M. Cristina Rabello Soares¹, Sarbani Basu², Richard S. Bogart¹, and Charles S. Baldner¹
1 Department of Physics, Stanford University, Stanford, CA94305
2 Department of Astronomy, Yale University, New Haven, CT06520

The Sun’s rotation varies with both latitude and depth, generally increasing inward through the near-surface shear layer (NSSL), which occupies roughly the outer five percent of the Sun. This radial shear may influence magnetic-field organization and angular-momentum transport, but its structure and solar-cycle variation closest to the photosphere remain poorly constrained because global helioseismology loses sensitivity in the upper few megameters. Building on the rotation measurements presented in Paper I^[1] (HMI Science Nugget 233), we use observations from the Helioseismic and Magnetic Imager (HMI) to measure the dimensionless radial shear, ∂ lnΩ/∂ lnr, from about 1 to 17 Mm below the photosphere. The data span May 2010 through early 2025. Carrington-rotation-averaged spectra from 15-degree ring diagrams were analyzed using OLA and RLS inversions, with a one-year running mean applied to suppress short-timescale variations.

The near-surface shear layer exhibits three distinct regions, including a middle layer of enhanced shear (Figure 1, left). The inversion results indicate that the strongest shear lies roughly 2–3 Mm below the surface, but finite radial resolution biases its position downward, so the actual peak is probably shallower^[2]. The depth-dependent latitude pattern also helps explain differences among earlier measurements.

Figure 1. Left: time-averaged radial shear versus latitude and radius (top), and the equatorial profile identifying layers D, M, and S (bottom). Right: north–south symmetric shear at 30° latitude versus depth and time (top), and residual shear after removing the time average (bottom). RLS results are shown; OLA gives similar results.

Solar-cycle variations are largest in shallow layers beyond the reach of most global-mode analyses. Figure 1 (right) shows that layers M and S generally vary in opposite directions. Figure 2 extends the comparison across latitude: residuals near 1–2 Mm are about five times larger than at 14 Mm. Their relation to surface activity also reverses with depth, with below-average shear near 2 Mm but enhanced shear near 1 Mm in active regions.

Figure 2. Residual shear versus time and latitude at approximately 1, 2, and 14 Mm, with MAI contours. The 14 Mm panel uses a different color scale.

We characterize layer M by the depth (d_max) and amplitude (A_max) of maximum shear and its width at 80% of that amplitude (FW80). We then examine how these quantities vary with surface magnetic activity (MAI) within the activity belts, from 22.5° S to 22.5° N. To separate this dependence from latitude variation, we combine seven latitudes and fit
,
where Y(θ, t) represents d_max, A_max, or FW80 at latitude θ and time t. The intercept c_0,Y(θ) is allowed to differ with latitude, whereas the slope c_1,Y is common to all latitudes. A logarithmic dependence on MAI is strongly preferred for d_max, while both forms describe A_max reasonably well.

Figure 3. Latitude-corrected dmax, Amax, and FW80 from RLS versus MAI. Gray curves fit ln(MAI); dark blue curves fit MAI directly. OLA shows the same trends.

Both inversion methods show the same systematic response to increasing activity: layer M moves toward the surface, becomes stronger, and narrows slightly (Figure 3). Baldner et al.^[3] inferred a cycle-varying toroidal field concentration of 1.4 ± 0.2 kG about 3 Mm below the surface (0.996 R_⊙), at the same depth as layer M. Kitchatinov^[4] predicted that strong toroidal fields enhance near-surface shear by suppressing turbulent viscosity more strongly than the Λ-effect. Our finding that layer M strengthens and moves upward toward activity maximum suggests that the strong toroidal field may likewise shift closer to the solar surface.

Together, these results indicate that near-surface rotational shear and magnetic fields are interconnected. The structure and evolution of layer M are consistent with a cycle-varying toroidal field. Finite radial resolution and instrumental calibration remain important when interpreting these small variations.

References

[1] Rabello Soares, M. C., Basu, S., & Bogart, R. S. 2026, ApJ, 1009, 108 (Paper I)
[2] Rabello Soares, M. C., Basu, S., & Bogart, R. S. 2024, ApJ, 967, 143
[3] Baldner, C. S., Antia, H. M., Basu, S., & Larson, T. P. 2009, ApJ, 705, 1704
[4] Kitchatinov, L. L. 2016, Astronomy Letters, 42, 339