We give a necessary criterion of validity for the Dupree–Weinstock theory of turbulence. In this connexion, we analyze two specific cases of high-frequency (HF) and low-frequency (LF) turbulence. In the HF case, we show that an appreciable modification of the growth rate of unstable modes cannot occur before trapping.

The oblique echo model for the resonance near the local plasma frequency fN observed by topside sounders involves the propagation of slow waves away from the sounder, which later return as echoes after reflexion due to an electron density gradient. The model is investigated using the WKB technique introduced by Fejer & Yu. The resulting set of saddle-point equations is solved by a method which can, unlike previous work, be applied to both the ‘strong’ resonance (for whichfN > fH wherefH is the electron gyrofrequency) and to the ‘weak’ (fN < fH) resonance. Throughout the calculations, the validity of the approximations made is checked. It is found that the weak resonance solution is not valid for typical topside parameters. For the strong resonance, the electric field of a small pulsed dipole is calculated. Expressions are found for the frequency of the two echoes that are received by the sounder and the frequency of the beating between them. Although these are only strictly valid for times longer than typical observation times, they yield results that agree well with the corresponding results from ray trajectory computations.

The oblique echo model for the resonance near the local upper hybrid frequency fT observed by topside sounders involves the propagation of slow waves away from the sounder, which later return as echoes after refiexion due to an electron density gradient. The model is investigated using the WKB technique introduced by Fejer & Yu. The resulting set of saddle-point equations is solved by a method which can, unlike previous work, be applied to both the ‘strong’ resonance (for which fT < 2fH where fH is the electron gyrofrequency) and to the ‘weak’ (fT 2fH) resonance. Throughout the calculations the validity of the approximations made is checked. It is found that the weak resonance solution is not valid for typical topside parameters. A condition is derived for the existence of intercepting ray paths and for the maximum time delay which occurs when they do exist. For the strong resonance the electric field of a small pulsed dipole is calculated. Expressions are found for the frequency of the two echoes which are received by the sounder and the frequency of the beating between them. Although they are only strictly valid for times longer than typical observation times, they yield results that agree well with the corresponding results from ray- trajectory computations.

The longitudinal behaviour of the electron ion ring in a stationary electron ring accelerator (ERA) is studied by means of a filament model which neglects the radial thickness of the ring. There are two widely different time scales (short or STS of the order of a bounce time in the ring, and long or LTS) that characterize the system. The Vlasov equations for a system which is stationary on the STS can be solved in the ion-pickup region by an approximation in which each species of particle essentially sees only a potential well formed by the other species. This gives a generalization of the classical Bernstein–Greene–Kruskal (BGK) problem, in which both species move in the same potential. The conditions under which this generalized BGK problem is well defined are given, and broad classes of quasi-equilibria (i.e. equilibria on the STS) are obtained. The time dependence on the LTS of some of these quasi-equilibria is then obtained by invoking charge conservation, momentum conservation and the adtained invariance of longitudinal action integrals. The stability of these quasi-equilibria (i.e. their behaviour under time-dependent perturbations on the STS) is deferred to a subsequent paper.

This paper analyses the dispersion relation for a collisionless moving electron plasma, when the direction of motion is normal to the magnetic field and that of the wave propagation along the magnetic field. It is shown that, in strong magnetic fields, the one continuous allowed band of the left-handed wave (of the stationary plasma) splits into two, and the right-handed wave shows a second resonance besides the cyclotron resonance. In weak magnetic fields, the lefthanded wave develops a backward wave band, which shows resonance at its low frequency edge, and the right-handed wave also develops an extra band of propagation. The effect of the motion of the plasma, on waves of frequency much lower than the plasma frequency, is identical to a doppler shift, but, on those of frequency much higher than that, is negligible.

This paper derives an expression for the velocity space diffusion tensor (time asymptotic) for resonant particles in the weak-coupling limit of strong plasma turbulence theory. The analysis is based on an improved approximation scheme for the determination of the actual particle trajectories, rather than the determination of &ensemble averaged particle orbits’ with the help of the &average Vlasov propagator’. Except for numerical factors of order unity, the results agree with those of Birmingham & Bornatici, both when Tc → 0 (Tc is the correlation width of the fluctuating field as seen by the resonant particles), and when Tc is finite but small. The paper indicates the limits of this method, as well as those of Birmingham & Bornatici.

Quantum field theory is used to investigate the nonlinear effects of two modified ordinary monochromatic waves propagating across a uniform magnetic field. Amplitude-dependent wavelength and frequency shifts are derived. Unlike previous work that employs the Krylov–Bogoliubov–Mitropolskii method, the investigation uses the diagrammatic technique of field theory. The calculations show that increase of the above-mentioned physical parameters due to nonlinear interaction and drifting electrons depends upon the ratio of drifting to phase velocity.

Wave propagation in a cold, collisionless, two-component plasma is analyzed by considering, first, the frame of reference in which the field has no space dependence, and then applying a Lorentz transformation to obtain a wave whose space-time dependence is a function of t — nz / c only, where n is a constant. Exact (nonlinear, relativistic) results for known special cases, and some others, are given; and it is shown that, when there is no ambient magnetic field, the general problem in essence reduces to the solution of one second-order, nonlinear differential equation. The desirabifity of introducing a free parameter representing a stream velocity in the direction of wave propagation is emphasized; and the significance of the choice of this parameter is discussed.

We consider the possibility of modifying the Sonnerup solution for incompressible fluid flow about an X-type re-connexion line, to include fields and flows parallel to the X line. We find that such fields and flows may change across the discontinuities of the Sonnerup solution. By considering the requirements imposed by a proper matching across the various regions of flow, and by the integral conservation properties of the diffusion region, we seek to find the restrictions that are imposed on this parallel field and flow, and on the arrangement of the discontinuity planes around the diffusion region. We find that four types of such arrangements are possible, each corresponding to a different set of restrictions on the parallel field and flow. In one case, where all the discontinuity planes intersect at a common line, the ‘ parallel’ parameters of the in-flow and out-flow regions may be arbitrarily and independently chosen. Of the remaining three cases, one contains solutions with uniform parallel fields ad flows, while the other two depend for their existence on large fluid flow or magnetic field shears across the two in-flow regions.

We consider the flow of a highly conducting incompressible fluid in the convection region surrounding an X-type neutral line. The Sonnerup solution is generalized to include cases in which the in-flowing anti-parallel magnetic fields are of unequal magnitude, and in which the plasmas of the two in-flow regions have unequal density. It is found that there exists no unique solution for a given set of in-flow plasma parameters, but that the out-flowing plasma may exit within a certain range of angles with respect to the in-flow direction, depending on in-flow parameters.