Surface waves and the heating of the corona

Abstract

Abstract The Sun's atmosphere is highly structured by magnetic fields, and this modifies the nature of the usual Alfvén waves which propagate in an everywhere uniform medium. Two simple magnetic field structures are considered: the interface profile (for which a discontinuity separates two media with different, but uniform, unidirectional magnetic fields), and the monotonic profile (for which the magnetic field is unidirectional and continuous but not everywhere constant). An initial-value problem is formulated in ideal MHD, and the perturbations from equilibrium are written in terms of a Bromwich integral. It is well known that the poles of the Bromwich kernel lead to the usual discrete normal mode solutions, and the branch-cuts of the kernel lead to a continuum. An analysis reveals that the interface profile supports Alfvén surface and body waves, whilst the monotonic profile allows only a continuous spectrum of waves. However, by suitably deforming the contour, spectral poles (not related to normal modes) can be located off the physical (principal) Riemann sheet. The residues from these poles yield solutions resembling the interfacial surface waves, but they decay on a timescale T. This property of temporal decay in a non-dissipative system has led lonson (1978) and Wentzel (1979b) to suggest that the spectral residues show the behaviour of a surface wave when the interface is made continuous, and they have interpreted T as the time-scale for dissipation of such a wave in a non-ideal medium. Here we argue that, on the contrary, the time T is in fact the scale for the build-up of the continuous spectrum of oscillations; the spectral residues indicate restructuring of wave motions. It is shown that the process of phase-mixing causes motions along the magnetic field to build-up, whilst those transverse to the field decay. However, the status of wave propagation in a highly structured medium as a means of heating the corona, by dissipating through irreversible processes in non-ideal MHD, remains open until a fully dissipative investigation of their character becomes available.