Saturation of the relativistic two-stream instability by electron trapping

Abstract

The single‐wave weak cold relativistic two‐stream instability has been studied in one dimension by numerical integration of the equations of motion, energy conservation, and Poisson’s equation. The dynamics is shown to depend only on the single parameter S = β₀²γ₀(n_b/2n_p)^1/3. The nondimensionalized wave energy at initial saturation due to beam trapping, W₁ = 〈E²〉/8πn_bmc²γ₀, reaches a maximum W₁ = 0.09 at S = 0.4 and then decreases with S, but if S≳0.5 the wave energy is larger at subsequent bounces of the trapped particles, so that W_max increases roughly monotonically with S, reaching 0.11 at S = 3. The drop‐off of W₁(S) is due to a relativistic tendency of beam electrons to bunch in the part of the orbit where the beam electron energy is largest. The self‐consistent nonlinear shifts in wave frequency and amplitude are shown to play an essential role in extracting energy from the beam.