Theory of Longitudinal Plasma Instabilities

A stability theory of longitudinal disturbances in a plasma is presented. The theory states necessary conditions and a necessary and sufficient condition of the instability in terms of the mathematical nature of linear plasma conductivities on the real $\omega$ axis, by introducing an idea of passive and active conductivities combined with Nyquist diagram techniques. It is applied to obtain conditions of instability in both collision-dominated and collision-free plasmas in the presence of a uniform magnetic field. Effects of drift, density gradient, and temperature anisotropy appear as the causes of the active conductivities, whose presence is shown to be the basic necessary condition for an instability. Each of these effects contributes differently to the conductivity, giving a different necessary and sufficient condition for the associated instability.