Magnetic-field structures in solar active regions are expected to be substantially force-free. A method is proposed for calculating such structures by numencal methods. The method is applied to the study of the magnetic-field pattern associated with a sunspot of one polarity surrounded by a magnetic region of opposite polarity when the sunspot rotates with respect to the surrounding region. Rotation introduces a toroidal component of magnetic field, and the associated pressure leads to inflation of the magnetic-field pattern. If the differential rotation exceeds about 1S0 , the force-free magnetic field has energy greater than that of an open magnetic-field configuration with the same photospheric boundary conditions. It is concluded that, beyond this point, the force-free field structure is metastable and can be converted into an open field structure by an explosive MHD instability. The resulting field pattern is then of the form previously proposed as the site of high-energy solar flares.
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Barnes et al. (1972) studied this question.