The solar meridional circulation — the axisymmetric flow that carries plasma poleward at the surface and returns it equatorward somewhere in the convection zone — is a key ingredient of flux-transport dynamo models of the solar cycle, yet its structure below about 0.9 R⊙0.9\,R_\odot remains undetermined. Böning, Roth, Jackiewicz and Kholikov (2017) inverted time-distance helioseismic travel times from GONG (the Global Oscillation Network Group) with spherical Born-approximation sensitivity kernels and showed that single-cell and multi-cell flow profiles fit the same data equally well, with the multi-cell solutions appearing or vanishing depending on the singular-value-decomposition truncation and other regularization choices. Later work sharpened but did not settle the question: the strongest single-cell claim (Gizon et al. 2020, Science) relies on physical constraints in layers that cannot be directly imaged, a re-inversion of the same travel times under a different physical constraint recovers a double-cell profile with equally good residuals (Hatta et al. 2024), forward-modeling and kernel-side-lobe studies show that present techniques may be unable to distinguish the two topologies, and instrument-level systematic artifacts in HMI (the Helioseismic and Magnetic Imager) and GONG remain uncorrected (Zhao et al. 2026). The open problem is a robust, systematics-controlled determination of the time-averaged deep meridional circulation — the number of radially stacked cells in each hemisphere and the depth of the equatorial return flow — that does not hinge on the inversion method, regularization, or imposed physical constraints.

Background

The Sun's meridional circulation is a slow (∼20\sim20 m/s at the surface, about 1% of the solar rotation rate), large-scale flow pattern in the meridional planes of the solar convection zone: plasma moves poleward in the near-surface layers and completes the circuit through an equatorward return flow at depth. This circulation is a central ingredient of flux-transport (Babcock–Leighton) dynamo models, in which the return flow transports magnetic flux toward the equator and its depth controls the memory, timing, and latitudinal drift of the magnetic activity cycle. Because the frequencies of the Sun's global acoustic oscillations (p-modes) are nearly insensitive to this flow at first order, the deep circulation must be probed with local helioseismology — most commonly time-distance helioseismology, which measures differences in the propagation time of acoustic waves traveling northward versus southward across the solar disk and relates these travel-time differences to the underlying flow through sensitivity kernels. Kernels computed in the ray approximation treat the waves as infinitely narrow rays; Born-approximation kernels additionally account for the waves' finite wavelengths, which matters for broad, weak flows such as the meridional circulation. An inversion then converts the finite set of measured travel times into a flow estimate; in the SOLA method (Subtractive Optimally Localized Averaging), linear combinations of the measurements are chosen so that the resulting averaging kernels — the spatial weighting functions that map the true flow onto the estimate — approximate a localized target. The inverse problem is severely ill-posed: it must be regularized, for instance by truncating a singular value decomposition (SVD) of the kernel matrix and by tuning a trade-off between spatial resolution and error propagation, and the inferred flow profile can depend on these choices. A physically important additional constraint is mass conservation: since the divergence of the mass flux ρ v\rho\,\mathbf{v} must vanish, the horizontal and radial flow components can be derived from a single stream function, which couples them in a joint inversion. Below r≈0.9 R⊙r\approx0.9\,R_\odot the published inversions have been contradictory: Zhao et al. (2013) reported evidence for a double-cell circulation from ray-kernel inversions of data from the Helioseismic and Magnetic Imager (HMI) aboard NASA's Solar Dynamics Observatory, while Jackiewicz et al. (2015), from ray-kernel inversions of data from the Global Oscillation Network Group (GONG), a six-station network providing continuous Doppler observations of the solar surface, favored a shallower, single-cell-like profile. Böning et al. (2017) performed the first inversions of these GONG travel times using spherical Born kernels, with the full measurement covariance matrix and a cross-talk regularization; they found that single-cell and multi-cell solutions both fit the travel times, that additional cells appear at lower SVD thresholds (mainly in the southern hemisphere), and that all their solutions show a return flow at a relatively shallow depth near 0.9 R⊙0.9\,R_\odot. Several systematic effects are known to contaminate such measurements at levels comparable to the weak deep-flow signal itself: the center-to-limb effect (a travel-time shift that varies with position on the solar disk, caused by near-surface physics and instrumental effects, which can mimic a meridional flow), surface magnetic fields, B- and P-angle errors (errors in the orientation of the solar rotation axis in the images), instrumental drifts, and the choice of observation period.

Problem Statement

Determine, from solar observations with quantified systematic uncertainties, the structure of the Sun's time-averaged deep meridional circulation: is it a single circulation cell per hemisphere, or multiple cells stacked in radius (in one or both hemispheres), and at what depth does the equatorial return flow close? Equivalently, establish whether the multi-cell features that appear in some inversions — for example, the additional southern-hemisphere cells that emerge at lower SVD truncation in the Born-kernel inversions of Böning et al. (2017) — are physical or artifacts of regularization choices and unmodeled systematic effects. A defensible answer must be robust: the discrimination between single-cell and multi-cell hypotheses must exceed the combined realization-noise (the random scatter caused by the stochastic excitation of the solar oscillations) and systematic error budget, and the conclusion must not hinge on the choice of inversion methodology (e.g., SOLA, regularized least-squares, Bayesian, or Pinsker inversions — the last a statistically regularized linear-estimator method), on the regularization parameters, or on imposed physical constraints such as mass conservation or angular-momentum transport, unless those constraints are independently justified. This entails quantifying the impact of the known observational systematics — center-to-limb variations, surface magnetic fields, B- and P-angle errors, instrumental differences and drifts, and time-period selection — on deep-flow inferences, and demonstrating that the inferred return-flow depth and cell count are stable against these effects across instruments and methods. The required deliverable is the determination of the structure itself — the number of radially stacked cells in each hemisphere and the depth of the equatorial return flow; the robustness conditions and the systematics checks are the required supporting evidence for that determination, not additional standalone goals.

References

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