Helioseismology shows that the Sun's rotation rate increases rapidly with depth in the outer ~30 Mm of the convection zone — the near-surface shear layer (NSSL) — with the normalized radial shear (r/Ω) ∂Ω/∂r≃−1(r/\Omega)\,\partial\Omega/\partial r \simeq -1 independent of latitude in the top ~10 Mm, and with the shear weakening and reversing sign near the layer base at ~30 Mm depth. A complete account of this layer — one that derives its full radial–latitudinal structure from the underlying rotating, anisotropic, stratified and likely magnetized convection, and establishes quantitatively how it drives the global meridional circulation — is still missing. Quasi-linear mean-field theory based on the Λ\Lambda-effect covers only the top ~10 Mm where the Coriolis number is small; thermal-wind, gyroscopic-pumping, and MHD force-balance explanations disagree with one another; and global simulations have only very recently (2025) reproduced the layer simultaneously with the fast equator and the poleward near-surface meridional flow, with remaining quantitative discrepancies against helioseismic inversions.

Background

Helioseismology — the inference of the Sun's internal structure and dynamics from its global oscillation modes — established in the 1990s that the solar differential rotation does not extend unchanged to the surface: in the outer few percent by radius the rotation rate increases rapidly with depth, defining the near-surface shear layer (NSSL), roughly 30 Mm deep, whose radial shear exceeds the latitudinal shear seen at the surface. Two seismic facts anchor the theory of this layer. First, using global-mode inversions, Barekat, Schou and Gizon (2014) found that in the outer ~10 Mm the normalized radial shear (r/Ω) ∂Ω/∂r≃−1(r/\Omega)\,\partial\Omega/\partial r \simeq -1 is essentially independent of latitude — a regularity any explanation must reproduce. Second, the radial shear weakens with depth and reverses sign near the base of the layer, at a depth conventionally placed near 30 Mm (about 0.95 R⊙0.95\,R_\odot), where the NSSL merges into the bulk convection zone.

The standard framework is mean-field hydrodynamics. Convective turbulence transports angular momentum through the Reynolds stress Rij=−ρ⟨uiuj⟩R_{ij}=-\rho\langle u_i u_j\rangle (the averaged correlated product of turbulent velocities). The part of this transport that cannot be reduced to diffusive viscous smoothing is the Λ\Lambda-effect (Lebedinsky effect): anisotropic turbulence — with different intensities of radial and horizontal mixing — exerts non-diffusive, non-conservative torques that can generate and sustain differential rotation. The key parameter is the Coriolis number Ω∗=2τΩ\Omega^{*}=2\tau\Omega, where τ\tau is the convective turnover time and Ω\Omega the rotation rate; Ω∗\Omega^{*} measures how strongly rotation organizes the convection. In the bulk of the convection zone Ω∗≫1\Omega^{*}\gg 1, and the theory must be nonlinear in Ω∗\Omega^{*}; in the top ~10 Mm, however, Ω∗\Omega^{*} is small because the near-surface convection is fast and granulation-scale. In this regime Kitchatinov's quasi-linear derivation shows that the latitude-independence of the normalized shear follows robustly from the stress-free upper boundary condition (zero external azimuthal force at r=R⊙r=R_\odot), independent of the particular closure used, and that the observed shear amplitude is reproduced when the radial and horizontal turbulent intensities are nearly equal — in agreement with radiative hydrodynamic simulations of the near-surface layers.

This linear account deliberately does not extend into the deeper part of the layer, where Ω∗\Omega^{*} grows large, the constancy of the normalized shear breaks down, and the radial shear reverses sign; a treatment of the Λ\Lambda-effect (or of turbulent angular-momentum transport generally) nonlinear in Ω∗\Omega^{*} is required there and is not available in validated form. Competing explanations of the layer compound the disagreement. Jha and Choudhuri attribute the NSSL to thermal wind balance — the meridional force balance, fundamental to the theory of meridional circulation, between the baroclinic term (arising from latitude-dependent temperature/entropy stratification) and the Coriolis term — arguing that convection in the top layer is rotationally unconstrained, forcing a large centrifugal term and hence large shear; Matilsky has argued instead that the classical thermal-wind balance cannot account for the NSSL and that a balance involving the centrifugal force, gravity, and the pressure gradient holds; Miesch and Hindman give a gyroscopic-pumping account (conservation of angular momentum requiring meridional circulation where turbulent angular-momentum stresses converge), in which non-diffusive, radially inward turbulent angular-momentum transport maintains the layer.

