Transverse instability and perpendicular electric field in two-dimensional electron phase-space holes

A multi-dimensional electron phase-space hole (electron hole) is considered to be unstable to the transverse instability. In this paper, we perform two-dimensional (2D) particle-in-cell (PIC) simulations to study the evolution of electron holes at different plasma conditions, and find that the evolution is determined by combined actions between the transverse instability and the stabilization of the background magnetic field. In very weakly magnetized plasma (Ωe<<ωpe, where Ωe andωpe are the electron gyrofrequency and plasma frequency, respectively), the transverse instability dominates the evolution of the electron holes. Accompanied by the kinking of the electron holes, the parallel cut of the perpendicular electric field (E⊥) has bipolar structures. Such structures last for only tens of electron plasma periods. With the increase of the background magnetic field, the evolution of the electron holes becomes slower. The bipolar structures of the parallel cut of E⊥ in the electron holes can evolve into unipolar structures. In very strongly magnetized plasma (Ωe>>ωpe), the unipolar structures of the parallel cut of E⊥ can last for thousands of electron plasma periods. At the same time, the perpendicular electric field (E⊥) in electron holes can also influence electron trajectories passing through the electron holes, which results in the variations of charge density along the direction perpendicular to the background magnetic field outside of the electron holes. When the amplitude of the electron hole is sufficiently strong, streaked structures of E⊥ can be formed outside of the electron holes, which then emit electrostatic whistler waves due to the interactions between the streaked structures of E⊥ and vibrations of the kinked electron hole.