Nonlinear force-free field (NLFFF) extrapolation from photospheric vector magnetograms is the standard route to reconstructing the coronal magnetic field; for fields of view spanning several active regions it must be formulated in spherical geometry, as demonstrated by Tadesse, Wiegelmann, MacNeice & Olson (2013b) with SOLIS/VSM data. That study, however, compared the modeled field lines with EUV coronal-loop images only by qualitative overlay, assumed current-free lateral and top boundaries, and left the impact of photospheric preprocessing and magnetogram data gaps on the reconstructed currents and free magnetic energy unquantified. Subsequent work answered parts of this agenda in neighboring settings: systematic quantitative loop-comparison metrics were established for Cartesian single-active-region extrapolations (Warren et al. 2018), loop-misalignment constraints were built into Cartesian NLFFF inversions (S-NLFFF, stereoscopy-guided NLFFF; Chifu et al. 2015, 2017) and machine-learning single-view 3D loop reconstruction was developed to widen the constraint supply (Chifu & Gafeira 2021); the sensitivity of spherical solutions to computational-domain size was quantified (Tadesse et al. 2015); and large-data performance was achieved through Yin-Yang-grid (a two-patch overlapping spherical grid that avoids polar coordinate singularities) and physics-informed neural-network codes rather than the multigrid scheme the source proposed. The surviving open problem is the combination at the source's core: a spherical-geometry NLFFF reconstruction for large multi-active-region domains that incorporates observed coronal-loop geometry as a constraint, is validated by objective quantitative comparison with EUV loop observations, and whose boundary and preprocessing uncertainties are quantified and reduced.

Background

The solar corona is a low-plasma-beta environment: away from flare cores, magnetic forces dominate pressure gradients and gravity, so the coronal magnetic field is approximated as force-free, satisfying ∇×B=αB\nabla\times\mathbf{B}=\alpha\mathbf{B} and ∇⋅B=0\nabla\cdot\mathbf{B}=0, where the parameter α\alpha varies across the field and is constant along each field line. When α\alpha varies spatially the field is called a nonlinear force-free field (NLFFF); the special case α=0\alpha=0 is the current-free potential field computable from the radial field alone. Because the coronal field itself cannot be measured routinely, it is reconstructed by extrapolation from photospheric vector magnetograms — maps of all three magnetic-field components at the solar surface, supplied for example by SOLIS/VSM (the Synoptic Optical Long-term Investigations of the Sun Vector Spectromagnetograph) and SDO/HMI (the Helioseismic and Magnetic Imager on the Solar Dynamics Observatory). The dominant reconstruction technique is the optimization method, which minimizes a functional measuring how far a trial three-dimensional field is from being force-free and divergence-free while matching the measured bottom boundary; its development and variants are reviewed by Wiegelmann & Sakurai (2021). Two standard practical complications are relevant here. First, the photosphere is not force-free, so measured boundary vectors are inconsistent boundary data for a force-free extrapolation and are commonly preprocessed toward force-free consistency, while noisy or undetermined pixels (for example regions of ambiguous transverse-field direction) enter the functional with reduced or zero weight (Wiegelmann et al. 2012; Tadesse et al. 2011). Second, side and top boundaries of a localized computational domain are not observed at all and are usually filled with a current-free (potential) field. Most NLFFF codes use Cartesian boxes, adequate for a single active region but not for fields of view spanning a large fraction of the solar surface, where curvature and inter-region connectivity cannot be neglected; Wiegelmann (2007) therefore reformulated the optimization method on a spherical grid, and the spherical line of work was developed with real vector data by Tadesse and collaborators (Tadesse et al. 2011; Tadesse et al. 2013a,b, 2014). Tadesse, Wiegelmann, MacNeice & Olson (2013b) applied the spherical code to SOLIS/VSM observations containing six active regions (and, in a second data set, three vertically aligned ones) and compared projections of the modeled field lines with extreme-ultraviolet (EUV) images from SDO/AIA (the Atmospheric Imaging Assembly). Since EUV-emitting plasma is believed to trace magnetic field lines, observed coronal loops are the natural independent check of an extrapolation; but the source study could only overlay field lines on loop images and judge the correspondence by eye. The paper explicitly flagged two gaps: its results and conclusion acknowledged the absence of a reliable quantitative, objective comparison, noting that the qualitative overlay "remains the best option in the absence of a more reliable quantitative comparison", and its outlook proposed adapting a multigrid technique to the spherical code as a computational-performance route for routine use with high-resolution, large-volume HMI data. Its introduction also noted that coronal loops "provide an additional constraint which is not currently being used due to the mathematical complications of incorporating such input into numerical models". In addition, its method documented assumptions whose impact was left unquantified: current-free lateral and top boundaries, and zero-weighting of pixels affected by preprocessing failures or magnetogram data gaps. How these assumptions affect the reconstructed field, its electric currents, its connectivity, and its free magnetic energy (the excess of the NLFFF energy over the potential-field energy, a key input to flare and coronal-mass-ejection estimates) was not raised explicitly as future work in the source, but became a quantified open question through Tadesse et al. (2015). The question asked here is the surviving core of that agenda after more than a decade of follow-up work: making spherical-geometry NLFFF modeling of multi-active-region domains constrained by observed loop geometry, quantitatively validated against EUV loop observations, and controlled in its boundary and preprocessing uncertainties.

