Oscillations and waves in anisotropic plasmas

The linearized Vlasov-Maxwell equations describing anisotropic plasma oscillations and waves are studied using an operator theoretic approach. The model considered is one dimensional so that after velocity averages perpendicular to this direction. have been taken, the equations can be naturally grouped into one set of equations for longitudinal modes and another set of equations for transverse modes.

The problems of longitudinal and transverse plasma oscillations are studied by Fourier transforming the equations in the space variable and analyzing the resulting operator equations using the theory of semigroups. Existence and uniqueness theorems are proved, and solutions are constructed by the resolvent integration technique. The solutions are put into the form of a generalized eigenfunction expansion with eigenmodes corresponding to zeros of the appropriate plasma dispersion function. The expansion coefficients for eigenmodes corresponding to simple and second order real zeros of the plasma dispersion function are also presented, and constitute some of the new results obtained by our analysis.

Existence and uniqueness of the solution to the longitudinal plasma wave boundary value problem is proved by writing the longitudinal equations in operator form and again using the theory of semigroups. The solution to the plasma wave boundary value problem is arrived at by a Fourier time transformation of the Vlasov equation coupled to Ampere's Law rather than Gauss‘ Law, and analyzing a scalar operator as opposed to the more complicated matrix operator that has previously been studied. Special care is used in constructing the half range transport operator whose resolution of the identity yields the solution in the form of a half range generalized eigenfunction expansion where again, new results are presented for the expansion coefficients for eigenfunctions corresponding to simple and second order real zeros of the fixed frequency longitudinal plasma dispersion function.

Since this study is concerned with anisotropic plasmas, a non-even plasma equilibrium distribution function is assumed with the direct result that more stable and unstable plasma modes corresponding to real and complex zeros of the plasma dispersion function are possible that has previously been considered. Also, for the longitudinal plasma wave problem, the Wiener-Hopf factorization of the fixed frequency longitudinal plasma dispersion function is presented and the coupled nonlinear integral equations for the Wiener-Hopf factors are studied. These Wiener-Hopf factors are required in the construction of the half range transport operator.