Magnetogram-matching Biot–Savart Law and Decomposition of Vector Magnetograms

We generalize a magnetogram-matching Biot–Savart law (BSl) from planar to spherical geometry. For a given coronal current density J, this law determines the magnetic field whose radial component vanishes at the surface. The superposition of with a potential field defined by a given surface radial field, Br, provides the entire configuration where Br remains unchanged by the currents. Using this approach, we (1) upgrade our regularized BSls for constructing coronal magnetic flux ropes (MFRs) and (2) propose a new method for decomposing a measured photospheric magnetic field as , where the potential, Bpot, toroidal, BT, and poloidal, , fields are determined by Br, Jr, and the surface divergence of B–Bpot, respectively, all derived from magnetic data. Our BT is identical to the one in the alternative Gaussian decomposition by P. W. Schuck et al., while Bpot and are different from their poloidal fields and , which are potential in the infinitesimal proximity to the upper and lower side of the surface, respectively. In contrast, our has no such constraints and, as Bpot and BT, refers to the same upper side of the surface. In spite of these differences, for a continuous J distribution across the surface, Bpot and are linear combinations of and . We demonstrate that, similar to the Gaussian method, our decomposition allows one to identify the footprints and projected surface-location of MFRs in the solar corona, as well as the direction and connectivity of their currents.