M. Cristina Rabello Soares1, Sarbani Basu2, Richard S. Bogart1
1 Department of Physics, Stanford University, Stanford, CA 94305-4013, USA
2 Department of Astronomy, Yale University, PO Box 208101, New Haven, CT 06520-8101, USA
The Sun does not rotate as a solid body: its rotation rate varies with both latitude and depth. Near the surface, its rotation rate sharply increases inward through a layer extending to roughly 0.95R☉. This near-surface shear layer (NSSL) is of particular interest because rotational shear and large-scale flows may influence the solar dynamo and the transport of magnetic flux. Although the NSSL has been studied extensively, the rotation rate in the immediate near-surface layers remains uncertain; the shallowest layer examined here, about 1 Mm below the photosphere, has not previously been probed in much detail.
In this study[1], we used observations from the Helioseismic and Magnetic Imager (HMI) on the Solar Dynamics Observatory to measure rotation from about 1 to 17 Mm below the photosphere. The data span May 2010 through January 2026, covering Solar Cycle 24 and much of Solar Cycle 25. We analyzed 15° ring diagrams, obtaining zonal velocities from Carrington-rotation-averaged power spectra and smoothing the resulting time series with a 1-year running mean. Two complementary inversion techniques – Optimally Localized Averages (OLA) and Regularized Least Squares (RLS) – provide an important consistency check.
Figure 1. Zonal-flow residuals (in m/s) versus time and latitude at four fractional radii. Left column: residuals were obtained by first subtracting the Snodgrass (1984) differential rotation profile, then subtracting the temporal and latitudinal mean at each depth to center the residuals around zero. Right column: residuals after subtracting the time-averaged flow at every depth and latitude. The contours show the magnetic activity index.
To study temporal changes in the rotation without making the result depend on the selected observing interval, we subtract a standard differential-rotation profile[2] and center the residuals by removing the temporal and latitudinal mean at each depth. Figure 1 (left) reveals clear equatorward-migrating bands of flow.
The flows differ significantly between the northern and southern hemispheres, particularly in the shallowest layers. This may reflect the hemispheric asymmetry of magnetic activity during the observed cycles. The hemispheric difference in the time-averaged rotation is only about one percent of the total rotation rate, and the available data cannot establish whether it is a persistent property of solar rotation or a consequence of the activity asymmetry. We also find an east-west antisymmetry that is likely observational or analysis-related, underscoring the importance of separating solar signals from center-to-limb systematics.
For direct comparison with measurements from global helioseismology and other techniques, we also subtract the time-averaged flow at every depth and latitude. The remaining zonal-flow bands, often called torsional oscillations, migrate toward the equator with the activity belts. Figure 1 (right) shows that this pattern is already present about 1 Mm below the surface and remains visible throughout the NSSL. At 0.98R☉, our results agree with global helioseismology (Figure 3 of Ref [3]). The bands occupy the same locations as in Figure 1 (left), but the retrograde flows are stronger. The Figure 1 (left) definition is nevertheless useful because it is insensitive to the exact observation interval.
Figure 2. Cumulative longitudinal displacement (in Mm) versus time and latitude. Columns show the full, north-south symmetric, and antisymmetric components; rows sample four depths. Magnetic activity contours are overplotted.
Surface magnetic activity is quantified by the magnetic activity index (MAI)[4], defined for each tracked tile as the unsigned HMI line-of-sight magnetic flux integrated above a 50 G threshold and averaged in the same manner as the flow time series. MAI contours are overplotted in Figures 1 and 2.
To explore what these small residual flows can produce over time, we introduce a new diagnostic: the cumulative longitudinal displacement, Δs, obtained by integrating the residual zonal velocity over time (Figure 2). It can be interpreted as the longitudinal displacement of a passive tracer carried by the residual zonal flow at a fixed latitude, while neglecting meridional circulation.
The displacement pattern lags the zonal-flow residuals by 2.7 years. After this shift, the north-south symmetric component reaches an average peak correlation of about 0.81, slightly higher than the unsymmetrized signal; at zero lag, the correlation is consistent with zero. This quarter-cycle lag is expected for an approximately 11-year sinusoidal torsional oscillation: integrating the velocity places the displacement in temporal quadrature, corresponding to a 90-degree phase shift.
The displacement also reveals pronounced high-latitude hemispheric asymmetry and solar-cycle variability (Figure 2 right column). At 75° latitude, it is largely antisymmetric, with the dominant hemisphere changing between successive half-cycles. The sign reversal occurs near 2017.3 in the shallowest layer and about one year later at a depth of 14 Mm (Figure 10 in Ref [1]). Some of these changes coincide with variations in the HMI polar field, raising the possibility that longitudinal advection by the residual zonal flows makes a secondary contribution to high-latitude magnetic-flux redistribution, alongside the dominant poleward transport.
To quantify the relative timing of the magnetic and flow signals, we performed a lagged-correlation analysis at each latitude. Significant correlations with MAI occur at only a limited number of latitudes and are more common for Δs than for ΔUx, consistent with Δs integrating the flow signal over time. The inferred lags span several years and hint at a hemispheric dependence: magnetic activity generally leads the cumulative displacement in the north but lags it in the south. A longer HMI time series is needed to determine whether these differences persist across solar cycles.
We show that rotation changes can be measured relative to a standard differential-rotation profile, avoiding an interval-dependent temporal mean. An innovation of this work is the cumulative longitudinal displacement, Δs, a new diagnostic whose 2.7-year lag behind the zonal-flow residuals matches the quarter-cycle offset expected for an approximately 11-year torsional oscillation. It reveals high-latitude hemispheric differences and cycle-scale variability. Its timing relative to magnetic activity appears to differ between hemispheres, but a longer HMI record is needed to determine whether this behavior recurs.
References
[1] Rabello Soares, M. C., Basu, S., & Bogart, R. S. 2026, arXiv:2608.19438 (doi.org/10.3847/1538-4357/ae985b)
[2] Snodgrass, H. B. 1984, SoPh, 94, 13
[3] Basu, S., & Antia, H. M. 2019, The Astrophysical Journal, 883, 93.
[4] Bogart, R. S., Baldner, C., Basu, S., et al. 2011, Journal of Physics: Conference Series, 271, 012008.