Sunspot models with Alfven wave emission

Sunspot models have been computed on the assumption that the missing flux is transported by undissipated Alfven waves. In order to estimate the flux of these waves, we propose an extension of Opik'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; this is assumed to be a source of Alfven 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 Alfven 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.