Trapping saturation of the bump-on-tail instability and electrostatic harmonic excitation in earth's foreshock

The Vlasov simulation is used to examine the trapping saturation of the bump-on-tail instability both with and without mode-mode coupling and subsequent harmonic excitation. It is found that adding the pumped harmonic modes leads to a significant difference in the behavior of the phase-space distribution function near the unstable bump at the saturation time of the instability. The pumped modes permit rapid plateau formation on the space-averaged velocity distribution, in effect preventing the onset of the quasi-linear velocity-diffusion saturation mechanism.