The shape of the solar tachocline: robust high-latitude determination, possible sharp latitudinal structure, and the implied interior magnetic field

Helioseismic forward modeling with 2304- and 4608-day mode-splitting time series shows that the solar tachocline — the shear layer at the base of the convection zone — is not simply prolate but bulges into the convection zone at mid latitudes, and its center migrates between the radiative and convective zones as a function of latitude. Three scientifically linked issues, however, remain open: the tachocline's position and width poleward of about 60 degrees are unconstrained and model-dependent because mode sensitivity degrades toward the poles; it is unsettled whether the tachocline position varies smoothly with latitude or is sharply structured, since a discontinuous model is disfavored by forward-fit chi-squared while a 2026 inversion study claims a clear bifurcation but explicitly requests independent confirmation; and the inference that a very strong (>200 kG) low-latitude toroidal magnetic field is present rests entirely on magnetohydrodynamic equilibrium modeling rather than any direct observational constraint. The problem is to determine the tachocline's latitudinal profile of position and width, including at high latitudes and for possible localized structure, with uncertainties robust across independent methodologies, and to establish the magnetic field strength and geometry that this shape implies.

Background

The Sun oscillates in millions of global acoustic modes (p-modes, pressure-driven standing sound waves). Because the solar interior rotates, modes with the same radial order but different azimuthal order are split in frequency; these rotational splittings are the primary data for inferring the internal rotation rate, a discipline called helioseismology. Inversions of splittings show that the convection zone (CZ) rotates differentially — faster at the equator, slower at the poles — while the radiative interior below rotates nearly as a solid body. The thin shear layer joining the two, centered near $0.7R_\odot$ ($R_\odot$ is the solar radius), is the tachocline, widely held to be the seat of the large-scale solar dynamo, where differential shear winds weak poloidal fields into strong toroidal ones. The tachocline's geometry matters physically: several dynamo models assume a toroidal field with a $\cos^2\vartheta$ colatitude profile ($\vartheta$ being colatitude), and the shape of the tachocline controls how much magnetic flux can be stored there. Early analyses (Antia et al. 1998; Charbonneau et al. 1999) modeled the tachocline position as a function with only a $\cos^2\vartheta$ dependence, which by construction can only be prolate (deeper/thicker at the poles), oblate, or spherical; they concluded the tachocline is prolate. Latitude-by-latitude (1D) fits by Basu & Antia (2003) hinted at a near-discontinuous jump in tachocline position around latitude $30^\circ$, but not at statistically significant levels. Basu & Korzennik (2025) re-analyzed the question with forward modeling (fitting parameterized 2D rotation models directly to splitting coefficients) using long-duration data sets from GONG (the Global Oscillation Network Group, a ground-based network of Doppler velocimeters), MDI (the Michelson Doppler Imager on the SOHO spacecraft), and HMI (the Helioseismic and Magnetic Imager on the Solar Dynamics Observatory spacecraft). The odd-order splitting coefficients $c_1$, $c_3$, $c_5$, $c_7$ are sensitive to successively higher-order Legendre components of the rotation profile — $c_7$ to a $P_7$-type term with $\cos^6\vartheta$ latitude dependence. Allowing a $\cos^4\vartheta$ term in the tachocline position significantly improves the fit and produces a mid-latitude bulge into the convection zone; the tachocline center lies in the radiative zone at low latitudes, in the convection zone at intermediate latitudes, and (model-dependently) back near the radiative zone at high latitudes. Adding the $\cos^6\vartheta$ term yields extremely large high-latitude uncertainties; a deliberately discontinuous 2D model is not favored by $\chi^2$; and results near the poles are acknowledged to be essentially extrapolation because the modes lack sensitivity there. The bulge carries a magnetic interpretation: multilayer magnetohydrodynamic shallow-water equilibrium modeling (a layer-thin MHD model balancing Coriolis, pressure-gradient, and magnetic-curvature stresses) by Dikpati & Gilman produces a local bump on the poleward side of a toroidal field band, and Basu & Korzennik quote their modeling as implying a mid-latitude bulge for a very strong (>200 kG) low-latitude toroidal field, suggesting — but only via that model — that such a field is present. Subsequently, Korzennik & Eff-Darwich (2026) pursued an alternative route of their own motivation — noting that it is necessary to implement inversion techniques capable of resolving sharp gradients — namely rotation inversions (regularized least-squares (RLS) smoothing inversions and SART, an algebraic iterative reconstruction technique) of splittings from 25.2-, 12.6-, and 6.3-year time series, with a sigmoid model of the tachocline fitted to the inverted rotation profiles. They find the tachocline location does not vary smoothly with latitude but jumps between the low-latitude branch (where the radial shear is positive) and the high-latitude branch (where it is negative, above about $32^\circ$), with the high-latitude tachocline coinciding with the convection-zone base — a claimed bifurcation that no current tachocline model explains, that they themselves flag as grid-dependent at high latitudes and as needing independent confirmation. Meanwhile Chatterjee & Hazra (2025) note that no observational technique, including helioseismology, currently estimates the interior magnetic field, and the GONG decadal review (Jain et al. 2026) presents the 2025 shape result and its 2026 follow-ups as the state of the art, with the high-latitude and magnetic-field questions left standing. These threads share one scientific objective: knowing what the tachocline physically is — its geometry as a function of latitude, whether that geometry is smooth or sharp, and the magnetic content it implies.

