Authors: Chitradeep Saha1,4, Sanghita Chandra2,3,4, and Dibyendu Nandy2,4
Affiliations:
1Department of Meteorology, University of Reading, Earley Gate, PO Box 243, Reading, RG6 6BB, UK
2Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, West Bengal, India
3Max-Planck-Institut für Sonnensystemforschung, Justus-Von-Liebig-Weg 3, 37077 Göttingen, Germany
4Center of Excellence in Space Sciences India, Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, West Bengal, India
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Introduction
The Sun’s 11-year sunspot cycle occasionally breaks down into extended quiescent phases known as grand minima, the most recent of which was the Maunder Minimum (1645–1715 CE). Kinematic flux transport solar dynamo simulations [1,2] have previously suggested that during these episodes the Sun’s polar fields weaken, and the polar field reversal may halt temporarily. The ceaseless meridional circulations eventually help accumulate sufficient polar flux to restart the reversal and bring back regular cyclic activity. However, the physics governing this recovery has remained poorly quantified. In this nugget, we summarise results from our recent multi-millennial solar dynamo simulations with stochastic forcing to pin down the polar flux threshold required for the Sun to achieve to recover from a grand minimum, and to test whether the onset, duration, and recovery of solar grand minima are causally linked.
Model
We use an axisymmetric, kinematic Babcock-Leighton solar dynamo model with prescribed flow and diffusivity profiles [3], with two magnetically quenched poloidal source terms [1]: a mean-field alpha effect operating in the bulk of the convection zone, and a Babcock-Leighton source operating near the surface. Stochastic fluctuations are introduced as uniform white noise in both poloidal sources, independently in each hemisphere, which is a simplified choice since more complex dynamical interactions can in principle produce autocorrelated “coloured” noise instead [4].
Model parameters are tuned to reproduce the occurrence statistics of grand minima from multi-millennial reconstructions of solar activity [5], and the dynamo is restricted to operate in the near-critical regime thought to characterise the present-day Sun [6]. To probe causal dependencies, we additionally ran simulation sets by varying the critical buoyancy threshold, and the mean-field quenching limit.
Results
Over the simulated ten-millennia of solar magnetic activity, we track the hemispheric polar flux and toroidal flux through grand-minima-like episodes (Figure 1). As a minimum develops, polar fluxes decline, toroidal flux weakens, and sunspots disappear; eventually the polar field stops reversing altogether and enters a “unipolar” phase, as have been suggested previously [7]. We find no specific amplitude threshold in our simulations that triggers the onset of these unipolar phases, implying their onset is not a gradual, predictable process.
Figure 1: An overview of the evolution of the simulated solar polar fields over the first thousand years (a representative segment of the 10,000 years simulation), with the top and bottom panels showing the northern and southern hemispheres, respectively, including their grand-minima-like quiet phases. Grey shading marks periods where the polar field stops reversing altogether. Cycle amplitudes are tracked with blue markers for negative polarity and red markers for positive polarity. The middle panel shows the corresponding butterfly diagram, combining a colour-coded signed radial field (saturated at ± 150 G) with sunspot eruption proxies overlaid in black. Over this particular 1000-year stretch, each hemisphere passes through two separate grand minima.
Recovery from a solar grand minimum appears to be more systematic. Tracking the signed hemispheric polar flux amplitude at the moment each unipolar phase ends, across ten millennia of simulation, reveals a striking clustering around a well-defined value (Figure 2) — roughly 63 ± 4% of the modal polar flux amplitude for our primary run. This recovery threshold holds for both polarities and hemispheres, pointing to a genuine, statistically robust threshold rather than a coincidence of our particular simulation run.
