Perpendicular electric field in two-dimensional electron phase-holes: A parameter study

Previous multi-dimensional particle simulations have shown that electron phase-holes can be formed during the nonlinear evolution of bi-stream instability. In these holes, the parallel cut of the parallel electric field (E-parallel to) has bipolar structures while the parallel cut of the perpendicular electric field (E-perpendicular to) has unipolar structures. In this paper, two-dimensional (2D) electrostatic particle-in-cell simulations are performed to investigate the evolution of E-perpendicular to in such electron holes for different plasma conditions, and the generation mechanism of the unipolar structures of E-perpendicular to is also discussed. The electrons trapped in electron holes bounce in the parallel direction, which leads to transverse instability (Muschietti et al., 2000). At the same time, they gyrate in the background magnetic field, which tends to stabilize electron holes. In this way, the trapped electrons are forced to accumulate locally, and the charge density has variations along the perpendicular direction inside the electron holes. The balance between these two effects leads to the following results: in weakly magnetized plasma (Omega(e) < omega(pe), but Omega(e) is comparable to omega(pe). Where Omega(e) and omega(pe) are the electron cyclotron frequency and electron plasma frequency, respectively), electron holes have two-dimensional structures (isolated along both the parallel and perpendicular directions). Within such holes the parallel cut of E-perpendicular to has unipolar structures. In strongly magnetized plasma (Omega(e) > omega(pe)), electron holes have one-dimensional structures along the direction perpendicular to the background magnetic field within which a series of islands (with alternate positive and negative E-perpendicular to) develop because of the variations of the charge density along the perpendicular direction. Therefore one recovers that a parallel cut of E-perpendicular to has unipolar structures at the location of the holes. Present results show that the unipolar structure of E-perpendicular to in electron holes is attributed to the balance between the electron transverse instability and the stabilization of the background magnetic field. The unipolar structures of E-perpendicular to in electron holes last for hundreds to thousands of electron plasma periods. They are destroyed and the streaked structures are formed in the whole simulation domain after the electrostatic whistler waves are excited and have sufficiently large amplitude. The influences of the initial perpendicular thermal velocity of electrons (via temperature anisotropy) and the drift speed of electron beam on the structures of E-perpendicular to are also analyzed in details. At last, the relevance between our simulation results and the unipolar structures of the parallel cut of E-perpendicular to observed in the auroral region is discussed.