A Lagrangian theory for nonlinear wave packets in a collisionless plasma

Abstract

A Lagrangian, for the slowly varying complex amplitude of an almost monochromatic electrostatic plasma wave in an unmagnetized plasma, is derived. The method is a variant of the averaged Lagrangian technique of Whitham, adapted for use in plasma physics by employing Low's Lagrangian, together with certain elimination procedures to simplify the Lagrangian to usable form. The expansion in powers of the wave amplitude is carried out to quartic terms, and non-adiabatic terms are also retained. Variations with respect to the amplitude lead to a nonlinear Schrödinger equation for the amplitude, while variations of the particle trajectories lead to a modified Vlasov equation, which includes the nonlinear reaction of the wave on the average trajectories. It is assumed that there are no resonant particles. These equations are coupled through the nonlinear frequency shift. It is found that the particle aspect of the plasma has a profound effect on the stability of the system, due to resonance of the wave envelope with particles moving at the group velocity. This effect, nonlinear Landau damping, is always destabilizing, leading to growth of modulations. In terms of the nonlinear Schrodinger equation, the effect changes the equation from a Hartree-Fock equation with a delta-function interaction to one with a non-momentum-conserving, non-local interaction. In the limit of fairly small wavelength, of the modulations, it is shown that the growth rate approaches that expected from plasma kinetic theory.