MAG waves in sunspots umbrae: Slow waves leaking to the corona

The linear oscillations of a stratified atmosphere embedded in a uniform vertical magnetic field are studied here. We use a simple theoretical model, formed by the superposition of two isothermal layers, representing, respectively, i) the photosphere and the chromosphere, and ii) the corona. The bottom layer behaves, for some modes, as a resonant cavity where MAG waves are semi-trapped. We find the existence of two types of modes: 1) Fast modes which are trapped below the transition layer, 2) Mixed modes which are resonant modes in the first layer and leak part of the energy to the corona. These mixed modes have been found to be damped in the horizontal direction and can explain the observed slow modes in the corona.