The fine structure features of the umbral magnetic fields (i.e. large field gradients and changes of polarity in local regions) are considered as evidence of strong non-linear interactions between magnetic and thermal or mechanical forces in the umbra. It is suggested that the umbral dots observed in white light are the optical manifestation of these interactions. A three-dimensional radiative transfer analysis of possible models for these bright features is discussed and this enables one to

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... ce limits on the geometry of these features and on the non-radiative energy requirements of the models. Of the several models considered those which were compatible with convection as the source of this energy were found to be quite inconsistent with the available data. The most likely model was found to have a diameter of 200 km, a height of 50 km and an average emission of non-radiative energy of 4 × 103erg s−1/cm−3throughout this region. It is shown that this is two orders of magnitude greater than the energy available from Joule heating by locally twisted magnetic fields. However, if the energy flux transported through the umbra by Alfvén waves is partially dissipated in regions of locally twisted fields it is shown that the emission into a volume of the above dimensions is of the right order of magnitude.