The Sun's meridional circulation — the large-scale poleward surface flow with a subsurface equatorward return flow — is a key unconstrained ingredient of flux-transport solar dynamo models, where its speed, depth of penetration, and single- versus multi-cellular structure set the cycle period and the equatorward drift of sunspot belts. Schad, Timmer & Roth (2013) reported global-helioseismic evidence, from perturbations of p-mode eigenfunctions, for a deeply penetrating, multi-cellular meridional flow, but could reconstruct the total flow only between 0.820.82 and 0.970.97 solar radii and acknowledged uncontrolled estimator bias and center-to-limb systematics. A decade of subsequent work — time-distance inversions from HMI, MDI and GONG, Fourier–Legendre analyses, forward modeling, and Bayesian methods — has produced discordant profiles: single-cell results coexist with double-cell results, and the differences trace to center-to-limb corrections, inversion priors, and instrumental miscalibrations. The open problem is the definitive, systematics-controlled determination of the meridional flow field throughout the convection zone — including the number of cells, the return-flow depth, and temporal variability — together with its forward-modeling consequences for the solar dynamo.

Background

The Sun's outer ~30% by radius is the convection zone, a turbulent layer that hosts three large-scale flows: differential rotation (rotation rate varying with latitude and depth), torsional oscillations (slow, solar-cycle-dependent bands of slightly faster and slower rotation), and the meridional circulation. The meridional circulation is a slow axisymmetric flow, poleward near the surface with horizontal speeds of about 20 m s−120\,\mathrm{m\,s^{-1}} at mid-latitudes; mass conservation requires a compensating equatorward return flow at depth, but the depth at which the flow returns and whether the circulation consists of one or several stacked cells in radius and latitude are not established. The answer matters far beyond flow cartography: in the flux-transport (Babcock–Leighton) dynamo — the standard paradigm for the 11-year activity cycle, in which the meridional circulation advects poloidal magnetic field poleward at the surface and equatorward at depth — the circulation speed and return-flow depth largely set the cycle period and the equatorward migration of the sunspot belts, so the internal flow profile directly conditions cycle physics and prediction.

Helioseismology infers the Sun's interior structure and dynamics from its acoustic oscillations. The Sun rings in millions of p modes — standing sound waves trapped by density stratification — observed as Doppler velocity oscillations. Local helioseismic techniques (time-distance helioseismology, which measures acoustic travel times between pairs of surface points, and ring-diagram analysis, which fits oscillation power rings in local patches) probe only the outer few percent of the radius. Deep inferences require global methods. In global mode coupling, a stationary flow slightly distorts the spatial eigenfunctions of the low- and medium-degree p modes, mixing modes of neighboring spherical-harmonic degree; the resulting coupling coefficients are first-order sensitive to the flow and can be inverted for it. Observational data come chiefly from the Michelson Doppler Imager (MDI) on SOHO (1996–2011), the Helioseismic and Magnetic Imager (HMI) on SDO (2010–present), and the ground-based Global Oscillation Network Group (GONG).

Before 2013, frequency-shift measurements suggested shallow flow reversals but were recognized to be only second-order sensitive to the flow. Zhao et al. (2013), using time-distance helioseismology on the first two years of HMI data, found a poleward flow extending to about 0.77 R0.77\,R (with RR the solar radius) and an equatorward flow below — evidence for a double-cell circulation. Schad, Timmer & Roth (2013) applied the mode-coupling method to MDI data from 2004–2010, expanding the flow in zonally symmetric spherical harmonics of degree ss. They found a significant poleward s=2s=2 component down to 0.82 R0.82\,R and an s=8s=8 component with flow reversals significant down to the base of the convection zone, concluding that the meridional flow is deeply penetrating and multi-cellular. However, the total vector flow was reconstructed only where all even components overlapped, 0.82≤r/R≤0.970.82\le r/R\le0.97; components s=1,3,5,6,7s=1,3,5,6,7 showed an unexplained systematic drop of the inferred radial coefficients in the near-surface layer 0.9<r/R<10.9<r/R<1, suspected to arise from non-linear estimation of the mode-coupling (bb) coefficients on insignificant couplings; and the analysis used sensitivity kernels computed from Solar Model S — the standard reference model of the Sun's interior — together with the MDI leakage matrix, which quantifies instrumental cross-talk between neighboring spherical-harmonic components, without accounting for the center-to-limb effect — an apparent Doppler shift that varies with viewing angle across the solar disk and contaminates flow measurements at a level comparable to the meridional-flow signal itself.

