Helioseismology shows that the Sun's angular velocity increases with depth by several percent within a thin (~30 Mm) near-surface layer, with normalized shear ∂ln⁡Ω/∂ln⁡r≈−1\partial\ln\Omega/\partial\ln r\approx-1 nearly independent of latitude. Kitchatinov (2024) explained this near-surface shear layer (NSSL) as the balance between the downward angular-momentum flux of the Λ\Lambda-effect (the non-diffusive Reynolds stress of anisotropic rotating turbulence) and eddy viscosity under a zero surface-stress condition, reproducing the observed shear for maximal radial turbulence anisotropy (S=−1S=-1). The account, however, rests on a quasi-linear theory valid only where the Coriolis number Ω∗=2τΩ≪1\Omega^{*}=2\tau\Omega\ll 1, uses an anisotropy parameter never measured inside the seismically probed layer, and its demonstration in 3D simulations was blocked by boundary conditions that exclude the Λ\Lambda-effect. As of August 2026: the NSSL has now been reproduced in a high-resolution 3D magnetohydrodynamic simulation (Hotta 2025), but direct rotating-convection measurements of the Λ\Lambda-effect (Käpylä 2026) conflict with mean-field assumptions at larger Coriolis number, and no observational determination of the turbulence anisotropy within the seismic layer exists. The problem of the NSSL's quantitative origin — extending and testing the Λ\Lambda-effect account across the full layer depth against realistic turbulence, observation, and simulation — remains open.

Background

Helioseismology — the interpretation of the Sun's global oscillation frequencies — shows that the solar angular velocity Ω\Omega increases with depth by several percent in a thin layer only about 30 Mm (roughly 5% of the solar radius) thick immediately below the photosphere (the Sun's visible surface layer), before the shear reverses sign near the bottom of the layer. This near-surface shear layer (NSSL) was discovered through inversions of the rotational splitting of solar oscillation modes (Schou et al. 1998). Two of its properties are especially constraining for theory: the normalized shear ∂ln⁡Ω/∂ln⁡r\partial\ln\Omega/\partial\ln r is close to −1-1, and this value is nearly independent of latitude (Barekat et al. 2014). The layer also matters beyond rotation theory: it is widely held to be where the global meridional circulation is driven, and that circulation largely controls the period of the solar cycle and the equatorward migration of sunspots in flux-transport dynamo models.

In mean-field theory the Reynolds stress Rij=−ρ⟨uiuj⟩R_{ij}=-\rho\langle u_i u_j\rangle — the statistical correlation of turbulent velocities uu that transports momentum in a rotating convective envelope — is split into a diffusive part, proportional to the large-scale velocity gradient through eddy viscosities, and a non-diffusive part, the Λ\Lambda-effect, by which rotating anisotropic turbulence transports angular momentum non-diffusively (for radially dominated turbulence the radial angular-momentum flux points downward, a consequence of the radial direction of the buoyancy forces that drive convection). The intensity of the interaction between convection and rotation is measured by the Coriolis number Ω∗=2τΩ\Omega^{*}=2\tau\Omega, where τ\tau is the turbulent correlation (turnover) time: Ω∗≪1\Omega^{*}\ll 1 means convection is effectively unaffected by rotation, so the stress can be expanded to first order in Ω\Omega.

Kitchatinov (2024) develops this account specifically for the NSSL. Near the photosphere the convective turnover time is short, so Ω∗≪1\Omega^{*}\ll 1 throughout the upper part of the layer and the theory can be linear in Ω\Omega. Imposing the physical surface condition that the total radial–longitudinal stress vanish at the outer boundary (Rrφ=0R_{r\varphi}=0, i.e. zero density of external tangential forces), the downward Λ\Lambda-flux is balanced by the opposing eddy-viscosity flux, and a quasi-linear closure (a second-order perturbation treatment of the turbulence, closed with the τ\tau-approximation — replacing the relaxation rate by 1/τ1/\tau — and an assumed spectral tensor for the turbulence) yields the simple relation

rΩ∂Ω∂r=S,S=⟨u2⟩⟨ur2⟩−3,\frac{r}{\Omega}\frac{\partial\Omega}{\partial r}=S,\qquad S=\frac{\langle u^{2}\rangle}{\langle u_{r}^{2}\rangle}-3,

