Physical origin of the invariant vertical magnetic field at the sunspot umbra–penumbra boundary and its joint reproduction with the Evershed flow in simulations

Spectropolarimetric observations show that the boundary between the dark umbra and the filamentary penumbra of a stable sunspot is marked by an approximately invariant vertical magnetic field component, $B_\mathrm{ver}\approx1.87$ kG in Hinode data (the Jurčák criterion), yet the physical mechanism that enforces this invariance is unexplained, and radiative magnetohydrodynamic (MHD) simulations cannot simultaneously reproduce this boundary property and the observed Evershed flow — the nearly horizontal, radially outward gas flow of a few km/s along penumbral filaments. The problem is to identify and quantitatively establish the physical mechanism — magneto-convective stability selection by the vertical field, as suggested by the Gough–Tayler criterion, or an alternative — that sets the umbra–penumbra boundary at a sharply defined vertical field strength, and, if such a mechanism is at work, to demonstrate it in sunspot simulations that at the same time reproduce the observed boundary magnetic properties, the direction and strength of the Evershed flow, the relative radial position of the boundary ($r/r_\mathrm{spot}\approx0.4$), and its observed smoothness, thereby constraining the subsurface magnetic flux-tube structure.

Background

A sunspot consists of a dark core (umbra) surrounded by a filamentary annulus (penumbra). Traditionally the umbra–penumbra (UP) boundary is defined operationally in continuum images as the contour where the intensity equals 50% of the quiet-Sun continuum, but this definition depends on wavelength, spatial resolution, and instrumental scattered light. Spectropolarimetric studies instead characterized the boundary magnetically: the vertical component of the magnetic field in the local reference frame, $B_\mathrm{ver}$ (the field component along the local surface normal, $B\cos\gamma$ with $\gamma$ the inclination to the vertical), is approximately constant along the umbral boundaries of stable sunspots, while the total field strength and inclination vary. A Bayesian analysis of 114 umbral boundaries observed with the Hinode solar optical telescope found a most likely value of 1867 G (99% interval 1849–1885 G), with method- and resolution-dependent offsets in other instruments (about 1693 G for SDO/HMI, the Helioseismic and Magnetic Imager on the Solar Dynamics Observatory, and 1787 ± 100 G for the GREGOR Infrared Spectrograph); regions with $B_\mathrm{ver}$ below this critical value are observationally prone to being colonized by the penumbral mode of magnetoconvection, i.e., convection modified by the presence of magnetic field. This empirical rule is called the Jurčák criterion. The Evershed flow is the well-known nearly horizontal, radially outward flow of a few km/s along penumbral filaments; its simulation counterpart is a standard test of sunspot models. The physical origin of the criterion is unclear. Classical linear stability analysis of convection in the presence of a vertical magnetic field shows that the vertical component is the field quantity that stabilizes against overturning, and the analysis of Gough and Tayler (1966) extends Schwarzschild's convective stability criterion for compressible gas to include the stabilizing effect of a vertical field; in its simplest form convection is inhibited where $1/\Gamma_1-d\ln\rho/d\ln p < B_\mathrm{ver}^2/(B_\mathrm{ver}^2+4\pi\Gamma_1 p)$, with $\Gamma_1$ the first adiabatic exponent, $\rho$ the density, and $p$ the gas pressure. Mullan and MacDonald (2019) connected this criterion to the observed boundary value, but only as an interpretation. On the simulation side, sunspots are modeled with radiative magnetohydrodynamics (MHD) codes such as MURaM (the Max Planck–University of Chicago Radiative MHD code); extended penumbrae are obtained only by artificially enhancing the horizontal magnetic field at the top boundary of the computational box, which produces boundary inclinations near 60° (observed values peak below about 45°) and Evershed flows, but then the magnetic properties at the UP boundary disagree with observations; runs with a self-consistent potential-field extrapolation at the top boundary reproduce boundary magnetic properties better but lack a proper penumbra and Evershed flow, and place the UP boundary too far out ($r/r_\mathrm{spot}$ of 0.52–0.64 versus about 0.4 observed), with $r_\mathrm{spot}$ the spot radius. The magnetopause (the interface between the sunspot's magnetic flux tube and the surrounding field-free plasma) is shaped by the subsurface expansion of the flux tube and, where it meets the surface, coincides with the equipartition field strength $B_\mathrm{eq}$, where gas and magnetic pressures are equal. Jurčák, Schmassmann, Rempel, Bello González and Schlichenmaier (2020) analyzed a set of such MURaM simulations, found that none is simultaneously consistent with the observed UP-boundary magnetic properties and the direction of the Evershed flow, that the simulated UP boundary is much more corrugated than observed, and concluded that a key physical ingredient or feedback is missing from the models — the concrete motivation for this problem.

