The toroidal-field amplitude problem of the solar flux-transport dynamo: reconciling saturation strength with active-region flux emergence

Mean-field flux-transport (Babcock–Leighton) solar dynamo models that include Lorentz-force feedback on the differential rotation and meridional flow saturate at mean toroidal field strengths of only about 1–1.5 T at the base of the convection zone, and energy-budget arguments suggest that dissipatively limited dynamos cannot sustain coherent fields of more than a few tesla without contradicting the observed amplitude of torsional oscillations. Yet models of buoyantly rising magnetic flux tubes require roughly 10 T (with later three-dimensional turbulent-convective tube simulations still requiring at least about 4 T) to reproduce the observed latitudes, tilts, and asymmetries of solar active regions. The problem is to reconcile these constraints — establishing the actual peak strength, intermittency (filling factor), latitudinal distribution, and per-cycle toroidal flux budget of the deep solar magnetic field, whether within the flux-transport paradigm or beyond it, by either a self-consistent theoretical or simulation determination or an observational determination (for example a helioseismic measurement or bound).

Background

The Sun's 11-year magnetic activity cycle is widely modeled as a flux-transport dynamo: differential rotation shears poloidal (north–south) magnetic field into toroidal (east–west) field concentrated near the base of the convection zone, and the Babcock–Leighton mechanism — poloidal field regenerated from the tilt of decaying active regions at the surface — closes the cycle, while a slow meridional flow (a large-scale circulation with poleward surface and equatorward deep branches) transports flux and sets the cycle period. Rempel (2006) studied such a model with the full Lorentz force fed back on the angular momentum and momentum equations, reproducing the solar butterfly diagram and the torsional oscillations (the ~nHz-amplitude, cycle-modulated perturbations of the differential rotation measured helioseismically). The model saturates with a mean toroidal field of only about 1.2–1.5 T (12–15 kG) at the base of the convection zone, and an energy argument shows this is near the maximum available: integrating the model's mean magnetic energy conversion rate (about 0.001 of the solar flux) over an 11-year cycle bounds the toroidal-field energy, and stronger fields would require a differential-rotation energy replenishment so fast that the predicted torsional oscillations would exceed what is observed. This conflicts with the classical emergence constraint: calculations of buoyantly rising magnetic flux tubes in the thin flux-tube approximation (a model treating an active-region-scale magnetic tube as a self-consistent 1D object in a 3D stratified, rotating envelope) found that fields of about 10 T (100 kG) at the base of the convection zone are needed for tubes to rise fast enough, at the observed low latitudes, and with tilts following Joy's law (the observed tendency of bipolar active-region axes to tilt with the leading polarity closer to the equator). Rempel (2006) noted two possible escapes: intermittency — since the model is a mean-field average, a filling factor f of coherent flux elements with element strength $B_f\sim B/\sqrt{f}$ (e.g., about 4 T for f = 0.1) could satisfy the Lorentz-force and energy constraints while providing stronger individual structures; and weaker-field emergence, hinted at by early three-dimensional rise simulations suggesting 2–3 T might suffice — but also that simulations of buoyant structures in turbulent convection questioned whether coherent flux ropes form and rise at all. Related open threads acknowledged in the same paper are whether the solar dynamo is in fact a flux-transport dynamo (a paradigm question, since competing mean-field and fully convective dynamo pictures remain live), and whether helioseismology (the inference of the Sun's internal structure and flows from solar oscillation frequencies) can detect or bound the deep toroidal field at all.

Problem Statement

Determine whether the solar dynamo can generate and store, at the base of the solar convection zone, coherent toroidal magnetic flux elements whose strength and total flux budget are sufficient to account for the observed emergence of solar active regions — i.e., reconcile the ~1–1.5 T mean-field saturation limit of Lorentz-force- and energy-constrained flux-transport dynamo models with the requirement of flux-rise calculations (about 10 T in the original thin flux-tube studies, at least ~4 T in modern three-dimensional turbulent-convective tube simulations), taking account of field intermittency (filling factor), the per-cycle emerged flux of about $10^{24}$–$10^{25}\ \mathrm{Mx}$ (maxwell, the CGS unit of magnetic flux; $10^{24}\ \mathrm{Mx} = 10^{16}\ \mathrm{Wb}$), and the constraint that any stronger dynamo must remain consistent with the observed amplitude of torsional oscillations and the solar energy budget. A complete answer must establish, with quantitative uncertainty, the actual peak toroidal field strength, its filling factor, its latitudinal distribution, and its per-cycle toroidal flux budget in the deep convection zone — either by a self-consistent theoretical/simulation reconciliation within a specified dynamo paradigm (or a demonstration that none can satisfy all constraints simultaneously), or by an observational determination (for example a helioseismic measurement or bound) of the deep toroidal field, these two routes being alternatives rather than cumulative requirements.

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