Dominant Effects of Coulomb Collisions on Maintenance of Landau Damping

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Abstract
An analytical criterion for the maintenance, by Coulomb collisions, of the damping of a weakly damped electron wave at the rate given by the theory of Landau when nonlinear effects are present is obtained. For sufficiently small values of ν , the collision frequency, the dominant collisional effects are effects of lowest order in (ν/ka) 1/3 on the linearized distribution function perturbation. The characteristics solution for the resonant component of the linearized distribution function perturbation differs from the collisionless result only in that the factor Re ([−i(kv−ω)] −1 ) is replaced by an integral in which collisional effects are embodied in the parameter κ = ([ν/(ka)] 1/3 /[ Im (−ω) / ka]) . Neglect of the deviation of the damping of the wave from its initial value permits determination of the spatial average distribution function perturbation in the range of time ([ Im (−ω)] −1 )≪t≪ν −1 . The principal contribution involves the resonant component of the linearized distribution function perturbation. The relative change in the slope of the spatial average distribution at v = [ Re (ω) / k] is determined by inspection of the integral containing κ , as well as by approximation of this integral by the method of steepest descents, to be negligible for κ≳O(1) . For this range of κ , therefore, Coulomb collisions can maintain the damping at the rate given by the theory of Landau. The physical mechanism for this effect is the inhibition by Coulomb collisions of the resonant wave‐particle interaction through which nonlinear modification of the spatial average distribution function occurs.