Thick penumbra in a magnetostatic sunspot model

The conjecture that energy transport in sunspot penumbrae occurs through convection by an interchange of magnetic flux tubes is used in order to construct a magnetostatic model for a deep penumbra which is part of a global sunspot model. The model describes an overall mechanical equilibrium between the three stratifications: of the umbra, the penumbra, and of the solar convection zone, all of them being treated as stellar-like envelopes. Local differences between the gas pressure in the neighbouring stratifications are balanced by the magnetic forces confined to two current sheets: the magnetopause and the peripatopause, whose shapes are determined by the solution of a free boundary problem. A self-consistent coupling between global magnetostatic and thermal equilibrium is achieved by the assumption of a perfect thermal isolation of the umbra and a parametrized inflow of the heat into the penumbra. Characteristic parameters observed in sunspots have been imposed as surface constraints for the model. Then only models with a total magnetic flux larger than about 3 1021 Mx turn out to be feasible. Smaller tubes would require more heat flux than is available in the solar convection zone if they contained an extended penumbra. The solutions obtained indicate that the penumbra contains 50-60% of the total magnetic flux of the spot and has a depth comparable with half the radius of the spot. About 60% of the solar heat flux impinging on the magnetopause seems to be transmitted into the penumbra and conveyed to the photosphere by the interchange convection which can be maintained if the average inclination of the magnetic field exceeds about 30 . Such a penumbra forming an extended buffer causes that the umbra widens with height much less rapidly than the whole flux tube. The efficiency of the convective energy transport in the umbra increases with depth and near the base it can be as strong as in the neighbouring convection zone.