Browsing by Author "Arthur, Michael D."
Now showing 1 - 3 of 3
Results Per Page
Sort Options
- Oscillations and waves in anisotropic plasmasArthur, Michael D. (Virginia Tech, 1979-05-05)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.
- Transverse plasma oscillationsArthur, Michael D.; Greenberg, William; Zweifel, Paul F. (AIP Publishing, 1979)An operator theoretic approach is used to solve the linearized Vlasov–Maxwell equations for transverse plasma oscillations. In particular, the special cases of simple and second‐order real zeros of the plasma dispersion function are treated and formulae for the amplitude of the plasma waves are presented. An existence and uniqueness theorem for the solution to the Vlasov–Maxwell transverse mode plasma equation is proved in an appendix. In a second appendix, a general characterization for the zeros of the plasma distribution function is presented for the case of any double humped equilibrium distribution.
- Vlasov theory of plasma oscillations: linear-approximationArthur, Michael D.; Greenberg, William; Zweifel, Paul F. (AIP Publishing, 1977)A functional analytic approach to the linearized collisionless Vlasov equation is presented utilizing a resolvent integration technique on the resolvent of the transport operator evaluated at a particular point. Formulae for the eigenfunction expansion are found for cases in which the plasma disperion function _ has first and second order zeroes. Special care is taken in the study of real zeroes of _ culminating in new results for this case. For a simple zero of _ with nonvanishing imaginary part the van Kampen-Case discrete modes are reproduced. The results are used to obtain the solution to the initial value problem.