Problem Statement

What physical mechanism establishes and maintains the umbra–penumbra boundary of a stable sunspot at a sharply defined vertical magnetic field component — the Jurčák criterion, with $B_\mathrm{ver}$ at the boundary invariant at about 1.87 kG in Hinode data (with known instrument-dependent offsets of roughly 1.6–1.8 kG for HMI and ground-based data) — and is the stabilizing effect of the vertical field on convection, formalized in the Gough–Tayler criterion, the responsible mechanism in penumbral magnetoconvection, or is another process at work? The overarching objective is a physically explained umbra–penumbra boundary: the answer must identify the mechanism, show quantitatively that it predicts an approximately invariant boundary vertical field with the observed magnitude (or, alternatively, demonstrate conclusively that the invariance is an artifact, e.g., of averaging over size-dependent properties of penumbral filaments and of spines, the stronger, more vertical field components between filaments), and account for the associated boundary observables — field inclinations reaching at most about 45°, boundary position at $r/r_\mathrm{spot}\approx0.4$, and a boundary substantially smoother than current simulations produce. Because the source's own evidence that the mechanism is not understood is that radiative MHD simulations fail jointly, a complete answer on the mechanism route must also demonstrate the mechanism in sunspot simulations: produce a simulation that simultaneously reproduces the observed magnetic properties at the umbra–penumbra boundary and the correct direction and strength of the Evershed flow, thereby identifying the missing ingredient or feedback, and use the matched boundary position and smoothness to constrain the subsurface magnetic flux-tube structure and its expansion with height. On the artifact route, a conclusive demonstration that the invariance is an artifact, together with an account of the associated boundary observables, is itself a resolving answer, and no joint simulation is required on this route. The two routes are alternative ways of resolving the same problem: within the mechanism route the identification of the mechanism, the quantitative prediction, the joint simulation and the subsurface flux-tube constraint are linked aspects of one objective — a self-consistent physical account of what defines the umbra–penumbra boundary — not independent targets.

