Thermal properties of magnetic flux tubes I. Solution of the diffusion problem

The heat flow and temperature structure within and surrounding a magnetic flux tube stored in mechan- ical equilibrium in a stellar convection zone are considered. The stationary thermal equilibrium state is determined through the analytical solution of a two-dimensional heat diusion problem for an innitely long cylinder with dierent thermal conductivities inside and outside the cylinder, both spatially variable. In the exterior of the cylinder, convective heat transport is approximated in terms of a linear diusive process, while in its interior convection is assumed to be suppressed and only the much smaller radiative conductivity remains. The results show that, under the conditions prevailing near the bottom of the solar convection zone and in the limit of small cylinder radius, the temperature disturbance (thermal shadow) in the exterior of the insulating cylinder is almost negligible due to the large eency of convective energy transport. The spatial dependence of the conductivities and the curvature of the external temperature prole lead to a temperature excess in the interior with respect to the undisturbed temperature prole far away from the cylinder. We show that, within the framework of the thin magnetic flux tube approximation, this temperature excess is due to a heating term equal to the negative divergence of the undisturbed radiative heat flow, as suggested earlier by Fan & Fisher (1996). These results are independent of the treatment of the convective transport in the exterior as long as the stratication is almost adiabatic. The consequences for the storage of magnetic flux in the solar convection zone, brought about by the enhanced buoyancy and caused by the heating eect, are discussed.