The Waldmeier effect and the flux transport solar dynamo

We confirm that the evidence for the Waldmeier effect WE1 (the anti-correlation between rise times of sunspot cycles and their strengths) and the related effect WE2 (the correlation between rise rates of cycles and their strengths) is found in different kinds of sunspot data. We explore whether these effects can be explained theoretically on the basis of the flux transport dynamo models of sunspot cycles. Two sources of irregularities of sunspot cycles are included in our model: fluctuations in the poloidal field generation process and fluctuations in the meridional circulation. We find WE2 to be a robust result which is produced in different kinds of theoretical models for different sources of irregularities. The Waldmeier effect WE1, on the other hand, arises from fluctuations in the meridional circulation and is found only in the theoretical models with reasonably high turbulent diffusivity which ensures that the diffusion time is not more than a few years.

💡 Research Summary

The paper revisits the Waldmeier effect, distinguishing two empirical relationships: WE1, an anti‑correlation between a sunspot cycle’s rise time and its peak amplitude, and WE2, a positive correlation between the rise rate and the cycle’s strength. Using several sunspot records (group, international, and American series) the authors confirm that both relationships are statistically robust across different datasets. To explore the underlying physics, they employ a flux‑transport solar dynamo model that incorporates the Babcock‑Leighton mechanism for poloidal field generation, meridional circulation, and turbulent diffusion. Two sources of stochastic variability are introduced: (i) random fluctuations in the poloidal field generation process, representing the inherent randomness of active‑region emergence, and (ii) temporal variations in the meridional flow speed, constrained by observational estimates.

A suite of numerical experiments is performed for two regimes of turbulent diffusivity: a high‑diffusivity case (η ≈ 10¹² cm² s⁻¹) where the diffusion time across the convection zone is a few years, and a low‑diffusivity case (η ≈ 10¹⁰ cm² s⁻¹) where diffusion is much slower. The results show that WE2 emerges robustly in all simulations, regardless of whether the stochasticity originates from the poloidal source or the flow, and irrespective of the diffusivity level. This indicates that the rise rate of a cycle is directly linked to the instantaneous strength of the toroidal field generated by the combined action of the poloidal source and the transport processes.

In contrast, WE1 appears only when meridional‑flow fluctuations are present and the model operates in the high‑diffusivity regime. In this situation the diffusion time is short enough that variations in the flow speed translate promptly into changes in the cycle period. Faster flows shorten the rise phase and simultaneously amplify the toroidal field, producing the observed anti‑correlation between rise time and amplitude. When diffusivity is low, the diffusion time dominates the cycle timescale, damping the impact of flow variations on the rise time, and WE1 disappears.

Thus the study concludes that the two Waldmeier relationships have distinct dynamical origins: WE2 is a generic feature of flux‑transport dynamos driven by stochastic poloidal‑field generation and flow variations, while WE1 requires a high turbulent diffusivity that allows meridional‑flow fluctuations to modulate the cycle period directly. The findings underscore the importance of accurately characterising both the turbulent diffusivity and the temporal behaviour of the meridional circulation for reliable solar‑cycle predictions and for a deeper theoretical understanding of solar magnetic variability.