Long-time simulation of the single-mode bump-on-tail instability

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Abstract
A Vlasov code developed by Denavit [Phys. Fluids 28, 2773 (1985)] was modified and used to study long-time behavior of the single-mode bump-on-tail instability. Large oscillations, which interrupted his original runs at ω0 T≊1300 shortly after reaching the O’Neil trapping level (OTL), were identified and controlled by heating the cold background electrons. A run to ω0T =30 000 showed growth well beyond the OTL and eventually saturated at about the level predicted by Simon and Rosenbluth (SR) [Phys. Fluids 19, 1567 (1976)], before oscillations set in. A run with warmer background electrons was again interrupted by oscillations. A weakly unstable case was followed to ω0T =60 000; it also showed growth well beyond the OTL but saturated at an amplitude that was significantly below that predicted by SR. In all cases, the velocity distribution developed strong ripples near v=0. These are shown to be caused by a numerical grid effect, and they may be responsible for the eventual development of the large oscillations and for the low level of saturation in the weakly unstable case.