3D BGK model for electron phase-space holes including polarization drift

Nonlinear kinetic structures are regularly observed in space and experimental magnetized plasmas, called electron phase-space holes (EH). The existence of EH is conditioned and varies according to the ambient magnetic field and the parameters of the electron beam that may generate them. The objective of this paper is to extend the 3D Bernstein-Greene-Kruskal (BGK) model with cylindrical geometry to include finite effects due to (i) the strength of the ambient magnetic field $\vec{B}_0$, by modifying the Poisson equation with a term derived from the electron polarization current, and (ii) the drift velocity $u_e$ of the background plasma with respect to the EH, by considering velocity-shifted Maxwellian distributions for the boundary conditions. This allows us to more realistically determine the distributions of trapped and passing particles interacting with the EH, as well as the width-amplitude relationships for their existence.
Comment: Submitted to The Astrophysical Journal on June, 4th 2021