The solar meridional circulation - the large-scale poleward-at-surface flow with an equatorward return flow somewhere in the convection zone - is a primary ingredient of flux-transport dynamo models of the solar cycle, yet helioseismic inversions still disagree on its depth profile: some find a single cell per hemisphere with the return flow near the base of the convection zone, others a double cell with a shallow return flow. The source paper (Mandal et al. 2018) performed a mass-conserving inversion with spherical Born kernels and found a single cell with return flow below 0.78 solar radii, while explicitly acknowledging that center-to-limb and magnetic-region systematics, noise-covariance treatment, kernel modeling (including line-of-sight effects), multi-year averaging versus solar-cycle variability, and the residual single-/multi-cell ambiguity at low latitudes all remain unresolved. The subsequent literature (2018–2026) leaves the core question open: competing flow morphologies still fit the existing travel-time data comparably, the dominant systematic errors remain unexplained at the level of the deep-flow signal, and cycle-resolved inversions disagree on morphology. The problem is to determine the deep meridional circulation profile - the number of radially stacked cells, the return-flow depth, and its solar-cycle dependence - with systematics, noise, and forward-modeling treatment adequate to exclude the competing morphologies at stated confidence.

Background

The Sun's meridional circulation is the axisymmetric (north–south) component of the large-scale internal plasma flow: plasma moves poleward near the surface in each hemisphere at 10–20 m/s and must return equatorward at depth to conserve mass. Where this return flow closes - near the base of the convection zone (a single cell per hemisphere) or at intermediate depths with one or more counter-cells stacked in radius - is a decisive input for flux-transport dynamo models, in which the meridional flow carries the poloidal magnetic field poleward at the surface and equatorward at depth, setting the pace and geometry of the 11-year sunspot cycle.

Because the flow is slow (a few m/s or less in the deep interior) and axisymmetric, it cannot be detected by conventional global-mode helioseismology; it is probed by local helioseismology. In time-distance helioseismology, one cross-correlates the Doppler velocity signal observed between pairs of points on the solar disk and fits a wavelet to measure the wave travel time between them; a subsurface flow shifts the travel time in the flow direction. North–south travel-time differences, measured across many distances (which sample different depths), are inverted to recover the flow as a function of depth and latitude. The inversion needs sensitivity kernels - functions that relate a flow perturbation at each interior point to the induced travel-time shift. Ray-approximation kernels (treating the waves as traveling along ray paths) are cheap but unreliable when the flow varies on scales below the acoustic wavelength; finite-frequency kernels computed in the first-Born approximation (a single-scattering perturbation treatment borrowed from quantum mechanics) remove that limitation, and spherical Born kernels appropriate for large-scale flows were developed around 2016–2017. Physically consistent inversions enforce mass conservation by inverting for a stream function (a scalar whose derivatives give both the radial and latitudinal velocity components) rather than for the two flow components independently.

The observational picture has long been contradictory. Zhao et al. (2013, ApJ 774, L29) inverted two years of data from the Helioseismic and Magnetic Imager (HMI) on the Solar Dynamics Observatory and reported a double-cell structure with the primary return flow at 0.82–0.91 solar radii; Chen & Zhao (2017, ApJ 849, 144), using seven years of HMI data with active-region masking and an empirical center-to-limb correction, found a qualitatively similar profile - but neither study inverted the radial flow or enforced mass conservation. Inverting instead with a mass-conserving stream function, Rajaguru & Antia (2015, ApJ 813, 114) and Mandal et al. (2018, ApJ 863, 39) found a single cell per hemisphere with the return flow below about 0.77–0.78 solar radii. Böning et al. (2017, ApJ 845, 2), inverting with spherical Born kernels, reported that both single- and multi-cell profiles were consistent with their measured travel times - an early statement of the non-uniqueness at the heart of the problem.

Three classes of limitation keep the question open. First, systematic errors: the center-to-limb effect (CLS) is a travel-time signal that depends only on the disk position of the measurement points and contaminates the north–south travel times at magnitudes several times the meridional-flow signal; it is corrected empirically (e.g., from east–west travel-time differences, following Zhao et al. 2012, ApJ 749, L5), but its physical origin is still unknown, and it differs between instruments. Strong surface magnetic fields (sunspots, active regions) also bias travel times and are handled by masking, at the cost of choices that propagate into the deep inversion (Liang & Chou 2015, ApJ 805, 165). Second, statistical and forward-modeling fidelity: inversions have mostly used diagonal error weighting, while the travel-time noise is correlated and its full covariance matrix changes regularization and uncertainty estimates; kernels neglect line-of-sight projection in the Doppler measurement, and ray versus Born treatments and truncations in spherical-harmonic degree and frequency all bias the result. Third, time dependence: averaging many years of data to beat down noise smears solar-cycle variations of the flow, so a time-averaged profile may conflate persistent structure with cycle-dependent structure. Gizon et al. (2020, Science 368, 1469) analyzed 1996–2019 data from MDI and GONG (the Michelson Doppler Imager aboard the SOHO spacecraft and the Global Oscillation Network Group of ground-based telescopes - two space- and ground-based Doppler imagers), corrected all known systematics, discarded the HMI data because of an unexplained 0.2 s travel-time offset, and found the time-averaged flow to be a single cell per hemisphere with an equatorward speed of about 4 m/s at 45° latitude near the base of the convection zone - but this remains one inference among conflicting ones, resting on empirical systematics corrections.

Problem Statement

Determine the depth profile of the Sun's meridional circulation in the convection zone - specifically the number of radially stacked circulation cells per hemisphere, the depth at which the equatorward return flow closes, and how this profile varies over the solar cycle - from helioseismic travel-time measurements, with a treatment of systematic errors (center-to-limb effect, surface magnetic regions, instrument-dependent travel-time offsets), noise correlations (full covariance rather than diagonal weighting), and forward modeling (finite-frequency, line-of-sight-aware sensitivity kernels with demonstrated convergence) that is adequate to distinguish single-cell from multi-cell solutions at a stated confidence level beyond the uncertainties of the published inferences. This subsumes the five issues explicitly acknowledged by Mandal et al. (2018): quantifying the bias of center-to-limb and magnetic-region corrections on deep-flow inferences; incorporating full noise covariance into the inversion; kernel-modeling convergence and line-of-sight effects; separating persistent deep structure from solar-cycle variations; and resolving the residual single-/multi-cell ambiguity at low latitudes. An acceptable answer must fix the number of cells and the closure depth and state how the profile varies over the solar cycle: it is either a systematics-controlled inference of these quantities that excludes the competing morphologies at stated confidence, or a demonstrated, quantitative proof that existing and foreseeable data cannot make that distinction (with the error budget that establishes the limit).

References

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