derived by Sazhin & Sazhina, for whistler-mode propagation in a hot anisotropic plasma, are clarified.