I H Hutchinson

The physics of isolated plasma potential structures sustained by a deficit of phase-space density on trapped orbits, commonly known as electron or ion holes, is reviewed. The principles of their equilibria are explained and illustrated, and contrasted with solitons. A literature review mostly prior to 2016 highlights the key historical developments of the field. Progress since, especially in hole acceleration, stability, and multi-dimensional effects, is summarized in more detail.

Slow solitary positive-potential peaks sustained by trapped electron deficit in a plasma with asymmetric ion velocity distributions are in principle asymmetric, involving a potential change across the hole. It is shown theoretically how to construct such asymmetric electron holes, thus providing fully consistent solutions of the one-dimensional Vlasov-Poisson equation for a wide variety of prescribed background ion velocity distributions. Because of ion reflection forces experienced by the hole, there is generally only one discrete slow hole velocity that is in equilibrium. Moreover the equilibrium is unstable unless there is a local minimum in the ion velocity distribution, in which the hole velocity then resides. For stable equilibria with Maxwellian electrons, the potential drop across the hole is shown to be $Δφ\simeq{2\over 9}f'''{T_e\over e}({eψ\over m_i})^2$, where $ψ$ is the hole peak potential, $f'''$ is the third derivative of the background ion velocity distribution function at the hole velocity, and $T_e$ the electron temperature. Potential asymmetry is small for holes of the amplitudes usually observed, $ψ\lesssim 0.5T_e/e$.

We analyse the kinematic properties of a plasma electron hole: a non-linear self-sustained localized positive electric potential perturbation, trapping electrons, that behaves as a coherent entity. When a hole accelerates or grows in depth, ion and electron plasma momentum is changed both within the hole and outside it, by an energization process we call jetting. We present a comprehensive analytic calculation of the momentum changes of an isolated general one-dimensional hole. The conservation of the total momentum gives the hole's kinematics, determining its velocity evolution. Our results explain many features of the behavior of hole speed observed in numerical simulations, including self-acceleration at formation, and hole pushing and trapping by ion streams.