Time–distance helioseismic inversions of Dopplergrams (full-disk maps of the line-of-sight velocity of the solar surface) from the Helioseismic and Magnetic Imager (HMI) on the Solar Dynamics Observatory (SDO), ensemble-averaged over 104 symmetric H-type sunspots, established that the shallow subsurface moat (0–3 Mm depth) around mature sunspots is a pure downflow region whose mass submergence rate is at least twice the mass circulation rate of an average supergranular cell, implying compensating upflow(s) within or adjacent to the strongly magnetised umbra and penumbra (Švanda, Sobotka & Bárta 2014). Because travel-time inversions are unreliable in strong magnetic fields, the location, geometry, and amplitude of these upflows remain unmeasured, and the quantitative confrontation with state-of-the-art magnetohydrodynamic (MHD) sunspot simulations that the source paper called for has not been performed. Later work supplies a candidate mechanism on the simulation side (Rempel 2015: a mass-flux-balance-driven overturning circulation) and documents the methodological obstacles (Braun 2019; DeGrave et al. 2018; Li et al. 2026), but no measurement of the vertical mass budget or like-for-like model comparison exists. The problem is to determine the subsurface moat-flow mass budget — in particular the compensating upflows — with methods validated in strong-field regions, and to confront the ensemble-averaged observables with radiative MHD simulations through identical analysis pipelines.

Background

Sunspots are dark, strongly magnetised regions of the solar photosphere. A mature, symmetric sunspot of McIntosh type H consists of a dark central umbra surrounded by a filamentary penumbra; the penumbra hosts the Evershed flow, a near-horizontal radial outflow of 2–5 km/s at the photospheric level. Beyond the penumbral boundary extends a slower horizontal outflow of a few hundred m/s, the moat flow, classically studied at the surface by tracking small magnetic features drifting away from the spot and by Doppler measurements. Supergranulation is the dominant convection pattern of the quiet Sun: roughly 30 Mm wide cells (1 Mm = 1000 km) with horizontal outflows from cell centres to the surrounding lanes, weak upflows (about 4 m/s) near the centres and downflows at the lanes.

Local helioseismology infers subsurface flows from the propagation of solar acoustic waves (p-modes). In time–distance helioseismology one measures the travel times of wave packets crossing between pairs of points on the surface and converts travel-time differences into subsurface flow vectors by an inversion; the inversion also returns an averaging kernel, a weight function characterising the horizontal resolution and depth sensitivity of each estimate. The Helioseismic and Magnetic Imager (HMI) on the Solar Dynamics Observatory (SDO) provides the full-disk Dopplergrams — maps of the velocity component along the observer's line of sight — used for such measurements. Švanda, Sobotka & Bárta (2014) applied an improved multichannel SOLA inversion (SOLA, subtractive optimally localised averaging: a standard scheme that linearly combines many travel-time measurements to estimate a local weighted average of the flow) to HMI data, ensemble-averaged over 104 H-type sunspots (with about 223,000 supergranular cells as a control sample), with averaging kernels of horizontal full width at half maximum of about 10 Mm, depth-averaged over 0–3 Mm (kernel centroid near 1.0 Mm). They found that the shallow moat replaces the supergranular circulation but differs from it in two ways: the moat flow is asymmetric in the east–west direction, attributed to the proper motion of the sunspots relative to the local frame of rest, and the whole moat is a downflow region, with a peak azimuthally averaged downflow speed of about 1.5 m/s and an estimated mass submergence rate at least twice the mass circulation rate of an average supergranular cell. By mass continuity, this sinking mass must be compensated by upflow(s) in or near the strongly magnetised sunspot region — for example, a homogeneous annular upflow of at least about 20 m/s under the umbra if confined to a 500-km-wide ring — but the paper shows that current helioseismic techniques cannot determine the actual structure: inside strong fields, wave absorption, mode conversion, and the Wilson depression (the geometric lowering of the visible sunspot surface within the magnetic field) corrupt the travel-time measurements, and the relevant magnetoacoustic interaction physics is not known well enough for reliable inversions in the umbra and penumbra. State-of-the-art radiative magnetohydrodynamic (MHD) sunspot simulations do produce moat-like outflows at all near-surface depths (Rempel 2011), and the source paper explicitly calls for confronting its ensemble-averaged measurements — the vertical mass fluxes, the east–west asymmetry, and the necessity of compensating upflows — with such simulations, noting that the coarse spatial and depth resolution also smoothes out small-scale, transient, and strongly localised flows, and that its sunspot drift-speed estimate of roughly 120 m/s, obtained by decomposing the ensemble-averaged flow field, is unreliable.

Problem Statement

Determine the subsurface structure and mass budget of the moat-flow circulation around mature symmetric (McIntosh H-type) sunspots in the shallow convection zone (the top roughly 3 Mm below the photosphere, the depth range probed by the ensemble-averaged inversions of Švanda, Sobotka & Bárta 2014), as follows. A complete resolution requires items (1) and (2) together; item (3) is an optional secondary extension.

(1) Establish, from observations of the real-Sun ensemble, or by an inference whose reliability in strong-field regions is demonstrated by the recovery test described below, the location, geometry, and amplitude of the upflow(s) that must compensate the shallow moat downflow: the downflow whose azimuthally averaged speed peaks near 1.5 m/s and whose mass submergence rate is at least twice the mass circulation rate of an average supergranular cell, with the upflow(s) located within or immediately adjacent to the strongly magnetised umbra and penumbra. Reliability in strong-field regions must be demonstrated, for example by applying the identical analysis pipeline to synthetic data from magnetohydrodynamic sunspot simulations with known flow fields and quantifying the recovery of the vertical component inside the penumbral boundary; and the recovered geometry must be resolved at scales finer than the roughly 10 Mm averaging kernels that motivated the original limitation.

