Correlations with Magnetic Activity in the Solar Near-Surface Shear Layer. II. Radial Shear

Abstract: Using Helioseismic and Magnetic Imager ring-diagram measurements and building on the rotation-rate $Ω$ inferences presented in Paper I for depths of 1-17 Mm, we examine the properties of the dimensionless radial shear $\partial\lnΩ/\partial\ln r$. In the radial range overlapping global-mode analyses, the inferred shear agrees with previous results. The near-surface shear layer exhibits a three-region shear structure with an enhanced-shear middle layer, and the largest residual variations occur in the two shallowest regions not accessible to global-mode analyses. We parameterize the enhanced-shear layer by the depth of maximum shear, its amplitude, and its width, and find all three to be strongly correlated with a magnetic activity index; increasing activity corresponds to a shallower, stronger, and modestly narrower layer, indicating that the flows and magnetic fields are interconnected in these layers. This behavior is consistent with expectations that sufficiently strong toroidal fields can enhance the near-surface rotational shear and with inferences of a near-surface toroidal-field concentration near the radius where we observe the strongest shear. Moreover, the observed strengthening and upward shift of the strong-shear layer toward solar-cycle maximum suggest a corresponding solar-cycle dependence in the location of the strong toroidal field. We also highlight the importance of finite-resolution effects and instrumental calibration in interpreting small variations in the results.

• The paper shows that HMI ring-diagram measurements from 2010–2025 reveal three near-surface shear regions and robust magnetic-activity correlations in the strongest-shear layer.
• Increasing magnetic activity shifts the shear maximum upward by about 0.3 Mm, increases its amplitude by 0.4–0.8, and narrows its width by roughly 0.05 Mm over a solar cycle.
• The findings support a link between near-surface shear and kG-scale toroidal magnetic fields, while inversion resolution, HMI calibration changes, and method-dependent slopes limit causal and quantitative conclusions.

This paper is the second of a two-part analysis of the solar near-surface shear layer (NSSL) based on Helioseismic and Magnetic Imager (HMI) ring-diagram measurements spanning 2010 May through 2025 February ("Correlations with Magnetic Activity in the Solar Near-Surface Shear Layer. II. Radial Shear" (2608.19450)). Building on the rotation-rate inferences of Paper I, it characterizes the dimensionless radial shear $\partial\ln\Omega/\partial\ln r$ between roughly $0.998$ and $0.975\,R_\odot$ and quantifies its coupling to surface magnetic activity over a full solar cycle. The central result is that three parameters describing the strongest-shear region — its depth $d_{\max}$, amplitude $A_{\max}$, and width $\mathrm{FW80}$ — are all strongly and significantly correlated with a magnetic activity index (MAI), with increasing activity producing a shallower, stronger, and modestly narrower strong-shear layer.

Data and methodology

The analysis uses HMI Dopplergrams tracked as 15°- and 30°-degree tiles along the central meridian and equator. Spatial-temporal power spectra are averaged over each Carrington rotation before fitting the mode-shift parameter $u_x$, a choice that suppresses realization noise and reduces oscillatory artifacts in inverted depth profiles that would otherwise be amplified when differentiating radially. Inversions of $u_x$ into zonal velocity profiles are performed independently with Optimally Localized Averages (OLA) and Regularized Least Squares (RLS); agreement between the two methods serves as an internal consistency check throughout. Time series are smoothed with a one-year running mean to remove short-timescale variations while preserving cycle trends. Magnetic activity is quantified by integrating unsigned flux above 50 G in the same tracked regions, yielding an MAI per tile.

Time-averaged shear structure

The time-averaged shear reproduces the three-region structure established by Rabello Soares et al.: a deep layer D with moderate shear ($\partial\ln\Omega/\partial\ln r \approx -0.5$ at $0.97\,R_\odot$, steepening to about $0.998$0 at $0.998$1), a narrow middle layer M containing the maximum shear ($0.998$2 near 2.4 Mm depth), and a shallow layer S near 1.6 Mm depth where the gradient nearly vanishes before steepening again toward the photosphere. In the radial range overlapping global-mode analyses the results agree with previous inversions, confirming consistency where comparisons are possible.

The latitudinal dependence changes sign with depth, which the authors argue explains discrepancies among earlier studies that sampled different radial ranges: below $0.998$3 the shear weakens toward high latitudes; in layers M the magnitude increases poleward; in layer S the trend reverses again, reaching near-zero or positive values by $0.998$4. The equatorially antisymmetric component is small — typically $0.998$5 of the symmetric amplitude — but reaches up to $0.998$6 significance, peaks close to the surface ($0.998$7), and, based on comparisons with east–west antisymmetric structure in Paper I, does not appear dominated by center-to-limb systematics.

Temporal variability

Shear residuals after removing the time average are largest in the shallowest layers inaccessible to global-mode analyses, approximately five times larger than at greater depths. Notably, variations in layer M generally have the opposite sign from those in layer S. At $0.998$8 the residuals reproduce the previously reported anticorrelation with sunspots (spots lie where the gradient is below average), whereas at the shallowest depths the behavior reverses, with residuals enhanced in regions of strong magnetic activity. This depth-dependent reversal implies that any comparison of shear–magnetic correlations across studies must account explicitly for the sampled depth range.

