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Radiative cooling of a hot flux tube in the solar photosphere
J.H.M.J. Bruls and
Received 7 June 1999 / Accepted 21 July 1999
Radiative energy transport is of key importance for the dynamics of slender magnetic flux tubes in the solar atmosphere, particularly so in connection with the filamentation of the sunspot penumbra. In investigations using the thin-flux-tube approximation of the MHD equations, the radiative exchange with the surrounding atmosphere has hitherto been described by the relaxation-time approach, also called `Newton's law of cooling'. The strongly nonlinear temperature-dependence of the radiative absorption coefficient and large temperature differences between the tube and its environment render this concept questionable. As a simple model of a bright penumbral filament we consider the cooling of a hot horizontal flux tube with a longitudinal flow, embedded in a non-stratified, homogeneous atmosphere at 4 800 K. We compare the results of the relaxation-time approach and of a nonlinear diffusion approximation with the numerical solution of the equation of (grey) radiative transfer. We find that the cooling times given by the relaxation-time method compare well with the results from radiative transfer as long as the initial temperature of the tube is below 7 500 K and its lateral optical depth does not exceed unity. Under these conditions, the tube cools more or less homogeneously over its cross section. For hotter and optically thick tubes, the strong temperature-dependence of the absorption coefficient leads to the formation of a cooling front, which migrates radially inward at approximately constant speed. Such inhomogeneous cooling is well represented by the nonlinear diffusion approximation. The self-similar evolution of the cooling front permits an analytical estimate of the cooling time, which provides a reasonable approximation of the result of the radiative transfer calculation. This estimate can be used to derive an improved radiative cooling term in the framework of the thin-flux-tube approximation, so that both optically thin and optically thick flux tubes can be treated adequately. The results of the radiative transfer calculations are applied to obtain an estimate of the length and brightness of penumbral bright grains.
Key words: Sun: photosphere Sun: sunspots radiative transfer diffusion
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Online publication: September 13, 1999