Ruizhu Chen1,2 and Junwei Zhao2
1. Department of Physics, Stanford University, Stanford, CA 94305-4060
2. W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305-4085
Meridional circulation is a crucial component of the Sun’s internal dynamics, playing an important role in driving the solar dynamo, transporting magnetic flux, and redistributing angular momentum. But its inference in the deep interior through helioseismic methods is complicated by a systematic center-to-limb (CtoL) effect1. Previously, an empirical method, removing travel-time shifts measured for east-west traveling waves in the equatorial area as a proxy of the CtoL effect from those measured for north-south traveling waves in the central meridian area2,3, was used, but its validity and accuracy need to be assessed. Here we develop a new method to separate the center-to-limb effect, δτCtoL, and meridional-flow-induced travel-time shifts, δτMF, in a more robust way with an attempt to obtain a more precise meridional-circulation profile.
In this study, we apply time-distance helioseismic measurements on the SDO/HMI Doppler data of 7 years. Our new method exhaustively measures the travel-time shifts between two surface locations along the solar disk’s radial direction for all azimuthal angles and all travel distances. The measured travel-time shifts are a linear combination of δτCtoL and δτMF determined by geometries, and the two shifts can be disentangled through solving the linear equation set. Our new method separates the CtoL effect using whole-disk measurements, and is more robust than the previous proxy-removal method, which measures the CtoL effect in a narrow band in the equatorial area. The solar B-angle variation is also reflected in the linear equation set and is handled analytically. Besides, we also apply a magnetic-region mask on the observation data to remove the surface magnetic effect in the travel-time shift measurements4.
Figure 1| (a) CtoL effect δτCtoL disentangled from our measurements. (b) Uncertainties of the δτCtoL. (c) Meridional-flow-induced travel-time shifts δτMF disentangled from our measurements. (d) Uncertainties of the δτMF.
The linear equation set relating δτCtoL and δτMF is solved in a least-square sense with a 2nd-order Tikhonov regularization on δτCtoL and a 1st-order regularization on δτMF. The disentangled δτCtoL and δτMF for the 7-yr period of measurements are shown in Figure 1. At large skip-distances the δτMF is one order of magnitude smaller than δτCtoL, making the CtoL-effect removal critical for meridional-flow determination. We then examine the symmetry of δτCtoL relative to the azimuthal angle for a few skip distances, as shown in Figure 2. The CtoL effects of different azimuthal angles agree well within error bars, indicating that the CtoL effect is indeed isotropic relative to the azimuthal angle. This confirms the validity of the previous method – removing a CtoL-effect proxy that is assessed along the equator. However, since a small error in the CtoL effect can cause a big error in the inverted flow, the advanced determination of the CtoL effect in our paper is still favored to improve the meridional flow inference.
The δτMF disentangled from the CtoL effect is then inverted for the meridional circulation, using ray-path approximation kernels for the acoustic travel-time shifts and solved in a least-square sense in the Fourier domain. Our inversion results are shown in Figure 3. In both hemispheres for all latitudes, the poleward flow, of about 10 m s−1, extends from the surface to about 0.91 R⊙. Below 0.91 R⊙, the flow turns equatorward with a speed of about 5 m s−1, extending to about 0.82 R⊙ for low latitude regions, but not as deep for higher latitudes. Beneath the layer of equatorward flow, the flow turns poleward again with a speed of lower than 5 m s−1. The averaged velocity profiles for 10° − 25° and 25° − 40° latitudinal bands, as shown in Figure 3b and 3c respectively, better illustrate the two direction reversals at depths of about 0.91 R⊙ and 0.82 R⊙ for 10° − 25° latitude, and 0.91 R⊙ and 0.85 R⊙ for 25° − 40° latitude. The overall pattern indicates a double-cell circulation in each hemisphere, although the northern and southern hemispheres exhibit an asymmetry in these inverted flow results. Compared with the results shown in Ref. 2, our results show a similar 3-layer flow structure but give a substantially slower speed in the deepest layer.
Figure 3| (a) Inverted meridional flow in the convection zone, with positive flow toward north. (b) Meridional-flow profile averaged from the 10° − 25° latitudinal bands in both hemispheres. (c) Same as panel (b), but for the 25° − 40° latitudinal bands.
For more details of this work, please refer to our recent publication Ref. 5.
 Zhao, J., Nagashima, K., Bogart, R. S., Kosovichev, A. G., Duvall, T. L., Jr. 2012, ApJL, 749, L5
 Zhao, J., Bogart, R. S., Kosovichev, A. G., Duvall, T. L., Jr., Hartlep, T. 2013, ApJL, 774, L29
 Rajaguru, S. P., & Antia, H. M. 2015, ApJ, 813, 114
 Liang, Z.-C., & Chou, D.-Y. 2015, ApJ, 805, 165
 Chen, R., & Zhao, J. 2017, ApJ, 849, 144