The so-called regularized Biot-Savart laws (RBSLs, Titov et al. 2018) provide an efficient and flexible method for modeling pre-eruptive magnetic configurations whose characteristics are constrained by observational image and magnetic-field data. This method allows one to calculate the field of magnetic flux ropes (MFRs) with small circular cross-sections and an arbitrary axis shape. The field of the whole configuration is constructed as a superposition of (1) such a flux-rope field, (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 deviations of the radial field caused by the axial current of the MFR. The RBSL kernels are determined from the requirement that the MFR field for a straight cylinder must be exactly force-free. For a curved MFR, however, the magnetic forces are generally unbalanced over the whole path of the MFR. To reduce this imbalance, we apply a modified Gauss-Newton method to minimize the magnitude of the residual magnetic forces per unit length and the unit axial current of the MFR. This is done by iteratively adjusting the MFR axis path and axial current. We then try to relax the resulting optimized configuration in a subsequent line-tied zero-beta MHD simulation toward a force-free equilibrium. By considering several examples, we demonstrate how this approach works depending on the initial parameters of the MFR and the ambient magnetic field. Our method will be beneficial for both the modeling of particular eruptive events and theoretical studies of idealized pre-eruptive magnetic configurations. This research is supported by NSF, NASA’s HSR, SBIR, and LWS Programs, and AFOSR

It has long been recognized that the energy source for major solar flares and coronal mass ejections (CMEs) is the solar magnetic field within active regions. Specifically, it is believed to be the release of the free magnetic energy (energy above the potential field state) stored in the field prior to eruption. For estimates of the free energy to provide a prognostic for future eruptions, we must know how much energy an active region can store – Is there a bound to this energy? The Aly-Sturrock theorem shows that the energy of a fully force-free field cannot exceed the energy of the so-called open field. If the theorem holds, this places an upper limit on the amount of free energy that can be stored. In recent simulations, we have found that the energy of a closely related field, the partially open field (POF), can place a useful bound on the energy of an eruption from real active regions, a much tighter constraint than the energy of the fully open field. A database of flare ribbons (Kazachenko et al., ApJ 845, 2017) offers us an opportunity to test this idea observationally. A flare ribbon mask is defined as the area swept out by the ribbons during the flare. It can serve as a proxy for the region of the field that opened during the eruption. In this preliminary study, we use the ribbon masks to define the POF for several large events originating in solar cycle 24 active regions, and compute the energy of the POF. We compare these energies with the X-ray fluxes and CME energies for these events. Work supported by NSF, NASA, and AFOSR.