Shocks and phase space vortices driven by a density jump between two clouds of electrons and protons

We study with 1D PIC simulations the expansion of a dense plasma into a dilute one for density ratios 2.5 ≤ α ≤ 20. Both are unmagnetized and consist of electrons and protons. Shocks form in all cases. We determine how α affects the speed of the shock, that of the trailing velocity plateau and the proton beam instabilities in its upstream region. The speed of the velocity plateau relative to the upstream plasma increases significantly with α. Faster shocks reflect more upstream protons and fewer protons make it downstream, which slows down the shock in the downstream frame. This slow-down reduces noticably the increase with α of the shock speed in the upstream frame. All simulations demonstrate that an ion acoustic instability develops between the shock-reflected proton beam and the ambient protons. We solve the linear dispersion relation for ion acoustic waves that have wave vectors which are parallel to the beam velocity vector. Upstream conditions, for which their growth rate is largest, lead to the most unstable upstream plasmas also in the simulation. Even though linear theory predicts the growth of sine waves, which reach a small amplitude in the simulations, solitary waves become the dominant ones upstream of the shock. They enforce the formation of new shocks and ion phase space vortices. We discuss the relevance of our findings to laser-plasma experiments.