UMBRAL OSCILLATIONS IN SUNSPOTS - ABSORPTION OF P-MODES AND ACTIVE-REGION HEATING BY MODE CONVERSION

The linear adiabatic oscillations of an infinite, stratified, perfectl y conducting atmosphere pervaded by a uniform vertical magnetic field are computed. The stratification is chosen to approximate the conditio ns appropriate for a sunspot umbra where strong reflection of the fast magnetoacoustic-gravity waves from the rapid increase of the Alfven s peed with height and the sound speed with depth is anticipated. Since the umbral oscillations are presumably excited by external p-mode forc ing, the angular frequency omega is prescribed-being set by the p-mode s-and it is required to solve for the allowed eigenvalues assumed by t he horizontal wavenumber kappa and the corresponding displacement eige nfunctions. We term these allowed solutions pi-modes in recognition of the crucial influence of the imposed magnetic field and to distinguis h them from their p-mode cousins present in the surrounding nonmagneti c quiet Sun. The wavenumber eigenvalues are complex and are consistent with the spatial decay of the pi-modes inward toward the center of th e sunspot from the umbral boundary. This spatial attenuation is a cons equence of the slow magnetoacoustic-gravity waves that propagate along the magnetic field lines and extract energy from the trapped fast wav es through mode coupling in the layers where the sound and Alfven spee ds are comparable. Through the consideration of several ancillary comp utations we argue that this salient attribute of the pi-modes may be e ssential 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.