Problem Statement

Determine, from helioseismic data, the latitudinal shape of the solar tachocline — the position $r_d(\varphi)$ of its center and its width $w_d(\varphi)$ as functions of latitude $\varphi$ — over the full latitude range, and the magnetic field that this shape implies. Concretely, the unresolved target inherited from Basu & Korzennik (2025) and sharpened by Korzennik & Eff-Darwich (2026) comprises three jointly linked parts, all of which are mandatory: (1) map $r_d$ and $w_d$ poleward of about $60^\circ$, where current forward fits are extrapolations and inversions are grid-dependent and systematics-dominated, to the point that the high-latitude shape is established rather than model-dependent; (2) establish whether the tachocline position varies smoothly with latitude or contains sharp, latitude-localized structure, the operative completion test being to confirm or refute the claimed discontinuous bifurcation of $r_d$ near the shear-reversal latitude (~30°) through analysis carried across independent methodologies, data sets, or mode selections, so that a verdict on that specific claim, reached consistently across those routes, resolves part (2) without requiring separate exclusion of sharp structure at every latitude; and (3) constrain, from the determined shape together with equilibrium magnetohydrodynamic modeling or from any independent observational signature, the amplitude and latitudinal geometry of the toroidal magnetic field in the tachocline — testing the current interpretation that a very strong (>200 kG) low-latitude field is required to produce the observed mid-latitude bulge. A field constraint obtained for part (3) through equilibrium magnetohydrodynamic modeling completes it provided its model dependence is quantified, and no direct observational field constraint is required, since, as the Background notes, no observational technique presently estimates the interior field. These parts form one objective: a determination of what the tachocline physically is — geometry, possible sharpness, and magnetic content — such that a qualified reviewer can decide whether the geometric determination is robust rather than an artifact of a single methodology and whether the magnetic determination's model dependence is adequately quantified.

References

1. Basu, S., & Korzennik, S. G. 2025, 'The Shape of the Solar Tachocline', The Astrophysical Journal 995, 91, arXiv:2511.07534, DOI 10.3847/1538-4357/ae1cc6, https://arxiv.org/abs/2511.07534
2. Korzennik, S. G., & Eff-Darwich, A. 2026, 'Resolving the Tachocline using Inversion of Rotational Splitting Derived from Fitting Very Long and Long Time Series', The Astrophysical Journal 999, 216, arXiv:2602.02884, DOI 10.3847/1538-4357/ae4026, https://arxiv.org/abs/2602.02884
3. Basu, S., Korzennik, S. G., & Tripathy, S. C. 2026, 'Latitude-Dependent Time Variations of the Solar Tachocline', The Astrophysical Journal 1000, 272, arXiv:2603.02321, DOI 10.3847/1538-4357/ae4c50, https://arxiv.org/abs/2603.02321
4. Dikpati, M., & Gilman, P. A. 2026, 'A Multilayer Magnetohydrodynamic Shallow-water Model of The Solar Tachocline: Equilibrium Shape and Thickness', The Astrophysical Journal 998, 134, DOI 10.3847/1538-4357/ae314e, https://doi.org/10.3847/1538-4357/ae314e
5. Chatterjee, S., & Hazra, G. 2025, 'Probing the large-scale magnetic field inside the Sun from three decades of observed surface magnetograms', arXiv:2509.23959, https://arxiv.org/abs/2509.23959
6. Jain, K., Harvey, J. W., Hill, F., Komm, R. W., Leibacher, J. W., Petrie, G. J. D., Pevtsov, A. A., & Tripathy, S. C. 2026, 'Three Decades of Global Oscillation Network Group (GONG): From Helioseismology to Magnetic Field and Space Weather (Invited Review)', Solar Physics 301, DOI 10.1007/s11207-026-02639-9, https://doi.org/10.1007/s11207-026-02639-9
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8. Charbonneau, P., Christensen-Dalsgaard, J., Henning, R., Larsen, R. M., Schou, J., Thompson, M. J., & Tomczyk, S. 1999, 'Helioseismic Constraints on the Structure of the Solar Tachocline', The Astrophysical Journal 527, 445, DOI 10.1086/308050, https://doi.org/10.1086/308050
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