Regularized Biot-Savart Laws for Modeling Magnetic Configurations with Flux Ropes

Many existing models assume that magnetic flux ropes play a key role in solar flares and coronal mass ejections (CMEs). It is therefore important to develop efficient methods for constructing flux-rope configurations constrained by observed magnetic data and the initial morphology of CMEs. For this purpose, we have derived and implemented a compact analytical form that represents the magnetic field of a thin flux rope with an axis of arbitrary shape and a circular cross-section. This form implies that the flux rope carries axial current I and axial flux F, so that the respective magnetic field is the curl of the sum of toroidal and poloidal vector potentials proportional to I and F, respectively. We expressed the vector potentials in terms of modified Biot-Savart laws whose kernels are regularized at the axis in such a way that these laws define a cylindrical force-free flux rope with a parabolic profile of the axial current density, when the axis is straight. For the cases we have studied so far, we determined the shape of the rope axis by following the polarity inversion line of the eruptions' source region, using observed magnetograms. The height variation along the axis and other flux-rope parameters are estimated by means of potential field extrapolations. Using this heuristic approach, we were able to construct pre-eruption configurations for the 2009 February13 and 2011 October 1 CME events. These applications demonstrate the flexibility and efficiency of our new method for energizing pre-eruptive configurations in MHD simulations of CMEs. We discuss possible ways of optimizing the axis paths and other extensions of the method in order to make it more useful and robust. Research supported by NSF, NASA's HSR and LWS Programs, and AFOSR.