Time-distance helioseismic inversions of Gizon et al. (2020) travel-time data yield qualitatively different deep meridional-circulation (MC) profiles — a single cell per hemisphere under conventional smoothness priors, a double cell when an angular-momentum (AM) transport constraint is enforced — at nearly the same residual level. Hatta, Hotta & Sekii (2024) showed that, with present data and the modeling choices used so far, this data–prior degeneracy remains unbroken — the authors found it difficult to determine the large-scale morphology at the moment — that the Reynolds-stress (RS) motivated constraint could only be implemented qualitatively because no explicit relation between turbulent-stress profiles and the meridional flow is known, and that no unique prescription exists for selecting inversion trade-off parameters. The open problem is to determine, in a prior-robust way, the large-scale radial morphology of the solar MC and, on that basis, which AM-transport regime — magnetic-redistribution (HK21-type) or turbulent Reynolds-stress transport — operates in the Sun.

Background

The Sun's meridional circulation (MC) is a large-scale axisymmetric flow pattern in the solar convection zone (the Sun's outer, convectively unstable envelope), transporting plasma and magnetic flux poleward near the surface; its depth profile is a primary uncertain ingredient of flux-transport dynamo models - the standard class of solar-cycle models in which the meridional flow carries remnant magnetic flux from low latitudes toward the poles, closing each roughly 11-year activity cycle - and hence of solar-cycle prediction. Time-distance helioseismology infers flows inside the Sun by measuring travel times of acoustic waves between surface points and inverting them, via spatially localized sensitivity functions called averaging kernels, into a flow profile; the inversion requires regularization with trade-off parameters that balance data fit against smoothness or other priors. Gizon et al. (2020, Science) inverted such travel times and concluded that the MC is a single cell in each hemisphere extending deep into the convection zone. Independently, Hotta & Kusano (2021, Nature Astronomy, 'Solar differential rotation reproduced with high-resolution simulation') reproduced the Sun's equator-fast differential rotation in a high-resolution simulation and attributed it to the strong magnetic field generated by a small-scale dynamo, which enhances an axisymmetric latitudinal entropy gradient at the base of the convection zone; in that regime (called HK21 here) the meridional circulation itself must transport angular momentum (AM) equatorward, which near the surface requires a return flow with a specific sign of AM flux. The competing regime attributes equatorward AM transport mainly to turbulent Reynolds stresses — the correlated fluctuations of the flow that appear as off-diagonal terms in the momentum equation, conventionally parameterized through a stream function — rather than to the mean MC. Hatta, Hotta & Sekii (2024, ApJ 972, 79) re-ran the inversion of the public Gizon et al. (2020) travel-time dataset under these two physically motivated priors. Enforcing the HK21-type equatorward-AM constraint (even with a very small enforced bias) produced a double-cell solution, while the conventional smoothness prior produced a single cell, both fitting the data at nearly the same squared residual; the AM constraint simultaneously degraded the depth localization of the averaging kernels, so it cannot be decided whether the deep counter-cell is resolved by the data or imposed by the prior. Because there is no known linear relation between the meridional flow and the Reynolds-stress profile, the RS-motivated constraint could only be implemented as a crude qualitative, linearized prescription (a cylindrical low-latitude stream function), which yields larger residuals and smaller AM-flux magnitudes. Regularization weights were chosen by grid search with objective criteria, but no unique prescription exists. The unresolved question is thus not a shortage of inversions but a degeneracy between data and prior that present observations — with low signal-to-noise ratio and no coverage of the poles — have not broken under the inversion approaches used so far.

Problem Statement

Determine, in a way robust to the choice of prior, the large-scale radial morphology of the solar meridional circulation (single versus multiple cells per hemisphere) and, on that basis, which angular-momentum-transport regime — magnetic redistribution with equatorward AM transport by the mean meridional circulation (HK21-type) or transport by turbulent Reynolds stresses (RS regime) — operates in the Sun. Following the criterion used in the source, the morphology counts as determined when the inversion distinguishes single- versus multiple-cell structure with 2-sigma significance, and the determination is prior-robust when this conclusion does not reverse across the physically motivated priors under comparison, namely conventional smoothness priors versus AM-transport constraints of the HK21-type and the RS-motivated prescription. The non-reversal requirement applies across the regularization trade-off weights that instantiate each prior, over the range of weights at which the data are fit at a comparable residual level, rather than at a single selected weight, the choice of weight being itself a source of the degeneracy. The regime inference rests on premises the source states explicitly: under mass conservation with vanishing latitudinal flow at the poles, a single-cell MC transports AM poleward and is therefore incompatible with HK21-type equatorward AM transport; and the HK21 and RS regimes are currently the only regimes in which the solar equator-fast rotation has been reproduced successfully. A multiple-cell morphology is compatible with, though not by itself conclusive proof of, HK21-type physics, so the regime counts as determined when the prior-robust morphology together with the stated premises leave a single regime not excluded, and otherwise the conclusion must report the set of regimes not excluded, in either case stated together with the residual uncertainty the source acknowledges.

Hatta, Hotta & Sekii (2024) identify as open future work relevant to this goal: (i) a physically based, quantitative relation between Reynolds-stress profiles and the meridional flow, replacing their qualitative, linearized cylindrical stream-function prescription, so that the RS regime can be tested on equal footing with the HK21-type constraint; (ii) a principled prescription for selecting the inversion trade-off/regularization parameters (for example, via Bayesian model comparison), so that the inferred deep topology is not an artifact of the weight choice; and (iii) inversions of travel-time data — present Gizon et al. (2020) data or improved future observations with higher signal-to-noise ratio and polar coverage — with the depth localization of the averaging kernels quantified. These directions represent partial progress toward the objective; none of them, alone or in combination, is imposed here as a completion criterion, and no particular method or data set is prescribed for determining the morphology and the regime in the sense above.

References

  1. Hatta, Hotta & Sekii (2024), 'Inversion for Inferring Solar Meridional Circulation: The Case with Constraints on Angular Momentum Transport inside the Sun', The Astrophysical Journal 972, 79; DOI 10.3847/1538-4357/ad596c; arXiv:2406.12077; https://arxiv.org/abs/2406.12077
  2. Hotta & Kusano (2021), 'Solar differential rotation reproduced with high-resolution simulation', Nature Astronomy 5, 1100-1102; DOI 10.1038/s41550-021-01459-0; https://doi.org/10.1038/s41550-021-01459-0
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