Nonlinear Landau damping and collisionless heating in bounded plasmas

Summary form only given. It is well known that linear Landau damping theory is not valid if the bounce frequency of resonant electrons, trapped in the wave, is larger than the linear decrement of wave. In this case (nonlinear Landau damping), pure collisionless heating vanishes after several bounces of resonance electrons in the potential well and the amplitude of the wave becomes stationary. The further decrement of the wave is only possible if rare collisions are taken into account. The decrement of nonlinear Landau damping has been calculated analytically taking electron collisions with neutrals into account. The approximation of analytical calculation for decrement of nonlinear Landau damping gives, within error less than 5%. /spl gamma//sub nl/=/spl gamma//sub l/ tanh(/spl nu//spl tau/), (1) where /spl gamma//sub l/ is the linear Landau damping, /spl nu/ is the total collision frequency, /spl tau/ is the bounce time of trapped electrons in the wave potential well. It follows from Eq. (1) that the nonlinear damping coincides with the linear damping for /spl nu//spl tau//spl Gt/1. In the opposite case /spl nu//spl tau//spl Lt/1, the nonlinear damping tends to zero with /spl nu/. The obtained result is applied to calculation of collisionless heating in a bounded plasma. As a practical example, the low-pressure radio-frequency gas discharges have been chosen. A comparison of analytical results and Monte Carlo simulations is presented. The importance of accounting for the nonuniform plasma density profile for computing the current density profile and the electron energy distribution function is demonstrated.