Origin of solar grand minima: the unconstrained Babcock–Leighton poloidal source for very weak cycles in the observation-based iterative map

The solar cycle is commonly modeled as a Babcock–Leighton (B-L) dynamo in which the amplitude of each 11-year cycle follows from the previous cycle's polar field through a nonlinear, stochastic recursion. Wang, Jiang & Wang (2025) reduced this dynamo to a one-dimensional iterative map whose observation-based poloidal-source term reproduces the probability density function (PDF) of normal cycle amplitudes and excludes deterministic chaos, but the map cannot generate the distinct grand-minimum mode (e.g., the Maunder Minimum) seen as a second peak in cosmogenic reconstructions of cycle strength. The reason is that the poloidal-source function is completely unconstrained for very weak cycles, where active regions are scarce. The open problem is to determine the weak-cycle poloidal source — observationally or theoretically — and to decide whether stochasticity alone within B-L dynamics, with the weak-cycle poloidal source still produced by the decay of tilted active regions (its efficiency or geometry possibly modified), reproduces the double-peaked PDF and occurrence statistics of grand minima, or whether a process beyond active-region decay is required.

Background

The Sun's magnetic activity varies in a roughly 11-year cycle whose amplitude, measured by the peak sunspot number, fluctuates strongly from cycle to cycle, occasionally collapsing into decades-long quiescent episodes called grand minima, of which the Maunder Minimum (c. 1645–1715) is the most famous. The accepted explanation is a magnetohydrodynamic dynamo in which two field components regenerate each other: the toroidal field (azimuthal, wound up by differential rotation via the so-called Ω-effect), whose eruption as tilted active regions (sunspot groups) produces the visible cycle, and the poloidal field (the large-scale north–south dipole component, whose axial dipole moment at cycle minimum, $DM$, seeds the next cycle). In the Babcock–Leighton (B-L) scenario, the poloidal field is rebuilt near the surface from the decay of tilted active regions: their leading-polarity flux cancels with the old polar field and their trailing-polarity flux migrates poleward. Because Hale's polarity law — the alternating magnetic polarity of successive sunspot cycles — makes the dipole moments of successive cycles opposite in sign, the map works with all-positive $DM$ values, the polarity alternation being folded into the plus sign, so that a cycle of amplitude $SN$ changes the dipole moment by a source term $\Delta DM = DM(n)+DM(n-1)$, which measures the net swing between the opposing dipoles of consecutive cycles and must first cancel out the residual poloidal field from the previous cycle. Statistical studies of active-region properties show B-L nonlinearity — the mean tilt angle decreases and emergence latitudes shift with stronger cycles, limiting poloidal-field production — and B-L stochasticity, the large scatter of tilts and latitudes buffeted by convection. Jiang (2020) quantified both and found that $\Delta DM$ follows an increase-then-saturate pattern as amplitude grows, well approximated by $\Delta DM = k_1\,\mathrm{erf}(SN/quench)$, where $\mathrm{erf}$ is the error function (the integral of a Gaussian, which rises smoothly and saturates), optionally multiplied by $(1+stoch\times X)$ with $X$ a random variable modeling the scatter. Combining this with the mostly linear toroidal generation $SN(n)=k_0\,DM(n-1)$ yields a one-dimensional iterative map for cycle amplitudes. Wang, Jiang & Wang (2025) analyzed this map and proved that deterministic chaos is absent for any parameters when the poloidal source is of increase-then-saturate type, so stochasticity must dominate cycle variability — in agreement with Cameron & Schüssler (2017), who described the cycle as a weakly nonlinear limit cycle driven by random noise, and with Cameron & Schüssler (2019), who showed that periodicities beyond 11 years are consistent with random forcing. However, the map's synthetic-amplitude probability density function (PDF) matches only the normal-cycle component of the reconstruction by Usoskin et al. (2014), which is derived from cosmogenic isotopes — nuclides such as $^{14}$C in tree rings and $^{10}$Be in polar ice cores, produced by cosmic rays whose flux is shielded by solar magnetic activity and therefore recording past cycle strength. That reconstruction is double-peaked: besides normal cycles there is a statistically distinct grand-minimum mode. Prior stochastic B-L dynamo models (Choudhuri & Karak 2012; Olemskoy & Kitchatinov 2013; Karak & Miesch 2017) do produce grand minima with realistic total occupancy, but they were only checked against the fraction of time spent in grand minima, not against the double-peaked amplitude PDF, because active regions are too scarce during weak activity to constrain the poloidal source there at all.

Problem Statement

Determine the origin and statistics of solar grand minima within the nonlinear-stochastic Babcock–Leighton cycle dynamics. Concretely: (i) establish, from observations, historical reconstructions, or physically grounded theory, the form of the poloidal-source function $\Delta DM(SN)$ for very weak cycles, where it is currently unconstrained because active regions are scarce; and (ii) decide, with that source specified, whether stochasticity alone within the B-L framework, with the weak-cycle poloidal source still produced by the decay of tilted active regions (even if its efficiency, tilt, or emergence-latitude dependence is modified for weak cycles), reproduces both components of the reconstructed solar-cycle amplitude distribution, namely the normal-cycle mode and the statistically distinct grand-minimum mode of the double-peaked PDF of Usoskin et al. (2014), together with the observed occurrence statistics of grand minima, or whether reproducing them instead requires a process beyond active-region decay (e.g., a non-B-L poloidal source, altered meridional circulation, or a bifurcation to a separate dynamo state). A complete answer must identify the weak-cycle poloidal source with stated uncertainties and demonstrate, by quantitative statistical comparison against paleoreconstructions, which of the two alternatives holds.

References

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