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