Yoshiki Hatta^1,2, Hideyuki Hotta¹, Takashi Sekii^2,3

1. Institute for Space-Earth Environmental Research, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, Japan
2. National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
3. Astronomical Science Program, The Graduate University for Advanced Studies, SOKENDAI, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

Meridional circulation (MC) of the Sun transports magnetic fields and angular momentum (AM) in the convective zone, thus playing important roles in the dynamo mechanism. We can observationally infer the internal meridional flow field by inverting measured wave travel times (time-distance helioseismology). Recent studies favor the single-cell MC profile (Ref [1], G20 hereafter), while the double-cell MC profile has not been ruled out yet^[2,3]; we have not reached an agreement on the solar MC profile.

The difficulty in determining the overall structure of the MC profile is mainly in that signals produced by MC are so small in measured travel times that it is hard to carry out inversion robustly; the information content may not be sufficient. In this light, physical constraints inspired by numerical simulations of solar convection may be useful as prior information in inversions. Hotta & Kusano [Ref. 4, HK21 hereafter] succeeded in reproducing the solar equator-fast rotation without any tweaking. Interestingly, MC in their model transports AM toward the equator, which supports equator-fast rotation. How about focusing on this physics, namely, equatorward AM transport by MC, to implement a physical constraint in the inversion to infer the MC profile? This is our main idea.

We used the data (travel times, observational uncertainties, kernels, and regularization matrices) that are made publicly available by G20. As an inversion method, we follow G20 and use the regularized least-squares method, where a sum of the residual term and the regularization term is minimized. The physical constraint that MC transports AM toward the equator (hereafter HK21-type constraint) can be expressed as additional regularization terms because AM flux by MC is linearly related to the meridional flow field. The minimization balance is controlled by the so-called trade-off parameters. If we put priority on minimization of the regularization terms that represent the HK21-type constraint, we may get an inversion result of equatorward AM transport by MC. More details on inversion procedures can be found in Ref [3].

Figure 1| Latitudinal component of the meridional flow field (U_θ) inferred without or with the HK21-type constraint (left and right panels, respectively). Red (blue) color indicates the northward (southward) flow. Contours of the stream functions are overplotted in both panels, where the solid (dashed) curve represent clockwise (counterclockwise) motion. The definition of the stream function can be found in H24.

Figure 1 shows a comparison of inversion results. The left (right) panel shows the MC profile inferred without (with) the HK21-type constraint. We have confirmed the G20 result (single-cell MC) in the case without the HK21-type constraint, and we have obtained a different result (double-cell MC) in the case with the constraint. Figure 2 shows radially averaged AM fluxes by MC that are computed for the MC profiles. We see poleward (equatorward) AM transport by MC in the case of the single-cell (double-cell) MC profile.

Figure 2| Latitudinal AM flux by MC radially averaged F_MC,θ, see H24 for the definition) as a function of the latitude. The red, blue, and green curves represent F_MC,θ computed with the double-cell solution, single-cell solution, and simulation result from Ref. [3], respectively.

Obtaining the double-cell solution under the HK21-type constraint is expected. This is because single-cell MC always transports AM toward poles if we assume the mass conservation in the convective zone as was assumed in Ref. [1] and [3]. (The theoretical argument can be found in Ref [3].) In other words, we cannot obtain a single-cell solution under the HK21-type constraint. Still, we would like to be conservative; we do not conclude that the MC profile is double cell. This is because the resolution of the averaging kernels for the single-cell solution is better than those for the double-cell solution (Figure 3), indicating that inversion works better for the single-cell solution. Note that this is also expected, as the number of constraints in inversion is smaller for the case of single-cell solution.

Figure 3| Averaging kernels computed with the double-cell solution (upper row) and those computed with the single-cell solution (lower row) for three different target depths (r/R_☉ ~ 0.82, 0.925, 0.975, from left to right). We see a clear difference in localization of the averaging kernels for the target depth r/R_☉ ~ 0.82; the averaging kernel for the double-cell solution fails to be localized around the target point. For more details, see Ref [3].

We thus have a dilemma so far: the constraint of equatorward AM transport by MC favors double-cell MC, but the localization of averaging kernels favors single-cell MC. One possible next step is to test another physical constraint that AM transport by Reynolds stress is equatorward, because it is generally considered that Reynolds stress can also contribute to the solar equator-fast rotation (e.g. Ref [5]). It should be relevant to keep investigating the solar MC profile from both theoretical and observational perspectives.

References

[1] Gizon, L., Cameron, R. H., Pourabdian, M., et al. 2020, Science, 368, 1469
[2] Zhao, J., Bogart, R. S., Kosovichev, A. G., Duvall, T. L. J., & Hartlep, T. 2013, ApJL, 774, L29
[3] Hatta, Y., Hotta, H., & Sekii, T. 2024, ApJ, 972, 79
[4] Hotta, H., & Kusano, K. 2021, Nature Astronomy, 5, 1100
[5] Featherstone, N., & Miesch, M. S. 2015, ApJ, 804, 67