Sunspot umbrae host umbral dots, small bright convective intrusions whose relationship to the subsurface structure of the spot is not established. Parker's cluster ('spaghetti') model posits that the umbral magnetic flux is partitioned into separate flux strands embedded in nearly field-free plasma — in Parker's (1979) original formulation, immediately below the visible surface — with umbral dots as the surface tops of field-free gaps; the competing monolithic picture is a single coherent flux tube in which umbral dots arise from shallow magnetoconvection. Shallow-domain simulations reproduce umbral dots without any field-free intrusion from below, from which Schüssler & Vögler (2006) inferred that a hypothetical cluster-type fragmentation, if it occurs, must lie significantly below their bottom boundary (about 1.2 Mm below the photospheric level), so these calculations constrain but cannot decide the question. The problem is to determine the subsurface magnetic configuration of real sunspot umbrae — monolithic or fragmented, and if fragmented at what depth — and whether umbral dots (in particular their tails and any opposite-polarity substructure) carry a detectable surface signature of that deeper structure.

Background

A sunspot consists of a dark central umbra, where a strong (1-3 kG) nearly vertical magnetic field suppresses normal convection, surrounded by a filamentary penumbra with more inclined fields. Even in the umbra the surface is not completely dark: it is dotted with umbral dots (UDs), bright point-like or elongated features a few hundred km across, conventionally divided into central umbral dots (CUDs) deep in the umbra and peripheral umbral dots (PUDs) near the umbra-penumbra boundary. Magnetoconvection — convection modified by an imposed magnetic field — is the accepted mechanism producing UDs, but the structure of the field below the visible surface remains contested.

Two idealized pictures frame the debate (Parker 1979; Solanki 2003). In the monolithic model the umbra is a single coherent flux tube; convection in its strong field is feeble, producing only the small, faint umbral dots observed at the photosphere. In Parker's cluster ('spaghetti') model the umbral flux is broken into separate, strongly compressed flux strands separated by field-free plasma; the spot is held together by subsurface converging flows, the visible umbra is the merged crown of the strand cluster, and umbral dots are the photospheric tops of hot field-free gaps between strands. In Parker's (1979) original formulation the field splits into many separate flux tubes immediately below the visible surface — within about 10310^3 km, with the umbral-dot gas rising in columns pushed up from depths of 500-1500 km (quoted as about 1 Mm below the photosphere by Strecker et al. 2021). Deep-anchored successor variants also exist — e.g. the Spruit (1981) 'tethered balloon'-type cluster described in Panja et al. (2021), in which the tube remains coherent up to a certain depth, beyond which it is fragmented into small individual flux tubes tied together at the base of the convection zone — without a specific numeric fragmentation depth.

Radiative magnetohydrodynamic (MHD) simulations with the MURaM code (which solves compressible MHD with realistic radiative transfer) reproduce umbral dots as narrow, cusp-shaped overturning convective plumes within a monolithic vertical field (Schüssler & Vögler 2006; Rempel, Schüssler & Knölker 2009; statistical properties of simulated UDs: Bharti, Beeck & Schüssler 2010). This demonstrates that umbral dots do not require a cluster-type subsurface structure. However, the umbral-simulation boxes of Schüssler & Vögler are only 1.6 Mm deep, covering roughly 1200 km below to 400 km above the photospheric level (optical depth τ=1\tau=1), and the whole-spot simulations extend to about 6 Mm. Because no intrusion of field-free plasma from the underlying convection zone was required to explain the simulated umbral dots, the authors argue that a hypothetical fragmentation of the magnetic field into a cluster of flux strands, if it occurs, has to lie significantly below their bottom boundary — an inferred lower bound that cannot exclude field-free intrusions rising from greater depth and leaves the relation of umbral dots to deeper fragmentation open (Schüssler & Vögler 2006).

Three later strands partially constrain the problem. (1) Deeper simulations: in 16-18 Mm deep MURaM domains, the subsurface field interacts with convection and partially separates from the main trunk, with flux separation in deeper layers surfacing within hours as light bridge and pore formation (Rempel 2011); stability analysis of an 18 Mm domain shows buoyancy-stabilized field above about 2 Mm, fluting instability (an interchange instability that corrugates the flux-tube boundary) peaking near 3 Mm, perimeter corrugation growing fastest near 8 Mm, and field-free intrusions at depth driving decay that is not yet visible at the surface (Strecker et al. 2021; fluting degree controlled by flux-tube curvature: Panja et al. 2021). Analysis with the Gough-Tayler criterion (a linear criterion for convective stability of a stratified medium with a vertical magnetic field) finds a deep umbral core below about 5 Mm that is stabilized against overturning convection, overlaid by slender unstable columns coupled to umbral dots that saturate near 4 Mm depth (Schmassmann et al. 2021). (2) The penumbral analogue: the field-free gap model of penumbral filaments (Spruit & Scharmer 2006) predicts opposite-polarity magnetic fields at filament tails, confirmed observationally in penumbrae (Scharmer et al. 2013) and in simulations (Bharti & Rempel 2019); the corresponding prediction for PUD tails remains untested — Hinode (a Japanese solar-observation satellite carrying a high-resolution spectropolarimeter) observations find no opposite-polarity BzB_z (the vertical component of the magnetic field) there, either because the reversal sits below the visible layer or because it is too weak to detect (Löptien et al. 2021). (3) Subsurface diagnostics: time-distance helioseismology (inferring subsurface flows and wave-speed perturbations from travel times of acoustic waves) has mapped flows beneath sunspots (Zhao, Kosovichev & Duvall 2001), and f-mode (surface gravity wave) scattering has been studied theoretically as a seismic discriminator between monolithic and clustered flux configurations (Daiffallah 2013), but local helioseismic inversions beneath strong, inclined fields carry large systematic uncertainties (Kosovichev 2012). A recent analysis tracking oscillation centers of umbral flashes (brief chromospheric brightenings above the umbra, produced by upward-propagating magnetoacoustic waves) below the photosphere was interpreted as indirect support for the cluster model, while the authors themselves note that no clear observational evidence for either model existed previously and call for further study (Cho, Chae & Madjarska 2021). Statistical studies find UD properties essentially uncorrelated with macroscopic spot properties, suggesting control by near-surface convection, but the depth extent and anchoring of that convection to deeper processes remain uncertain (Yadav, Louis & Mathew 2018).

Problem Statement

For real sunspots, determine the subsurface magnetic configuration of the umbra and its observable relationship to umbral dots, preserving the natural generality of the question (no fixed depth, spot size, or feature class is presupposed). Concretely: (1) Is the umbral magnetic field below the photosphere fragmented into discrete flux strands separated by field-free or weak-field plasma — the cluster-type configuration Parker proposed — or does it remain a coherent (monolithic) flux tube down through the upper convection zone? (2) If the field is fragmented, at what depth below the photosphere does the fragmentation set in, and how does the fragmentation level relate to spot size, field strength, and evolutionary phase? (3) Do umbral dots carry any detectable surface signature of this deeper structure — in particular, is there opposite-polarity magnetic field in the tails of peripheral umbral dots or beneath umbral dots at depths below the visible surface — or are umbral dots purely the photospheric expression of shallow magnetoconvection (roughly the top few megameters) dynamically decoupled from whatever lies deeper? Answers may combine deep-domain radiative MHD simulations, helioseismic inversions of subsurface field and flow topology, and high-resolution, depth-sensitive spectropolarimetry of umbral dots; a complete answer must state which subsurface configuration holds for real sunspots (the determination may be depth-stratified and may vary with spot parameters), over what depth range, and why the observed umbral-dot properties are (or are not) a signature of it.

References

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