Envelope equation for the linear and nonlinear propagation of an electron plasma wave, including the effects of Landau damping, trapping, plasma inhomogeneity, and the change in the state of wave

TL;DR

This paper derives a comprehensive envelope equation for electron plasma wave propagation that includes effects like Landau damping, trapping, plasma inhomogeneity, and state changes, applicable in both linear and nonlinear regimes.

Contribution

It introduces a new envelope equation derived from Vlasov-Poisson and variational methods, capturing complex plasma effects in three dimensions with smooth transitions between linear and nonlinear states.

Findings

Predicts abrupt transitions in wave coefficients in 1D

Provides a simple analytic expression for nonlinear Landau damping

Shows smoother transitions in 3D compared to 1D

Abstract

This paper addresses the linear and nonlinear three-dimensional propagation of an electron wave in a collisionless plasma that may be inhomogeneous, nonstationary, anisotropic and even weakly magnetized. The wave amplitude, together with any hydrodynamic quantity characterizing the plasma (density, temperature,...) are supposed to vary very little within one wavelength or one wave period. Hence, the geometrical optics limit is assumed, and the wave propagation is described by a first order differential equation. This equation explicitly accounts for three-dimensional effects, plasma inhomogeneity, Landau damping, and the collisionless dissipation and electron acceleration due to trapping. It is derived by mixing results obtained from a direct resolution of the Vlasov-Poisson system and from a variational formalism involving a nonlocal Lagrangian density. In a one-dimensional situation, abrupt transitions are predicted in the coefficients of the wave equation. They occur when the state of the electron plasma wave changes, from a linear wave to a wave with trapped electrons. In a three dimensional geometry, the transitions are smoother, especially as regards the nonlinear Landau damping rate, for which a very simple effective and accurate analytic expression is provided.