Linear Time-Dependent Vlasov Equation; Case-Van Kampen Modes

The multiple-time-scale method is used to find a succinct solution for the linearized Vlasov equation when the background state of the plasma is a slowly varying function of time. It is found to be necessary to consider normal modes of the equation because the slow and fast time variables synchronize separately for each mode. The asymptotic series for each normal mode and for its associated frequency are computed to all orders in the expansion parameter. The lowest-order results are the Case-Van Kampen modes in WKB form. The corrections represent a diagonalization of the results of Baldwin, Trocheris, and Imre. In particular, the dispersion function is corrected by higher-order terms and the frequencies of the discrete spectrum have complex-valued shifts.