Resonant oscillations in sunspot umbrae

The Uchida and Sakurai (1975), Thomas and Scheuer (1982), and Scheuer and Thomas (1981) theory of umbral oscillations as resonant modes of magneto-acoustic-gravity waves is re-examined. For an isothermal atmosphere it is found that the quasi-Alfvén approximation is not a good approximation to the complete linearized wave equations. The new results presented here show that 3 min umbral oscillation periods are fairly insensitive to magnetic field strength above some critical value. For a detailed model umbra (Thomas and Scheuer, 1982) the calculations presented here show that 3 min umbral oscillations do not depend to any great extent on the level of forcing of the oscillations for those magnetic field strengths which are observed in sunspot umbras. Modes outside the 3 min range appear, as the lowest mode, as the level of forcing is placed at deeper and deeper levels in the solar atmosphere.