Several conclusions regarding the stability of inhomogeneous Vlasov equilibria are drawn from earlier work. A technique is presented for generating first-order formulae for the change δω in the frequency of any normal mode, when a parameter λ characterizing the equilibrium is changed slightly. Several applications are given, including a first-order calculation of the growth rate or damping of an electromagnetic mode due to the presence of plasma. A condition is derived for the existence of a normal mode with real frequency. When there are ignorable co-ordinates, the normal modes can be written in the form of waves propagating in the ignorable directions. The character of the modes depends on certain symmetries in the dynamic spectral matrix. Special cases arise when the orbits can be approximated in certain ways.

The time-dependent behaviour of a homogeneous plasma in a weak electric field near the onset of runaway of electrons is studied for astrophysical applications. The relativistic Boltzmann equation is used to calculate the particle distribution function. It is shown that the plasma is stable against electric field fluctuations in the onset region for runaway, provided that the global behaviour of the System is determined by the magnetic field structure.

An elementary but rigorous derivation is given for a variational principle for guiding centre motion. The equations of motion resulting from the variational principle (the drift equations) possess exact conservation laws for phase volume, energy (for time-independent systems), and angular momentum (for azimuthally symmetric systems). The results of carrying the variational principle to higher order in the adiabatic parameter are displayed. The behaviour of guiding centre motion in azimuthally symmetric fields is discussed, and the role of angular momentum is clarified. The application of variational principles in the derivation and solution of gyrokinetic equations is discussed.

In the third part of this paper devoted to the problem of the thermal self-focusing of an electromagnetic wave in an underdense plasma, we extend the lineary theory of part 1 to include the effect of ambient drift motion perpendicular to the plane of the axis of the beam and the magnetic field. We show that the amplitude of the density perturbation satisfies a third-order hyperbolic equation, when this effect is included, and we apply Riemann's method to obtain its solution in integral form. This solution shows that a state of stationary self-focusing is established within the beam after a time equal to the transit time for a perturbation to drift across it. The form of the axial growth in the stationary state is derived from an asymptotic evaluation of the integral representation for large axial distances. It is found that the axial growth increases across the beam starting from zero at the edge through which plasma enters the beam and reaching its maximum attainable value at or before the opposite edge, depending on the magnitude of the drift velocity. The maximum growth decreases as the drift velocity increases and only occurs at the rates deduced in parts 1 and 2 when the beam width is several times larger than the scale length which describes the decay of a density perturbation within the beam due to the combined effect of drift longitudinal ambipolar diffusion, recombination and the stabilizing influence of perturbations to the electrical conductivity. The factor by which the amplitude of a convected density perturbation is reduced relative to the value it would have in the absence of drift motion is evaluated and the results are applied to the case of ionospheric modification by a transmission from the proposed solar power satellite systems.

It is shown that drift dissipative instabilities can occur in the frequency regime Ωi ≪ ω ≪ Ωe. These instabilities are associated with the electron-acoustic and lower-hybrid waves.

In the fourth part of this paper devoted to the problem of the thermal self-focusing of an electromagnetic wave in an underdense plasma, we examine the spatial nonlinear development of the instability, assuming that a stationary state of the type described earlier has been established. Our treatment considers the interaction of the pump wave with the Stokes side-band waves and the density perturbation corresponding to the most unstable linear mode. We show that the system of equations describing their interaction possesses three invariants. These invariants are used to reduce the problem to a single differential equation for the axial behaviour of the amplitude of the density perturbation. The form of the invariants differs according as the inclination of the beam to the magnetic field is above or below the critical value defined earlier. Above the critical orientation, the amplitude of the density perturbation can be expressed algebraically in terms of the amplitudes of the pump and side-band waves. We show that this case is integrable in terms of the Jacobean elliptic function dn and we use the results to find the saturation level. Below the critical orientation, the response of the plasma becomes non-local, above a certain threshold power. The algebraic relation is then replaced by a differential equation which results in a significantly more complicated differential equation for the amplitude of the density perturbation. A particular solution of this equation is found for a special choice of initial conditions and from this the saturation level is found. The results of these calculations are applied to the problem of ionospheric modification by microwave transmissions from the proposed solar power satellite systems.

