Modeling Magnetic Flux Ropes with the RBSL Method

We describe progress in developing a method for smoothly embedding magnetic flux ropes (MFRs) with various internal structures into ambient potential magnetic fields. The method uses the so-called regularized Biot-Savart laws (RBSLs, Titov et al. 2018), which enable one to calculate the magnetic field produced by axial and azimuthal currents flowing in a channel with a circular cross-section and an axis path of arbitrary shape. In the latest version of our method, the whole configuration is a superposition of the following three fields: 1) the MFR field determined by the RBSLs, 2) an ambient potential field determined, for example, by the radial field component of an observed magnetogram, and 3) a so-called compensating potential field that counteracts perturbations of the radial field at the surface by the MFR field. To make the configuration as force-free as possible, the method aims to optimize the MFR characteristics in two ways. First, for a cylindrical force-free MFR, we determine the corresponding kernels of the RBSLs for different profiles of the axial current density and use those to calculate the magnetic field of a thin, curved MFR, which has slightly imbalanced magnetic forces. Second, we minimize this imbalance by iteratively adjusting the shape of the MFR, based on the line density of magnetic forces that we calculate along the MFR at every iteration. If the resulting optimized MFR is stable, a subsequent line-tied zero-beta MHD relaxation will typically yield a force-free MFR whose parameters are quite close to the initial ones. MFR configurations produced in this way should be very useful for modeling solar eruptions, because the initial MFR parameters are largely determined by the specific properties of the source-region. Our efficient method also facilitates parametric studies and the stability analysis of pre-eruptive configurations. We demonstrate this by using an idealized model of a toroidal-arc MFR, for which we derive the critical decay index of its ambient field as a function of the MFR parameters.