Using Coronal Loops to Reconstruct the Magnetic Field of an Active Region Before and After a Major Flare

The shapes of solar coronal loops are sensitive to the presence of electrical currents that are the carriers of the nonpotential energy available for impulsive activity. We use this information in a new method for modeling the coronal magnetic field of AR 11158 as a nonlinear force-free field (NLFFF). The observations used are coronal images around time of major flare activity on 2011/02/15, together with the surface line-of-sight magnetic field measurements. The data are from the Helioseismic and Magnetic Imager and Atmospheric Imaging Assembly (HMI and AIA, respectively) onboard the Solar Dynamics Observatory (SDO). The model fields are constrained to approximate the coronal loop configurations as closely as possible, while also subject to the force-free constraints. The method does not use transverse photospheric magnetic field components as input, and is thereby distinct from methods for modeling NLFFFs based on photospheric vector magnetograms. We validate the method using observations of AR 11158 at a time well before major flaring, and subsequently review the field evolution just prior to and following an X2.2 flare and associated eruption. The models indicate that the energy released during the instability is about $1\times10^{32}$ erg, consistent with what is needed to power such a large eruptive flare. Immediately prior to the eruption the model field contains a compact sigmoid bundle of twisted flux that is not present in the post-eruption models, which is consistent with the observations. The core of that model structure is twisted by $\approx0.9$ full turns about its axis.Comment: ApJ, in pres

Modeling Magnetic Field Structure of a Solar Active Region Corona using Nonlinear Force-Free Fields in Spherical Geometry

We test a nonlinear force-free field (NLFFF) optimization code in spherical geometry using an analytical solution from Low and Lou. Several tests are run, ranging from idealized cases where exact vector field data are provided on all boundaries, to cases where noisy vector data are provided on only the lower boundary (approximating the solar problem). Analytical tests also show that the NLFFF code in the spherical geometry performs better than that in the Cartesian one when the field of view of the bottom boundary is large, say, $20^\circ \times 20^\circ$. Additionally, We apply the NLFFF model to an active region observed by the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO) both before and after an M8.7 flare. For each observation time, we initialize the models using potential field source surface (PFSS) extrapolations based on either a synoptic chart or a flux-dispersal model, and compare the resulting NLFFF models. The results show that NLFFF extrapolations using the flux-dispersal model as the boundary condition have slightly lower, therefore better, force-free and divergence-free metrics, and contain larger free magnetic energy. By comparing the extrapolated magnetic field lines with the extreme ultraviolet (EUV) observations by the Atmospheric Imaging Assembly (AIA) on board SDO, we find that the NLFFF performs better than the PFSS not only for the core field of the flare productive region, but also for large EUV loops higher than 50 Mm.Comment: 34 pages, 8 figures, accepted for publication in Ap

Gauge Invariant Framework for Shape Analysis of Surfaces

This paper describes a novel framework for computing geodesic paths in shape spaces of spherical surfaces under an elastic Riemannian metric. The novelty lies in defining this Riemannian metric directly on the quotient (shape) space, rather than inheriting it from pre-shape space, and using it to formulate a path energy that measures only the normal components of velocities along the path. In other words, this paper defines and solves for geodesics directly on the shape space and avoids complications resulting from the quotient operation. This comprehensive framework is invariant to arbitrary parameterizations of surfaces along paths, a phenomenon termed as gauge invariance. Additionally, this paper makes a link between different elastic metrics used in the computer science literature on one hand, and the mathematical literature on the other hand, and provides a geometrical interpretation of the terms involved. Examples using real and simulated 3D objects are provided to help illustrate the main ideas.Comment: 15 pages, 11 Figures, to appear in IEEE Transactions on Pattern Analysis and Machine Intelligence in a better resolutio