The effect of the nonlinear changes of the dispersion characteristics of highfrequency azimuthally symmetric surface waves in a plasma column is investigated theoretically. Both (ω + ω) – ω and (ω – ω) + ω nonlinear interactions of the third order (in relation to the fundamental wave amplitude) are considered here. These two types of interactions influence the wave phase velocity and the collisional damping rate in opposite ways. When ω2Pi/v2 < 1, the contribution of the (ω – ω) – ω interaction is negligible and the (ω + ω) – ω interaction results in an increase of the phase velocity and a decrease of the time damping rate. When ω2pi/v2 > 1, both interactions are involved; the (ω – ω) + ω interaction associated with the ponderomotive force becomes more important, decreasing the phase velocity and increasing the time damping rate (ωpi and v are ion plasma and electron-neutral collision frequencies, respectively).

We show that under a wide range of conditions easily achievable in the laboratory, the dispersion of the radial eigenmodes of electromagnetic whistler mode waves in a cylindrical magneto-plasma is dominated by the electron inertia and becomes synonymous with the electrostatic Trivelpiece-Gould modes. A similar situation exists close to, but just below, the ion gyrofrequency.

The dispersion of the radial eigenmodes of a cylindrical m = 1 whistler wave with Ωi; ≪ ω < Ωe ≪ ωpe is investigated for both conducting and insulating boundaries. The effects of electron inertia and resistivity on the modes are discussed.

The self-focusing of whistler waves in a low-β plasma is considered. Axially symmetric solutions of the MHD and Schrödinger equations are found that describe stationary and homogeneous wave beams together with the plasma ducts in which they are trapped.

The self-focusing occurs both at ω < ½ωc and ω > ½ωc where ωc is the electron cyclotron frequency. The wave leakage which takes place in some types of duct and pre vents self-focusing from developing is investigated in detail.

The excitation of whistler wave instability due to slow cyclotron (m = – 1) interaction in an inhomogeneous plasma penetrated by an inhomogeneous beam of electrons is studied. Expressions are obtained for the elements of the plasma and beam dielectric tensors. It is shown that the inhomogeneity in both beam and plasma number densities affects the growth rate of the instability.

Linear and nonlinear mechanisms for generating convective cells with finite but small parallel (to the external magnetic field B0) wavelength are presented. The problems of mode-coupling as well as quasi-steady nonlinear mode structures are analytically studied. Possible applications in space plasmas are discussed.

The coupling of volume, long-wavelength TM electromagnetic and longitudinal space charge (electrostatic) waves by the rippling of magnetically focused electron beams is examined analytically. The dispersion relation is obtained and then solved for these types of wave. Instability, with growth rates proportional to the relative ripple amplitude of the beam, is found and discussed.

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 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.

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.

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.

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.

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.

Simple algebraic approximations for the plasma dispersion function Z(s) and its derivative are obtained by the Aitken's Δ2 or Shanks transformation. Using the approximate function Z'(s), higher-order dispersion relations for the electron wave and ion-acoustic wave are derived.