The Sun's near-surface shear layer (NSSL), a thin layer at the top of the convection zone in which the angular velocity decreases rapidly outwards, was modeled by Jha & Choudhuri (2021) as a consequence of thermal wind balance: below a radius rcr_c rotation-influenced convection maintains a latitude-dependent temperature perturbation ΔT(r,θ)\Delta T(r,\theta), while above rcr_c rotationally unconstrained convection keeps ΔT\Delta T radially constant, forcing a large thermal wind term that requires a strong radial shear to balance it. The model matches helioseismic data for rc≈0.96 R⊙r_c\approx0.96\,R_\odot but leaves its own validation program incomplete. This problem asks to complete that program: quantify how replacing the assumed sharp transition at rcr_c with a gradual one changes the predicted ΔT(r,θ)\Delta T(r,\theta) and the structure and thickness of the NSSL; establish the physical reality and origin of the anomalous high-latitude ΔT\Delta T annulus near r≈0.925 R⊙r\approx0.925\,R_\odot associated with the submerged polar jet reported by helioseismology; identify the dynamical mechanisms behind any small but finite breakdown of thermal wind balance in the outermost ∼0.005 R⊙\sim0.005\,R_\odot; and obtain an independent, higher-precision measurement of the pole–equator temperature difference (PETD, ≈2.5\approx2.5 K from the only existing measurement by Rast et al. 2008) that would fix rcr_c observationally and confirm or refute the thermal-wind origin of the NSSL.

Background

Helioseismology, the inference of the Sun's internal structure from the frequencies of its acoustic oscillation modes, shows that the solar angular velocity Ω(r,θ)\Omega(r,\theta), after being nearly radius-independent through most of the convection zone, decreases sharply with radius in a thin near-surface shear layer (NSSL) occupying the outer ∼5%\sim5\% by radius (∼30\sim30–35 Mm) of the convection zone. Explaining why this layer exists is a long-standing problem in the theory of the Sun's large-scale flows, and the NSSL matters beyond rotation itself: it is the source region of the meridional circulation — the large-scale flow circulating between the equator and the poles — used in flux-transport models of the solar dynamo (models of the solar magnetic cycle in which that circulation transports magnetic flux).

Jha & Choudhuri (2021) built on an earlier order-of-magnitude proposal by Choudhuri (2021) that the NSSL follows from the thermal wind balance equation, a steady-state meridional force balance stating that the latitudinal entropy gradient drives the vertical shear of the rotation rate: rsin⁡θ ∂∂zΩ2=1rgγCV∂S∂θ,r\sin\theta\,\frac{\partial}{\partial z}\Omega^{2}=\frac{1}{r}\frac{g}{\gamma C_{V}}\frac{\partial S}{\partial\theta}, where θ\theta is co-latitude, zz the distance from the equatorial plane, gg gravity, γ\gamma the adiabatic index, and SS and CVC_V the specific entropy and specific heat per unit mass. Written for temperature perturbations ΔT(r,θ)\Delta T(r,\theta) relative to a non-rotating standard solar model, and evaluated over isochoric surfaces (surfaces of constant density, whose solar oblateness is negligibly small), this becomes r2sin⁡θ ∂∂zΩ2=gγT(ddθΔT)isochore.r^{2}\sin\theta\,\frac{\partial}{\partial z}\Omega^{2}=\frac{g}{\gamma T}\left(\frac{d}{d\theta}\Delta T\right)_{\rm isochore}. The model's physical premise is the Rossby-number stratification of the convection zone: the Rossby number, the ratio of the convective turnover time to the solar rotation period, is small (rotationally influenced convection, latitude-dependent heat transport) in the deep convection zone but large in the near-surface layers, where small granular convection cells turn over in minutes and are essentially unaffected by the Coriolis force. The authors idealize this change of regime as a sharp transition at a single radius rcr_c: below rcr_c they invert the thermal wind equation to compute ΔT(r,θ)\Delta T(r,\theta) from the helioseismically inferred Ω(r,θ)\Omega(r,\theta); above rcr_c, ΔT\Delta T is independent of radius, the steep outward temperature drop makes the thermal wind term very large, and the balance then forces a large radial shear — the NSSL. For rc≈0.96 R⊙r_c\approx0.96\,R_\odot the predicted Ω(r>rc,θ)\Omega(r>r_c,\theta) matches helioseismic inversions well, and the pole–equator temperature difference (PETD) mapped to the surface equals ≈2.5\approx2.5 K, the value measured by Rast, Ortiz & Meisner (2008) from the latitudinal variation of the photospheric intensity (a 2.5 K temperature difference implies a pole-to-equator intensity contrast of only ∼0.16%\sim0.16\%, which is why the measurement is difficult: instrumental systematics and bright polar magnetic faculae compete with the signal).

