Generalized Case—van Kampen modes in a multidimensional non-uniform plasma with application to gyroresonance heating

The generalization of the Case—van Kampen analysis to a multidimensional non-uniform plasma is presented. Application of this analysis is made to minority-ion gyrolesonant heating in an axisymmetric tokamak. In previous work the Case—van Kampen analysis, in conjunction with the Bateman—Kruskal algorithm, was used in a one-dimensional slab model to compute the collective wave spin-off (to the minority-ion Bernstein wave) and the gyroballistic continuum for minority gyroresonant absorption. The generalization to many gyroresonant dimensions and non-trivial geometries requires several important new developments: In tokamak geometry particles can be trapped, an effect that is absent in the slab model. Also, the ray propagation dynamics for both the flee gyroballistic waves and the collective minority-ion Bernstein wave is far more complicated than in the slab model. In particular, a resonance zone is identified wherein the gyroballistic waves interact strongly and cannot be treated as free. We use the 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 generalization of the Fourier tiansformation). After this simplification, the equation is shown to be of Case—van Kampen type with weak non-uniformities; hence there are no true Case—van Kampen eigenfunctions. Using the Bateman—Kruskal approach, a local Case—van Kampen basis can be constructed and the initial-value problem solved. The self-consistent interactions of the gyroballistic continuum lead to a collective wave, the minority-ion Bernstein wave. The Bernstein wave is extracted by the spectral deformation approach of Crawford and Hislop. The relevance of this work to the theory of collective phenomena in nonlinear oscillator ensembles is briefly discussed.