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 $\bf{J}$, this law determines the magnetic field $\tilde{\bf B}$ whose radial component vanishes at the surface. The superposition of $\tilde{\bf B}$ 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 BSLs for constructing coronal magnetic flux ropes (MFRs) and (2) propose a new method for decomposing a measured photospheric magnetic field as ${\bf B} = {\bf B}_{\rm{pot}} + {\bf B}_T + {\bf B}_{\tilde{S}}$, where the potential, ${\bf B}_{\rm{pot}}$, toroidal, ${\bf B}_T$, and poloidal, ${\bf B}_{\tilde{S}}$, fields are determined by $B_r$, $J_r$, and the surface divergence of ${\bf B} - {\bf B}_{\rm{pot}}$, respectively, all derived from magnetic data. Our ${\bf B}_T$ is identical to the one in the alternative Gaussian decomposition by Schuck et al. (2022), while ${\bf B}_{\rm{pot}}$ and ${\bf B}_{\tilde{S}}$ are different from their poloidal fields ${\bf B}^{<}_{\rm{P}}$ and ${\bf B}^{>}_{\rm{P}}$, which are $potential$ in the infinitesimal proximity to the upper and lower side of the surface, respectively. In contrast, our ${\bf B}_{\tilde{S}}$ has no such constraints and, as ${\bf B}_{\rm{pot}}$ and ${\bf B}_T$, refers to the $same$ upper side of the surface. In spite of these differences, for a continuous $\bf{J}$ distribution across the surface, ${\bf B}_{\rm{pot}}$ and ${\bf B}_{\tilde{S}}$ are linear combinations of ${\bf B}^{<}_{\rm{P}}$ and ${\bf B}^{>}_{\rm{P}}$. 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.