Sunspot-group measurements over 1874–2004 (Javaraiah and Ulrich, 2006) show that the Sun's differential rotation and meridional flow vary with the solar cycle in ways that disagree — in amplitude, latitude dependence, and north–south symmetry — with Doppler and helioseismic measurements, and exhibit unexplained features such as an anomalously low latitude-gradient rotation coefficient around 1962 and an approximately 11-year phase lag between the hemispheres' cycle-to-cycle meridional variations. The open problem is to determine the depth- and hemisphere-resolved structure of these axisymmetric flows and their solar-cycle variations so that all diagnostics are quantitatively reconciled: attributing the discrepancies among depth-dependent anchoring of magnetic tracers, tracer selection effects, and data systematics; explaining the anomalous epochs and hemispheric phase relationships; and connecting the result quantitatively to dynamo and torsional-oscillation theory. Later work has lengthened the data series, reframed part of the discrepancy as anchoring-depth evidence, and sharpened — but not settled — the question: the depth profile of the meridional circulation remains actively contested (single-cell and double-cell inversions published in 2013 and 2020 contradict each other, and Bayesian and constrained inversions in 2023–2024 still cannot decide between the two profiles), and no mechanism yet reproduces the observed hemispheric phase lags.

Background

The Sun does not rotate as a solid body: its angular velocity is well described, at the surface and per hemisphere, by a differential-rotation law ω(λ)=A+Bsin⁡2λ+Csin⁡4λ\omega(\lambda)=A+B\sin^{2}\lambda+C\sin^{4}\lambda, where λ\lambda is latitude, AA is the equatorial rotation rate, and BB (with CC) measures how much rotation slows toward the poles. Superposed on this mean rotation is a meridional flow — a slow (few m s−1^{-1} to tens of m s−1^{-1}) north–south circulation that is poleward at the surface in each hemisphere — and torsional oscillations, alternating bands of slightly faster and slower zonal rotation that migrate with the 11-year activity cycle. These two large-scale flows matter because in flux-transport dynamo models — the standard picture in which the solar cycle is driven by cyclonic convection winding up magnetic fields while meridional circulation transports flux along the convection zone — the meridional flow's depth profile and the rotation profile set the cycle period and shape, so knowing them is a prerequisite for modeling and predicting the cycle.

Several diagnostics measure these flows, and they do not fully agree. Tracer measurements follow the apparent motions of long-lived features, classically sunspot groups in the Greenwich Photoheliographic Results (GPR, 1874–1976) continued by the Debrecen Photoheliographic Data (DPD); fitting their motions yields AA, BB, and the mean meridional velocity vv per hemisphere. Doppler measurements obtain the plasma rotation directly from spectral-line shifts (e.g., the Mount Wilson program). Helioseismology — the study of solar oscillations, via networks such as GONG (Global Oscillation Network Group) and space instruments MDI and HMI — infers flows below the surface; local techniques such as ring-diagram and time-distance analysis resolve the near-surface layers where sunspots are thought to be anchored. Classical comparisons had already shown magnetic tracers rotating faster than surface plasma, usually interpreted as tracers being anchored at depth.

Javaraiah and Ulrich (2006) analyzed 131 years of sunspot-group data and found: strong solar-cycle-phase variation of the meridional flow that differs markedly between the northern and southern hemispheres; an approximately 11-year phase difference between the hemispheres' cycle-to-cycle meridional variations; an approximately 180° phase difference between the hemispheres' BB variations; a correlation between the mean meridional flow and BB (good in the north, weak in the south); a northern-hemisphere lead of vv over AA by about 11 years; and predominantly poleward mean meridional motion in long cycles, equatorward in short ones. They flagged as unresolved: the closeness of several correlations to the statistical uncertainties; discrepancies with Doppler and helioseismic amplitudes, latitude dependence, and north–south symmetry; the unexplained abnormally low BB around 1962; the physical origin of the hemispheric phase lags; and the need for multi-depth inversions and dynamo-linked modeling to decide which features are surface phenomena and which reflect deeper flow structure. Subsequent helioseismology sharpened rather than settled the depth question: Zhao et al. (2013) inferred a double-cell meridional circulation (an equatorward return flow deep in the convection zone), while Gizon et al. (2020) concluded from a different analysis that the flow is a single cell per hemisphere — a contradiction that remains unresolved in the current literature, with Bayesian and angular-momentum-constrained inversions in 2023–2024 still unable to decide between the two profiles — reviewed against the longer-standing method-dependent rotation differences by Howe (2009).

Problem Statement

Determine the depth- and hemisphere-resolved structure of the Sun's axisymmetric flows — the differential rotation and the meridional circulation — and of their solar-cycle variations, to the point where the tracer-based (sunspot-group), Doppler, and local/global helioseismic measurements are quantitatively consistent. Concretely, building on the 1874–2004 sunspot-group analysis of Javaraiah and Ulrich (2006) and its successors: (a) quantify and attribute the discrepancies between tracer-derived and Doppler/helioseismic results — in amplitudes, latitude dependence, and north–south symmetry — among depth-dependent anchoring of the magnetic tracers, tracer selection effects, and instrumental/data systematics, establishing which observed cycle-varying features are surface phenomena and which reflect deeper flow structure; (b) determine the depth profile and time dependence of the meridional circulation (resolving the current contradiction between double-cell and single-cell inversions) and of the torsional-oscillation zonal flows, including how deep the cycle variations penetrate; (c) explain the anomalous epochs in the rotation latitude-gradient coefficient BB, such as the abnormally low BB around 1962; (d) determine the physical origin of the approximately 11-year hemispheric phase lag in the cycle-to-cycle meridional variations and of the northern-hemisphere lead of the mean meridional velocity vv over the equatorial rotation rate AA; and (e) connect these flow fields and phase relationships quantitatively to dynamo and torsional-oscillation theory, so that the observed cycle-dependent transport is predicted rather than merely parameterized.

Completion is bounded: a documented, well-supported result on the reconciliation core — the depth- and hemisphere-resolved flow description together with the attribution of the tracer/Doppler/helioseismic differences (a) and the meridional depth profile and time dependence (b) — is accepted as a valid resolution, while (c)–(e) are additional deliverables rather than individually required parts. The uncertainties and epochs of the cited measurements — including the 1874–2004 sunspot-group record and any extension of it that is adopted — may be taken from the cited sources and stated explicitly, and any tolerance the answer defines must be justified. Consistency between the flow description and the diagnostics is judged from an explicit per-diagnostic residual budget — the flow description, its residuals against each diagnostic, and the attribution of those residuals — rather than from objective equality with every measurement; part (e) is an interpretation deliverable presenting candidate mechanisms and their residual misfits.

References

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