Alfvén ‘resonance’ reconsidered: Exact equations for wave propagation across a cold inhomogeneous plasma

Previous discussions of Alfvén wave propagation across an inhomogeneous plasma predicted that shear Alfvén waves become singular (resonant) at the ω=kzvA layer and that there is a strong wave absorption at this layer giving localized ion heating. In this paper the three standard derivations of the Alfvén ‘resonance’ (incompressible magnetohydrodynamics, compressible magnetohydrodynamics, and two-fluid) are re-examined and shown to have errors and be mutually inconsistent. Exact two-fluid differential equations for waves propagating across a cold inhomogeneous plasma are derived; these show that waves in an ideal cold plasma do not become ‘resonant’ at the Alfvén layer so that there is no wave absorption or localized heating. These equations also show that the real ‘shear’ Alfvén wave differs in substance from both the ideal MHD and earlier two-fluid predictions and, in the low density, high field region away from the ω=kzvA layer, is actually a quasielectrostatic resonance cone mode. For ω≪ωci and ky=0, the ω=kzvA layer turns out to be a cutoff (reflecting) layer for both the ‘shear’ and compressional modes (and not a resonance layer). For finite ω/ωci and ky=0 this layer becomes a region of wave inaccessibility. For ω≪ωci and finite ky there is strong coupling between shear and compressional modes, but still no resonance.