DYNAMICS OF SOLAR CORONAL MAGNETIC-FIELDS

Many aspects of solar activity are believed to be due to the stressing of the coronal magnetic field by footpoint motions at the photosphere. We present the results of a numerical simulation which, to our knowledge, is the first three-dimensional time-dependent simulation of footpoint stressing in a geometry appropriate for the corona. The numerical code generalizes to dissipative, incompressible MHD, a time-split method previously used for Navier-Stokes flows in channel geometry by Zang and Hussaini. We discretize space by a Fourier-Chebyshev collocation method, with a staggered grid used to avoid imposition of pressure boundary conditions. The velocity-vector potential intermediate step is performed with a low-storage third-order Runge-Kutta-Crank-Nicolson scheme. We consider an arcade that is initially current-free and impose a smooth footpoint motion that produces a twist in the field of approximately 2-pi. We then stop the photospheric motion and follow the evolution until the field relaxes, primarily by resistive diffusion, to another current-free state. We see no evidence for any instability, either ideal or resistive, during the complete simulation, and no evidence for current-sheet formation. The most striking feature of the evolution is that, in response to photospheric motions, the field expands rapidly upward to minimize the stress. We conclude that the expansion has two important effects. First, it suppresses the development of "dips" in the field that would be capable of supporting dense, cool material. At least for the motions that we assume, the magnetic field does not develop a geometry suitable for explaining prominence formation. Second, the expansion inhibits ideal instabilities such as kinking. Our results indicate that simple shearing of a single arcade is unlikely to lead to any significant manifestation of solar activity such as flares or prominences. We discuss effects that have not been included in the simulation and may possibly lead to such activity.