Asymmetric One-Dimensional Slow Electron Holes

TL;DR

This paper develops a theoretical framework for constructing asymmetric slow electron holes in plasmas, analyzing their stability, and deriving potential drop formulas, with implications for understanding plasma solitary structures.

Contribution

It provides the first fully consistent solutions of asymmetric electron holes in the Vlasov-Poisson system for various ion velocity distributions.

Findings

Stable equilibria occur at specific hole velocities influenced by ion reflection.

Potential asymmetry across the hole is generally small for typical amplitudes.

Derived a formula relating potential drop to ion distribution curvature and electron temperature.

Abstract

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 $\Delta\phi\simeq{2\over 9}f'''{T_e\over e}({e\psi\over m_i})^2$, where $\psi$ 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, $\psi\lesssim 0.5T_e/e$.