Electron holes in laboratory and space plasmas are quasi-stationary functions of the parallel adiabatic invariant

Electrostatic coherent structures such as electron and ion holes are ubiquitous features in laboratory and space plasma observations. They are long-lived structures often thought to be well-described by Bernstein-Greene-Kruskal (BGK) modes. As a model problem, we revisit the long-time behavior of the two-stream instability analytically and numerically. Using the Gkyell code, we show from direct numerical simulation of the self-consistent Vlasov-Poisson system that the initial-value problem of the two-stream instability leads to an electron hole, which, however, after several hundreds of electron cyclotron times, still does not relax to the stationary state predicted by standard BGK theory. We demonstrate that such modes are better characterized as quasi-stationary states that are functions of J, the parallel adiabatic invariant. As a function of J, the distribution function allows us to overcome the problem of non-existence of higher-dimensional BGK modes that are functions only of energy. Finally, we address the question of stability and the effects of weak collisionalities on such quasi-steady states.