Electrostatic Instabilities of a Uniform Non-Maxwellian Plasma

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 ${\mathrm{F(u)\equiv \Sigma }}_j{\omega }_j^2{\mathrm{\int d}}^3\mathbf{v}{g}_j(\mathbf{v}\mathrm{)\; \delta (}\mathbf{e·v}\mathrm{-u)}$ , where g_i(v) is the normalized unperturbed distribution function, and ${\omega }_j{\text{≡(4πn}}_j{e}_j^2{\mathrm{/m}}_j{)}^{\frac{1}{2}}$ 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 ${\int }_{\mathrm{-\infty }}^{\infty }{\mathrm{du(u-\xi )}}^{\text{−2}}\text{[F(u)−F(ξ)]>0}$ . 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.