Evolution of the electron phase space of the two-stream and bump-on-tail instabilities in collisional plasma

Particle collisions can have significant effects on plasma instabilities, especially in dense and/or low temperature plasmas. To understand the influence of collisional effects on the plasma waves, the Vlasov-Poisson system with Krook collisions is applied to study the long-term evolution of the two-stream (TS) and bump-on-tail (BOT) instabilities. The system is solved numerically with the fourth-order Runge-Kutta scheme and the Thomas algorithm. It is found that collisions can enhance the wave damping and mitigate the energy of the characteristic slow evolving nonlinear Landau damping oscillations associated with the wave-trapped electrons, especially if the collision rate nu is higher than 0.01 omega(p), where omega(p) is the plasma frequency of the background plasma. Collisions can also decrease the growth rate and saturation level of the TS and BOT unstable waves and tend to shrink the phase space vortex and narrow the phase-mixed region of the trapped electrons. However, our simulations show that collisions cannot readily prevent the nonlinear Landau damping oscillations. In fact, only with v >0.001 omega(p) for the TS instability and v>0.01 omega(p) for the BOT instability, as well as evolution times greater than several hundred omega(-1)(p), the vortex structure of the wave-trapped electrons can be undetectable. The corresponding growth rates also drop dramatically, and the maximum wave energy can be one or two orders lower than that of the collisionless limits.