References

1. J. Jurčák, M. Schmassmann, M. Rempel, N. Bello González, R. Schlichenmaier (2020), 'A distinct magnetic property of the inner penumbral boundary III. Analysis of simulated sunspots', A&A 638, A28, DOI 10.1051/0004-6361/202037852, arXiv:2004.03940, https://arxiv.org/abs/2004.03940
2. J. Jurčák, R. Rezaei, N. Bello González, R. Schlichenmaier, J. Vomlel (2018), 'The magnetic nature of umbra-penumbra boundary in sunspots', A&A 611, L4, DOI 10.1051/0004-6361/201732528, arXiv:1801.08983, https://arxiv.org/abs/1801.08983
3. M. Schmassmann, R. Schlichenmaier, N. Bello González (2018), 'Magnetic properties of a long-lived sunspot', A&A 620, A104, DOI 10.1051/0004-6361/201833441, https://doi.org/10.1051/0004-6361/201833441
4. D. O. Gough, R. J. Tayler (1966), 'The Influence of a Magnetic Field on Schwarzschild's Criterion for Convective Instability in an Ideally Conducting Fluid', MNRAS 133, 85, DOI 10.1093/mnras/133.1.85, https://doi.org/10.1093/mnras/133.1.85
5. D. J. Mullan, J. MacDonald (2019), 'The umbral-penumbral boundary in sunspots in the context of magnetoconvection', ApJ 873, L10, DOI 10.3847/2041-8213/ab0a0f, arXiv:1902.09431, https://arxiv.org/abs/1902.09431
6. P. Lindner, R. Schlichenmaier, N. Bello González (2020), 'Characterization of the umbra-penumbra boundary by the vertical component of the magnetic field — Analysis of ground-based data from the GREGOR Infrared Spectrograph', A&A 638, A25, DOI 10.1051/0004-6361/202037716, arXiv:2004.09956, https://arxiv.org/abs/2004.09956
7. B. Löptien, A. Lagg, M. van Noort, S. K. Solanki (2020), 'No universal connection between the vertical magnetic field and the umbra-penumbra boundary in sunspots', A&A 639, A106, DOI 10.1051/0004-6361/202037974, arXiv:2006.02346, https://arxiv.org/abs/2006.02346
8. M. Schmassmann, M. Rempel, N. Bello González, R. Schlichenmaier, J. Jurčák (2021), 'Characterization of magneto-convection in sunspots — The Gough-Tayler stability criterion in MURaM sunspot simulations', A&A 656, A92, DOI 10.1051/0004-6361/202141607, https://doi.org/10.1051/0004-6361/202141607
9. B. Löptien, A. Lagg, M. van Noort, S. K. Solanki (2021), 'Similarities of magnetoconvection in the umbra and in the penumbra of sunspots', A&A 655, A61, DOI 10.1051/0004-6361/202141440, arXiv:2110.01352, https://arxiv.org/abs/2110.01352
10. V. Efremov, L. Parfinenko, A. Soloviev (2021), 'Vertical magnetic field on boundary of sunspot umbra', arXiv:2101.11675, https://arxiv.org/abs/2101.11675
11. M. García-Rivas, J. Jurčák, N. Bello González (2021), 'Magnetic properties on the boundary of an evolving pore', A&A 649, A129, DOI 10.1051/0004-6361/202039661, arXiv:2102.08459, https://arxiv.org/abs/2102.08459
12. M. García-Rivas, J. Jurčák, N. Bello González (2024), 'Magnetic properties of the umbral boundary during sunspot decay. Comparative study of multiple datasets', A&A 689, A160, DOI 10.1051/0004-6361/202348068, arXiv:2406.15272, https://arxiv.org/abs/2406.15272
13. M. García-Rivas, J. Jurčák, N. Bello González, J. M. Borrero, R. Schlichenmaier, P. Lindner (2024), 'Onset of penumbra formation', A&A 686, A112, DOI 10.1051/0004-6361/202348764, arXiv:2403.18455, https://arxiv.org/abs/2403.18455
14. P. Lindner, C. Kuckein, S. J. González Manrique, N. Bello González, L. Kleint, T. Berkefeld (2023), 'The role of the chromospheric magnetic canopy in the formation of a sunspot penumbra', A&A 673, A64, DOI 10.1051/0004-6361/202245702, arXiv:2303.07112, https://arxiv.org/abs/2303.07112
15. I. Chifu, N. Bello González, J. Jurčák (2025), 'Advancing understanding of sunspot penumbra formation with non-linear force-free field extrapolations', A&A 699, A218, DOI 10.1051/0004-6361/202554738, https://doi.org/10.1051/0004-6361/202554738
16. M. Schmassmann, N. Bello González, R. Schlichenmaier, J. Jurčák (2026), 'Sunspot simulations with MURaM — I. Parameter study using potential field initial conditions', A&A 708, A123, DOI 10.1051/0004-6361/202558785, arXiv:2601.23189, https://arxiv.org/abs/2601.23189