The physical origin of the Sun's near-surface shear layer

The Sun's angular velocity decreases outward by about 5% within the outer ~30 Mm (1 Mm = 1000 km) of the convection zone, forming the near-surface shear layer (NSSL) revealed by helioseismology. Its key properties — a logarithmic radial shear $\partial\ln\Omega/\partial\ln r$ close to $-1$ and independent of latitude in the top ~10 Mm, a sign reversal of the radial shear near 30 Mm depth, a latitude-dependent base, and solar-cycle-varying substructure — are well measured, but the physical mechanism that establishes and maintains the layer is not settled. Competing accounts attribute the NSSL to the Lambda-effect of radially anisotropic, weakly rotating near-surface turbulence; to thermal-wind (baroclinic) balance above the radius where convection ceases to feel rotation; or to inward angular-momentum transport by rotationally unconstrained downflow plumes, coupled to meridional circulation and, in the latest simulations, to magnetic stresses — and these accounts disagree even about the leading-order force balance in the layer. The problem is to identify the mechanism or combination of mechanisms actually operating in the Sun and to account quantitatively for the observed depth profile, including the shear sign reversal at the layer's base.

Background

The Sun rotates differentially: the angular velocity $\Omega(r,\theta)$ varies with radius $r$ and latitude $\theta$, with the equatorial photosphere rotating roughly 25% faster than the poles. Helioseismology — the inversion of the frequencies of the Sun's global oscillation modes, including their rotational splittings, into profiles of interior angular velocity — showed in the 1990s that this differential rotation changes character near the surface (Schou et al. 1998). Within a layer about 30 Mm deep (1 Mm = 1000 km; the outer ~5% by radius), the angular velocity decreases rapidly outward by ~5%, in contrast to the nearly radius-independent rotation in the bulk of the convection zone. This is the near-surface shear layer (NSSL). Two of its properties are especially constraining for theory. First, the logarithmic radial gradient $\partial\ln\Omega/\partial\ln r$ in the top ~10 Mm is close to $-1$ and independent of latitude from the equator to about 60° (Barekat, Schou and Gizon 2014). Second, at a depth of roughly 30 Mm the radial shear changes sign, marking the base of the layer; the layer is deeper at low latitudes than at high latitudes (Antia and Basu 2022). Local helioseismic techniques (ring-diagram analysis, which infers horizontal flows beneath localized patches of the surface from the distortion of solar oscillation power spectra) have further resolved a three-region radial substructure with an enhanced-shear middle layer at ~1.5–3 Mm depth (Rabello Soares, Basu and Bogart 2024), solar-cycle-related changes of the gradient (Antia and Basu 2022), and activity-correlated strengthening and upward migration of the enhanced-shear layer (Rabello Soares et al. 2026).

The NSSL is a problem of angular-momentum transport by rotating turbulent convection. Correlations between fluctuating velocity components (Reynolds stresses) exert stresses on the mean flow. Mean-field theory splits the radial–longitudinal stress into a diffusive viscous part and a non-diffusive part, the Lambda-effect (the term proportional to the angular velocity in the mean stress tensor, named for its coefficient $\Lambda$), which drives rotation away from uniformity and depends on the anisotropy of the turbulence and on the Coriolis number $\mathrm{Co}=2\Omega\tau_c$ — the ratio of the Coriolis force to inertial forces for an eddy with correlation time $\tau_c$. Near the surface, convection is fast and small-scale (granulation with correlation times of minutes and scales of ~1–2 Mm), so $\mathrm{Co}\ll1$ and the convection is rotationally unconstrained; deeper down, $\mathrm{Co}$ grows past unity and rotation organizes the flow. The NSSL straddles this transition. A second key concept is the thermal wind: in a stratified rotating star, the steady azimuthal force balance in the meridional plane (the thermal wind equation) links the latitudinal variation of angular momentum to baroclinicity, i.e. to latitude-dependent temperature and density surfaces. Relatedly, gyroscopic pumping denotes the constraint of angular-momentum conservation by which convergent meridional flow tends to spin up the layer it converges on, forcing turbulent or magnetic stresses to balance the Coriolis torque; Maxwell stresses (the stresses exerted by magnetic fields on the flow) can supply part of that balance.

