Correlations with Magnetic Activity in the Solar Near-Surface Shear Layer. I. Rotation

Abstract: We used data from the Helioseismic and Magnetic Imager to determine the rotation rate of the near-surface shear layer and its time variation. We applied the ring-diagram analysis technique allowing us to probe the layer between the depths of 1 Mm and 17 Mm. We find that the rotation rate increases inwards; it reaches values consistent with those inferred from global helioseismic analyses in the deeper layers, however, there are differences in the rotation rate of the northern and southern hemispheres. We show that the time variation of the rotation rate can be determined even without subtracting the time-averaged rotation rate from each epoch; however, such a subtraction is needed to get the canonical ``torsional oscillation'' signal. We find that even at depths as shallow as 1 Mm, the rotation rate shows the typical torsional oscillation pattern. The cumulative zonal displacement inferred from the residual flows exhibits a pronounced high-latitude hemispheric asymmetry and varies on solar-cycle timescales; at $75<sup>\circ$ it shows an apparent temporal association with the polar magnetic field. We find significant correlations between the cumulative displacement and magnetic activity at a subset of latitudes, with multi-year lags: the displacement leads activity by ~5 years near $15<sup>\circ$, whereas at higher latitudes activity leads by ~4 years. At mid to high latitudes, the inferred lags show a hemispheric dependence, with activity tending to lead in the north and lag in the south, suggesting possible hemispheric differences in the timing of cycle evolution and motivating longer time series to test cycle-to-cycle variation.

• The paper extends helioseismic measurements of the solar near-surface shear layer from about 1 to 17 Mm depth, using HMI ring diagrams and consistent OLA and RLS inversions to resolve rotation and zonal flows.
• The paper finds torsional-oscillation bands migrating with magnetic activity, including north–south flow asymmetries and a rotation increase of up to roughly 10 nHz below the shallowest 2 Mm.
• The paper identifies significant, latitude-dependent correlations between cumulative zonal displacement and magnetic activity at lags of approximately ±6 years, while emphasizing that the short observational record prevents causal or cycle-independent conclusions.

Overview and motivation

This paper presents a helioseismic analysis of the Sun's near-surface shear layer (NSSL) using ring-diagram inversions of Helioseismic and Magnetic Imager (HMI) data spanning Carrington rotations 2096–2307 (2010 May through 2026 January). The NSSL, the region where the solar rotation rate increases inward from the surface to roughly $0.95\,R_\odot$, is of dynamo interest: it has been invoked in distributed-dynamo models, in the equatorward migration of activity belts, and more recently as the site of a magnetorotational instability proposed to drive the solar cycle itself (Vasil et al., 2024). Despite extensive prior work, the rotation rate within the shallowest few megameters remains poorly constrained; prior local-helioseismic studies either avoided the topmost layers or averaged over depth ranges of several Mm.

The authors use custom-created $15^\circ$ ring-diagram tiles tracked for 28.8 hours with the HMI pipeline module mtrack, restricted to tiles on the central meridian and equator to avoid center-to-limb systematics. Power spectra are averaged per Carrington rotation, fitted for zonal-flow shifts $u_x$, smoothed with a 1-year running mean, and inverted independently with two complementary techniques—Optimally Localized Averages (OLA) and Regularized Least Squares (RLS)—whose agreement serves as an internal consistency check. The analysis probes depths from about 1 Mm ($0.9987\,R_\odot$) to 17 Mm ($0.975\,R_\odot$). Using HMI data alone avoids instrument-to-instrument cross-calibration corrections that are often ad hoc.

Mean rotation rate and its substructure

The time-averaged rotation rate exhibits the familiar latitudinal differential rotation at all depths, with uncertainties growing at high latitudes due to foreshortening and at the deepest radius because high-$n$ modes are unavailable. After subtracting the Snodgrass differential-rotation profile,

$$\Omega(\theta)=451.36 - 54.75\sin^2\theta - 80.21\sin^4\theta \ \ \mathrm{nHz},$$

the residuals reveal that the NSSL has internal structure: the inferred rate tracks the Snodgrass value down to $\sim 2$ Mm and then rises rapidly by up to $\sim 10$ nHz at greater depths, consistent with global helioseismic results in the deeper part of the layer. A notable finding is a persistent north–south asymmetry, with the southern hemisphere rotating slightly faster; at high latitudes this antisymmetric component reaches about 1% of the rotation rate. The authors caution that the asymmetry could reflect the known hemispheric imbalance of sunspot activity during Cycle 24 and the rising phase of Cycle 25 rather than a permanent feature of solar rotation, and they cannot fully exclude residual systematic effects—determining whether the southern hemisphere always rotates faster requires a longer record covering additional cycles.

An east–west antisymmetry along the equator, of similar magnitude to the north–south one, reproduces the "center-to-limb" effect previously reported in time-distance analyses (Vieira et al., 2012). The authors attribute it to spatial variation in observations or analysis, plausibly linked to asymmetric granulation as suggested by Baldner & Schou, but decline to apply any empirical correction because no physics-informed basis exists for doing so. This is a deliberate methodological choice whose consequence is that absolute rotation rates may carry this small systematic bias.

Time variation and torsional oscillations

A central methodological result is that the migrating zonal-flow bands—the torsional oscillation—are visible even without subtracting the time-averaged flow at each latitude; removing only the Snodgrass profile suffices. This makes the inferred time dependence robust against the arbitrary choice of averaging interval, a dependency inherent to standard global-mode analyses. When the conventional temporal-mean subtraction is applied for comparison, the resulting residuals agree closely with updated GONG-based global-mode results of Basu & Antia at $r=0.98\,R_\odot$, validating the local measurement while extending the pattern to depths as shallow as 1 Mm—shallower than any previous local-helioseismic detection of the full torsional pattern, and achieved without depth averaging.

