The solar coronal magnetic field must be inferred from photospheric magnetograms, and standard extrapolations assume a current-free field (PFSS) or a force-free field (LFFF/NLFFF), assumptions violated in the low corona where measurable Lorentz forces exist. Wang et al. (2020) introduced the first global, full-sphere forced (non-force-free) extrapolation based on the minimum-dissipation-rate principle: the field decomposes into one potential and two linear force-free components whose bottom boundaries are determined iteratively from a single-layer vector magnetogram. Validated only against two analytic magnetohydrostatic equilibria, the method reconstructs the field well up to about 1.5 solar radii but degrades toward 2.0 solar radii, with unquantified outer-boundary effects, a costly iterative decomposition, unrecovered Lorentz-force magnitudes, and no test on realistic data or comparison with established codes. These four issues constitute one coherent problem — maturing the method into a validated, practical global reconstruction tool — and remain unaddressed as of 2026: the paper has essentially no methodological follow-up, while parallel non-force-free developments (numerical magnetohydrostatic extrapolation, multi-height physics-informed neural networks, global NLFFF codes) proceed in different frameworks.

Background

Direct measurements of the solar magnetic field exist routinely only at the photosphere (the Sun's visible surface), yet the three-dimensional magnetic field of the corona (the Sun's hot outer atmosphere) governs coronal structure, flares, eruptions, and the heliosphere. The coronal field is therefore inferred by extrapolating photospheric magnetograms — maps of the vector magnetic field B\mathbf{B} on the solar disk — upward under physical assumptions. The potential-field source-surface (PFSS) model assumes a current-free field below a spherical source surface (commonly near 2.5R⊙2.5R_\odot, where the field is forced to be purely radial); it cannot represent current-carrying, sheared configurations. Force-free extrapolations instead assume the current density J=(1/μ0)∇×B\mathbf{J}=(1/\mu_0)\nabla\times\mathbf{B} is parallel to B\mathbf{B}, so the Lorentz force J×B\mathbf{J}\times\mathbf{B} vanishes; in the linear version (LFFF) ∇×B=αB\nabla\times\mathbf{B}=\alpha\mathbf{B} with a constant force-free parameter α\alpha, and in the nonlinear version (NLFFF) α\alpha varies with position. Both assumptions fail in the low atmosphere, where the plasma beta (ratio of gas pressure to magnetic pressure) is high and the observed photospheric field carries net Lorentz forces; tests against reference fields document the resulting failures of force-free extrapolations driven by single-layer photospheric data (Peter et al. 2015), and the March 2015 eclipse comparison of global non-potential models illustrates the spread of competing global methods (Yeates et al. 2018).

One route beyond force-free fields is the minimum-dissipation-rate (MDR) principle: treating the corona as an open, externally driven system, a variational ansatz of minimum Joule dissipation yields a third-order field equation whose solutions decompose uniquely into one potential field plus two LFFF fields with distinct constants α1\alpha_1 and α3\alpha_3 (Hu & Dasgupta 2006; Hu et al. 2008). The resulting forced (non-force-free field, NFFF) extrapolation can carry finite Lorentz forces. In local Cartesian geometry the method was developed and tested against MHD simulations and real magnetograms (Hu et al. 2008, 2010), with a critical case evaluation by Gary (2009). Wang et al. (2020) extended the approach to full-sphere geometry: using a fast global potential/LFFF solver of the type developed by Jiang & Feng (2012), an iterative scheme determines the bottom (photospheric) boundary conditions of the three component fields from a single-layer vector magnetogram alone, with the pair (α1,α3)(\alpha_1,\alpha_3) chosen by minimizing the mismatch of the transverse field. Validation was performed only against two analytic magnetohydrostatic (MHS) equilibria — self-consistent static solutions balancing Lorentz, pressure-gradient, and gravitational forces — of the Neukirch (1995) family, in the domain 1.01.0–2.5R⊙2.5R_\odot with the field forced radial at the top boundary Rt=2.5R⊙R_t=2.5R_\odot. Below about 1.5R⊙1.5R_\odot the reconstruction is excellent (vector correlation ≥0.9985\ge 0.9985, normalized vector-error complement ≈0.94\approx 0.94–0.970.97) and clearly outperforms a potential-field extrapolation, but accuracy degrades toward 2.0R⊙2.0R_\odot; the iterative decomposition dominates the computational cost (about 74 s per iteration on a 107×360×720107\times360\times720 spherical grid); the spatial pattern but not the magnitudes of the Lorentz-force components is recovered; and no test on realistic magnetograms or comparison with established codes was performed. Related non-force-free routes developed in parallel include optimization-based, Grad–Rubin-type, and MHD-relaxation numerical MHS extrapolations (Miyoshi et al. 2020; reviewed by Zhu, Neukirch & Wiegelmann 2022; applied to an active region by Yu et al. 2023), analytical MHS solution families (Neukirch & Wiegelmann 2019; Nadol & Neukirch 2025), multi-height data-driven approaches (Jarolim et al. 2024), and new global NLFFF codes on overset spherical grids (Koumtzis & Wiegelmann 2023). None of these implements or evaluates the global MDR-based forced extrapolation of Wang et al. (2020).

Problem Statement

Extend the global forced (non-force-free) extrapolation method of Wang et al. (2020) from proof-of-concept status to a validated, practical tool for reconstructing the full-sphere coronal magnetic field from a single-layer photospheric vector magnetogram, by resolving the four issues the authors explicitly left open:

(1) Top-boundary error. Quantify, and remedy, the impact of the outer-boundary treatment — the source surface at Rt=2.5R⊙R_t=2.5R_\odot where the field is forced to be radial — on reconstruction accuracy, which is good below about 1.5R⊙1.5R_\odot but degrades toward 2.0R⊙2.0R_\odot: determine how much of this degradation is caused by inconsistencies between the extrapolation top-boundary conditions and the reference/true field rather than by intrinsic limitations of the MDR decomposition, and identify boundary choices that reduce it.

(2) Iterative single-layer decomposition. The iterative determination of the three component bottom boundaries from one-layer data is the most time-consuming step; accelerate or replace it — for example by a direct two-layer formulation using chromospheric vector magnetograms if and when such data become available, or by an algorithmically faster scheme — while preserving the method's operability from routinely available single-layer data.

(3) Lorentz-force fidelity. The method reproduces the spatial pattern of the Lorentz force, but the magnitudes of individual force components differ considerably from the reference field; diagnose the causes (boundary treatment, the fixed two-parameter (α1,α3)(\alpha_1,\alpha_3) representation, or the MDR ansatz itself), establish their implications for force-balance diagnostics and energetics, or remedy them.

(4) Validation on real data. Demonstrate and quantify the method's behavior on realistic photospheric (e.g., synoptic) vector magnetograms — including noise, 180-degree azimuthal ambiguity, and polar-field gaps — and benchmark its accuracy against established NLFFF, PFSS, and other global extrapolation approaches over the heliocentric range relevant to the low corona.

All four deliverables are mandatory and jointly answer one question: is MDR-based global forced extrapolation a reliable, practically usable method for routine global coronal magnetic-field reconstruction? Within deliverables (1)–(3) either stated branch suffices — quantifying or remedying the outer-boundary impact in (1), accelerating or replacing the iterative single-layer decomposition in (2), diagnosing or remedying the Lorentz-force magnitude discrepancies in (3) — while deliverable (4) requires both testing on realistic magnetograms and benchmarking against established methods. A complete answer establishes, or refutes, this reliability and practical usability with quantified evidence going beyond analytic test equilibria; a fully evidenced negative verdict of the same scope and specificity likewise completes the task.

References

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