An improved asymptotic matching technique to trace the wave amplitude of rays across singularities: Application to lower-hybrid wave propagation in tokamaks

A most persistent limitation of the geometrical-optics (GO) approximation is the difficulty in integrating the focusing/defocusing term in the equation for the wave amplitude when rays go through singularities (i.e., caustics and cutoffs), points where GO fails and wavelengths and other wave field-related quantities (e.g., the wave amplitude and energy density) become arbitrarily large. A new asymptotic matching (AM) technique is thus presented which allows one to recover the wave amplitude of rays crossing singularities and which improves on a previous approach [A. H. Glasser and A. Bravo-Ortega, Phys. Fluids 30, 797 (1987)]: it goes higher in the order of the asymptotic expansion about the singular point and eventually leads to a less critical violation of the GO ordering when a ray approaches and crosses a caustic or a cutoff. The implementation of this new AM technique is verified and validated against analytical solutions in slab geometry for the lower-hybrid (LH) cutoff and is illustrated with numerical examples of LH wave propagation in a tokamak plasma, using parameters characteristic of a LH current drive experiment. The new and previous approaches are compared regarding several computed ray quantities (e.g., the second derivatives of the eikonal phase, which contribute to the focusing/defocusing term, as well as the wave amplitude and energy density), the inclusion of more terms in the asymptotic expansion about the singularity making it possible to start, say, jumping the latter in a region where GO is less severely challenged. Practical criteria for AM implementation and for testing the validity of GO and of the asymptotic expansion are also provided and discussed.