By reconsidering the general theory for the resonant interaction of three waves in a plasma, we find explicit expressions for the coupling coefficients for three MHD waves. In particular we demonstrate that the interaction between two magnetosonic waves and one Alfvén wave, as well as the interaction between two Alfvén waves and one magnetosonic wave, can be described by very simple formulae for the coupling coefficients.

The time evolution of the distribution function of the beam-injected particles in the presence of ICRH in a two-component plasma is determined. Consideration is restricted to the time development during two complementary time periods: (i) the early time period, i.e. 0 ≤t ≪ TS, and (ii) the quasi-steady state, i.e. t > Ts, where Ts is the slowing-down time for beam-ion-electron collisions. Explicit analytical solutions are obtained for anisotropic as well as isotropie beam injection.

The theory of the slow reaction of a charged particle in the combined presence of a strong quasi-static magnetic field and a high-frequency electromagnetic field (generalized oscillation-centre motion) is constructed by using a Hamiltonian formalism with non-canonical variables and pseudo-canonical transformations. The theory combines the features studied in our previous works for the case in which only one of the previously mentioned fields is present. The new averaging transformation is based on the fact that the Larmor frequency of the quasi-static field is of the same order as the external frequency of the high-frequency field. Our theory is manifestly gauge-invariant and involves only physical quantities (particle velocity and electromagnetic fields). Explicit expressions for the drift velocity of the oscillation centre and for the ponderomotive force are derived.

Analytical solutions are presented for the linear growth rate of oblique plasma waves in a magnetized plasma due to resonant interactions with a model ringbeam distribution. Explicit closed-form solutions for the angular dependence are obtained in terms of modified Bessel functions of the first kind. In the limits of either quasi-longitudinal or quasi-transverse propagation the analytical solutions take the form of simple algebraic expansions, which allow an immediate comparison of the relative contributions from different harmonic resonances, and which also determine the conditions for marginal stability for any specific resonance. The results can be applied, for instance, to the growth of waves following ionization of neutrals originating from cometary, planetary, or interstellar material in the solar wind. In a weakly unstable plasma the analytical results also provide an important check on the complex numerical codes that hitherto constituted the only method available for evaluating the growth of oblique plasma waves.

It is shown that a plasma in which the background magnetic field varies in a direction perpendicular to its line of action can support ‘Rossby-type’ electrostatic waves at frequencies very much less than the ion gyrofrequency. The intrinsic wave propagation mechanism at work is structurally similar to that in the atmospheric Rossby wave, which comes about from fluid perturbations being in quasi-geostrophic equilibrium (i.e. the Coriolis force nearly balances the pressure gradient) and the latitudinal variation of the vertical component of rotational frequency vector (the β-effect) so that the time rate of change of the vertical component of the fluid vorticity is equal to the northward transport of the planetary vorticity. In a plasma this ‘geostrophic balance’ arises from the near-vanishing of the Lorentz force on the ion motion while the β-effect is provided by the transverse spatial variation of the ambient magnetic field. Unlike the atmosphere, however, such a magnetized plasma is capable of supporting two distinct types of Rossby wave. The interesting dispersive and anisotropic features of these waves are revealed by the properties of their wave operators and described in terms of the geometry of their wavenumber surfaces. Since these surfaces intersect, inhomogeneity or nonlinearity will give rise to strong mode-mode coupling in regions where the phases of both modes nearly match.

An eigenvalue equation is derived for electrostatic waves with frequencies below the ion bounce frequency in a plasma confined by a magnetic quadrupole. This integral equation for the electrostatic potential is developed to second order in the curvature and contains the effects of electron and ion temperature gradients. A dispersion equation for flute modes is derived from it. The present theory applied to the private flux region of the UMIST quadrupole predicts that low-frequency flute modes will propagate in the electron and ion diamagnetic drift directions.

By using the average-Lagrangian method (average variational principle), a nonlinear dispersion relation has been derived for the cubic nonlinear Schrödinger equation. It is found that the size of the instability region in wavenumber space decreases with increasing field amplitude in comparison with the linear theory.

Grad's 13-moment theory is used to model thermal transport in laser-produced plasmas where conditions are such that one may not assume that the plasma is collision-dominated. The equations are presented for a multi-fluid model in slab geometry and are solved numerically for various cases. Comparisons are made with classical theory and with simulations of collisionless plasmas. Results show that, although there are differences between 13-moment and classical predictions, the two theories agree on the general behaviour of the plasma. In particular, 13-moment theory fails to explain the heat-flux inhibition implied by some experimental observations. The theory has also been applied to model thermal transport in plasmas with a population of suprathermal electrons.

