We show what corrections have to be made to the equations of ideal magneto-hydrodynamics when there is fast-time-scale turbulence present in a magnetized plasma. We show how the dispersion relations for the ideal Alfvén and magnetoacoustic MHD normal modes are modified when such turbulence is present. Finally, we discuss the relation of our work to that of other authors.

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.

The main properties of a kinetic wave equation in an isothermal plasma, Te ≈ Ti, are discussed and a quick method of numerical solution based on factorization of the kernel of an integral equation is outlined. As an illustration the method is applied to the problem of plasma heating with a relativistic electron beam and it is shown how the evolution of a spectrum of Langmuir turbulence is the principal contributor to the heating of the plasma. The technique allows an estimate of the error which is present in the stationary solution, and this is made in the final section.

The theory for potential surface wave fluctuation spectra in cold, inhomogeneous, magnetized plasmas is reconsidered, adopting a transition probability approach. A fluctuation dissipation relation is derived, which is valid for general density profiles and which reduces to the familiar fluctuation dissipation theorem for homogeneous plasmas. Significant corrections to previous work are found. Particular spectra are calculated for the special case of a magnetized plasma with a one-dimensional inhomogeneity.

The propagation characteristics of right-hand circularly polarized whistler mode waves propagating parallel to the external magnetic field in an anisotropic plasma have been reformulated including the effect of a parallel electric field. Analytical expressions for the real frequency and growth rate have been obtained for the full range of the parameters β (the ratio of particle pressure to magnetic pressure of the hot particles), A (temperature anisotropy) and P ( = βA(A + 1)2) without any restriction on the magnitude of the imaginary part of the wave frequency. The effect of cold plasma injection on the marginal instability has also been studied. Possible application of the present theory to the atmospheres of Earth and Jupiter has been discussed.

The transport of neutrals in the outermost region of a tokamak plasma has been studied using kinetic theory. Analytical expressions for the density profiles of hydrogen and impurity neutrals are obtained. It is shown that the inclusion of recombination increases the density of hydrogen neutrals in the blanket region by orders of magnitude. It is also found that the major contribution to the impurity neutral density, for impurities which have a resonant mutual charge exchange with hydrogen, comes from mutual charge exchange for moderately dense plasmas whereas for dense plasmas this comes from recombination.

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.

Hydrodynamical equations describing the mutual interaction of cosmic rays, thermal plasma, magnetic field and Alfvén waves scattering the cosmic rays used in cosmic ray shock acceleration theory (e.g. McKenzie & Völk 1982; Drury 1983; Webb 1983) are analysed for long-wavelength linear compressive instabilities. The Alfvén wave field may contain a pre-existing component as well as a component excited by the cosmic ray streaming instability. In the case of no Alfvén wave damping, adiabatic wave growth and Alfvén wave generation by the cosmic ray streaming instability, it is found that the backward propagating slow magneto-acoustic mode is driven convectively unstable by the pressure of the self-excited Alfvén waves, provided the thermal plasma β is sufficiently large. The equations are also analysed for the case where the Alfvén wave growth is balanced by some nonlinear damping mechanisms. In the latter case both the forward and backward propagating slow magneto-acoustic modes may be driven unstable if the plasma β is sufficiently small. The conditions under which the instabilities occur are delineated, and sample calculations of growth rates given. Possible applications of the instabilities to astrophysical situations are briefly discussed.

