The deep structure of the Sun's meridional circulation: a physically consistent helioseismic determination of the radial and latitudinal flow robust against the center-to-limb systematic

The Sun's meridional circulation — the large-scale axisymmetric flow of plasma in meridional planes, poleward at the surface — is a central ingredient of flux-transport dynamo models of the solar cycle, yet its depth profile remains contested: after two decades of helioseismic measurements there is still no consensus on the number of circulation cells stacked in radius, the depth at which the equatorward return flow closes, or the amplitude of the radial flow component. Jackiewicz, Serebryanskiy & Kholikov (2015) analyzed GONG Doppler data and found a shallow equatorward flow near 65 Mm depth, no strong evidence for multiple cells, and, critically, that the inferred circulation badly violates mass conservation — while the largest contaminating systematic, the center-to-limb (CTL) travel-time effect of unknown physical origin, could only be removed empirically. The open problem is to determine the meridional flow $v_\theta(r,\theta)$ and $v_r(r,\theta)$ throughout the convection zone — the cell count, return-flow depth and radial-flow amplitude — by inversion that jointly recovers both components, enforces physical consistency such as mass or angular-momentum conservation, and is demonstrably robust against (or physically explains) the CTL systematic, so that independent data sets and independent physical priors converge on one profile. Later work has supplied each methodological ingredient in isolation, but the deep profile is still doubly degenerate: the same travel times admit single-cell (Gizon et al. 2020) and double-cell (Hatta et al. 2024) solutions depending on the imposed prior, and the CTL effect still lacks an accepted physical explanation (Waidele et al. 2023; Kitiashvili 2026).

Background

The solar convection zone, the outer roughly 30% of the Sun's radius, is pervaded by two large-scale plasma flows in addition to convection: differential rotation and the meridional circulation. The meridional circulation is a slow (about 10–20 m/s at the surface, weaker at depth) flow within meridional planes that is poleward in each hemisphere near the surface; mass conservation requires that this transported plasma return equatorward at some depth, closing the loop. This flow is a cornerstone of flux-transport dynamo models — the class of models in which the 11-year sunspot cycle is explained by a bipolar magnetic dynamo loop whose poloidal field is advected poleward near the surface and whose toroidal field is carried equatorward at depth, so that the sunspot-emergence latitudes drift toward the equator over the cycle. Whether the return flow closes in a single cell reaching the base of the convection zone, or in two or more cells stacked in radius, fundamentally changes the dynamo's period, memory, and field geometry.

Because the solar interior cannot be seen directly, the meridional circulation is probed with helioseismology. In time-distance helioseismology, one measures the travel time of acoustic (p-mode) waves between pairs of points on the surface: a flow along the connecting ray path slightly advances or delays the wave, and the difference between travel times in opposite directions isolates the flow contribution. Inverting these travel-time differences for an interior flow requires sensitivity kernels — functions describing how deeply a given surface point pair senses the interior. The ray approximation treats the wave as traveling along a narrow geometrical ray and has limited sensitivity to deep, weak flows; the Born approximation instead describes scattering from the full finite-frequency wave field and yields more realistic deep kernels. Inversions (e.g. the SOLA 'subtractive optimally localized averages' method) combine many measured travel times into local averages of the flow; the price is a strong depth–noise trade-off, and unobserved realizations of solar convection excite 'realization noise' in the modes that mimics small flow signals. Only the horizontal (latitudinal) flow component is directly sensed by standard point-to-point travel times; the radial component $v_r$ enters weakly, which is why several studies invert only $v_\theta$ and why Jackiewicz et al. (2015) could not produce a mass-consistent three-dimensional flow.

Two obstacles dominate the deep problem. First, travel-time measurements suffer a center-to-limb (CTL) systematic: travel times depend on the position of the measurement patch between disk center and the solar limb, in a way unrelated to the meridional flow but of comparable amplitude; since the CTL pattern varies across the disk much as a meridional-flow signal does, it biases inversions. Jackiewicz, Serebryanskiy & Kholikov (2015, ApJ 805, 133) removed it by the empirical expedient of subtracting east–west (E–W) travel-time differences measured along the equator, and flagged its unknown physical origin as a key uncertainty; the CTL effect is commonly conjectured to arise from how solar convection and spectral-line formation change with viewing angle, but no explanation is accepted. Second, anelastic mass conservation — the requirement $\nabla\cdot(\rho_0 \mathbf{v})=0$ for the density-stratified convection zone, where $\rho_0$ is the spherically symmetric background density — links $v_r$ and $\theta$-colatitude flow $v_\theta$; Jackiewicz et al. (2015) found their inversions of GONG data (GONG, the Global Oscillation Network Group, is a ring of six ground-based Doppler imagers) and published inversions of HMI data (HMI, the Helioseismic and Magnetic Imager, flies aboard NASA's Solar Dynamics Observatory satellite) satisfy this continuity equation poorly unless implausibly large radial flows are invoked.

The landscape was already conflicting when Jackiewicz et al. (2015) appeared: Zhao et al. (2013) reported seismic evidence from HMI for a double-cell circulation with a return flow near 0.82 solar radii, while Rajaguru & Antia (2015), also using HMI, found no double cell. Jackiewicz et al. (2015) added GONG: a shallow equatorward flow roughly 65 Mm below the surface agreeing with HMI in the outer 15% of the interior radius, no strong evidence for multiple cells ('yet the whole phase space has not yet been explored'), and the mass-conservation failure noted above. They posed, as the paper's open problems: the unknown physical origin and imperfect removal of the CTL effect; the absence of inversions that recover $v_r$ together with $v_\theta$ or impose mass conservation directly; the need for finite-frequency kernels and non-diagonal noise covariance; and the need for longer, multi-instrument data sets — GONG, MDI (the Michelson Doppler Imager aboard the SOHO spacecraft), and HMI — to beat realization noise. These are not separate curiosities but the load-bearing components of one scientific objective: a trustworthy map of the Sun's internal meridional circulation.

Problem Statement

Determine the meridional circulation of the solar convection zone as a physically consistent two-component flow — the latitudinal velocity $v_\theta(r,\theta)$ and the radial velocity $v_r(r,\theta)$ as functions of radius and (co)latitude, with quantified uncertainties — settling in particular the number of circulation cells stacked in radius in each hemisphere, the depth and latitude dependence of the equatorward return flow, and the amplitude of the radial flow, by means of helioseismic inversion that (a) recovers $v_r$ and $v_\theta$ jointly rather than the horizontal component alone, (b) enforces or tests physical consistency such as anelastic mass conservation $\nabla\cdot(\rho_0\mathbf{v})=0$ or angular-momentum-transport constraints, and (c) is demonstrably robust against the center-to-limb travel-time systematic — either because that systematic is removed by a method validated across instruments and observables, or because its physical origin is identified and modeled. Finite-frequency sensitivity kernels and non-diagonal noise covariance are among the methodological improvements identified in the literature toward these ends; no particular inversion technique is prescribed. The problem is solved when at least two independent data sets drawn from GONG, MDI, and HMI and at least two independent defensible physical priors converge, to within their stated uncertainties, on the same profile — cell count, return-flow depth, and radial-flow amplitude — that, with method conditions (a), (b), and (c) satisfied jointly, satisfies mass conservation and reproduces the measured travel times; it is not solved by any single inversion whose output depends materially on the choice of prior, regularization, or systematic-removal recipe, which is the situation as of the literature reviewed through 2026.

References

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