Linear mode conversion and singularities of geometric optics approximation in plasmas

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Abstract
The general, unreduced, one‐dimensional, linear wave propagation problem in weakly inhomogeneous plasmas is considered. Conventional geometric optics perturbation expansion reduces this high‐order, multimode problem to the solution of a first‐order ordinary differential equation for the wave amplitude along the rays given by εi=0, where εi is one of the eigenvalues of the generalized tensor E characterizing the problem. In some plasma regions, however, this perturbation expansion, based on following only one mode at a time, fails, predicting fast variation of the zero‐order amplitude and local wave vector k of the wave, or leading to large first‐order corrections to the amplitude. Typically, in this region either (a) an additional eigenvalue of E becomes small and couples a new mode into the problem, or (b) all the eigenvalues except εi remain large; however, ∂εi/∂kx→0, x being the direction of the inhomogeneity. Following an earlier work on case (a), a renormalization technique is employed in formulating a consistent, nonsingular, eikonal type perturbation expansion in case (b). The method yields a single, second‐order, ordinary differential equation for amplitude of the wave, in contrast to case (a), more naturally described by a system of two first‐order equations for the amplitudes of the coupled modes. Both theories thus comprise a complete general description of pairwise, linear mode conversion in the aforementioned singular regions. The approach is illustrated by the example of wave interaction in a cold magnetized plasma with plane parallel stratification. The importance of studying the unreduced problem is demonstrated by showing that in case (a) the information on the type of singularity can be lost by formulating the problem in terms of the conventional, reduced dielectric tensor.