Problem Statement

For coronal magnetic-field modeling over domains that contain several active regions, where NLFFF extrapolation must be formulated in spherical geometry, the open problem is to make the reconstruction observationally constrained and quantitatively validated. Concretely, for wedge-shaped or global spherical domains driven by photospheric vector magnetograms: (1) incorporate observed coronal-loop geometry — stereoscopically or otherwise reconstructed 3D loops, or loop tangents projected in EUV images — as a constraint term in the spherical-geometry NLFFF inversion, generalizing the loop-misalignment constraint so far realized in Cartesian active-region codes (Aschwanden 2013; Chifu et al. 2015, 2017) and, with synthetic loops, in a global non-potential spherical model (Antonio et al. 2026); (2) establish quantitative, objective comparison between extrapolated field lines and EUV coronal-loop observations — for example field-line-to-loop distances and misalignment angles over statistically meaningful loop samples, in the spirit of Warren et al. (2018) — as the standard validation for spherical-geometry models, replacing qualitative overlay; and (3) quantify and reduce the influence of the unmeasured lateral and top boundaries, currently prescribed as current-free fields, and of photospheric preprocessing and magnetogram data gaps, on the reconstructed magnetic connectivity, electric currents, and free magnetic energy over the large domain, whose sensitivity is documented by Tadesse et al. (2015). These components jointly serve one scientific objective: trustworthy, observationally validated coronal magnetic-field models over multi-active-region domains, as needed for trans-equatorial connectivity studies and large-scale free-energy budgets. An answer may advance the components separately, but each claim must be scoped to what was actually demonstrated (domain sizes, dates, instruments, loop-sample sizes).

