Whether a sunspot is magnetically coherent below the visible solar surface — a single monolithic flux tube — or fragments into a cluster ('spaghetti') of separated flux strands interspersed with almost field-free plasma remains observationally unsettled. The question controls our understanding of sunspot stability, energy transport, and decay. Helioseismic inversions, magnetohydrodynamic simulations, and oscillation-source diagnostics give partly conflicting indications, and proposed seismic discriminators have not yet been applied to a decisive observational test.

Background

A sunspot is a long-lived dark region on the solar photosphere (the visible surface layer of the Sun) where a strong magnetic field, typically 2000–3500 G, blocks convective heat transport. Its dark central umbra is surrounded by a filamentary penumbra. What lies directly beneath the visible surface — the subsurface magnetic topology that anchors the spot — cannot be imaged directly, and two competing idealized pictures have coexisted since the 1970s. In the monolithic model the umbra is the top of a single coherent magnetic flux tube (a bundle of magnetic field lines threading a tube of plasma), possibly widening flask-like with depth to stay stable against the interchange (or 'fluting') instability, in which the boundary between magnetized and weakly magnetized plasma buckles into fingers when the field lines curve too steeply. In the cluster or 'spaghetti' model, proposed by Parker (1979), the umbral flux below the surface is split into many separate, strongly compressed flux strands with nearly field-free plasma between them; the strands are held together by converging plasma flows rather than by magnetic coherence (Parker 1979, https://doi.org/10.1086/157150; Solanki 2003, https://doi.org/10.1007/s00159-003-0018-4). The distinction matters physically: it decides how sunspots remain thermally and mechanically stable for weeks, how magnetoconvection transports energy inside them, why they decay, and what flux-emergence models should assume about their roots. The main observational probe is local helioseismology, which uses solar oscillation waves that traverse the interior; in the time-distance variant, travel-time shifts of waves crossing a sunspot are inverted for subsurface flow and wave-speed perturbations. Zhao, Kosovichev and Duvall (2001, https://doi.org/10.1086/321491) inferred depth-stratified flows beneath a sunspot — converging inflows in the near-surface layers and downward, vortex-like motions at greater depth — which Kosovichev (2006, https://doi.org/10.1016/j.asr.2005.04.091) judged to give empirical support to Parker's cluster picture. However, as Kirichek and Solov'ev (2009, https://doi.org/10.1017/S1743921310009737) emphasize, the effects of temperature and of magnetic field on wave propagation speed have not yet been separated, so the deep magnetic structure of sunspots is still constrained by theory more than by observation. The theory itself is split: realistic radiative magnetohydrodynamic (MHD) simulations — numerical models of the solar plasma as an electrically conducting, magnetized fluid including radiative energy transport — initialized with a monolithic trunk reproduce observed sunspot properties (Rempel 2011, https://doi.org/10.1088/0004-637x/740/1/15), and convective-dynamo-driven emergence simulations form mostly monolithic spots anchored in deep downdraft lanes (Chen, Rempel and Fan 2017, https://doi.org/10.3847/1538-4357/aa85a0); yet fluting-instability experiments show curved subsurface tubes fragmenting from below into strand-like structures (Panja, Cameron and Solanki 2021, https://doi.org/10.3847/1538-4357/abccbf), and current realistic simulations essentially all assume a monolithic configuration for lack of an observational verdict (Bhatia et al. 2025, https://doi.org/10.1051/0004-6361/202452747). Two diagnostic lines have been developed toward an observational test. First, forward wave-propagation computations show that monolithic and spaghetti sunspot models leave measurably different seismic signatures — mode absorption coefficients, the behavior of phase shift with mode degree (the mode's horizontal angular wavenumber), and mode mixing (Felipe, Crouch and Birch 2014, https://doi.org/10.1088/0004-637x/788/2/136), and, most recently, the location and pattern of caustics behind the spot in f-mode travel-time maps (Daiffallah and Boumia 2025, https://doi.org/10.1007/s11207-025-02436-w), where the f-mode is the oscillation mode trapped at the solar surface. Second, Cho, Chae and Madjarska (2021, https://doi.org/10.1051/0004-6361/202141500) tracked umbral flashes — recurrent brightenings of the sunspot chromosphere produced by magnetoacoustic shock waves launched by convection cells below the photosphere — and used the inferred oscillation centers to map those convection cells, finding preferential location under dark umbral cores, which they read as tentative evidence for the cluster model while stating that no clear observational evidence supports either model to date.

