Umbral Oscillations in Sunspots: Absorption of p-Modes and Active Region Heating by Mode Conversion

The linear adiabatic oscillations of an infinite, stratified, perfectly conducting atmosphere pervaded by a uniform vertical magnetic field are computed. The stratification is chosen to approximate the conditions appropriate for a sunspot umbra where strong reflection of the fast magnetoacoustic-gravity waves from the rapid increase of the Alfven speed with height and the sound speed with depth is anticipated. Since the umbral oscillations are presumably excited by external p-mode forcing, the angular frequency omega is prescribed -- being set by the p-modes -- and it is required to solve for the allowed eigenvalues assumed by the horizontal wavenumber k and the corresponding displacement eigenfunctions. We term these allowed solutions pi-modes in recognition of the crucial influence of the imposed magnetic field and to distinguish them from their p-mode cousins present in the surrounding nonmagnetic quiet Sun. The wavenumber eigenvalues are complex and are consistent with the spatial decay of the pi-modes inward toward the center of the sunspot from the umbral boundary. This spatial attenuation is a consequence of the slow magnetoacoustic-gravity waves that propagate along the magnetic field lines and extract energy from the trapped fast waves through mode coupling in the layers where the sound and Alfven speeds are comparable. Through the consideration of several ancillary computations we argue that this salient attribute of the pi-modes may be essential both in explaining the observed absorption of (the forcing) p-modes by sunspots and in providing a source of mechanical energy for the overlying active regions. However, more realistic calculations are clearly called for before these notions may be confirmed.