Nonlinear evolution of the electron two‐stream instability: Two‐dimensional particle simulations

Abstract

We studied nonlinear evolution of the electron two‐stream instability in a two‐dimensional system. Electron two‐stream and bump‐on‐tail instabilities are considered to be the most probable generation mechanisms for electrostatic solitary waves and electron holes observed in various regions of the Earth's magnetosphere. We performed two‐dimensional particle‐in‐cell simulations for various sets of electron cyclotron frequencies and initial electron thermal velocities. We found that the nonlinear evolution falls into four groups. When the electron cyclotron frequency is smaller than the bounce frequency of electrons trapped by electron holes, the electron holes become unstable as reported in previous simulations. When the electron cyclotron frequency is larger than the bounce frequency, the stability of electron holes is controlled by their amplitude. In the case of the cold two‐stream instability where the potential energy of excited electrostatic waves becomes larger than the thermal energy of background electrons, electron holes decay into electrostatic whistler waves. In another case, the two‐stream instability develops to form electron holes in runs with high initial electron thermal velocities. When the electron cyclotron frequency is much larger than the electron bounce frequency, we found the formation of stable one‐dimensional electron holes through coalescence. When the electron cyclotron frequency is smaller than twice the bounce frequency, we found formation of two‐dimensional electron holes isolated in both directions parallel and perpendicular to the ambient magnetic field.