Saturation of a single mode driven by an energetic injected beam. I. Plasma wave problem

A formalism is established for calculating the saturation level of a discrete mode that is destabilized by the distribution function formed by a high-energy injected beam. The electrostatic plasma wave interaction is studied here for two problems. In one the distribution function is formed by injection of a source with a velocity spread and a steady-state bump-on-tail instability is established with only particle annihilation taken into account. In the second problem particle drag as well as particle annihilation is accounted for. In both problems the self-consistent distribution function in the presence of a finite amplitude wave needs to be calculated. By calculating the power transfer between particles and a finite amplitude wave, the saturation level of the discrete mode can be predicted. The drag problem with annihilation has the interesting feature that in steady state holes in phase space are formed for a large enough amplitude wave and the power transferred from particles to waves can be greatly enhanced as a result of the drag force on the holes.