Global numerical simulations long failed to reproduce the layer: high-resolution calculations of Hotta, Rempel and Yokoyama produced the NSSL preferentially at high latitudes only, and the density-contrast study of Matilsky, Hindman and Toomre obtained near-surface shear only at low latitudes, concluding that a rotationally unconstrained outer fluid layer alone is insufficient. Local simulations designed to match NSSL conditions (Barekat, Käpylä, Käpylä, Gilson and Ji) did not validate the mean-field explanation that neglects meridional components of the Reynolds stress. In 2025, Hotta reported the first simulation to simultaneously reproduce the fast equator, the near-surface poleward meridional flow, and the NSSL, in an unprecedented high-resolution magnetohydrodynamic (MHD) calculation; the maintenance mechanism identified there involves radially inward angular-momentum transport by rotationally unconstrained convection, the sheared poleward flow it builds, and a balance of turbulent viscous and magnetic-tension stresses against the Coriolis force — with classical thermal wind balance shown not to hold inside the NSSL. The simulated meridional-flow shear, however, exceeds what helioseismic inversions indicate, and the author flags validation against observations as the outstanding issue.

The layer's significance is dynamical. It is a leading candidate for the main driver of the global meridional circulation — the large-scale flow in the meridional plane that is the conveyor belt of flux-transport dynamo models of the solar activity cycle — and seismic inversions indeed find the meridional flow concentrated near the top and bottom boundaries of the convection zone. Local helioseismic techniques sharpen the observational target: ring-diagram analysis (inferring flows beneath small patches of the surface from the distortion of acoustic waves) resolves a three-region substructure of the NSSL, including a narrow strong-shear layer near ~3 Mm depth or shallower, part of the shallow leptocline where the seismic and thermodynamic structure of the outermost layers changes rapidly and varies over the solar cycle.

Problem Statement

Derive, and validate against helioseismic observations, a theory of the solar near-surface shear layer that explains its full radial-latitudinal structure from the underlying rotating, anisotropic, stratified, and possibly magnetized convection, and establish quantitatively how the layer drives the global meridional circulation. Concretely, the theory should: (1) reproduce the latitude-independent normalized radial shear (r/Ω) ∂Ω/∂r≃−1(r/\Omega)\,\partial\Omega/\partial r \simeq -1 and its observed amplitude in the outer ~10 Mm, as a consistency requirement on the full-depth theory rather than an independent re-derivation; (2) extend through the full depth of the layer (~30 Mm), predicting the breakdown of the constant normalized shear and the sign reversal of the radial shear near the layer base, where the Coriolis number Ω∗=2τΩ\Omega^{*}=2\tau\Omega is large and the quasi-linear treatment fails — that is, provide a nonlinear treatment of the Λ\Lambda-effect or an equivalent account of turbulent angular-momentum transport valid at large Coriolis number; and (3) determine the meridional circulation generated by the layer, with amplitudes and depth profiles consistent with helioseismic inversions. These three deliverables form one objective: a single validated dynamical account of the layer's origin and structure, since the same turbulent stresses, force balances, and boundary conditions that set the shear also set the circulation it drives. Validation is quantitative: the predicted radial-shear profile, and the amplitudes and depth profiles of the meridional circulation, must agree with the cited helioseismic inversions — those of Barekat, Schou and Gizon (2014) and Gizon et al. (2020) — within their observational uncertainties, not merely qualitatively. The broader literature additionally flags extension of the analysis to other late-type stars as an open challenge. This task targets the Sun, whose NSSL is the only one resolved in observational detail.

