Nonlinear force-free field (NLFFF) extrapolation from photospheric vector magnetograms is the standard tool for reconstructing the three-dimensional coronal magnetic field of solar active regions. Kawabata, Inoue and Shimizu (2020) showed that, within a single magnetohydrodynamic-relaxation NLFFF code, different initial 3D field guesses (linear force-free fields with different constant alpha, including the potential field) converge to solutions that agree well below about 10 Mm but diverge increasingly with height. The unresolved question is whether this initialization dependence reflects a genuine multiplicity of coronal force-free equilibria compatible with the same boundary data - a fundamental non-uniqueness of the NLFFF reconstruction problem - or an artifact of incomplete relaxation, residual Lorentz forces, divergence errors, and the non-force-free photospheric boundary. The problem asks to adjudicate between these alternatives under controlled conditions, to identify which physical properties of an active region control the sensitivity, to quantify how much of the ambiguity a more force-free (e.g., chromospheric) lower boundary removes, and to determine which additional observational constraints select the physically correct solution when several mathematically acceptable equilibria exist.

Background

In the solar corona the plasma beta (the ratio of gas pressure to magnetic pressure) is low, so magnetic forces dominate and the field is approximately force-free: the Lorentz force vanishes, which for a solenoidal field means ∇×B=αB\nabla\times\mathbf{B}=\alpha\mathbf{B} with B⋅∇α=0\mathbf{B}\cdot\nabla\alpha=0 and ∇⋅B=0\nabla\cdot\mathbf{B}=0, where α\alpha is a scalar that is constant along each field line. If α\alpha is the same everywhere the field is a linear force-free field (LFFF); if α\alpha varies from field line to field line it is a nonlinear force-free field (NLFFF). NLFFF extrapolation reconstructs the 3D coronal field from a photospheric vector magnetogram - a 2D map of all three components of the magnetic field measured at the solar photosphere - imposed as the lower boundary of a computational box.

Mathematically, the NLFFF equations are of mixed elliptic-hyperbolic type, and the reconstruction is not covered by a general uniqueness theorem: existence of solutions in three-dimensional domains was proved by Boulmezaoud and Amari (2000), but uniqueness results cover only weakly nonlinear regimes (small α\alpha), as recounted by Wiegelmann and Sakurai (2021). The boundary-value formulation of the reconstruction problem goes back to Aly (1989). A further physical complication is that the photosphere, where the boundary data are measured, is itself not force-free: the field there carries significant Lorentz forces, so the boundary data are inconsistent with an exactly force-free coronal solution. Preprocessing (Wiegelmann, Inhester and Sakurai 2006) mutates the measured boundary toward a smoother, chromospheric-like, nearly force-free state before extrapolation, but this only enforces necessary, not sufficient, force-free consistency conditions.

Practical NLFFF codes fall into three families: Grad-Rubin-type iteration (which prescribes the vertical field and the distribution of α\alpha on one magnetic polarity); variational optimization of a field-averaged force-free and solenoidal error functional; and magnetohydrodynamic (MHD) relaxation (magnetofrictional) schemes, in which an initial 3D field is evolved by a reduced, friction-damped MHD system (in the scheme of Valori, Kliem and Keppens 2005 used by the source paper, with pseudo-density ρ=∣B∣\rho=|\mathbf{B}| and an effective resistivity) until the Lorentz force is quenched. Community benchmarks have long shown that different codes, and the same code with different boundary treatments, return different coronal fields for the same active region: Schrijver et al. (2006), Metcalf et al. (2008), and DeRosa et al. (2009) found large spreads in field-line geometry and free magnetic energy (the excess of the field energy over that of the potential field - the current-free field with α=0\alpha=0 - i.e., the energy available to power flares), attributing them to small fields of view, boundary inconsistency, and noisy transverse-field data, and recommending preprocessing and verification against coronal observations. For Grad-Rubin methods with noisy data, even the two solutions obtained by prescribing α\alpha on one or the other polarity can differ; self-consistency iterations that reconcile the two polarities were developed by Wheatland and Regnier (2009), Amari and Aly (2010), and Mastrano, Yang and Wheatland (2020).