The banded flows migrate equatorward in step with the magnetic activity index (MAI), with high activity concentrated at the boundary between the low-latitude prograde band and the adjacent retrograde band, mirroring global-mode findings. A substantial north–south asymmetry appears in the flows, strongest in the shallowest layers, consistent with hemispheric differences in magnetic activity. At $15^\circ$0 there is some indication that larger northern MAI coincides with larger northern zonal-flow residuals, though this correspondence is absent at shallower radii.

Cumulative longitudinal displacement and polar-field association

The paper introduces a cumulative longitudinal displacement $15^\circ$1, obtained by integrating the zonal-flow residuals over time at fixed latitude, interpreted kinematically as the displacement of a passive tracer advected by the residual flow (neglecting meridional transport). Because integration suppresses short-term variability, $15^\circ$2 reveals the migratory pattern more clearly than the velocity residuals themselves, lagging them by 2.7 yr—consistent with quadrature between a sinusoidal torsional oscillation and its integral. Near $15^\circ$3, nearly stationary alternating bands persist for roughly half a cycle; toward higher latitudes $15^\circ$4 becomes strongly antisymmetric between hemispheres.

At $15^\circ$5 latitude the displacements are nearly antisymmetric, reversing sign around 2017.3 in the shallow layer and about a year later at $15^\circ$6. The authors report an apparent temporal association with the polar field: enhanced southern displacement during approximately 2011–2014 coincides with the epoch when the southern polar field reversed markedly faster than the northern one, whereas the enhanced northern displacement during 2019–2023 coincides instead with a weakening southern field and a near-constant northern maximum. They note explicitly that these episodes occur at different cycle phases—late rising phase in Cycle 24 versus early rising phase in Cycle 25—which may explain why the correspondence with polar-field evolution is weaker in Cycle 25. The interpretation of $15^\circ$7 as tracer advection is an assumption; meridional circulation, which dominates polar-field evolution, is neglected, so the result establishes a temporal association rather than a demonstrated causal contribution to flux redistribution.

The authors also verify that the October 2018 HMI focus-control change—which shifts the inferred shear layer by $15^\circ$8 Mm in their companion analysis of radial shear—leaves no signature in the zonal flows or displacements, and that the $15^\circ$9 sign reversal predates it.

Lagged correlations with magnetic activity

Lagged cross-correlations between MAI and the flow diagnostics were computed at four radii, with significance assessed via 5000 phase-randomized surrogates preserving each series' power spectrum. Two robust conclusions emerge:

• MAI versus velocity residuals: correlations with $u_x$0 are generally not statistically significant at any radius.
• MAI versus cumulative displacement: significant, OLA/RLS-consistent correlations occur at a subset of latitudes, with multi-year lags spanning roughly $u_x$1 yr.

At $u_x$2—within the activity belts, where sunspots sit near the shear boundary between prograde and retrograde bands—$u_x$3 leads MAI by approximately 4.2–6.5 yr depending on hemisphere and depth, with the lead increasing toward the surface in the north. At mid-to-high latitudes the lags show a clear hemispheric dependence: positive lags (activity leading) of about +3.5 yr at $u_x$4N and +4.6–5.6 yr at $u_x$5N, versus negative lags of about −5 yr at $u_x$6S. At $u_x$7S the preferred lag is roughly half a solar cycle, placing $u_x$8 in antiphase with the unsigned MAI but approximately in phase with the signed south polar-field polarity throughout 2014–2024. Radial dependence of the lag is generally weak ($u_x$9 yr), except at $0.9987\,R_\odot$0S where it reverses sign by $0.9987\,R_\odot$1 yr between the deepest and shallowest layers.

These detections are sparse: many latitudes yield poorly constrained lags ($0.9987\,R_\odot$2), edge-of-range lags near the 7.6 yr search limit, or OLA/RLS disagreement. The authors suggest that hemispheric offsets in cycle progression and phase-dependent coupling between magnetic evolution and large-scale flows may underlie both the asymmetry and the limited number of significant detections—an explanation that remains untested without a longer baseline.

Limitations and open questions

Several caveats bound the results. The dataset covers only Cycle 24 and the rising phase of Cycle 25, so the north–south rotational asymmetry cannot be established as a long-term property, and the multi-year lags cannot yet be tested for cycle-to-cycle persistence. The east–west "center-to-limb" systematic is left uncorrected by design. The tracer interpretation of $0.9987\,R_\odot$3 neglects meridional advection, and the polar-field correspondences rest on two episodes occurring at different cycle phases. The physical mechanism linking cumulative zonal displacement to polar-field evolution or to the timing of activity emergence is not identified; the paper leaves open whether the inferred lags persist across cycles, depend on cycle phase, or reflect genuine hemispheric differences in flow–field coupling.

Conclusion

Using HMI ring-diagram inversions with dual OLA/RLS validation, this work extends the measured NSSL rotation rate and its solar-cycle variation down to 1 Mm below the photosphere, shows that the torsional-oscillation signal can be recovered without time-averaging subtraction, and quantifies a strongly antisymmetric high-latitude cumulative zonal displacement whose sign reversals track the activity cycle. The statistically significant multi-year lags between cumulative displacement and magnetic activity—at negative lag near $0.9987\,R_\odot$4 and hemispherically split lags at high latitudes—provide quantitative constraints on the timing of flow–activity coupling in the upper convection zone, motivating extension of the analysis over subsequent cycles.