Cristina Rabello Soares1, Sarbani Basu2, Rick Bogart1
1. W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305-4085, U.S.A.
2 Department of Astronomy, Yale University, PO Box 208101, New Haven, CT 06520-8101, USA
Introduction
The gradient of rotation in the near-surface shear layer (NSSL) of the Sun provides valuable insights into the dynamics associated with the solar activity cycle and the dynamo. Results obtained with global oscillation mode splitting lack resolution near the surface, prompting the use of the local helioseismic ring-diagram method. While the Helioseismic and Magnetic Imager ring-analysis pipeline has been used previously for analyzing this layer, default pipeline parameters limit the accuracy of the near surface gradients. To address these challenges, we fitted the flow parameters to power spectra averaged over one-year periods at each location, followed by additional averaging over 12 years. The complete analysis can be found in Ref [1].
Figure 1| We identified three distinct regions within the NSSL: a deeper layer (D), a middle layer (M), and one closest to the surface (S). Rotational gradient obtained using OLA and RLS inversion techniques for the 30-degree tiles at the equator are shown in blue and red, respectively. Global helioseismology results from Ref. [2] for HMI are symbolized by stars at 0.97 and 0.99 R⊙. The horizontal dotted line corresponds to a dimensionless gradient of −1.
Three Distinct Regions
We find that the NSSL can be divided into three fairly distinct regions (Figure 1).
• A deeper, larger region with small shear, where the radial gradient of the logarithmic rotation rate increases in amplitude from -0.5 at 21 Mm to -1 at 4 Mm.
• A narrow middle layer with a strong shear, with a gradient approximately three times larger than in the deeper layers. This layer appears to be centered at 3 Mm, the poor resolution in these layers implies that this is potentially located closer to the surface, around 1.5 Mm deep.
• Additionally, a layer very close to the surface, where the logarithmic gradient is close to zero but becomes steeper again towards the surface.
Figure 2| The North-South symmetric component of the rotational
gradient plotted against latitude at three depths: 20.88 Mm (0.970 R⊙), 4.66 Mm (0.9933 R⊙), and 2.30 Mm (0.9967 R⊙) for both inversion methods and tile sizes. The solid lines represent the linear fits against cosine(latitude) for latitudes smaller than 60◦. While values for 60◦ latitude are shown, they were not included in the fitting. Dotted lines indicate the confidence level of the fitting to the OLA results. Those for the RLS fitting have similar values. The black crosses in the top right panel represent results from Ref. [2].
Latitudinal Variations
To investigate the latitude-dependent variation of the rotational gradient in our results, we fitted a straight line to the north-south symmetric gradient as a function of the cosine of latitude (Fig. 2). The North-South symmetric component rotational gradient is calculated as the weighted average of the northern and southern components, denoted as (N+S)/2. For both OLA and RLS inversion results, we could identify three depth ranges that show Pearson correlation coefficients that indicate a probability of lower than 0.05 for no correlation: 19.5 ± 2.1 Mm, 4.11 ± 0.70 Mm, and 2.23 ± 0.14 Mm. However, it is essential to acknowledge the possibility of small latitude variations elsewhere given the uncertainties inherent in gradient determination.
Figure 3| Comparison of solar rotation rates and their gradients. Left: Rotation rates from 30-degree tile inversions (filled circles), a simple model for 15 and 30-degree tiles, and simulations, with the Carrington rate subtracted. Green dotted line shows rates from RHD simulations[3]. Solid green line represents their results weighted by OLA averaging kernels for 30-degree equatorial tiles. Right: Rotation rate gradient at 30◦ latitude. Green dotted line shows results from Ref. [3]; solid green line shows gradient after applying OLA averaging kernels. Blue filled circles and red triple-dot-dashed line represent OLA and RLS inversion results, respectively. Error bars shown for OLA only; RLS uncertainties are comparable. Results for 30◦S (not shown) closely mirror 30◦N. Black star indicates global helioseismology results [2] at 0.99 R⊙.
Within the depth range of 21.6 to 17.4 Mm (0.969-0.975 R⊙), a substantial increase in the logarithmic gradient with latitude is noted, aligning with the findings of Ref. [4] and [5] when analyzing radii closer to the surface than ours (∼ 7 Mm deep).
Simple Model
To assess the impact of the width of the averaging kernels on our estimates of the rotation rate, we used a simple model and examined how it would be reproduced in our analysis. The model consisted of a linear component and a Gaussian with a negative amplitude. This analysis suggests that the solar rotation rate declines towards the surface more sharply than the inversion results, starting at a shallower depth, around 2 Mm deep, with a decrease of 5 nHz at 1.5 Mm depth (Fig. 3 – left panel). After reaching this minimum, the rotation rate appears to increase again; note that at this point, we are either within, or in close proximity to, the super-adiabatic layer.
We also applied the averaging kernels to the rotation rate calculated from Kitiashvili et al. (2023)’s 3D radiative simulation. Their simulation too exhibits a second shear layer (layer M) whose rotational gradient has similar amplitude and location to that observed in this work, but with a much broader width.
References
[1] Rabello Soares, M. C., Basu, Sarbani, and Bogart, R. S. 2024, ApJ, 967, 143
[2] Antia, H. M., and Basu, S. 2022, ApJ, 924, 19
[3] Kitiashvili, I. N. et al. 2023, MNRAS, 518, 504
[4] Corbard, T., and Thompson, M. J. 2002, Solar Phys, 205, 211
[5] Zaatri, A., and Corbard, T. 2009, ASPCS, 416, 99