The Sun transports plasma and magnetic flux through a large-scale axisymmetric flow, the meridional circulation, which runs poleward at the surface with a speed of about 15-20 m/s and must return equatorward somewhere in the convection zone to close the mass circuit. Two decades of local helioseismology have produced mutually incompatible inferences of where this return flow occurs: a shallow equatorward flow near 0.9 solar radii with a second counter-cell below it, a single cell closing below about 0.77-0.78 solar radii, or a single cell extending to the base of the convection zone at about 0.71 solar radii. Reviews in 2022 and 2025 still classify the return-flow location and the number of radially stacked circulation cells as open, debated problems, and analyses of systematic errors (center-to-limb effects, inversion constraints, kernel side lobes, instrument disagreements) show the competing profiles are not yet discriminated by the data. The problem is to determine the reversal depth or depths, the radial cell structure, and the full radial extent of the time-averaged meridional circulation without imposing a cell geometry, with all known systematics controlled well enough to decide among the published candidate profiles.

Background

The solar convection zone (roughly the outer 0.29 of the solar radius, from the base at about 0.713 R⊙0.713\,R_\odot to the surface, where R⊙≈696R_\odot \approx 696 Mm is the solar radius) is the turbulent outer envelope in which plasma flows transport angular momentum, heat, and magnetic flux. Its axisymmetric (longitude-averaged) flow part consists of differential rotation plus a slow overturn called the meridional circulation (MC): plasma moves poleward at the surface at about 15-20 m/s and must return equatorward at depth, with radial flows closing each circulation cell. Although the MC is weak compared with rotation, it is the transport agent in flux-transport (Babcock-Leighton) dynamo models, the prevailing framework for the 11-year sunspot cycle, in which the circulation itself carries the magnetic field: poloidal field is transported poleward at the surface, while the equatorward return flow carries the toroidal field that spawns new sunspots toward the equator. The depth of the return flow therefore fixes how long magnetic flux dwells at what depth while being advected toward the equator, and hence the cycle period and the shape of the butterfly diagram of sunspot emergence (the pattern, resembling butterfly wings, in which sunspots appear at mid-latitudes early in each cycle and migrate toward the equator as the cycle progresses). Whether the MC is a single cell reaching the base of the convection zone, or two or more cells stacked in radius with a mid-zone reversal, leads to qualitatively different dynamo behavior.

Local helioseismology images this flow: solar acoustic (p) oscillations, excited by near-surface convection, traverse the interior, and waves that travel with or against a flow arrive with slightly shifted travel times. Time-distance helioseismology measures these north-south travel-time differences and inverts them, through sensitivity kernels relating flow to travel-time shift, into a flow profile. Kernels computed in the ray approximation treat the wave as traveling along a geometrical ray; Born-approximation kernels account for finite wavelengths and are needed for broad deep-focused sensitivity. The inversion is ill-conditioned: mass conservation is commonly imposed by inverting for a stream function (a scalar whose derivatives give the flow components), the solution is projected onto spline and Legendre-function bases, and choices of regularization, lower boundary, and physical constraints influence the result. A dominant systematic is the center-to-limb (CtoL) effect: waves measured near the solar limb suffer frequency-dependent shifts from near-surface physics unrelated to the deep flow, which contaminate precisely the large-distance travel times that probe the deep convection zone; different CtoL corrections are known to flip the inferred profile between one and two cells. Additional obstacles are realization noise from the stochastic wave excitation (deep flows of a few m/s are near the noise floor even after averaging two decades of data), side lobes of the finite averaging kernels that leak the strong surface-flow signal into the deep interior at the 1-2 m/s level, and unexplained disagreements between the Helioseismic and Magnetic Imager (HMI, on the Solar Dynamics Observatory satellite), the Michelson Doppler Imager (MDI, on the Solar and Heliospheric Observatory), and the Global Oscillation Network Group (GONG, a six-station ground-based telescope network).

Problem Statement

Determine, from the publicly archived helioseismic observations of HMI, MDI and GONG — the HMI and MDI data products served by the Joint Science Operations Center (JSOC, jsoc.stanford.edu) and the GONG data products served by the National Solar Observatory (gong.nso.edu) — the structure of the Sun's internal meridional circulation through the convection zone (from 0.713 R⊙0.713\,R_\odot to 1.0 R⊙1.0\,R_\odot): the depth or depths at which the poleward surface flow of each hemisphere reverses to equatorward return flow, the number of circulation cells stacked in radius (one cell closing at the base of the convection zone versus two or more cells with interior reversals), and the full radial extent of the circulation. The determination must not presuppose a cell geometry: it may not fix the return-flow depth or lower boundary by construction, and the outcome must be shown robust against the systematics that currently separate the published inversions — the frequency-dependent center-to-limb effect, ray versus Born kernel approximations and averaging-kernel side lobes, realization noise, inversion constraints (mass conservation, imposed boundaries, angular-momentum transport), and instrument-to-instrument disagreement between HMI, MDI, and GONG — so that at least the competing profiles on record (double cell reversing near 0.9 R⊙0.9\,R_\odot; single cell returning below 0.770.77-0.78 R⊙0.78\,R_\odot; single cell closing at the base of the convection zone) are discriminated with quantified uncertainties. The required outcome is the time-averaged structure — the reversal depth or depths, the number of radially stacked cells, and the full radial extent of the circulation — robust against the systematics above and reported with quantified uncertainties; whether that time-averaged structure is itself stationary or varies over the solar cycle is a secondary aim, not a precondition for a completed determination. The ray- and Born-approximation sensitivity kernels and the inversion machinery to be used are those documented in the cited works.

References

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