Growth of phase-space density holes

Phase-space density holes are shown to grow in a plasma for any nonzero electron–ion drift velocity. As a hole grows, its depth, velocity width, and electrostatic potential increase. For a hole with velocity u, the growth rate is of order −v2e f′0e (u)v2i f0i(u) times the bounce frequency of a particle trapped in the hole. The theoretical predictions agree reasonably well with a recent computer simulation. The results call into question the role of linear stability theory. Energy and momentum conservation are analyzed in detail, and the relationship to the clump instability is discussed.