The model entered an existing debate about whether thermal wind balance holds at all near the top of the convection zone: Vasil, Julien & Featherstone (2020) argued that the dominant balances change in the near-surface boundary layer, and earlier mean-field theories (theories that model the collective effect of turbulence through averaged transport coefficients; Kitchatinov and coworkers) attribute the NSSL instead to the radial anisotropy of rotationally unconstrained near-surface turbulence rather than to a thermal wind. The paper also acknowledges that the high-latitude parts of helioseismic rotation inversions are the least reliable, and its computed ΔT\Delta T shows an anomalous annular region near r≈0.925 R⊙r\approx0.925\,R_\odot at high latitudes, connected with the 'submerged polar jet' — a persistent high-latitude prograde flow feature reported as consistent across methods by Schou et al. (1998) — whose physical reality and origin the authors leave unaddressed. Subsequent simulations have added new constraints but not adjudicated the question: Kitchatinov (2023) sharpened the mean-field alternative; Hotta (2025) reproduced the NSSL, the fast equator, and poleward surface flow in a high-resolution magnetohydrodynamic simulation in which turbulent viscosity and magnetic tension balance the Coriolis force in the NSSL; Teweldebirhan, Curt & Featherstone (2026) showed in three-dimensional simulations that thermal wind balance is sustained at low Rossby number but breaks down at sufficiently high Rossby number; and Bekki, Cameron & Gizon (2024) showed that baroclinically unstable high-latitude inertial modes (global oscillations of the rotating fluid driven by the latitudinal entropy gradient) cap the pole-to-equator temperature difference at ≤7\le7 K theoretically. No independent measurement of the PETD has been published since Rast et al. (2008), and high-latitude helioseismic results remain unreliable above roughly 70∘70^{\circ} latitude (Antia & Basu 2022; Mandal et al. 2025).

Problem Statement

Complete the validation program of the thermal-wind model of the Sun's near-surface shear layer proposed by Jha & Choudhuri (2021). The single scientific objective is to determine whether the NSSL is genuinely produced by thermal wind balance acting on the latitude-dependent temperature perturbation ΔT(r,θ)\Delta T(r,\theta) that rotation-influenced convection maintains below a transition radius rcr_c, by removing the model's acknowledged weak points. Concretely: (1) generalize the model to a gradual (finite-width) transition between rotation-influenced and rotation-ineffective convection instead of a sharp jump at a single rcr_c, and quantify how this changes the predicted ΔT(r,θ)\Delta T(r,\theta), the PETD–rcr_c relation, and the detailed structure and thickness of the NSSL; (2) establish the physical reality, or otherwise, of the high-latitude anomalous ΔT\Delta T annulus near r≈0.925 R⊙r\approx0.925\,R_\odot associated with the submerged polar jet reported by Schou et al. (1998), and either identify its physical origin or show it to be an artifact of high-latitude inversion uncertainties; (3) determine the dynamical mechanisms, and the sign and magnitude, of any small but finite violation of thermal wind balance within the outermost ∼0.005 R⊙\sim0.005\,R_\odot of the convection zone (candidate mechanisms including enhanced Reynolds stresses — the turbulent transport of momentum by correlated velocity fluctuations — radiative effects, and magnetic back-reaction); and (4) measure the pole–equator temperature difference of the solar photosphere independently and at higher precision than Rast, Ortiz & Meisner (2008), so that rcr_c is fixed observationally and the thermal-wind-derived origin of the NSSL is quantitatively confirmed or refuted. An answer that resolves the overall question may resolve these components jointly (for example, a validated simulation or extended calculation that simultaneously predicts the transition profile, the near-surface balance, and the PETD) or may settle individual components decisively as partial progress toward it; the objective is met when the thermal-wind explanation of the NSSL is either validated against the relaxed assumptions and new observations, or decisively refuted — whether by a demonstrated alternative balance that accounts for the same observations, including the PETD, or by measurements (for example a PETD inconsistent with the value the model requires for its fitted rcr_c) that contradict its predictions.

References

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