Slow electrostatic fluctuations generated by beam-plasma interaction
  Eulerian simulations of the Vlasov-Poisson equations have been employed to analyze the excitation of slow electrostatic fluctuations (with phase speed close to the electron thermal speed), due to a beam-plasma interaction, and their propagation in linear and nonlinear regime. In 1968, O'Neil and Malmberg [Phys. Fluids {\bf 11}, 1754 (1968)] dubbed these waves "beam modes". In the present paper, it is shown that, in the presence of a cold and low density electron beam, these beam modes can become unstable and then survive Landau damping unlike the Langmuir waves. When an electron beam is launched in a plasma of Maxwellian electrons and motionless protons and this initial equilibrium is perturbed by a monochromatic density disturbance, the electric field amplitude grows exponentially in time and then undergoes nonlinear saturation, associated with the kinetic effects of particle trapping and phase space vortex generation. Moreover, if the initial density perturbation is setup in the form of a low amplitude random phase noise, once the most unstable mode has reached its nonlinear saturation amplitude after the linear growth, the whole Fourier spectrum of wavenumbers is excited. As a result, the electric field profile appears as a train of isolated pulses, each of them being associated with a phase space vortex in the electron distribution function. At later times, these vortical structures tend to merge and, correspondingly, the electric pulses collapse, showing the tendency towards a time asymptotic configuration with a single phase space structure associated to an electric soliton-like pulse. This dynamical evolution is driven by purely kinetic processes, possibly at work in many space and laboratory plasma environments. 
Karen Pommois, Francesco Valentini, Oreste Pezzi and Pierluigi Veltri
physics.plasm-ph