Figure 2: The figure shows how simulated polar flux amplitudes are distributed across ten millennia, with positive values in blue and negative values in red, separately for the northern (top) and southern (bottom) hemispheres. Accompanying histograms summarise this distribution over the full simulated timescale; since most of the time the Sun is in its regular activity phase, these histograms are best described by skewed Gaussian fits rather than symmetric ones. The green diamonds mark the polar flux amplitude at the point each unipolar phase ends, and the fact that these values cluster tightly around a common mean (the green line) points to a consistent threshold that the polar flux must reach before the cycle can recover from a grand minimum. Importantly, low polar flux values alone do not necessarily signal a grand minimum and even during otherwise normal activity, individual cycles can occasionally fall below this recovery threshold.
We further investigate how this threshold responds to two nonlinear ingredients in the dynamo model, namely the critical buoyancy threshold and the mean-field magnetic quenching limit. The modal polar flux amplitude correlates strongly with both parameters, but the recovery threshold itself is independent of physically sensible critical buoyancy, since during unipolar phases the dynamo mostly generates weak toroidal fields that rarely approach the buoyancy limit. In contrast, the recovery threshold correlates strongly with the intensity of mean-field quenching, i.e., a higher quenching limit allows stronger toroidal fields to be inducted, raising the modal polar flux, the recovery threshold, and the toroidal flux together, implying a harder, slower recovery from an ongoing grand minimum.
Figure 3: This figure illustrates whether the duration of a simulated grand minimum is correlated to how quickly it sets in (left panel) or how quickly the cycles recover (right panel), with the correlation coefficient and significance level (r and p, respectively) reported in each case. Neither comparison shows a statistically meaningful correlation.
We also tested, across all 22 simulated grand minima, whether the rate of onset or rate of recovery correlates with how long a minimum lasts (Figure 3). We find no statistically significant correlation in either case, i.e, the duration of grand minima appears to be independent of both onset and recovery rates in our simulations.
Conclusion
Our simulations show that recovery from a solar grand minimum is governed by a well-defined, model-robust polar flux threshold, and thus offers a potential observational proxy for forecasting the end of an ongoing grand minimum. The mean-field poloidal source emerges as the key physical ingredient that controls how difficult the process of recovery is. However, despite an exhaustive search, we could not identify any precursor or signature that predicts the onset of a grand minimum.
Publication link: Saha, C., Chandra, S., Nandy, D., 2025, Recovery of the Solar Cycle from Maunder-Like Grand Minima Episodes: A Quantification of the Necessary Polar Flux Threshold Through Solar Dynamo Simulations, Solar Physics, 300, 125. https://doi.org/10.1007/s11207-025-02538-5
References
[1] Passos, D., Nandy, D., Hazra, S., Lopes, I.: 2014, A solar dynamo model driven by mean-field alpha and Babcock-Leighton sources: fluctuations, grand-minima-maxima, and hemispheric asymmetry in sunspot cycles. Astron. Astrophys. 563, A18.
[2] Saha, C., Chandra, S., Nandy, D.: 2022, Evidence of persistence of weak magnetic cycles driven by meridional plasma flows during solar grand minima phases. Mon. Not. R. Astron. Soc. 517, L36.
[3] Chatterjee, P., Nandy, D., Choudhuri, A.R.: 2004, Full-sphere simulations of a circulation-dominated solar dynamo: exploring the parity issue. Astron. Astrophys. 427, 1019.
[4] Saha, C., Mukhopadhyay, S., Nandy, D.: 2025, On the origin of long-term modulation in the Sun’s magnetic activity cycle. Astrophys. J. Lett. 984, L5.
[5] Usoskin, I.G.: 2023, A history of solar activity over millennia. Living Rev. Sol. Phys. 20, 2.
[6] Reinhold, T., Shapiro, A.I., Solanki, S.K., Montet, B.T., Krivova, N.A., Cameron, R.H., Amazo-Gómez, E.M.: 2020, The Sun is less active than other solar-like stars. Science 368, 518.
[7] Makarov, V.I., Tlatov, A.G.: 2000, Polar magnetic field reversals of the Sun in Maunder Minimum. J. Astrophys. Astron. 21, 193.