In this paper we have studied the refraction of a hydromagnetic shock wave in an aligned field, when it encounters an interface between two streams of gases at different mach and Alfvén numbers. The hodograph mapping technique is employed in all the discussions. Both regular and irregular refractions are considered for the fast and slow shocks, but in the case of the fast shock, the discussions have been restricted to the first type. A brief account of the refraction of transverse shocks is also given. A detailed discussion is not necessary because of the similarity between this problem and that of the type 1 fast shock.

We consider the effects of resistive dissipation on energy absorption by spatially localized Alfvén waves which form a part of the continuous spectrum of ideal magnetohydrodynamics (MHD). We demonstrate that the strong absorption rate found in ideal MHD is unaltered for plasma resistivities of the order found in tokamaks, thus implying the possibility of effective heating of these machines by Alfvén waves.

The generation of magnetic fields by currents driven by the electron pressure gradient in non-uniform plasmas is considered. The equations for field generation and energy balance are derived under the MHD approximation. It is shown that there exists a class of waves, thermal magnetic waves, which propagate along surfaces of constant density. The properties of such waves are examined. In addition, a class of self-similar solutions are presented. These represent an unstable growth of magnetic field in localized plasma regions, i.e. magnetic hot spots.

A stationary collisionless and neutral plasma flowing along magnetic lines of force (B-lines) is considered and the variation of electric potential along the B-lines is described. A steep magnetic field aligned gradient of electric potential is generated at a certain location whenever plasma moves along the lines of force.

We study nonlinear effects associated with large amplitude electron plasma waves propagating at an arbitrary angle to the external magnetic field. It is shown that the nonlinear coupling of high-frequency plasma waves with static density perturbations leads to a convective instability. The growth length of the latter is obtained. The evolution of the wave electric field is also investigated.

A number of reconnexion concepts and experiments are briefly reviewed in order to re-examine the present interpretation of these experiments. In particular, we offer explanations as to why some experiments appear to develop Petschek modes, tearing modes, or netural current sheets. The explanations require an understanding of the proper role of magnetic Reynolds numbers, the limits of the frozen-in concept, and the importance of natural importance of natural boundary conditions. We find that netural current sheets usually from in experiments with highly symmetrical (and therefore unnatural) boundary conditions. The classical tearing mode develops from perturbations of a neutral current sheet. In less constrained geometries multiple neutral points may appear but the classical tearing mode theory needs modification to explain these cases rigorously. A Petschek mode develops in even less constrained systems although the theoretical description is highly idealized. We offer explanations as to why some experimenters appear to find neutral current sheets in quadrupole fields and examine the usefulness of concepts derived from neutral current sheet theory.

The gravitational instability of a two-component plasma has been studied here to include simultaneously the effects of neutral gas friction, finite ion Larmor radius, magnetic resistivity and Hall currents. The viscosities of the two components of the plasma have also been taken into account. The mode of the transverse as well as the longitudinal wave propagation have been discussed. The dispersion relations have been obtained for both these cases and numerical calculations have been performed to obtain the dependence of the growth rate of the gravitationally unstable mode on the various physical parameters involved. For the transverse mode of propagation, it is found that the growth rate of the unstable mode increases with magnetic resistivity and with the ratio of the densities of two components. The influence of the magnetic resistivity is, therefore, destabilizing on this mode of wave propagation. The viscosities of the two components are found to have a stabilizing influence on the growth rate in this case since it is found that the increase of hte viscosity effects reduces the growth rate. For the longitudinal mode also it is found that the effects of viscosities as well as that of neutral gas friction are stabilizing. The magnetic resistivity does not affect the growth rate since the equation determining the growth rate is found to be independent of this effect.

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

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.

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.

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.

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.

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.

An investigation is made of the effect of an external growth-rate modulation on the saturation of a linearly unstable plasma wave by means of nonlinear Landau damping. Different ways of exciting the saturated system by suitable choices of modulation frequency are discussed and the characteristics of the resulting instabilities, like thresholds, growth-rates, maximum amplitudes and final saturation levels are determined.

We study the dispersion equation of drift waves in the presence of electron- plasma turbulence in a plasma with magnetic shear. We show that the shear stabilization can be cancelled out by a hot turbulent spectrum, even for a low level of turbulence, W/n0Te≪ 1.