Problem Statement

Determine, from helioseismic observations, the total meridional flow field of the Sun throughout the convection zone — the radial and horizontal components (or an equivalent mass-conserving stream function) as functions of radius from the base of the convection zone to the photosphere and of latitude in both hemispheres — with quantified uncertainties, thereby settling (i) the number of meridional circulation cells in depth and latitude, (ii) the depth at which the equatorward return flow closes, and (iii) where the observational record permits, the temporal variation of the profile over the solar cycle. To count as a determination, the result must address the systematics identified by Schad et al. (2013) as limiting their analysis — the center-to-limb effect and, for global mode-coupling measurements, the estimator bias on insignificant couplings — and the inferred cell structure may not be imposed by the assumed depth of the return flow. As the final, required component, use the inferred profile in time-dependent flux-transport (Babcock–Leighton) dynamo forward modeling to assess how the layering and counterflows of the circulation influence magnetic-flux transport, cycle timing, and the equatorward drift of activity belts. All components serve one objective: replacing the current discordant inferences with a definitive internal flow profile and its established dynamo consequences.

References

  1. A. Schad, J. Timmer, and M. Roth, 'Global helioseismic evidence for a deeply penetrating Solar meridional flow consisting of multiple flow cells', The Astrophysical Journal Letters 778, L38 (2013). DOI: 10.1088/2041-8205/778/2/L38. https://doi.org/10.1088/2041-8205/778/2/L38 (arXiv:1311.7623)
  2. J. Zhao, R. S. Bogart, A. G. Kosovichev, T. L. Duvall Jr., and T. Hartlep, 'Detection of Equatorward Meridional Flow and Evidence of Double-Cell Meridional Circulation inside the Sun', The Astrophysical Journal Letters 774, L29 (2013). DOI: 10.1088/2041-8205/774/2/L29. https://doi.org/10.1088/2041-8205/774/2/L29 (arXiv:1307.8422)
  3. S. P. Rajaguru and H. M. Antia, 'Meridional Circulation in the Solar Convection Zone: Time-Distance Helioseismic Inferences from Four Years of HMI/SDO Observations', The Astrophysical Journal 813, 114 (2015). DOI: 10.1088/0004-637X/813/2/114. https://doi.org/10.1088/0004-637X/813/2/114 (arXiv:1510.01843)
  4. J. Jackiewicz, A. Serebryanskiy, and S. Kholikov, 'Meridional Flow in the Solar Convection Zone. II. Helioseismic Inversions of GONG Data', The Astrophysical Journal 805, 133 (2015). DOI: 10.1088/0004-637X/805/2/133. https://doi.org/10.1088/0004-637X/805/2/133
  5. R. Chen and J. Zhao, 'A Comprehensive Method to Measure Solar Meridional Circulation and the Center-to-limb Effect Using Time-Distance Helioseismology', The Astrophysical Journal 849, 144 (2017). DOI: 10.3847/1538-4357/aa8eec. https://doi.org/10.3847/1538-4357/aa8eec (arXiv:1709.07905)
  6. R. Chen and J. Zhao, 'Frequency Dependence of Helioseismic Measurements of the Center-to-Limb Effect and Flow-induced Travel-time Shifts', The Astrophysical Journal 853, 161 (2018). DOI: 10.3847/1538-4357/aaa3e3. https://doi.org/10.3847/1538-4357/aaa3e3
  7. C.-H. Lin and D.-Y. Chou, 'Solar-cycle Variations of Meridional Flows in the Solar Convection Zone Using Helioseismic Methods', The Astrophysical Journal 860, 48 (2018). DOI: 10.3847/1538-4357/aac026. https://doi.org/10.3847/1538-4357/aac026
  8. Z.-C. Liang, L. Gizon, A. C. Birch, T. L. Duvall Jr., and S. P. Rajaguru, 'Solar meridional circulation from twenty-one years of SOHO/MDI and SDO/HMI observations. Helioseismic travel times and forward modeling in the ray approximation', Astronomy & Astrophysics 619, A99 (2018). DOI: 10.1051/0004-6361/201833673. https://doi.org/10.1051/0004-6361/201833673
  9. S. P. Rajaguru and H. M. Antia, 'Time-Distance Helioseismology of Deep Meridional Circulation', in Dynamics of the Sun and Stars, Astrophysics and Space Science Proceedings 57, 107–113 (2020). DOI: 10.1007/978-3-030-55336-4_11. https://doi.org/10.1007/978-3-030-55336-4_11
