Congruent reduction in geometric optics and mode conversion

Abstract

Standard eikonal theory reduces, to N=1, the order of the system of equations underlying wave propagation in inhomogeneous plasmas. The condition for this remarkable reducibility is that only one eigenvalue of the unreduced N×N dispersion matrix D(k,x) vanishes at a time. If, in contrast, two or more eigenvalues of D become simultaneously small, the geometric optics reduction scheme becomes singular. These regions are associated with linear mode conversion and are described by higher-order systems. A new reduction scheme is developed based on congruent transformations of D, and it is shown that, in degenerate regions, a partial reduction of order is still possible. The method comprises a constructive step-by-step procedure, which, in the most frequent (doubly degenerate) case, yields a second-order system, describing the pairwise mode conversion problem in four-dimensional plasmas. This N=2 case is considered in detail, and dimensionality arguments are used in studying the characteristic ordering of the elements of the reduced dispersion tensor in mode conversion regions. The congruent reduction procedure is illustrated by classifying pairwise degeneracies in cold multispecies magnetized plasmas.