Electrostatic Instabilities of a Uniform Non-Maxwellian Plasma

TLDR

Instabilities propagating parallel to a unit vector in a uniform non‑Maxwellian plasma are linked to the function F(u) derived from the unperturbed distribution, and the paper discusses the mathematical assumptions underlying this analysis. The authors derive a stability criterion from Vlasov’s collision‑free kinetic equations and use a Nyquist‑criterion–based method to identify exponential growth conditions, applying it to examples such as a current‑carrying plasma with electrons and ions at different Maxwellian temperatures. They find that exponential growth occurs only when F(u) has a minimum satisfying a specific integral inequality, and that while most distribution functions yield normal stability, an exceptional class leads to a breakdown of linearized stability theory.

Abstract

A stability criterion is obtained starting from Vlasov's collision-free kinetic equations. Possible instabilities propagating parallel to an arbitrary unit vector e are related to a function F(u)≡Σjωj2∫d3v gj(v) δ(e·v−u), where gi(v) is the normalized unperturbed distribution function, and ωj≡(4πnjej2/mj)12 the plasma frequency, for the jth type of particle. By using a method related to the Nyquist criterion, it is shown that plasma oscillations growing exponentially with time are possible if and only if F(u) has a minimum at a value u = ξ such that ∫−∞∞du(u−ξ)−2[F(u)−F(ξ)]>0. A study of the initial-value problem confirms that the plasma is normally stable if no exponentially growing modes exist; but there is an exceptional class of distribution functions (recognizable by means of an extension of the above criterion) for which linearized stability theory breaks down. The method is applied to several examples, of which the most important is a model of a current-carrying plasma with Maxwell distributions at different temperatures for electrons and ions. The meaning of the mathematical assumptions made is carefully discussed.