Magneto-atmospheric oscillations in a model umbra

The behaviour of magneto-acoustic-gravity oscillations in a model umbra is investigated by means of a normal mode analysis. The normal modes are evaluated for different types of boundary conditions holding at various levels in the model atmosphere and for a range of magnetic field strengths and horizontal wave numbers. For certain boundary conditions complex eigenfrequencies are present. For atmospheric cavities whose lower boundary is located at the bottom of the photosphere or in the upper convective region, the coupling between horizontal and vertical motions leads to spectra which are sensitive to the value of the horizontal wave number, k, and the strength of the vertical magnetic field, B-0. For cavities located above the temperature minimum, the eigenfrequencies are independent of both k and B-0, supporting the conclusions of Zugzda, Locans, and Staude (1987) that the motion is longitudinal along the magnetic field lines. The frequencies evaluated from the normal mode analysis correspond closely to the observed 2-3-minute band found in sunspot umbrae.