Sunspot Models with Alfvin Wave Emission
Abstract
Sunspot models have been computed on the assumption that the missing flux is transported by undissipated Alfve'n waves. In order to estimate the flux of these waves, we propose an extension of C)pik's cellular model of convection to include the effects of a vertical magnetic field on horizontal gas flow. Horizontal motions, not vertical motions, are impeded more or less severely depending on the electrical conductivity, and this reduces the convective flux. These motions, however, shake the field lines; and this is assumed to be a source of Alfve'n waves, compensating for the reduction in the convective flux. The free parameter D/H (ratio of cell diameter to cell depth) is adjusted such that the total sum of radiative, convective, and Alfve'n wave fluxes remains constant at all depths and equals the undisturbed solar flux entering the spot from below. Our treatment is speculative to some extent, but it leads to a unique spot model for a given field strength. We derive the depthdependence of the effective temperature T, in a spot, and find that it increases nonmonotonically from a low value at the surface (2750 K in a spot with magnetic field B = 3000 gauss) to the solar value (5780 K) at depth z. The latter turns out to lie within 1 percent of the depth of the solar convection zone in all cases of interest. Convection cells in sunspots are found to be narrow cylinders (DIH < 0.2 and decreasing, the stronger the field) aligned along the field lines. The diameter of a cell at the top of a spot with B = 3000 gauss is about 80 km. The depth ZA where the Alfve'n wave flux has a maximum is found to be 841 km in a spot with B = 3000 gauss, and ZA decreases with decreasing B. The values of ZA seem to correspond with the "Wilson depression. Radiative influx of heat from the walls of a spot is found to be minimal for B = gauss, and increases rapidly outside this range. Thus spots with B in this range are expected to live longest, and the most common observed field strengths are expected to lie in this range. If radiative heating is responsible for the lower limit on the permissible field strength in spots (1200 gauss), then there should also be an upper limit on spot field strength at 5300 gauss. Subject headings: hydromagnetics  sunspots
 Publication:

The Astrophysical Journal
 Pub Date:
 February 1974
 DOI:
 10.1086/152672
 Bibcode:
 1974ApJ...187..621M