Helioseismic measurements of the Sun's meridional circulation - the axisymmetric north-south flow that closes the large-scale mass transport of the convection zone and advects magnetic flux in the solar-cycle dynamo - give conflicting answers for its deep structure. Time-distance inversions of SDO/HMI data find a double cell per hemisphere (Zhao et al. 2013); a global-mode perturbation analysis finds multiple cells in depth and latitude (Schad et al. 2013); GONG-based time-distance analyses split between multi-cell and single-cell pictures (Kholikov et al. 2014; Jackiewicz et al. 2015); while 21-year averaged travel times (Liang et al. 2018) and a two-cycle MDI+GONG analysis (Gizon et al. 2020) favor a single cell. Work through 2026 shows the disagreement persists: averaging-kernel side lobes can mimic or hide weak deep cells (Fuentes et al. 2024), inversions of the same travel times return single- or double-cell profiles depending on imposed physics constraints (Hatta et al. 2024), and both HMI and GONG carry instrument systematics that still defy accurate correction (Zhao et al. 2026). The open problem is to identify and quantify the systematics responsible for the mutually conflicting inferences and to establish a consistent, validated profile of the interior meridional circulation - the number of cells per hemisphere and the depth of the equatorward return flow.

Background

The Sun transports mass and magnetic flux across latitudes by the meridional circulation, an axisymmetric (north-south) flow of plasma in the meridional planes of the solar convection zone, the outer roughly 30% of the solar radius (0.7≲r/R⊙≲10.7 \lesssim r/R_\odot \lesssim 1, with R⊙R_\odot the solar radius) where convection carries energy outward. Near the surface the flow is poleward in each hemisphere at about 20 m/s, roughly 1% of the solar rotation rate; mass conservation then requires an equatorward return flow at depth, but its depth - and whether the circulation closes in one cell per hemisphere or in several cells stacked in depth - is unknown. This matters because flux-transport (Babcock-Leighton) dynamo models, the leading explanation of the 11-year sunspot cycle, rely on the meridional circulation to carry poloidal magnetic field poleward at the surface and toroidal field equatorward at depth; single-cell and double-cell profiles imply qualitatively different cycle behavior and predictability.

Because the solar interior cannot be observed directly, interior flows are inferred by helioseismology, the study of the Sun's acoustic oscillations (p modes, pressure-driven waves trapped in the convection zone). In time-distance helioseismology one measures the difference in acoustic travel time between waves propagating northward and southward between pairs of points on the visible surface and converts these travel-time differences into flow maps by solving an inverse problem: the sensitivity of each travel time to the flow at each interior point is described by a kernel, and because the kernels overlap the inversion is ill-posed and must be regularized, often with a mass-conservation constraint expressed through a stream function (a scalar whose contours are the flow lines). In global-mode analysis one instead exploits the small shifts and distortions the flow induces in the frequencies and spatial patterns (eigenfunctions) of the p modes. The data come from Doppler imagers that measure the line-of-sight velocity of the solar surface: SOHO/MDI (Michelson Doppler Imager, 1996-2011), SDO/HMI (Helioseismic and Magnetic Imager, 2010-present), and the ground-based GONG network (Global Oscillation Network Group, 1995-present). A dominant obstacle is the center-to-limb systematic: the apparent velocity of the quiet Sun varies between disk center and limb, mainly through wavelength-dependent convective blueshift, shifting travel times by amounts comparable to the deep-flow signal itself; further systematics include spacecraft roll-angle (P-angle) errors, surface magnetic fields, and instrument-specific image artifacts.

Against this background the conflicting results appeared. Time-distance inversions of two years of HMI data (Zhao et al. 2013), after a center-to-limb correction, showed a double-cell profile: poleward flow to 0.91 R⊙0.91\,R_\odot, equatorward flow between 0.820.82 and 0.91 R⊙0.91\,R_\odot, and evidence for a second, deeper poleward cell below 0.82 R⊙0.82\,R_\odot. A global-mode eigenfunction-perturbation analysis of MDI data (Schad et al. 2013) found a deeply penetrating flow consisting of multiple cells in depth and latitude. GONG time-distance analyses split: the travel-time measurements of Kholikov et al. (2014) were reported as strengthening the multi-cell picture, while the inversions of Jackiewicz et al. (2015) found a shallow equatorward flow about 65 Mm below the surface and no strong evidence for multiple cells. A mass-conserving inversion of four years of HMI data (Rajaguru and Antia 2015) placed the return flow possibly below 0.77 R⊙0.77\,R_\odot. Twenty-one years of merged MDI and HMI travel times (Liang et al. 2018), and the two-cycle analysis of MDI plus GONG data by Gizon et al. (2020), in which the time-averaged flow is a single cell in each hemisphere with an equatorward flow of about 4 m/s at the base of the convection zone, both favor a single cell. The discrepancy has persisted: forward tests show that broad kernel side lobes can mimic or hide weak deep cells (Fuentes et al. 2024), inversions of the same travel times return single- or double-cell profiles depending on the imposed physics constraints (Hatta et al. 2024), and direct HMI-GONG comparisons reveal instrument systematics that still defy accurate correction (Zhao et al. 2026). What remains unresolved is the cause of the method- and instrument-dependent disagreement and a consistent, validated inference of the deep circulation profile.

Problem Statement

Determine the structure of the Sun's time-averaged meridional circulation in the convection zone - the number of circulation cells per hemisphere, their extent in depth and latitude, and the depth and speed of the equatorward return flow - and establish which instrumental and methodological systematics caused the mutually conflicting published inferences. This requires (1) identifying and quantifying the dominant error sources behind the disagreement among time-distance inversions of HMI data (double cell; Zhao et al. 2013), 21-year averaged MDI/HMI travel times (single cell; Liang et al. 2018), the two-cycle MDI/GONG analysis (single cell; Gizon et al. 2020), the split GONG time-distance results (Kholikov et al. 2014; Jackiewicz et al. 2015), and global-mode perturbation analysis (multiple cells; Schad et al. 2013); documented candidate error sources include the center-to-limb effect, instrument-dependent phase and image artifacts, P-angle errors, magnetic-field contamination, averaging-kernel side lobes, and the dependence of inversions on regularization and on imposed constraints such as mass conservation or angular-momentum transport; and (2) establishing the deep-flow structure, with quantified uncertainties, for explicitly stated epochs and averaging windows, validated with independent evidence - for example artificial-data tests, null tests, or comparisons of overlapping instruments and analysis methods - as justified by the analysis. No fixed composition of instruments or method families is prescribed for this validation, and a resolution that explains one method's or instrument's conflicting results as biased or artifact-driven, with the bias quantified, meets the same goal. Where conflicting inferences are attributed in whole or in part to genuine solar-cycle or hemispheric variability rather than to systematics, that variability must itself be demonstrated; a validated cycle-dependent description of the deep flow, with the systematics quantified, resolves the same problem.

References

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