A general investigation of the electrical conductivity for radio-frequency (RF) fields in strongly magnetized plasmas with small-scale density fluctuations is performed within the cold plasma hydrodynamical approximation. It is shown that in such plasmas an RF phenomenon similar to the Bohm diffusion exists: the presence of stochastic RF electric field in a turbulent plasma can lead to a strong enhancement of the RF currents flowing in the direction of the applied electric field components transverse to the magnetic field. The appearance of these turbulent drift currents favours energy transfer from the RF fields to the plasma and thus leads to their stronger damping. This effect allows us to interpret quantitatively the enhanced damping of the magnetosonic waves observed in several experiments. The magnetized radially inhomogeneous cylindrical plasmas in these experiments are characterized by density fluctuations due to drift instabilities. The theory has also a number of other applications; an example is given of the whistlers damped by the ionospheric density fluctuations.

An investigation is made of the effect of RF-induced velocity diffusion on the distribution function, f(v), of the heated ion species in minority heating schemes using the fast Alfvén wave. The solution for the distorted distribution function is expressed in a Legendre series from which deviations from isotropy can be assessed. Solutions are found in the limits of high and low velocities and the elongation of the constant f(v) surfaces is also given.

The energy loss rate, Ė, of test ions injected with velocity V into a Maxwellian electron-ion plasma in the presence of an external magnetic field, is studied. Most of the integrals appearing in the expression for Ė are evaluated analytically and the remaining integrals are evaluated numerically for various values of the parameters η = Ωe / ωe, x = V/ve, r = Te/Ti, and the angle of injection θ. It is shown that the effect of the magnetic field is rather small except for η > 1, the temperature ratio is important for small x (x ≤ 0·2), and the injection angle is not important.

In the fifth part of this paper devoted to the problem of the thermal self-focusing of an electromagnetic wave in an underdense plasma, we extend the nonlinear theory of part 4 to include the effect of cubic nonlinearities. These terms describe the stabilizing influence of perturbations to the electrical conductivity. We assume that a stationary state of the type described earlier has been established and consider the spatial nonlinear interaction of the pump wave with the Stokes side-band waves and the density perturbation corresponding to the most unstable linear mode. We show that the system of єquations describing their interaction possesses three invariants. These invariants are used to reduce the problem to quadratures and an implicit solution is obtained in terms of incomplete elliptic integrals of the first and third kinds. From this solution we derive explicit expressions for the saturation level and the nonlinear growth length. The results are applied to the problem of ionospheric modification by microwave transmissions from the proposed solar power satellite systems.

The development of anisotropy in an initially isotropie spectrum is studied numerically for two-dimensional magnetohydrodynamic turbulence. The anisotropy develops through the combined effects of an externally imposed d.c. magnetic field and viscous and resistive dissipation at high wavenumbers. The effect is most pronounced at high mechanical and magnetic Reynolds numbers. The anisotropy is greater at the higher wavenumbers.

Rescaling transformations bringing friction terms in the new equation are used to obtain the asymptotic solution of a one-dimensional, one-species beam. It is shown that for all possible initial conditions this asymptotic solution coincides with the self-similar solution.

The present series of papers is concerned with the problem of the thermal selffocusing of an electromagnetic wave in an underdense plasma which is examined within the context of ionospheric modification by high-power radio waves. We formulate this problem as a coherent four-wave interaction between a pump wave, Stokes side-band waves and a thermally driven density perturbation. Our treatment assumes that the heating of the electrons in the interference pattern of the pump and side-band waves is in equilibrium with longitudinal heat conduction and energy transfer to the ions and, in addition, that the density perturbation is subject to decay on account of longitudinal ambipolar diffusion, ambient plasma drift motion, recombination and the stabilizing influence of perturbations to the electrical conductivity.

After giving a higher-order solution of the equation describing the nonlinear equilibrium of rippled electron beams, in the second part of the paper we derived mathematical expressions for the growth rates of electrostatic perturbations for the resonant coupling of (i) fast and slow and (ii) fast—fast or slow—slow waves. Actually, viewed from a physical point of view, only type (i) of wave coupling may lead to the kind of instability considered in the paper.

A new theory is developed for the transport of energy across a strong magnetic field. The mechanism is micro-convection by sheared fluid elements, and it proves to be much more effective in heat transport than classical conduction. The equivalent thermal conductivity is calculated for the electron gas in a cylindrical plasma carrying an axial current. When the plasma pressure can be neglected, this conductivity is about 400 times greater than the classical value under certain typical conditions.

The motion of electrons and ions in a system which consists of a uniform rotating magnetic field and a steady uniform magnetic field which is aligned with the axis of rotation is re-examined. Stable orbits are identified and explicit expressions for these orbits are provided. The more realistic situation of motion in the non-uniform self-consistent fields appropriate to a cylindrical plasma equilibrium maintained by the rotating field is investigated. Again, stable orbits are found. The stability of orbits in this more complex situation is examined using an equivalent potential concept.