The cooling of a sunspot

This paper considers the recent criticism by Mullan (1973) of sunspot models and the cooling mechanism which I have proposed in Papers I, II and III of this series. The discussion of the cooling produced by an idealized flow cycle has been extended to include vertical temperature gradients which are consistent with a convectively unstable atmosphere. This leads to an expression for Mullan's parameter f (the ratio in which estimates of the energy flux based on an idealized Carnot cycle should be reduced) which is appropriate to this situation. It is shown that, for a cycle similar to that of Paper III, f = 0.82, while for one which has a vertical extent of order 5 Mm, f= 0.4. Hence the energy flux which, in principle, can be transported away from a sunspot by such a cycle is conservatively estimated to be 1.1 × 10²⁹ erg s^?1 compared with a typical sunspot energy deficit of 2.2 × 10²⁹ erg s^?1. Other criticisms relating to the magnetic field amplification and the ‘cool one’ model are discussed. It is concluded that the essential features of these models remain valid and that the modifications suggested by Mullan's criticism greatly increase their applicability to the sunspot problem.