  10. L. Gizon, R. H. Cameron, M. Pourabdian, Z.-C. Liang, D. Fournier, A. C. Birch, and C. S. Hanson, 'Meridional flow in the Sun's convection zone is a single cell in each hemisphere', Science 368, 1469–1472 (2020). DOI: 10.1126/science.aaz7119. https://doi.org/10.1126/science.aaz7119
  11. D. C. Braun, A. C. Birch, and Y. Fan, 'Probing the Solar Meridional Circulation using Fourier Legendre Decomposition', The Astrophysical Journal 911, 54 (2021). DOI: 10.3847/1538-4357/abe7e4. https://doi.org/10.3847/1538-4357/abe7e4 (arXiv:2103.02499)
  12. A. M. Stejko, A. G. Kosovichev, and V. V. Pipin, 'Forward Modeling Helioseismic Signatures of One- and Two-cell Meridional Circulation', The Astrophysical Journal 911, 90 (2021). DOI: 10.3847/1538-4357/abec70. https://doi.org/10.3847/1538-4357/abec70 (arXiv:2101.01220)
  13. S. M. Hanasoge, 'Surface and interior meridional circulation in the Sun', Living Reviews in Solar Physics 19, 3 (2022). DOI: 10.1007/s41116-022-00034-7. https://doi.org/10.1007/s41116-022-00034-7
  14. A. Herczeg and J. Jackiewicz, 'Inferring the Solar Meridional Circulation Flow Profile by Applying Bayesian Methods to Time-distance Helioseismology', The Astrophysical Journal 954, 187 (2023). DOI: 10.3847/1538-4357/acea7b. https://doi.org/10.3847/1538-4357/acea7b (arXiv:2308.11035)
  15. J. R. Fuentes, B. W. Hindman, J. Zhao, C. C. Blume, M. E. Camisassa, N. A. Featherstone, T. Hartlep, L. Korre, and L. I. Matilsky, 'Assessing the Observability of Deep Meridional Flow Cells in the Solar Interior', The Astrophysical Journal 961, 78 (2024). DOI: 10.3847/1538-4357/ad13f3. https://doi.org/10.3847/1538-4357/ad13f3 (arXiv:2308.07513)
  16. S. G. Kashyap and S. M. Hanasoge, 'Modeling the Center-to-limb Systematic in Normal-mode Coupling Measurements', The Astrophysical Journal 978, 78 (2024). DOI: 10.3847/1538-4357/ad9476. https://doi.org/10.3847/1538-4357/ad9476 (arXiv:2411.07717)
  17. A. Schad and M. Roth, 'Inference of solar rotation from perturbations of acoustic mode eigenfunctions', The Astrophysical Journal 890, 32 (2020). DOI: 10.3847/1538-4357/ab65ec. https://doi.org/10.3847/1538-4357/ab65ec (arXiv:2002.06114)
  18. A. R. Choudhuri, 'The meridional circulation of the Sun: Observations, theory and connections with the solar dynamo', Science China Physics, Mechanics & Astronomy 64, 239601 (2021). DOI: 10.1007/s11433-020-1628-1. https://doi.org/10.1007/s11433-020-1628-1 (arXiv:2008.09347)
  19. G. Hazra, B. B. Karak, and A. R. Choudhuri, 'Is a Deep One-cell Meridional Circulation Essential for the Flux Transport Solar Dynamo?', The Astrophysical Journal 782, 93 (2014). DOI: 10.1088/0004-637X/782/2/93. https://doi.org/10.1088/0004-637X/782/2/93
  20. B. Belucz, M. Dikpati, and E. Forgács-Dajka, 'A Babcock–Leighton Solar Dynamo Model with Multi-cellular Meridional Circulation in Advection- and Diffusion-dominated Regimes', The Astrophysical Journal 806, 169 (2015). DOI: 10.1088/0004-637X/806/2/169. https://doi.org/10.1088/0004-637X/806/2/169
  21. V. V. Pipin and A. G. Kosovichev, 'On the Origin of the Double-cell Meridional Circulation in the Solar Convection Zone', The Astrophysical Journal 854, 67 (2018). DOI: 10.3847/1538-4357/aaa759. https://doi.org/10.3847/1538-4357/aaa759
  22. S. Cloutier, R. H. Cameron, and L. Gizon, 'A Babcock-Leighton dynamo model of the Sun incorporating toroidal flux loss and the helioseismically-inferred meridional flow', Astronomy & Astrophysics 680, A42 (2023). DOI: 10.1051/0004-6361/202347022. https://doi.org/10.1051/0004-6361/202347022 (arXiv:2309.17328)
  23. J. Zhao, R. Chen, S. P. Rajaguru, and S. Kholikov, 'Oscillatory Phase and Acoustic Travel-Time Inconsistencies Measured between SDO/HMI and GONG Dopplergrams' (2026). arXiv:2606.06703. https://arxiv.org/abs/2606.06703
  24. A. V. Getling and A. G. Kosovichev, 'Variation of global and local flows in the solar convection zone during activity cycles 24 and 25' (2025). arXiv:2501.02607. https://arxiv.org/abs/2501.02607