Abstract
The nonlinear saturation of the single‐mode, bump‐on‐tail instability is studied by numerical solutions of the Vlasov equation. Modes close to marginal stability are found to saturate with a field amplitude E∝Δ2, where the small parameter Δ=2(ω−ω’)/ω’ defines the difference between the mode frequency ω and the frequency ω’ of the marginally stable mode. This scaling agrees with O’Neil’s theory [Phys. Fluids 1 4, 1204 (1971)] of the two‐stream instability, and does not confirm the time asymptotic analysis of Simon and Rosenbluth [Phys. Fluids 1 9, 1567 (1976)], which predicts larger amplitudes with E∝Δ1/2.

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