The propagation of a strong normal shock wave into a quiescent mixture of nitrogen gas seeded with small, spherical inert dust particles is studied. While crossing the shock front, the gaseous phase of the suspension experiences a sudden change in temperature, pressure, density and velocity. (These changes can easily be evaluated using the Rankine-Hugoniot relations.) The solid phase of the suspension (dust) is initially unaffected by the shock wave. As a result, immediately behind the shock front, one phase of the suspension (the nitrogen gas) is in a state of relatively high temperature and low velocity while the other (the dust) is in a state of relatively low temperature and high velocity. Owing to these differences in temperature and velocity, intense heat transfer and viscous interactions between the two phases take place leading eventually to a new state of equilibrium that is reached farther downstream of the shock front. The flow field where these interactions take place, the relaxation zone, is solved numerically. It is shown that the spatial extent of this zone is strongly affected by the mass concentration of the dust in the suspenson and its physical properties (size, density and specific-heat capacity). These parameters also affect the post-shock equilibrium suspension properties. It was found that increasing the dust concentration results in a shorter kinematic relaxation zone, higher post-shock suspension pressure, density and temperature, and lower velocity, as compared to a similar pure-gas case. Increasing the dust particle density or its diameter results in a longer relaxation zone and a higher post-shock equilibrium suspension pressure, density and temperature. Changes in the dust specific-heat capacity affect the extent at the thermal relaxation length and the suspension temperature and density; they do not affect the extent of the kinematic relaxation length or the post-shock suspension pressure and velocity. For the range of dust concentration, size, density, specific-heat capacity and shock-wave Mach number investigated, the kinematic relaxation zone is always longer than the thermal relaxation zone.

Stationary wave interactions are considered in a non-uniform plasma involving a cold electron beam propagating along the plasma density gradient. The character of the interactions is shown to depend on the beam-to-plasma energydensity ratio. The coupling of the beam space-charge waves and slow Langmuir waves gives rise to anomalous mode conversion and to the excitation of intensive fields near the resonance point. The regime of amplification is proved to be universal but is found to be sensitive to the boundary conditions.

The plasma models used in analytical or numerical studies exhibit, mainly for practical reasons, symmetry reducing the dimensions of the space. It is not obligatory that the restrictions on the dynamics and the chosen form of the electrostatic interactions should correspond to the same symmetry and this introduces the concept of a d (dynamic), d' (electrostatic) model with d' ≥ d. We show that the crucial parameter is d' – d. For d' – d = 0, the model exhibits both at low and at high temperature the same characteristic plasma frequency. At high temperature we recover the Vlasov limit and the plasma acts as a high-pass filter. For d' – d = 1, the two frequencies (purely collective at high temperature and ‘crystal’ frequency at very low temperature) are different, while in its high temperature (Vlasov) limit the plasma acts as a low-pass filter. Finally if d' – d ≥ 2 there is no longer a Vlasov limit for all values of temperature and density. These properties are obtained through a systematic use of the homothety transformations and a search for the corresponding invariants.

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.

An asymptotic analysis is carried out to calculate the effects of a small thermal spread in the injection energy of an electron beam on its space charge limit. It is found that the space charge limit is lowered proportionally to the beam temperature T near T = 0.

The propagation of small-amplitude hydromagnetic waves in a cold plasma mixed with hot electrons is investigated using the first order CGL equations for electrons. It is assumed that in an equilibrium state the electrons consist of two components, cold electrons and hot electrons with bi-Maxwellians. Propagation properties of hydromagnetic waves are analysed by use of phase speed and refractive index surfaces, polarization, and the amplitude ratio between perturbed density and magnetic field. It is shown that the existence of cold electrons affects the properties of hydromagnetic waves through finite frequency corrections only when the temperature anisotropy exists; and that the existence of cold electrons diminishes the resonance angle and the critical angle at which the polarization sense changes from left-handed to right-handed, and also weakens the tendency of intermediate waves to follow the lines of force of the static magnetic field.

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.

Lindhard's relation between the transverse and longitudinal permittivities and the magnetic susceptibility is shown to be incorrect. The tensorial nature of the wave-vector- and frequency-dependent dielectric permittivity is emphasized. A Vlasov plasma is used as an illustrative counter-example.

General properties of the eigenvalues of Schaper (1983), concerning the continuum of eigenvalues on p. 7, needs correction. The solutions in the distributional sense of the eigenvalue problem (5.3), will be given for two cases. The first case has been solved by A. Lifshitz and E. Fedorov and concerns continuous eigenvalues. In the second case, the solution for the points of accumulation of discrete eigenvalues is discussed.