Streaming instabilities driven by mildly relativistic proton beams in plasmas

The interaction between relativistic ion beams,electrostatic waves and particles in plasmas is studied numerically. This beam-particle interaction is important for the acceleration of electrons above the threshold, where they can perform Fermi acceleration across shocks perpendicular to the magnetic field direction. The electrons form Bernstein–Greene–Kruskal (BGK) modes, which are unstable due to a sideband instability. This process heats the electrons to relativistic temperatures and the electrons finally attain a flat-top momentum distribution with exponentially decreasing tails of the momentum distribution function. A scaling law, depending on the ion beam speed, is found for the electrondistribution function. Furthermore, in the fully nonlinear phase, ion phase space vortices, or proton BGK modes are formed in the beam, which continue to interact with the electrons after the electron BGK modes have collapsed.