Problem Statement

Determine, from observations of the Sun, the subsurface magnetic structure of sunspots: beneath the photosphere, does a sunspot's umbral magnetic flux remain a single coherent monolithic flux tube (at most a flask-like widening trunk), or is it partitioned into a cluster/'spaghetti' of separated flux strands interspersed with almost field-free plasma, and how is the topology distributed with depth? An admissible resolution is a determination, using any subsurface-sensitive observational diagnostic (for example local helioseismology, or mapping of subsurface convection cells via umbral-flash oscillation sources), that discriminates the monolithic and cluster topologies — or establishes where the actual structure lies between these idealizations as a function of depth — with quantified confidence. The problem is posed for sunspots in their natural generality: no restriction on depth range, field strength, or geometry is imposed, and a conclusion limited to particular spots or depths must be stated with that scope, as partial progress toward the general determination rather than a complete resolution of it.

References

  1. Cho, Kyuhyoun; Chae, Jongchul; Madjarska, Maria S. (2021), 'Investigation of the subsurface structure of a sunspot based on the spatial distribution of oscillation centers inferred from umbral flashes', Astronomy & Astrophysics 656, A86, DOI 10.1051/0004-6361/202141500, https://doi.org/10.1051/0004-6361/202141500 (arXiv:2109.11185, https://arxiv.org/abs/2109.11185)
  2. Kosovichev, Alexander G. (2006), 'Subsurface characteristics of sunspots', Advances in Space Research 38, 876, DOI 10.1016/j.asr.2005.04.091, https://doi.org/10.1016/j.asr.2005.04.091
  3. Kirichek, Elena A.; Solov'ev, Alexandr A. (2009), 'Magnetic structure of sunspot under the photosphere', Proceedings of the International Astronomical Union 5 (IAU Symposium S273: Physics of Sun and Star Spots), 351, DOI 10.1017/S1743921310009737, https://doi.org/10.1017/S1743921310009737
  4. Zhao, Junwei; Kosovichev, Alexander G.; Duvall, Thomas L., Jr. (2001), 'Investigation of Mass Flows beneath a Sunspot by Time-Distance Helioseismology', The Astrophysical Journal 557, 384, DOI 10.1086/321491, https://doi.org/10.1086/321491
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  7. Rempel, Matthias (2011), 'Subsurface Magnetic Field and Flow Structure of Simulated Sunspots', The Astrophysical Journal 740, 15, DOI 10.1088/0004-637x/740/1/15, https://doi.org/10.1088/0004-637x/740/1/15 (arXiv:1106.6287, https://arxiv.org/abs/1106.6287)
  8. Chen, Feng; Rempel, Matthias; Fan, Yuhong (2017), 'Emergence of Magnetic Flux Generated in a Solar Convective Dynamo. I. The Formation of Sunspots and Active Regions, and The Origin of Their Asymmetries', The Astrophysical Journal 846, 149, DOI 10.3847/1538-4357/aa85a0, https://doi.org/10.3847/1538-4357/aa85a0 (arXiv:1704.05999, https://arxiv.org/abs/1704.05999)
  9. Panja, Mayukh; Cameron, Robert H.; Solanki, Sami K. (2021), 'Sunspot Simulations: Penumbra Formation and the Fluting Instability', The Astrophysical Journal 907, 102, DOI 10.3847/1538-4357/abccbf, https://doi.org/10.3847/1538-4357/abccbf (arXiv:2011.11447, https://arxiv.org/abs/2011.11447)
  10. Felipe, T.; Crouch, A. D.; Birch, A. C. (2014), 'Evaluation of the Capability of Local Helioseismology to Discern between Monolithic and Spaghetti Sunspot Models', The Astrophysical Journal 788, 136, DOI 10.1088/0004-637x/788/2/136, https://doi.org/10.1088/0004-637x/788/2/136 (arXiv:1405.0036, https://arxiv.org/abs/1405.0036)
  11. Daiffallah, Khalil; Boumia, Omar (2025), 'Travel Time Estimation from Numerical Simulations of f-Mode Propagation Through Sunspot Models and Plages', Solar Physics 300, 18, DOI 10.1007/s11207-025-02436-w, https://doi.org/10.1007/s11207-025-02436-w
  12. Bhatia, Tanayveer Singh; Panja, Mayukh; Cameron, Robert H.; Solanki, Sami K. (2025), '3D radiative MHD simulations of starspots - II. Large-scale structure', Astronomy & Astrophysics 693, A264, DOI 10.1051/0004-6361/202452747, https://doi.org/10.1051/0004-6361/202452747 (arXiv:2412.16921, https://arxiv.org/abs/2412.16921)