Comparing electrostatic instabilities driven by mildly and highly relativistic proton beams

An electrostatic instability driven by counter-propagating tenuous proton beams that traverse a bulk plasma consisting of electrons and protons is considered. The system is spatially homogeneous and is evolved in time with a one-dimensional particle-in-cell simulation, which allows for a good statistical plasma representation. Mildly and highly relativistic beam speeds are modeled. The proton beams with a speed of 0.9c result in waves that saturate by the trapping of electrons. The collapse of the phase space holes in the electron distribution scatters these to a flat-top momentum distribution. The final electric fields are weak and the proton beams are weakly modulated. No secondary instabilities are likely to form that could thermalize the proton beams. The proton beams moving with 0.99c initially heat the bulk plasma through a three-wave interaction. Coalescing phase space holes in the bulk proton distribution arising from the saturation of ion acoustic waves transport wave energy to low wavenumbers. Highly relativistic phase space holes form in the electron distribution, which are not spatially homogeneous. The spatial envelope of these electron phase space holes interacts with the fluctuations driven by the phase space holes in the bulk protons, triggering a modulational instability. A Langmuir wave condensate forms that gives rise to strong and long electrostatic wave packets, as well as to a substantial modulation of the proton beams. The final state of the system with the highly relativistic proton beams is thus more unstable to further secondary instabilities that may transfer a larger beam energy fraction to the electrons and thermalize the proton beams more rapidly.