where SS measures the directional anisotropy of the turbulence (S<0S<0 for radially dominated flows; S=−1S=-1 is the maximum radial anisotropy). The helioseismically measured normalized shear ≃−1\simeq-1 is then reproduced for S=−1S=-1, the same anisotropy found in the three-dimensional radiative-hydrodynamic (RHD) simulations of solar near-surface convection by Kitiashvili et al. (2023). Directional anisotropy is essential: shape anisotropy of the turbulence spectrum (its dependence on μ\mu, the cosine of the angle to the radial direction) alone produces no Λ\Lambda-driven shear. Earlier mean-field modeling with the same stress condition reproduced the shear increase in only about the lower third of the NSSL, together with the global meridional flow (Kitchatinov & Olemskoy 2011). Competing origins have been proposed — pumping by the large-scale meridional flow, and a thermal-wind/baroclinic balance driven by a pole-equator temperature difference (Jha & Choudhuri 2021) — and no consensus exists.

Kitchatinov (2024) is explicit that the account is incomplete in four linked respects: the transport coefficients and numerical shear values rest on the quasi-linear closure, the τ\tau-approximation, and an assumed spectral form; the turbulence anisotropy entering the result has not been measured observationally inside the seismically probed layer, because different velocity components are accessed by different techniques (line-of-sight Doppler shifts versus horizontal motions of tracers); the derivation holds only where Ω∗≪1\Omega^{*}\ll 1, while Ω∗\Omega^{*} grows and the latitude-independent shear relation breaks down toward the deeper part of the layer; and common 3D convection simulations, which impose vr=0v_r=0 on the total velocity at the upper boundary without separating large-scale and turbulent components, exclude the Λ\Lambda-effect altogether and have historically failed to produce the layer.

Problem Statement

Determine, quantitatively and testably, the mechanism that maintains the Sun's near-surface shear layer — in particular, whether the balance between the Λ\Lambda-effect of anisotropic near-surface convection and turbulent (eddy) viscosity, under the surface stress condition Rrφ=0R_{r\varphi}=0, is the origin of the observed rotation profile. An answer must resolve the following linked questions left open by the source account:

(1) Extension beyond the small-Coriolis-number regime. The relation ∂ln⁡Ω/∂ln⁡r=S\partial\ln\Omega/\partial\ln r=S and its latitude independence were derived for Ω∗=2τΩ≪1\Omega^{*}=2\tau\Omega\ll 1. Does the Λ\Lambda-effect account survive the nonlinear-in-rotation regime Ω∗≳1\Omega^{*}\gtrsim 1 entered toward the deeper part of the NSSL, where the normalized shear decreases, changes sign (near ~30 Mm depth), and loses its latitude independence — or does another mechanism take over there?

(2) Observational determination of the turbulence anisotropy. Is the required directional anisotropy real? A direct, reliable measurement of SS (equivalently, the ratio of mean-square horizontal to radial turbulent velocities) within the seismically probed NSSL, with the systematic differences between line-of-sight Doppler and horizontal-tracer/helioseismic techniques controlled, is needed to test the inference S≃−1S\simeq-1.

(3) Quantitative robustness of the transport coefficients. Do the predicted shear values survive recomputation of the transport-coefficient integrals with realistic turbulence spectra (including shape anisotropy, i.e. spectral μ\mu-dependence) instead of the quasi-linear/τ\tau-approximation model, and are they consistent with Λ\Lambda-effect values measured in rotating-convection simulations?

(4) Demonstration in 3D simulations. Can global convection simulations whose boundary conditions retain the physically motivated stress condition (large-scale Vr=0V_r=0 together with turbulent Rrφ=0R_{r\varphi}=0) — or otherwise adequately resolved near-surface layers — reproduce the NSSL, with a term-by-term angular-momentum diagnosis establishing which transport (Reynolds stress, viscosity, meridional flow, Maxwell stress) actually maintains the layer?

These four questions jointly support one objective: a validated physical account of the NSSL whose predictions for the depth- and latitude-dependence of the shear agree with helioseismic inversions across the full layer; a complete answer resolves all four linked questions above.

References

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