Against this backdrop, Kawabata, Inoue and Shimizu (2020) isolated a cleaner axis of ambiguity: holding both the boundary data and the numerical method fixed, they varied only the initial 3D guess (LFFFs with constant α0=0\alpha_0=0 and ±\pm values, i.e., including the potential field standardly used) for two flare-productive regions, the simple bipolar NOAA 11692 and the complex multipolar NOAA 11967. They found the solutions robust below about 10 Mm height (1 Mm = 1000 km; pairwise vector-field correlation coefficients C>0.9C>0.9) but increasingly initialization-dependent higher up; the total magnetic energy was weakly affected while the free energy varied more; the complex multipolar region was less sensitive than the bipolar one; a residual Lorentz force remained concentrated below 10 Mm, so the relaxed states are not ideal force-free solutions; and choosing an initial α0\alpha_0 whose sign matches the photospheric global α\alpha (the flux-weighted average of α\alpha over the boundary) produced field lines that better matched the observed soft X-ray sigmoid (an S-shaped bright structure in soft X-ray images of the corona, associated with strongly sheared or twisted fields). The authors could not decide whether the height-dependent divergence of solutions is a genuine property of the force-free equilibrium problem or a technical artifact, and noted that only two active regions were studied.

Problem Statement

For a flare-productive solar active region whose lower-boundary vector magnetic field is fixed, determine whether the initialization dependence observed in magnetohydrodynamic (MHD)-relaxation nonlinear force-free field (NLFFF) extrapolations - reconstructions that evolve an initial 3D field guess toward a force-free state consistent with the boundary data - is a genuine multiplicity of force-free equilibria or a removable artifact. Concretely, with one common scientific objective - establishing the determinacy and observational grounding of NLFFF coronal field reconstructions - an answer must resolve the following linked questions:

  1. Multiplicity versus artifact. Using a single, controlled relaxation framework and boundary data that are demonstrably force-free-consistent (synthetic data from a known solution, or preprocessed or chromospheric data with reported consistency metrics), either (a) exhibit two or more initial 3D field guesses (e.g., the potential field and linear force-free fields with constant α\alpha of both signs) whose relaxations converge to distinct solutions that each satisfy standard force-free and solenoidal quality metrics (current-weighted field-current misalignment angle - the current-weighted average angle between B\mathbf{B} and ∇×B\nabla\times\mathbf{B} - and fractional divergence error) while differing materially at coronal heights - establishing genuine multiplicity; or (b) exhibit a relaxation and boundary treatment under which all such initial guesses converge to one and the same field, within stated height-resolved tolerances, at equal solution quality - establishing that the dependence is an artifact of incomplete relaxation, divergence errors, or the non-force-free boundary.

  2. Controlling region properties. Identify, across a sample of active regions large enough to separate correlated covariates, which physical property (magnetic flux content, flux imbalance, boundary force-freeness, characteristic loop length scale, or prevalence of strong horizontal fields) primarily controls the initialization sensitivity, replacing the two-region comparison of the source study.

  3. Boundary and observational disambiguation. Quantify how much of the height-dependent ambiguity is removed by a more force-free lower boundary (measured chromospheric vector fields or preprocessing toward them), and determine which additional observational constraints - chromospheric or coronal imaging, or coronal magnetography (direct measurement of coronal vector magnetic fields) - select the physically correct solution when several mathematically acceptable equilibria exist, using objective correspondence metrics rather than visual inspection.

Claims of multiplicity must hold for genuinely force-free, divergence-free solutions, not merely for relaxed states with residual Lorentz force; claims of uniqueness must cover the height range where the initialization dependence was observed (above roughly 10 Mm), not only the well-constrained low corona.

References

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