Magnetic-field effects on nonlinear electrostatic-wave Landau damping

A numerical code, which solves the Vlasov-Poisson system of equations for an electron magnetized plasma with motionless ions, is presented. the numerical integration of the Vlasov equation has been performed using the "splitting method" and the cylindric geometry in the velocity space is used to describe the motion of the particles around the external field. The time evolution of an electrostatic wave,, propagating perpendicularly to the background magnetic field, is numerically studied in both the linear and nonlinear regimes, for different values of the ratio gamma between, the electron oscillation time in a sinusoidal potential well and the electron cyclotron period. It is shown that the external magnetic field plays very different roles, depending on the values of the initial wave amplitude. When the initial amplitude is less than some threshold, the magnetic field prevents the Landau damping of the electrostatic wave (Bemstein-Landau paradox). When the wave amplitude is above the threshold, for intermediate values of gamma the presence of a background magnetic field allows for the electric energy dissipation at variance with the behavior of electrostatic wave in unmagnetized plasma, while for high gamma values once again the magnetic field prevents the damping.