At least three mechanisms have been proposed for the NSSL, each with quantitative support but none accepted. (1) Lambda-effect: Kitchatinov (Astronomy Letters 49, 2023) shows within mean-field hydrodynamics that radial (vertically elongated) anisotropy of near-surface convection combined with its short correlation time relative to the rotation period predicts a latitude-independent normalized shear matching the observed value of Barekat et al. (2014); he notes this is not a generally accepted explanation and restricts the theory to depths where the scaling holds, leaving the deeper layer and the ~30 Mm sign reversal unexplained. (2) Thermal wind: Choudhuri (2021) and Jha and Choudhuri (2021) argue that above a rotation-influence radius $r_c\approx0.96\,R_{\odot}$, where convection is unaffected by rotation, the temperature must fall with depth at the same rate at all latitudes; the resulting large thermal-wind (baroclinic) term forces a strong radial shear, and solving the balance equation with helioseismic input reproduces the NSSL 'reasonably well'. (3) Inward angular-momentum transport by rotationally unconstrained downflows, first suggested by Foukal and Jokipii (1975) and tested by Matilsky, Hindman and Toomre (2019), who found that fast downflow plumes do deplete angular momentum from the outer layers but that a rotationally unconstrained layer alone cannot produce a solar-like (all-latitude) NSSL — the coupled meridional circulation, whose dynamics Miesch and Hindman (2011) analyzed helioseismically, is an inseparable part of the mechanism.

Numerical simulations have progressively reproduced parts of the layer without settling the question. Hotta, Rempel and Yokoyama (2015), extending global rotating convection to $0.99\,R_{\odot}$, obtained NSSL-like shear at high and low latitudes but failed at mid-latitudes with amplitudes below observed values. Kitiashvili et al. (2022), using realistic three-dimensional radiative hydrodynamics in a local 80 Mm × 25 Mm domain, found a shallow 10-Mm high-shear substructure (the 'leptocline', seated in the hydrogen ionization zone) with self-organized meridional flows, in qualitative agreement with observations. Most recently, Hotta (2025), in the highest-resolution global magnetohydrodynamic simulation to date, reproduced the fast equator, the poleward near-surface meridional flow, and the NSSL simultaneously, and concluded that in the NSSL the Coriolis force is balanced by turbulent viscosity and magnetic tension rather than by the classical thermal wind. The proposed accounts thus disagree about even the leading-order force balance in the layer, and no mechanism or combination has been established as the one operating in the Sun.

Problem Statement

Identify the physical mechanism or combination of mechanisms that establishes and maintains the Sun's near-surface shear layer, and demonstrate quantitatively that it accounts for the observed properties of the layer. Specifically, from the dynamics of rotating, stratified, magnetized convection, explain:

(i) why the layer is roughly 30 Mm deep, with the angular velocity decreasing outward by about 5%;

(ii) why the normalized radial shear $\partial\ln\Omega/\partial\ln r$ is close to $-1$ and independent of latitude in the top ~10 Mm, from the equator to at least 60°;

(iii) why the radial shear changes sign near 30 Mm depth, and why the base of the layer is deeper at low latitudes than at high latitudes;

(iv) which of the proposed balances is the operative one in the Sun — non-diffusive angular-momentum transport by the Lambda-effect of radially anisotropic, weakly rotating turbulence; thermal-wind (baroclinic) balance above the rotation-influence radius; inward angular-momentum transport by rotationally unconstrained downflow plumes; or a balance involving meridional circulation and magnetic (Maxwell) stresses — and in what proportions the mechanisms act together where more than one contributes.

An acceptable answer must be validated against helioseismic constraints: the rotation profile and its radial shear as functions of depth and latitude, including the sign reversal near the base of the layer, the substructure of the upper ~20 Mm, and the known solar-cycle variation of the gradient.

References

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