A class of force-free magnetic fields for modeling pre-flare coronal magnetic configurations

Three-dimensional, quasi-static evolutions of coronal magnetic fields driven by photospheric flux emergence are modeled by a class of analytic force-free magnetic fields. Our models relate commonly observed photospheric magnetic phenomena, such as the formation and growth of sunspots, the emergence of an X-type separator, and the collision and merging of sunspots, to the three-dimensional magnetic fields in the corona above. By tracking the evolution in terms of a continuous sequence of force-free states, we show that flux emergence and submergence along magnetic neutral lines in the photosphere are essential processes in all these photospheric phenomena. The analytic solutions we present have a parametric regime within which the magnetic energy attained by an evolving force-free field may be of the order of 10³⁰ ergs to several 10³¹ ergs, depending on the magnetic environment into which an emerging flux intrudes. The commonly used indicators of magnetic shear in magnetogram interpretation are discussed in terms of field connectivity in our models. It is demonstrated that the crossing angle of the photospheric transverse magnetic field with the neutral line may not be a reliable indicator of the magnetic shear in the coronal field above, due to the complexity of three-dimensionality. The poorly understood constraint of magnetic-helicity conservation on the availability of magnetic free energy for a flare is briefly discussed.The National Center for Atmospheric Research is sponsored by the National Science Foundation.