GENERALIZED CASE-VANKAMPEN MODES IN A MULTIDIMENSIONAL NONUNIFORM PLASMA WITH APPLICATION TO GYRORESONANCE HEATING

The generalization of the Case-van Kampen analysis to a multidimension al non-uniform plasma is presented. Application of this analysis is ma de to minority-ion gyroresonant heating in an axisymmetric tokamak. In previous work the Case-van Kampen analysis, in conjunction with the B ateman-Kruskal algorithm, was used in a one-dimensional slab model to compute the collective wave spin-off (to the minority-ion Bernstein wa ve) and the gyroballistic continuum for minority gyroresonant absorpti on, The generalization to many dimensions and non-trivial geometries r equires several important new developments: In tokamak geometry partic les can be trapped: an effect that is absent in the slab model. Also, the ray propagation dynamics for both the free gyroballistic waves and the collective minority-ion Bernstein wave is far more complicated th an in tile slab model, In particular, a resonance zone is identified w herein the gyroboallistic waves interact strongly and cannot be treate d as free, We use tile Weyl calculus to construct a local form of the self-consistent gyroballistic equation within the resonance zone. This reduced equation is: simplified via a metaplectic transformation (a g eneralization of the Fourier transformation). After this simplificatio n, the equation is shown to be of Case-van Kampen type with weak non-u niformities; hence there are no true Case-van Kampen eigenfunctions. U sing the Bateman-Kruskal approach, a local Case-van Kampen basis can b e constructed and the initial-value problem solved. The self-consisten t inter actions of the gyroballistic continuum lead to a collective wa ve, the minority-ion Bernstein wave, The Bernstein wave is extracted b y the spectral deformation approach of Crawford and Hislop, The releva nce of this work to the theory of collective phenomena in nonlinear os cillator ensembles is briefly discussed.