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

A multidimensional 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; we find that the evolution is determined by combined actions between the transverse instability and the stabilization by the background magnetic field. In very weakly magnetized plasma (Omega(e) << omega(pe), where Omega(e) and omega(pe) are the electron gyrofrequency and plasma frequency, respectively), the transverse instability dominates the evolution of the electron holes. The parallel cut of the perpendicular electric field (E-perpendicular to) has bipolar structures, accompanied by the kinking of the electron holes. 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-perpendicular to in the electron holes can evolve into unipolar structures. In very strongly magnetized plasma (Omega(e) >> omega(pe)), the unipolar structures of the parallel cut of E-perpendicular to can last for thousands of electron plasma periods. At the same time, the perpendicular electric field (E-perpendicular to) in the electron holes can also influence electron trajectories passing through the electron holes, which results in 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-perpendicular to can be formed outside of the electron holes, which then emit electrostatic whistler waves because of the interactions between the streaked structures of E-perpendicular to and vibrations of the kinked electron holes.