Longitudinal Ion Oscillations in a Hot Plasma

TLDR

In a hot plasma, longitudinal waves include electron plasma oscillations and ion plasma oscillations with frequencies ω ≈ ωp(m/M)½, the latter exhibiting acoustic‑wave characteristics at small k. The ion oscillations are investigated via a Vlasov equation description of the plasma. For equal ion and electron temperatures the ion oscillations are strongly damped, but increasing Te/Ti or driving a current can reduce damping or even make the waves grow, revealing the two‑stream instability as an unstable ion oscillation.

Abstract

Linearized, longitudinal waves in a hot plasma include, besides the familiar electron plasma oscillations, in which the frequency ω is of order ωp = (4πne2/m)½, also ion plasma oscillations with ω ≈ ωp(m/M)½. The properties of the latter are explored using a Vlasov equation description of the plasma. For equal ion and electron temperatures, Te = Ti, there exists a discrete sequence of ion oscillations, but all are strongly damped, i.e., have -Im ω/Re ω ⪞ 0.5, and hence are not likely to be observable. The ratio Im ω/Re ω can be made to approach zero (facilitating detection of the waves) by either increasing Te/Ti or by producing a current flow in the plasma. In the latter case, Im ω can even be made positive (corresponding to growing waves), the current required for this being smaller the larger the value of Te/Ti. This growing wave is just the familiar two-stream instability which is thus seen to be an unstable ion oscillation. It is also noteworthy that the ion oscillations, which for small k have the properties usually associated with an acoustic wave (longitudinal polarization, ω ∝ k), are obtained using a formalism which is sometimes designated as ``collisionless.''