Photospheric Imprints of Coronal Electric Currents, I. Magnetic Structure Near Polarity Inversion Lines

Flares and coronal mass ejections (CMEs) are powered by magnetic energy stored in coronal electric currents. Here, we use photospheric vector magnetic field observations to study currents in active regions 10930 and 11158, which both produced eruptive, X-class flares. We employ Gauss's separation method in Cartesian geometry to partition the photospheric field into three distinct components: (i) the toroidal field, $\vec B_T$, from vertical currents, $J_z$, passing through the photosphere; (ii) $\vec B^<$, from currents, $\vec J^<$, flowing below it; and (iii) $\vec B^>$, from currents, $\vec J^>$, flowing above it. We refer to $\vec B^>$ as the photospheric imprint of coronal currents. We give two representations of $\vec B^<$ and $\vec B^>$: (i) as second-order derivatives of poloidal potentials $P^<$ and $P^>$, respectively; and (ii) in terms of gradients of scalar potentials, with $\vec B^< = -\vec \nabla \psi^<$ and $\vec B^> = -\vec \nabla \psi^>$. The central polarity inversion line (PIL) in each region possesses magnetic structure similar to that reported in AR 12673 by Schuck et al. (2022): (i) $\vec B_T$, which arises from $J_z$, produces sheared fields along the PIL; (ii) $\vec B^>$ exhibits large-scale, spatially coherent structure, consistent with $\vec J^>$ flowing horizontally above and along the PIL; and (iii) the near-PIL, horizontal currents $\vec J_h^<$ and $\vec J_h^>$ are roughly parallel. Because parallel currents attract, such parallel-current configurations are likely more stable than misaligned-current configurations. Horizontal current $\vec J_h^>$ flowing along a PIL increases $|\vec \nabla B_z|$ across the PIL, providing a physical explanation for previously reported empirical associations of strong, cross-PIL gradients in $B_z$ with flare and CME occurrence.