| Magneto-atmospheric waves, which involve the combined restoring forces due to buoyancy, compressibility, and magnetic field, are studied in several model atmospheres permeated by a uniform vertical magnetic field. The results are applied to understand oscillations observed in the umbra of sunspots and to evaluate the possibility of the cooling of sunspots by these waves.; The umbral atmosphere is idealized as a three-layer model with the chromosphere and corona taken as separate isothermal layers and the convection zone taken as a superadiabatic layer with a linear temperature profile. Magneto-atmospheric waves are studied in model atmospheres consisting of different combinations of these three layers.; In the simple isothermal atmosphere, two types of problems are considered: an eigenvalue problem with perfectly conducting, rigid boundaries and a forcing problem in which a harmonic forcing is applied at the base of the atmosphere and outward going waves are considered. In either case, the linearized magneto-atmospheric wave equation is solved numerically using a generalized Newton-Raphson iteration technique. The horizontal velocity component of the wave is found to be trapped by the combination of the frozen-in magnetic field lines and the strong density gradient in the compressible atmosphere. This trapping leads to a set of eigenfrequencies in the eigenvalue problem and a set of resonant frequencies in the forcing problem. Using values for the density, temperature, gravitational acceleration, etc. appropriate for the sun, the lowest eigenfrequency (.043 s('-1)) and the lowest resonant frequency (.041 s('-1)) lie comfortably in the range of frequencies associated with umbral oscillations (.034 s('-1) - .045 s('-1)).; By assuming that the horizontal velocity component is independent of the vertical velocity component, an approximate dispersion relation is found. The resonant frequencies predicted by this "quasi-Alfven" approximation agree closely with the results of the full numerical solutions.; Although the horizontal velocity component is trapped, the vertical velocity component is not completely trapped and a small amount of energy can escape to infinity along the magnetic field lines. However, the vertical velocity is found to be reflected strongly downward by the presence of an upper isothermal layer representing the corona above the sunspot.; In the lower superadiabatic atmosphere, the velocity components of the wave persist to large depths but they are eventually reflected upwards by the compressibility and the increasing sound speed. This, combined with the strong downward reflection of the wave in the upper layers of the atmosphere, shows that it is unlikely that sunspots are cooled by the generation and propagation of magneto-atmospheric waves.; Finally, a more detailed model umbra is constructed, based on several published observational models, and used to determine the depths at which umbral oscillations are excited. The results are in excellent agreement with studies involving the growth of these waves in an overstable convective layer. |
