A New Normal Form for Multidimensional Mode Conversion

Abstract

Linear conversion occurs when two wave types, with distinct polarization and dispersion characteristics, are locally resonant in a nonuniform plasma [1]. In recent work, we have shown how to incorporate a ray‐based (WKB) approach to mode conversion in numerical algorithms [2,3]. The method uses the ray geometry in the conversion region to guide the reduction of the full N×N‐system of wave equations to a 2×2 coupled pair which can be solved and matched to the incoming and outgoing WKB solutions. The algorithm in [2] assumes the ray geometry is hyperbolic and that, in ray phase space, there is an ‘avoided crossing’, which is the most common type of conversion. Here, we present a new formulation that can deal with more general types of conversion [4]. This formalism is based upon the fact (first proved in [5]) that it is always possible to put the 2×2 wave equation into a ‘normal’ form, such that the diagonal elements of the dispersion matrix Poisson‐commute with the off‐diagonals (at leading order). Therefore, if we use the diagonals (rather than the eigenvalues or the determinant) of the dispersion matrix as ray Hamiltonians, the off‐diagonals will be conserved quantities. When cast into normal form, the 2×2 dispersion matrix has a very natural physical interpretation: the diagonals are the uncoupled ray hamiltonians and the off‐diagonals are the coupling. We discuss how to incorporate the normal form into ray tracing algorithms.