Landau damping and transit-time damping of localized plasma waves in general geometries

Abstract

Landau’s original derivation of the collisionless damping of small-amplitude Langmuir waves in an infinite homogeneous plasma relied on the introduction of complex velocities and was therefore somewhat difficult to interpret physically. This has inspired many subsequent derivations of Landau damping that involve only real physical quantities throughout. These “physical” derivations, however, have required the calculation of quantities to second order in the wave field, whereas Landau’s approach involved only first-order quantities. More recent generalizations of Landau damping to localized fields, often called “transit-time damping,” have followed the physical approach, and thus also required second-order calculations, which can be quite lengthy. In this paper it is shown that when the equilibrium distribution function depends solely on the energy, invoking the time-reversal invariance of the Vlasov equation allows transit-time damping to be analyzed using only first-order physical quantities. This greatly simplifies the calculation of the damping of localized plasma waves and, in the limit of an infinite plasma, provides a derivation of Landau damping that is both physical and linear in the wave field. This paper investigates the transit-time damping of plasma waves confined in slabs, cylinders, and spheres, analyzing the dependence on size, radius, and mode number, and demonstrating the approach to Landau damping as the systems become large. It is also shown that the same approach can be extended to more general geometries. A companion paper analyzes transit-time damping in a cylinder in more detail, with applications to the problem of stimulated Raman scattering in self-focused light filaments in laser-produced plasmas.