Threshold conditions for electron trapping by nonlinear plasma waves

Numerical solutions of the nonlinear Vlasov equation show that the long time behavior of an electrostatic plasma wave depends critically on the value of the parameter γτ (γ is Landau's damping coefficient and τ is the electron bounce time), both for large and small initial wave amplitudes. When the initial wave amplitude is large, the initial nonlinear damping is stronger than Landau damping; for smaller initial amplitudes, the behavior is initially described by Landau's theory. For both types of problems if γτ > 0.5, the wave damps monotonically; if γτ < 0.5, the numerical results indicate that the plasma performs amplitude oscillations. When γτ ≈ 0.5, the plasma displays its critical behavior; here, after some initial damping the wave amplitude levels off at a constant value. The critical behavior observed for mild nonlinear problems is in good agreement with the experimental results of Franklin, Hamberger, and Smith. For a moderately strong nonlinear wave displaying the critical behavior, the wave performs periodic oscillations with a frequency smaller than Landau's linear frequency; this frequency shift is explained because the uniform electron distribution develops a plateau centered at Landau's phase velocity.