The solar cycle is driven by a roughly 22-year conversion between a poloidal magnetic field and a toroidal magnetic field. The poloidal component is routinely observed as the longitudinally averaged photospheric radial field, but the dynamo-relevant axisymmetric toroidal field resides below the photosphere and, despite four cycles of longitudinally averaged azimuthal-field observations and a series of model-based calibrations, still has no direct or non-circular observational determination; Babcock–Leighton flux-transport dynamo models have accordingly never been subjected to a formal quantitative test against the joint time–latitude evolution of both observed components. The open problem is to establish quantitative observational access to the subsurface axisymmetric toroidal field — by a direct measurement technique or by a rigorously calibrated, uncertainty-quantified proxy relation that does not circularly rely on the dynamo model class being tested — and to use it, together with the observed poloidal field, to test Babcock–Leighton scenarios quantitatively over multiple activity cycles with observationally constrained flows.

Background

The Sun's global magnetic cycle has a period of about 11 years in surface activity and about 22 years in magnetic polarity. In the standard poloidal–toroidal decomposition, the poloidal field is the component in the meridional (north–south, radial) plane and the toroidal field is the azimuthal (east–west) component. The dynamo is understood as a loop: differential rotation shears poloidal field into toroidal field, and the emergence of toroidal flux through the surface, together with the tilt of the emerging regions, regenerates poloidal field of the opposite parity. The poloidal component is directly observed: magnetographs measure the line-of-sight photospheric field, and its longitudinal average yields time–latitude ('butterfly') maps of the radial field going back to the mid-1970s. The toroidal component, however, resides in the convection zone below the photosphere and is not directly measurable; it is only seen when loops of it emerge. Observationally it leaves a trace through Hale's law — the statistical preference of emerging bipolar magnetic regions for the same east–west polarity orientation within a hemisphere, reversing between hemispheres and between cycles: most large active regions comply, and even small ephemeral regions (short-lived bipoles below active-region size), which comply at only about 60%, retain the same preference strongly enough for it to appear in longitudinal averages. A technique introduced by Duvall et al. (1979) recovers the longitudinally averaged surface azimuthal field from the east–west inclination of the photospheric field: a yearly average of full-disk line-of-sight magnetograms is split into components symmetric and antisymmetric about the solar meridian, and the antisymmetric part is proportional to the yearly, longitudinally averaged azimuthal field. In the Babcock–Leighton scenario, named after the phenomenological models of H. Babcock and R. Leighton from the 1960s, the poloidal field is rebuilt not by a turbulent alpha-effect in the convection zone but by the observed emergence of tilted bipolar regions: Joy's law (leading polarity closer to the equator) plus poleward meridional flow (a large-scale circulation from the equator to the poles at the surface with a return flow at depth) carries leading-polarity flux across the equator where it cancels, building the next cycle's polar field. Flux-transport dynamos add the near-surface shear layer (the strongly radially sheared rotation in the outer ~5% of the Sun known from helioseismology, the inference of the interior's sound-speed and rotation profiles from oscillation frequencies) as the main toroidal-field generator and the meridional circulation as the transport agent closing the loop. Cameron & Schüssler (2015) showed that the net toroidal flux of a half-cycle is quantitatively accounted for by the winding up of the observed poloidal flux threading the surface. Cameron et al. (2018) then presented, for the first time, time–latitude maps of both components over almost four cycles: the longitudinally averaged radial field (synoptic maps from the Kitt Peak National Solar Observatory vacuum telescope, KPNSO/VT, and the Synoptic Optical Long-term Investigations of the Sun facility, SOLIS) and the longitudinally averaged azimuthal field (the Wilcox Solar Observatory, WSO, and the Michelson Doppler Imager, MDI, on the Solar and Heliospheric Observatory, SOHO, via the Duvall technique). They showed the observed azimuthal field is quantitatively consistent, at the order-of-magnitude level, with what flux emergence should produce (about 0.1 G in the quiet Sun from ephemeral regions, 1–2 G in the activity belts from active regions), arguing that it is therefore a proxy of the subsurface axisymmetric toroidal field. They further showed that an updated two-equation Babcock–Leighton model (Cameron & Schüssler 2017), driven by observationally constrained meridional flow and differential-rotation profiles, reproduces the large-scale joint evolution of both components, including the extended activity cycle — though with three parameters (a source strength α0=1\alpha_0=1 m/s, an effective return-flow speed V0=2V_0=2 m/s, a turbulent diffusivity η0=90\eta_0=90 km²/s) chosen by hand, with the correspondence assessed visually rather than by a formal statistical fit, and with cycle-to-cycle variability explicitly out of scope. What the later literature has and has not settled defines the remaining problem. The flux loss through the photosphere during emergence — flagged by Cameron et al. (2018) only as an upper-limit 'at most modest' effect — was quantified by Cameron & Schüssler (2020): emergence of bipolar regions removes net toroidal flux at a rate corresponding to a decay time of about 12 years, consistent with independent in-situ solar-wind estimates, and this loss mechanism has since been incorporated into Babcock–Leighton models (Biswas et al. 2022; Cloutier et al. 2023). Cameron & Jiang (2019) derived calibration curves relating a modeled subsurface toroidal flux density to the observed surface toroidal field and sunspot areas — but the calibration runs through a one-dimensional mean-field evolution equation driven by the observed radial field, so it is model-based rather than an independent measurement. Liu & Scherrer (2022) extended the observed surface toroidal-field record to 44 years. Gizon et al. (2020) constrained the meridional circulation to a single cell per hemisphere with a deep return flow whose speed is consistent with the migration of the activity belts. Yet, as stated explicitly in 2024 and 2026 work, the subsurface part of the dynamo still 'lacks direct observational constraints' and 'none of the techniques, including helioseismology, are able to provide an estimation of the interior magnetic field' (Zhong et al. 2024; Chatterjee & Hazra 2026); the most recent reconstruction of the interior large-scale field (Chatterjee & Hazra 2026) does so by assimilating three decades of surface magnetograms into a 3D Babcock–Leighton dynamo — a model-dependent reconstruction in which the model class under test is part of the measurement chain. The authors' own 2023 review likewise notes that the nature of the equatorward transport of the subsurface toroidal flux is not directly constrained.

Problem Statement

Establish quantitative observational access to the Sun's dynamo-relevant axisymmetric subsurface toroidal magnetic field as a function of time and latitude — either by a direct measurement technique (for example helioseismic or polarimetric) or by a calibration of the longitudinally averaged photospheric azimuthal field, or another photospheric observable, against the subsurface toroidal field in which the assumptions and uncertainties are quantified and the calibration does not circularly rely on the dynamo model class being tested — and use that determination, together with the observed longitudinally averaged radial (poloidal) field, to carry out a quantitative test of Babcock–Leighton flux-transport dynamo models with observationally constrained flows over multiple activity cycles. A complete answer specifies the measurement or calibration with its uncertainty budget, and a statistically defined comparison of the model's predicted joint time–latitude evolution of both components against the observed record (including the phase relations, the latitudinal migration, and the extended-cycle behavior), of a strength sufficient to confirm or to discriminate between Babcock–Leighton model variants. The two deliverables form one objective: giving the toroidal half of the dynamo loop an observational handle, and using it to turn the Babcock–Leighton scenario from a visually plausible reproduction into a quantitatively tested model.

References

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