Advancing understanding of sunspot penumbra formation with non-linear force-free field extrapolations

Although sunspots have been extensively studied, the mechanism behind the formation of penumbrae is still not fully understood. In this work, we investigate the process of sunspot penumbrae formation from a novel approach based on the analysis of magnetic fields from non-linear force-free field (NLFFF) extrapolations aiming to identifying the key parameters driving this process. We calculate NLFFF extrapolations from HMI/SDO data sampling the development of the active region NOAA 12757 before, during, and after penumbra formation. We analyse the resulting magnetic field inclination, magnetic field tension, and the current density evolution in the leading spot. The analysis focuses on the stable part of the sunspot, away from the AR opposite polarity. The analysis of the extrapolations has revealed that: (1) the magnetic field inclination measured at the footpoints of the extrapolated field lines agrees with the results inferred from inversions, confirming the consistency of the methodology. (2) Penumbra formation is preceded by the continuous emergence of magnetic flux as outlined by serpentine fields observed in the low layers. After emerging, these fields further rise shaping the active region field topology formed by high-lying loops. (3) The emergence of local patches of new flux is characterised by high magnetic tension and significant current densities, primarily concentrated at the protospot boundary. Both the magnetic tension and current densities gradually dissipate as the magnetic loops continue to rise. This flux emergence occurs in regions where the penumbra is not yet formed and the magnetic canopy is not yet developed. (4) With the increase of emerged flux, a magnetic canopy develops and gradually expands around the spot in unison with the forming underlying penumbra. (5) As the penumbra and canopy expand, the surrounding network field, initially present close to the spot boundary, gradually migrates outward, reaching a maximum distance of about 9,Mm. This and the rise of serpentine fields, footpointed by migrating moving magnetic features provide evidence of the connectivity between the sunspot's core magnetic field and the surrounding network field. This case study provides clear evidence that the formation of stable penumbra results from a bottom-up approach: the continuous emergence and upward rise of serpentine field lines into the corona during which a sunspot magnetic canopy develops.