Linear wave action decay entailed by Landau damping in inhomogeneous, nonstationary and anisotropic plasma

Abstract: This paper addresses the linear propagation of an electron wave in a collisionless, inhomogeneous, nonstationary and anisotropic plasma. The plasma is characterized by its distribution function, $f_H$, at zero order in the wave amplitude. This distribution function, from which are derived all the hydrodynamical quantities, may be chosen arbitrarily, provided that it solves Vlasov equation. Then, from the linearized version of the electrons equation of motion, and from Gauss law, is derived an envelope equation for the wave amplitude, assumed to evolve over time and space scales much larger than the oscillation periods of the wave. The envelope equation may be cast into an equation for the the wave action, derived from Whitham's variational principle, that demonstrates the action decay due to Landau damping. Moreover, the Landau damping rate is derived at first order in the variations of the wave number and frequency. As briefly discussed, this paper generalizes numerous previous works on the subject, provides a theoretical basis for heuristic arguments regarding the action decay, and also addresses the propagation of an externally driven wave.