(2) Quantitatively confront the ensemble-averaged helioseismic observables of the shallow moat — the vertical mass flux as a function of distance from the spot centre, the east–west asymmetry of the horizontal moat flow, and the necessity of compensating upflows — with state-of-the-art radiative magnetohydrodynamic sunspot simulations, by processing synthetic data from the simulations through the same helioseismic and ensemble-averaging pipeline as the observations so that the comparison is like-for-like, and stating whether the simulations reproduce the measured mass fluxes, asymmetry, and required upflows within uncertainties.

(3) Optionally, as a secondary extension tied to the interpretation of the asymmetry: provide a direct and robust determination of sunspot proper motions at the analysed depths, replacing the indirect decomposition estimate of roughly 120 m/s that the source authors themselves judged unreliable.

Answers that address only the surface manifestation of the moat flow, or only its horizontal component, do not answer this problem: the open core is the vertical mass budget beneath and around the magnetised region and its verification against simulations.

The observational and simulation data required by (1) and (2), and the common analysis pipeline applied to both, are not supplied with this statement; obtaining and processing them is part of the task.

References

  1. Švanda, M., Sobotka, M., Bárta, T. 2014, 'Moat flow system around sunspots in shallow subsurface layers', The Astrophysical Journal 790, 135. DOI 10.1088/0004-637X/790/2/135, arXiv:1406.2482, https://doi.org/10.1088/0004-637X/790/2/135
  2. Rempel, M. 2015, 'Numerical simulations of sunspot decay: On the penumbra–Evershed flow–moat flow connection', The Astrophysical Journal 814, 125. DOI 10.1088/0004-637X/814/2/125, arXiv:1511.01410, https://doi.org/10.1088/0004-637X/814/2/125
  3. Rempel, M. 2011, 'Subsurface magnetic field and flow structure of simulated sunspots', The Astrophysical Journal 740, 15. DOI 10.1088/0004-637X/740/1/15, arXiv:1106.6287, https://doi.org/10.1088/0004-637X/740/1/15
  4. Braun, D. C. 2019, 'Flows around Averaged Solar Active Regions', The Astrophysical Journal 873, 94. DOI 10.3847/1538-4357/ab04a3, arXiv:1902.02298, https://doi.org/10.3847/1538-4357/ab04a3
  5. DeGrave, K., Braun, D. C., Birch, A. C., Crouch, A. D., Javornik, B. 2018, 'Validating Forward Modeling and Inversions of Helioseismic Holography Measurements', The Astrophysical Journal 863, 34. DOI 10.3847/1538-4357/aacffd, arXiv:1807.03841, https://doi.org/10.3847/1538-4357/aacffd
  6. Verma, M., Kummerow, P., Denker, C. 2018, 'On the extent of the moat flow in axisymmetric sunspots', Astronomische Nachrichten 339, 268. DOI 10.1002/asna.201813482, arXiv:1805.04356, https://doi.org/10.1002/asna.201813482
  7. Strecker, H., Bello González, N. 2018, 'Evolution of the flow field in decaying active regions: Transition from a moat flow to a supergranular flow', Astronomy & Astrophysics 620, A122. DOI 10.1051/0004-6361/201732164, arXiv:1811.01607, https://doi.org/10.1051/0004-6361/201732164
  8. Strecker, H., Bello González, N. 2022, 'Evolution of the flow field in decaying active regions II. Converging flows at the periphery of naked spots', Astronomy & Astrophysics 664, A195. DOI 10.1051/0004-6361/202142564, arXiv:2208.14272, https://doi.org/10.1051/0004-6361/202142564
  9. Rabello-Soares, M. C., Bogart, R. S., Scherrer, P. H. 2018, 'Statistical Analysis of Acoustic Wave Power and Flows around Solar Active Regions', The Astrophysical Journal 859, 7. DOI 10.3847/1538-4357/aabba5, arXiv:1804.07791, https://doi.org/10.3847/1538-4357/aabba5
  10. Kosovichev, A. G., Sadykov, V. M. 2023, 'Subsurface Flows Associated with Formation and Flaring Activity of Solar Active Regions', Proceedings of the International Astronomical Union 19, 275–295. DOI 10.1017/S1743921324000991, arXiv:2401.17598, https://doi.org/10.1017/S1743921324000991
  11. Gottschling, N., Schunker, H., Birch, A. C., Löptien, B., Gizon, L. 2021, 'Evolution of solar surface inflows around emerging active regions', Astronomy & Astrophysics 652, A148. DOI 10.1051/0004-6361/202140324, arXiv:2105.10501, https://doi.org/10.1051/0004-6361/202140324
  12. Švanda, M., Jurčák, J., Schmassmann, M. 2025, 'Average solar active region: I. Intensities, velocities, and the photospheric magnetic field', Astronomy & Astrophysics 700, A40. DOI 10.1051/0004-6361/202554440, arXiv:2506.11713, https://doi.org/10.1051/0004-6361/202554440
  13. Li, H., Felipe, T., Khomenko, E., Tian, H., Rajaguru, P., Gao, Y. 2026, 'Understanding the Travel-time Asymmetry of Acoustic Waves in Sunspots With Time-distance Helioseismology', The Astrophysical Journal 1008, 41. DOI 10.3847/1538-4357/ae90ac, arXiv:2607.26006, https://doi.org/10.3847/1538-4357/ae90ac