Fourier-Hermite Solutions of the Vlasov Equations in the Linearized Limit

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Abstract
The properties of the Fourier-Hermite transformation of the one-dimensional Vlasov equation for the motion of electrons against a uniform positive neutralizing background are examined in the linearized limit. The general dispersion relation is presented for the transformed equations. Numerical and analytical results are given which show the recovery of Landau damping for Maxwell's distribution even though the corresponding eigenfrequencies are always real. An example of double streaming is treated to show the appearance of imaginary eigenfrequencies in the analysis. Introduction of a small number of collisions is shown to improve the long-term behavior. Nonlinear numerical results for the nearly cold plasma are shown to agree with the exact treatment of Kalman.