Time-asymptotic field amplitudes in nonlinear Landau damping

The behavior of an electrostatic plasma wave of initial amplitude Ei in a collisionless electron plasma depends crucially on the ratio δ = γL/ωB, where γL is the damping coefficient calculated from linear plasma kinetic theory and ωB = (ekEi/m)1/2 is the open-quotes bounceclose quotes frequency for trapped electrons. For sufficiently small Ei, i.e., δ much-gt 1, the wave damps away completely; conversely, a wave of large initial amplitude, for which δ much-lt 1, saturates at a finite final amplitude due to the nonlinear effects of particle trapping. In this paper, we report results concerning the transition region δ ∼ 0(1), which separates these two regimes. For the case of weak damping, i.e., |γL/ωp|much-lt 1, where ωp is the plasma frequency, we calculate analytically the critical value δi,e below which complete damping occurs and derive the first explicit analytic relationship between Ei and Ef, the time-asymptotic amplitude, for Ei above the critical value Ei,c = (m,ek)(γL/δi,c). The results predict an interesting phenomenon - a finite jump δE in the final field amplitude as Ei is increased through Ei,c. We have corroborated the reality of this phenomenon through large-scale numerical simulations of the Vlasov-Poisson system, carried out using the spectral splitting code of Klimas. These important new results establish a detailed picture of the fundamental transition between linear and nonlinear Landau damping and, in particular, show that the onset of essentially nonlinear behavior occurs very abruptly as the initial wave amplitude increases through its critical value