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

TL;DR

This paper extends the Biot-Savart law to spherical geometry for better modeling of solar magnetic fields and introduces a new decomposition method for vector magnetograms, aiding in understanding coronal magnetic structures.

Contribution

It generalizes the magnetogram-matching Biot-Savart law to spherical geometry and proposes a novel decomposition of magnetic fields into potential, toroidal, and poloidal components from observational data.

Findings

Enhanced modeling of coronal magnetic flux ropes.

New method for decomposing surface magnetic fields.

Ability to identify magnetic flux rope footprints and currents.

Abstract

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.