Longitudinal Electrostatic Oscillations in Velocity-gradient Plasmas I. The Slipping-stream Model

Electrostatic oscillations in velocity-gradient plasmas are studied with the aid of two simplifying models: (i) the slipping-stream model; and (ii) the adjacent-stream model. The slipping-stream model is dealt with in the present paper, and the adjacent-stream model in the following paper. In the simplest slipping-stream plasma the mean longitudinal velocity u, parallel to the z axis, varies linearly in a transverse direction: du/dx = α = const. Various assumptions are made in the treatment: collisions are neglected; it is assumed that a longitudinal magnetic field is sufficiently intense to suppress transverse oscillatory electron motions; it is assumed that the boundaries are sufficiently conducting for the electric potential to vanish at these boundaries; and finally only electron electrostatic interactions are considered. It is first shown that in the case of a zero-temperature slipping stream, longitudinal electron oscillations are stable, in the sense that there are no modes which are exponentially time-growing, when ω e 2 /α 2 < ¼, where ω e is the electron plasma frequency. When the longitudinal velocity u is an arbitrary function of the transverse coordinate x, general considerations show that ω e 2 < ¼( du/dx ) max 2 is a sufficient condition for stability, where ( du/dx ) max 2 is the maximum value of the velocity gradient squared in the stream. It is then shown that in the case of a finite temperature the slipping stream has increased stability and larger velocity gradients are necessary to cause instability.