Correlation of strong-shear parameters with magnetic activity

For each latitude band and Carrington rotation, the authors locate the radius of maximum shear within $0.998$9 and define its depth $0.975\,R_\odot$0, amplitude $0.975\,R_\odot$1, and width $0.975\,R_\odot$2 (the radial extent above $0.975\,R_\odot$3). Within the activity belts ($0.975\,R_\odot$4), a pooled log-linear regression,

$0.975\,R_\odot$5

with latitude-dependent intercepts but a common slope, yields highly significant negative slopes for all three quantities. After removing the latitude offsets, $0.975\,R_\odot$6 alone explains $0.975\,R_\odot$7 of the variance in $0.975\,R_\odot$8 for RLS (Pearson $0.975\,R_\odot$9; slope $d_{\max}$0-ratio $d_{\max}$1) and $d_{\max}$2 for OLA; the corresponding values for $d_{\max}$3 are $d_{\max}$4 (RLS) and 0.39 (OLA), and for $d_{\max}$5, $d_{\max}$6 (RLS) versus only 0.08 (OLA). All HAC-corrected $d_{\max}$7-values are far below $d_{\max}$8. Akaike Information Criterion comparisons decisively favor logarithmic over linear dependence on MAI for $d_{\max}$9 ($A_{\max}$0 to $A_{\max}$1 against the linear model) and for $A_{\max}$2, while the evidence for nonlinearity in $A_{\max}$3 is inconsistent between inversion methods and therefore not established.

Over a full cycle ($A_{\max}$4 G), the implied changes are of order $A_{\max}$5 Mm in $A_{\max}$6, $A_{\max}$7–0.8 in $A_{\max}$8, and $A_{\max}$9 Mm in $\mathrm{FW80}$0: the strong-shear layer rises, strengthens, and narrows toward activity maximum. The latitude-specific intercepts show that, relative to the equator, the strong-shear layer deepens poleward — by $\mathrm{FW80}$1 Mm (28% ± 4%) between the equator and $\mathrm{FW80}$2 — and becomes stronger and broader by tens of percent.

Physical interpretation

The correlations indicate that flows and magnetic fields are interconnected in these layers, although the paper is careful not to claim causality. The interpretation offered rests on two external results: Kitchatinov's prediction that sufficiently strong toroidal fields (~kG) enhance near-surface rotational shear because magnetic quenching affects turbulent viscosity more than the $\mathrm{FW80}$3-effect, and Baldner et al.'s helioseismic inference of a kG-scale toroidal-field concentration centered at $\mathrm{FW80}$4 — precisely where layer-M shear peaks. Since toroidal fields should be strongest near activity maximum, the observed strengthening and upward shift of the strong-shear layer with activity suggests a corresponding solar-cycle migration of the strong toroidal field toward the surface.

Limitations and open questions

Several caveats qualify these results. Finite inversion resolution biases inferred properties: broad averaging kernels shift the apparent location of the strong-shear feature to greater depths by roughly 0.7 Mm at the equator, so the true $\mathrm{FW80}$5 lies closer to the photosphere. Kernel-convolution tests suggest the equator-to-high-latitude increase in $\mathrm{FW80}$6 is if anything underestimated, but the high-latitude rise in $\mathrm{FW80}$7 is more resolution-sensitive and should be interpreted cautiously; notably, the simple parametric flow model fails to reproduce that amplitude increase at all. Instrumental systematics also matter: a step-like change in the HMI plate scale associated with the October 2018 focus adjustment coincides with an abrupt change in $\mathrm{FW80}$8 residuals, and plate-scale variations of a few mas per pixel correspond to shifts of a few tenths of a Mm. A tentative quasi-periodic $\mathrm{FW80}$9 yr signal in the $u_x$0 residuals appears in a Lomb–Scargle periodogram, but given the 13.8 yr record length and the plate-scale coincidence, the authors regard it as possibly instrumental. Slope magnitudes differ systematically between OLA and RLS — by nearly a factor of two for $u_x$1 — reflecting sensitivity to regularization choices, so the qualitative signs of the trends are robust while quantitative slopes are method-dependent. Finally, statistically meaningful MAI correlations are confined to latitudes $u_x$2, likely because the MAI varies too little outside the belts to constrain any dependence there.

Conclusion

Using 15 years of HMI ring-diagram data and dual inversion methods, this work demonstrates that the radial shear of the NSSL is structured in three distinct regions whose latitudinal gradients reverse sign with depth, and that the strongest-shear layer responds systematically and strongly to the solar magnetic cycle: higher activity drives it shallower, stronger, and narrower. Combined with prior evidence for a near-surface toroidal-field concentration at the same radius, the results support models in which kG-strength subsurface fields enhance near-surface rotational shear, and they imply a cycle-dependent vertical migration of that field. The study also provides a concrete demonstration that finite-resolution effects and instrumental calibration must be controlled when interpreting variations at the level of tenths of a Mm in local helioseismic inversions (2608.19450).