Umbral oscillations in the presence of a spreading magnetic field

A simple umbral model incorporating spreading magnetic field lines is introduced, and the equations governing Magneto-Acoustic-Gravity (MAG) and Alfvén modes are derived. In the absence of stratification analytic solutions may be found. For the MAG modes, under coronal conditions, these consist of a travelling (slow) wave and an evanescent (fast) mode, whilst the Alfvén solution may be expressed in terms of a Hankel function. These are matched onto numerical solutions in a stratified model photosphere-chromosphere, and the vertical energy fluxes are calculated. Resonant frequencies are generally found to be shifted downwards compared with the vertical field case, though the MAG flux is virtually unaltered otherwise; however, the Alfvén flux is increased by up to two orders of magnitude. In an unstratified medium, the usual formula for the Alfvén flux, F = ρ{variant}a 〈δv²〉, is shown to be generally incorrect in the presence of field spread, and must be replaced by F = N^-2/3(4π)^-1ρ{variant}a 〈δv²〉, where N is a spread parameter which usually satisfies N ≪ 1.