The generalization of the Balescu-Lenard collision operator to its fully electromagnetic counterpart in Kaufman's action-angle formalism is derived and its properties investigated. The general form may be specialized to any particular geometry where the unperturbed particle motion is integrable, and thus includes cylindrical plasmas, inhomogeneous slabs with non-uniform magnetic fields, tokamaks and the particularly simple geometry of the standard operator as special cases. The general form points to the commonality between axisymmetric, turbulent and ripple transport, and implies properties (e.g. intrinsic ambipolarity) that should be shared by them, under appropriate conditions. Along with a turbulent ‘anomalous diffusion coefficient’ calculated for tokamaks in previous work, an ‘anomalous pinch’ term of closely related structure and scaling is also implied by the generalized operator.

є-Expansion renormalization-group theory is applied to a model Alfvén-wave turbulence equation. In particular, the effect of small ‘unresolvable’ subgrid scales on the large scales is computed. It is found that the removal of the subgrid scales leads to a renormalized response function v. The Lorenzian wavenumber spectrum of Chen & Mahajan can be recovered for finite є, but the nonlinear coupling constant still remains small, fully justifying the neglect of higher-order nonlinearities introduced by the renormalization-group procedure.

Using first-order smoothing theory, Fourier analysis and perturbation methods, a new equation is derived governing the evolution of the spectrum tensor (including the energy and helicity spectrum functions) of the random velocity field as well as the ponderomotive and mean electromotive forces generated by random Alfvén waves in a plasma with weak magnetic diffusion. The ponderomotive and mean electromotive forces are expressed as series involving spatial derivatives of mean magnetic and velocity fields whose coefficients are associated with the helicity spectrum function of the random velocity field. The effect of microscale random Alfvén waves, through ponderomotive and mean electromotive forces generated by them, on the propagation of large-scale Alfvén waves is also investigated by solving the mean-field equations, including the transport equation of the helicity spectrum function.

A study of the ideal magnetohydrodynamic linear eigenvalue spectrum for free-boundary modes in the Z-pinch is presented. The application of a variational method to estimate eigenvalues is described and limitations imposed by the nature of the spectrum are discussed. An analytic expression for the long-wavelength m = 0 growth rate is derived

A two-point boundary-value problem has been formulated that describes the conversion between ordinary (O) and extraordinary (X) wave modes in a cold inhomogeneous plasma. Numerical solutions to this problem have been obtained for various values of the WKB parameter k0L; where k0 is the vacuum wavenumber and L the density-gradient scale length. The results are compared with three different theoretical expressions for the O-X mode conversion efficiency derived by others in the WKB limit of k0 L ≫ l. Most of the results presented in this paper are obtained for a collisionless plasma with finite density near the plasma cut-off density. However, some examples are also given of wave propagation from vacuum. In these examples, collision effects are added to the equations in order to remove the singularity otherwise present at the position of the upper hybrid resonance layer.

A spectral-method numerical code is used to compute mass-density fluctuation spectra in turbulent magnetofluids. The computations are used to test and extend a recent analytical theory of density variations in slightly compressible magnetofluids given by Montgomery, Brown and Matthaeus, and used by them to infer inertial-range density-fluctuation spectra for the nearby interstellar medium and solar wind. A local equation of state is assumed, relating density to pressure. Constant, scalar resistivities and viscosities are used. In the limit of low Mach numbers and high mechanical-to-magnetic pressure ratios, the fit of the computations to the analytical theory is seen to be close.

The nonlinear behaviour of short-wavelength perturbations in the two-fluid cosmic-ray hydrodynamical model is examined. We show that such a perturbation leads to shock formation and derive the appropriate wave equation. We show that a discontinuous perturbation incident on a weak cosmic-ray shock destabilizes, in a time-asymptotic sense, the shock.

Drift-Alfvén vortices are investigated, taking into account the nonlinear ion dynamics parallel to the external magnetic field. It is found that the parallel ion motion restricts the vortex speed.

The influence of nonlinearities and plasma resistivity on the so-called trivial marginal modes of ideal MHD is investigated. It turns out that to lowest significant order of a reductive perturbation analysis nonlinearities have no influence, while in toroidal confinement configurations resistivity induces a local plasma diffusion across the magnetic surfaces. This is demonstrated for tokamaks with zero poloidal current density. In addition, the appearance of singularities in the plasma motion parallel to the magnetic field lines suggests a profile condition at the edge of the plasma. Heat conduction would directly tend to keep these effects at lower level, while indirectly it creates a reinforcement mechanism through the coupling to usual cross diffusion.

A nonlinear drift mode equation for finite ion temperature plasma is derived. Possibilities of explosive instabilities for the drift wave are obtained. To discuss the modulational instability of drift wave, a nonlinear evolution equation for the drift mode is deduced from a third-order secularity elimination condition using the multiple time and space scales technique. The effect of low-frequency localized structure on particle transport has been investigated and the diffusion tensor is calculated using Ficks law.