References

  1. Kitchatinov, L. 2023, 'Near-surface shear layer of solar rotation: origin and significance', Proceedings of the International Astronomical Union 19 (Symposium S365), 54-58, DOI 10.1017/S174392132300491x, https://doi.org/10.1017/S174392132300491x (also arXiv:2311.07875)
  2. Kitchatinov, L. 2023, 'Origin of the Near-Surface Shear Layer of Solar Rotation', Astronomy Letters 49, 754-761, DOI 10.1134/S106377372311004X, https://arxiv.org/abs/2402.07421
  3. Barekat, A.; Schou, J.; Gizon, L. 2014, 'The radial gradient of the near-surface shear layer of the Sun', Astronomy & Astrophysics 570, L12, DOI 10.1051/0004-6361/201424839, https://doi.org/10.1051/0004-6361/201424839
  4. Miesch, M. S.; Hindman, B. W. 2011, 'Gyroscopic Pumping in the Solar Near-Surface Shear Layer', The Astrophysical Journal 743, 79, DOI 10.1088/0004-637x/743/1/79, https://arxiv.org/abs/1106.4107
  5. Hotta, H.; Rempel, M.; Yokoyama, T. 2015, 'High-resolution calculation of the solar global convection with the reduced speed of sound technique: II. Near surface shear layer with the rotation', The Astrophysical Journal 798, 51, DOI 10.1088/0004-637X/798/1/51, https://arxiv.org/abs/1410.7093
  6. Matilsky, L. I.; Hindman, B. W.; Toomre, J. 2019, 'The Role of Downflows in Establishing Solar Near-surface Shear', The Astrophysical Journal 871, 217, DOI 10.3847/1538-4357/aaf647, https://arxiv.org/abs/1810.00115
  7. Jha, B. K.; Choudhuri, A. R. 2021, 'A theoretical model of the near-surface shear layer of the Sun', Monthly Notices of the Royal Astronomical Society 506, 2189, DOI 10.1093/mnras/stab1717, https://arxiv.org/abs/2105.14266
  8. Barekat, A.; Käpylä, M. J.; Käpylä, P. J.; Gilson, E. P.; Ji, H. 2021, 'Generation of mean flows in rotating anisotropic turbulence: The case of solar near-surface shear layer', Astronomy & Astrophysics 655, A79, DOI 10.1051/0004-6361/202040052, https://arxiv.org/abs/2012.06343
  9. Kitchatinov, L. L.; Olemskoy, S. V. 2011, 'Differential rotation of main-sequence dwarfs and its dynamo efficiency', Monthly Notices of the Royal Astronomical Society 411, 1059, DOI 10.1111/j.1365-2966.2010.17737.x, https://doi.org/10.1111/j.1365-2966.2010.17737.x
  10. Gizon, L.; Cameron, R. H.; Pourabdian, M.; Liang, Z.-C.; Fournier, D.; Birch, A. C.; Hanson, C. S. 2020, 'Meridional flow in the Sun's convection zone is a single cell in each hemisphere', Science 368, 1469, DOI 10.1126/science.aaz7119, https://doi.org/10.1126/science.aaz7119
  11. Matilsky, L. I. 2023, 'The stellar thermal wind as a consequence of oblateness', Monthly Notices of the Royal Astronomical Society: Letters 526, L100, DOI 10.1093/mnrasl/slad121, https://doi.org/10.1093/mnrasl/slad121
  12. Choudhuri, A. R.; Jha, B. K. 2023, 'The Near-surface Shear Layer (NSSL) of the Sun: A Theoretical Model', Proceedings of the International Astronomical Union 19 (Symposium S365), 59-62, DOI 10.1017/S1743921323004969, https://arxiv.org/abs/2311.04197
  13. Hotta, H. 2025, 'Simultaneous Construction of Fast Equator, Poleward Meridional Flow, and Near-surface Shear Layer in Solar Magnetohydrodynamic Calculation', The Astrophysical Journal 985, 163, DOI 10.3847/1538-4357/adca3b, https://arxiv.org/abs/2504.05680
  14. Rabello Soares, M. C.; Basu, S.; Bogart, R. S. 2024, 'Exploring the Substructure of the Near-surface Shear Layer of the Sun', The Astrophysical Journal 967, 143, DOI 10.3847/1538-4357/ad3d59, https://arxiv.org/abs/2404.02321
  15. Rozelot, J.-P.; Kosovichev, A.; Kitiashvili, I. N. 2024, 'Improving Our Knowledge of the Solar Near-Surface Shear Layer: The Special Case of the Leptocline', Astronomy in Focus XXXII (Proceedings of the International Astronomical Union 20), 360-370, DOI 10.1017/S1743921325001541, https://arxiv.org/abs/2501.08021
  16. Sen, A.; Rajaguru, S. P.; Iyer, A. G.; Chen, R.; Zhao, J.; Kholikov, S. 2025, 'Solar-cycle variations in meridional flows and rotational shear within the Sun's near-surface shear layer', The Astrophysical Journal Letters 984, L1, DOI 10.3847/2041-8213/adc919, https://arxiv.org/abs/2512.12345