Introduction

As already mentioned in earlier chapters, the eikonal approximation can become invalid in local regions of the plasma. The most common problems are caustics (see Chapter 5), tunneling, and mode conversion. Both tunneling and mode conversion are processes where one incoming ray splits into two outgoing rays, a transmitted ray and a converted ray. The matched asymptotic methods are therefore more complicated than for caustics. Tunneling concerns only one eigenvalue of the N × N dispersion matrix, while mode conversion entails two. It follows that tunneling involves only one polarization, while mode conversion is associated with a pair. Therefore, tunneling can be reduced by Galerkin projection locally to a scalar formulation, while mode conversion is inherently a vector problem. An important point we should emphasize is the following: For caustics, it is always possible to find a local representation where the eikonal approximation is valid. In contrast, in tunneling and mode conversion regions, there is no representation in which the eikonal approximation is valid. It is only when we consider points far from the conversion region that we recover eikonal behavior. This leads to two important questions:

1. If the eikonal approximation is not valid within the conversion region, why persist in using ray tracing there?

2. Although the eikonal approximation is valid for the incoming wave field (by assumption), what justifies the assumption that the transmitted and converted wave fields become eikonal once more?