Nonlinear and Collisional Effects on Landau Damping

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Abstract
In the collisionless case, nonlinear effects considerably reduce Landau damping after times of the order of the period of oscillations of trapped electrons. A perturbation analysis of the nonlinear Vlasov equation, modified by the addition of a simplified Fokker‐Planck collision term, is carried out following Montgomery's expansion technique. The modified Landau poles for Maxwellian plasmas are determined. The first‐order number density and the second‐order spatially homogeneous distribution function are evaluated. A numerical solution of the same equation, using the method of velocity Fourier transforms, is also presented and the results of the two methods are compared. Several values of the collision parameter β, wavelength λ, and initial amplitude ε are considered, showing the competing effects of nonlinearities and collisions on the formation of a plateau in the spatially homogeneous distribution function. Values of β which prevent the formation of the plateau and maintain Landau damping close to the value prescribed by the linear theory are determined.