Magnetogram-matching Biot–Savart Law and Decomposition of Vector Magnetograms - Viacheslav S. Titov; Cooper Downs; Tibor Török; Jon A. Linker; Michael Prazak; Jiong A. Qiu (The Astrophysical Journal)
We generalize a magnetogram-matching Biot–Savart law (BS l ) from planar to spherical geometry. For a given coronal current density J , this law determines the magnetic field $\widetilde{{\boldsymbol{B}}}$ whose radial component vanishes at the surface. The superposition of $\widetilde{{\boldsymbol{B}}}$ with a potential field defined by a given surface radial field, B _r , provides the entire configuration where B _r remains unchanged by the currents. Using this approach, we (1) upgrade our regularized BS l s for constructing coronal magnetic flux ropes (MFRs) and (2) propose a new method for decomposing a measured photospheric magnetic field as ${\boldsymbol{B}}={{\boldsymbol{B}}}_{{\rm{pot}}}+{{\boldsymbol{B}}}_{T}+{{\boldsymbol{B}}}_{\tilde{S}}$ , where the potential, B _pot , toroidal, B _T , and poloidal, ${{\boldsymbol{B}}}_{\tilde{S}}$ , fields are determined by B _r , J _r , and the surface divergence of B – B _pot , respectively, all derived from magnetic data. Our B _T is identical to the one in the alternative Gaussian decomposition by P. W. Schuck et al., while B _pot and ${{\boldsymbol{B}}}_{\tilde{S}}$ are different from their poloidal fields ${{\boldsymbol{B}}}_{{\rm{P}}}^{\lt }$ and ${{\boldsymbol{B}}}_{{\rm{P}}}^{\gt }$ , which are potential in the infinitesimal proximity to the upper and lower side of the surface, respectively. In contrast, our ${{\boldsymbol{B}}}_{\tilde{S}}$ has no such constraints and, as B _pot and B _T , refers to the same upper side of the surface. In spite of these differences, for a continuous J distribution across the surface, B _pot and ${{\boldsymbol{B}}}_{\tilde{S}}$ are linear combinations of ${{\boldsymbol{B}}}_{{\rm{P}}}^{\lt }$ and ${{\boldsymbol{B}}}_{{\rm{P}}}^{\gt }$ . 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.