A phenomenological model for the interaction of solar p-modes with active regions

We adopt a phenomenological approach to study the interaction of solar p-mode waves with magnetic regions. We assume that the interaction can be described by a complex index of refraction. The modification of the index of refraction due to magnetic fields is defined as a complex interaction parameter, which results in a change of the p-mode wave amplitude and phase shift. The effect of the complex index of refraction appears as a source term in the wave equation, which is a function of the interaction parameter. The wave equation with the source is solved with a method of Green's functions and the Born approximation to obtain the scattered waves, if the interaction parameter is small. In general, an incident mode would be converted into the degenerate modes, which have the same frequency as the incident mode but different wavenumbers, after interacting with a magnetic region. Since the incident waves consist of many modes, there is coupling between different incident modes through the interaction with the sunspot. The coupling is the greatest for the adjacent modes. The range of coupling depends on the distribution of sigma. This mode coupling would complicate the calculation of the absorption coefficient and phase shift of each mode from a model, since all incident modes within the coupling range would contribute to the outgoing mode. The mode coupling would also make the comparison of the computed absorption coefficients and phase shifts with the observed values difficult. For uniform axisymmetric sunspots, the mode coupling is small, and the absorption coefficient and phase shift of each mode can be calculated with a given interaction parameter. We use a solar model to calculate absorption coefficients and phase shifts for several horizontally uniform axisymmetric sunspots. The comparison of observed values with computed values suggests that (1) the horizontal size of the regions responsible for the absorption and phase shift does not change much with depth (2) the overall size of an active region does not change much with depth, though the size of each small flux tube might decrease with depth over a few pressure scale heights, and (3) the regions responsible for the absorption and phase shift probably have a depth of about 15-40 Mm.