The Sun's meridional circulation — the large-scale north–south overturning flow in the solar convection zone — transports magnetic flux and angular momentum and is a central ingredient of flux-transport dynamo models of the 11-year activity cycle. Zhao et al. (2013) used time-distance helioseismology on two years of data from the Helioseismic and Magnetic Imager (HMI) on the Solar Dynamics Observatory (SDO) to report a double-cell circulation profile in each hemisphere, with an equatorward flow between about 0.82 and 0.91 solar radii, and listed the limitations of that detection: the physical origin of the center-to-limb travel-time systematic was unknown, the radial flow was neglected, reliable inversions stopped near 0.75 solar radii and below 60° latitude, and the temporal stability of the profile was untested. Over the following decade these limitations became the core of an unresolved controversy: independent analyses of Global Oscillation Network Group (GONG), Michelson Doppler Imager (MDI), and longer HMI time series have variously recovered single-cell, double-cell, and cycle-dependent profiles, and it has been demonstrated that single- and double-cell flows both fit the measured travel times. The problem is to determine the meridional circulation profile throughout the convection zone — cell number, return-flow depth, mass-conserving radial structure, latitudinal extent, and temporal and hemispheric variability — with the center-to-limb effect and instrumental phase systematics controlled on physical grounds, so that the competing profiles are discriminated at quantified significance.

Background

Helioseismology uses the Sun's acoustic oscillations — standing sound waves trapped in the solar interior — to probe conditions below the visible surface. In time-distance helioseismology, one cross-correlates the Doppler velocity oscillations recorded at pairs of points on the solar surface; a flow along the connecting path advances or delays the wave packet, producing a travel-time difference of order one second, and these differences can be inverted for the interior flow. The Sun's convection zone, the outer roughly 30% by radius where energy is transported by convection, hosts the meridional circulation: a slow poleward flow near the surface (about 15–20 m/s, directly measurable by tracking surface features) in each hemisphere, with a compensating equatorward return flow somewhere in the interior. The circulation matters because it acts as the conveyor belt of the flux-transport (Babcock–Leighton) dynamo, the leading model of the solar cycle: the depth of the equatorward return flow sets how long magnetic flux takes to migrate toward the equator, and hence the cycle period and the equatorward drift of sunspot-emergence latitudes (the butterfly diagram).

Before 2013 the standard picture was a single circulation cell per hemisphere with the return flow near the base of the convection zone. Using the first two years of observations by the Helioseismic and Magnetic Imager (HMI) on the Solar Dynamics Observatory, Zhao et al. (2013) inverted the travel times for the depth dependence of the flow after applying an empirical correction for the center-to-limb effect — a systematic apparent travel-time shift that depends on where on the solar disk a wave is observed, arising in the solar atmosphere and in the spectral line used to measure the velocity, which otherwise mimics a flow signal. They found instead a double-cell profile: poleward flow of about 15 m/s from the photosphere down to about 0.91 R⊙0.91\,R_\odot (solar radii), an equatorward flow of about 10 m/s between 0.820.82 and 0.91 R⊙0.91\,R_\odot, and a return to poleward flow below 0.82 R⊙0.82\,R_\odot. The inversion neglected the radial component of the flow, was reliable only above about 0.75 R⊙0.75\,R_\odot and below about 60° latitude, and could not test whether the profile persists over time or is sensitive to the empirical center-to-limb correction, whose physical origin was unknown.

Two technical aspects explain why the deep profile has stayed contentious. First, travel times are converted to flows by solving an inverse problem using averaging kernels — spatial sensitivity functions that map the true flow to the measured travel time. Kernels for deep flows carry extended side lobes near the surface, so the strong surface flow can leak into an apparent deep-flow signal of the same magnitude as the genuine deep flow (a few m/s). Second, because the flow is nearly incompressible in the convection zone, a physically admissible circulation must satisfy mass conservation, which is conveniently encoded in a stream function whose contours are the flow lines; the radial flow component, however, produces travel-time signatures much smaller than the horizontal component. The amplitude of genuine deep flows is comparable to both the realization noise (random error from the stochastic convective excitation of the acoustic wave field) and the systematic uncertainties, so different published analyses of overlapping datasets recover different cell topologies — single cell with a deep return flow (Gizon et al. 2020), double cell (Zhao et al. 2013; Chen & Zhao 2017), or profiles that vary with the solar cycle (Herczeg & Jackiewicz 2023).

Problem Statement

Determine the structure of the Sun's meridional circulation throughout the convection zone, resolving the open questions left by the double-cell detection of Zhao et al. (2013): (1) the number of circulation cells per hemisphere in depth — a single cell whose equatorward return flow lies near the base of the convection zone, versus the double-cell profile with an equatorward flow between about 0.820.82 and 0.91 R⊙0.91\,R_\odot and a deeper counter-cell, versus additional cells below about 0.75 R⊙0.75\,R_\odot or poleward of about 60° latitude, where current inversions are unreliable; (2) the depth and latitude dependence of the equatorward return flow(s); (3) the radial-flow component and a mass-conserving reconstruction of the full circulation — both the latitudinal velocity vθv_\theta and the radial velocity vrv_r as functions of radius and latitude; and (4) the temporal evolution and north–south asymmetry of the deep flow over the solar cycle. The determination must be made with the center-to-limb travel-time systematic and the instrumental phase anomalies of the observing instruments removed on the basis of their physical origin rather than an empirical proxy, so that the single-cell and double-cell (or multi-cell) hypotheses are discriminated at quantified statistical significance rather than left as a degeneracy of the travel-time data. These four deliverables jointly support one objective — establishing the Sun's interior circulation profile — and all four are required: an answer is complete only when that full profile is fixed, including the cell number and the depth and latitude dependence of the return flow(s) (1–2), the radial-flow component and mass-conserving reconstruction of vθv_\theta and vrv_r (3), and the temporal evolution and north–south asymmetry of the deep flow (4), with its uncertainty and systematics budget, well enough to decide between the competing published topologies.

References

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