References

  1. Tadesse, T., Wiegelmann, T., MacNeice, P. J., Olson, K. (2013b). Modeling coronal magnetic field using spherical geometry: cases with several active regions. Astrophysics and Space Science 347, 21-27. DOI: 10.1007/s10509-013-1493-3. URL: https://doi.org/10.1007/s10509-013-1493-3 (arXiv:1301.7522, https://arxiv.org/abs/1301.7522)
  2. Wiegelmann, T. (2007). Computing Nonlinear Force-Free Coronal Magnetic Fields in Spherical Geometry. Solar Physics 240, 227. DOI: 10.1007/s11207-006-0266-3. URL: https://doi.org/10.1007/s11207-006-0266-3
  3. Tadesse, T., Wiegelmann, T., Inhester, B., Pevtsov, A. (2011). Nonlinear force-free field extrapolation in spherical geometry: improved boundary data treatment applied to a SOLIS/VSM vector magnetogram. Astronomy & Astrophysics 527, A30. DOI: 10.1051/0004-6361/201015491. URL: https://doi.org/10.1051/0004-6361/201015491
  4. Wiegelmann, T., Thalmann, J. K., Inhester, B., Tadesse, T., Sun, X., Hoeksema, J. T. (2012). How Should One Optimize Nonlinear Force-Free Coronal Magnetic Field Extrapolations from SDO/HMI Vector Magnetograms? Solar Physics 281, 37-51. DOI: 10.1007/s11207-012-9966-z. URL: https://doi.org/10.1007/s11207-012-9966-z
  5. Tadesse, T., Wiegelmann, T., Inhester, B., MacNeice, P., Pevtsov, A., Sun, X. (2013a). Full-disk nonlinear force-free field extrapolation of SDO/HMI and SOLIS/VSM magnetograms. Astronomy & Astrophysics 550, A14. DOI: 10.1051/0004-6361/201220044. URL: https://doi.org/10.1051/0004-6361/201220044
  6. Tadesse, T., Wiegelmann, T., Gosain, S., MacNeice, P., Pevtsov, A. A. (2014). First use of synoptic vector magnetograms for global nonlinear, force-free coronal magnetic field models. Astronomy & Astrophysics 562, A105. DOI: 10.1051/0004-6361/201322418. URL: https://doi.org/10.1051/0004-6361/201322418
  7. Tadesse, T., Wiegelmann, T., MacNeice, P. J. (2015). Effect of the Size of the Computational Domain on Spherical Nonlinear Force-Free Modeling of a Coronal Magnetic Field Using SDO/HMI Data. Solar Physics 290, 1159. DOI: 10.1007/s11207-015-0664-5. URL: https://doi.org/10.1007/s11207-015-0664-5
  8. Aschwanden, M. J. (2013). Nonlinear force-free magnetic field fitting to coronal loops with and without stereoscopy. The Astrophysical Journal 763, 115. DOI: 10.1088/0004-637X/763/2/115. URL: https://doi.org/10.1088/0004-637X/763/2/115
  9. Malanushenko, A., Schrijver, C. J., DeRosa, M. L., Wheatland, M. S. (2014). Using coronal loops to reconstruct the magnetic field of an active region before and after a major flare. The Astrophysical Journal 783, 102. DOI: 10.1088/0004-637x/783/2/102. URL: https://doi.org/10.1088/0004-637x/783/2/102
  10. Chifu, I., Inhester, B., Wiegelmann, T. (2015). Coronal magnetic field modeling using stereoscopy constraints. Astronomy & Astrophysics 577, A123. DOI: 10.1051/0004-6361/201322548. URL: https://doi.org/10.1051/0004-6361/201322548
  11. Chifu, I., Wiegelmann, T., Inhester, B. (2017). Nonlinear force-free coronal magnetic stereoscopy. The Astrophysical Journal 837, 10. DOI: 10.3847/1538-4357/aa5b9a. URL: https://doi.org/10.3847/1538-4357/aa5b9a (arXiv:1709.04177, https://arxiv.org/abs/1709.04177)
  12. Warren, H. P., Crump, N. A., Ugarte-Urra, I., Sun, X., Aschwanden, M. J., Wiegelmann, T. (2018). Toward a Quantitative Comparison of Magnetic Field Extrapolations and Observed Coronal Loops. The Astrophysical Journal 860, 46. DOI: 10.3847/1538-4357/aac20b. URL: https://doi.org/10.3847/1538-4357/aac20b (arXiv:1805.00281, https://arxiv.org/abs/1805.00281)
  13. Chifu, I., Gafeira, R. (2021). 3D Solar Coronal Loop Reconstructions with Machine Learning. The Astrophysical Journal Letters 910, L10. DOI: 10.3847/2041-8213/abed53. URL: https://doi.org/10.3847/2041-8213/abed53 (arXiv:2103.09960, https://arxiv.org/abs/2103.09960)
  14. Chifu, I., Inhester, B., Wiegelmann, T. (2022). Coronal magnetic field evolution over cycle 24. Astronomy & Astrophysics 659, A174. DOI: 10.1051/0004-6361/202038001. URL: https://doi.org/10.1051/0004-6361/202038001 (arXiv:2201.03853, https://arxiv.org/abs/2201.03853)
  15. Koumtzis, A., Wiegelmann, T. (2023). A New Global Nonlinear Force-Free Coronal Magnetic-Field Extrapolation Code Implemented on a Yin-Yang Grid. Solar Physics 298, 20. DOI: 10.1007/s11207-023-02109-6. URL: https://doi.org/10.1007/s11207-023-02109-6
  16. Jiang, C., Feng, X., Xiang, C. (2012). A New Code for Nonlinear Force-Free Field Extrapolation of the Global Corona. The Astrophysical Journal 755, 62. DOI: 10.1088/0004-637X/755/1/62. URL: https://doi.org/10.1088/0004-637X/755/1/62 (arXiv:1206.1989, https://arxiv.org/abs/1206.1989)
  17. Jarolim, R., Thalmann, J. K., Veronig, A. M., Podladchikova, T. (2023). Probing the solar coronal magnetic field with physics-informed neural networks. Nature Astronomy 7, 1171-1179. DOI: 10.1038/s41550-023-02030-9. URL: https://doi.org/10.1038/s41550-023-02030-9
  18. Wiegelmann, T., Sakurai, T. (2021). Solar force-free magnetic fields. Living Reviews in Solar Physics 18, 1. DOI: 10.1007/s41116-020-00027-4. URL: https://doi.org/10.1007/s41116-020-00027-4
  19. DeRosa, M. L., Wheatland, M. S., Leka, K. D., Barnes, G., Amari, T., Canou, A., Gilchrist, S. A., Thalmann, J. K., Valori, G., Wiegelmann, T., Schrijver, C. J., Malanushenko, A., Sun, X., Régnier, S. (2015). The Influence of Spatial Resolution on Nonlinear Force-Free Modeling. The Astrophysical Journal 811, 107. DOI: 10.1088/0004-637x/811/2/107. URL: https://doi.org/10.1088/0004-637x/811/2/107
  20. Costa, M., Barros, F. S., Chifu, I., Restivo, A. (2026). Solar coronal loop detection in EUV images: Computational methods, dataset limitations, and open problems. Astronomy and Computing 57. DOI: 10.1016/j.ascom.2026.101169. URL: https://doi.org/10.1016/j.ascom.2026.101169
  21. Antonio, C., Chifu, I., Gafeira, R., Lima, J. J. G. (2026). A new global non-potential magnetic field model. Astronomy & Astrophysics (in press). DOI: 10.1051/0004-6361/202558738. URL: https://doi.org/10.1051/0004-6361/202558738