Modelling the Solar Cycle Nonlinearities into the Algebraic Approach

TL;DR

This work presents an algebraic, physically grounded framework that integrates Latitude Quenching (LQ) and Tilt Quenching (TQ) to quantify how active-region dipole contributions regulate the solar cycle and saturate polar-field growth. By employing a Synthetic Active Region Cycle Model and an error-function–based dipole calculation across a range of dynamo effectivity $λ_R$, it shows that LQ dominates at low $λ_R$ and TQ dominates at high $λ_R$, with a transition near $λ_R ≈ 12^\circ$ and an overall shallow scaling $R(λ_R) ≈ C_1 + C_2/λ_R^{n}$ with $n ≈ 0.36$. The study also finds that tilt and morphological asymmetries in ARs have minimal impact on the main trend, reinforcing that transport controls the LQ–TQ balance more than AR geometry. The algebraic approach thus provides a fast, interpretable complement to full surface flux transport simulations for understanding solar-cycle saturation and has potential for data-driven calibration of solar-cycle forecasts.

Abstract

Understanding and predicting solar-cycle variability requires accounting for nonlinear feedbacks that regulate the buildup of the Sun's polar magnetic field. We present a simplified but physically grounded algebraic approach that models the dipole contribution of active regions (ARs) while incorporating two key nonlinearities: tilt quenching (TQ) and latitude quenching (LQ). Using ensembles of synthetic cycles across the dynamo effectivity range $λ_R$, we quantify how these mechanisms suppress the axial dipole and impose self-limiting feedback. Our results show that (i) both TQ and LQ reduce the polar field, and together they generate a clear saturation (ceiling) of dipole growth with increasing cycle amplitude; (ii) the balance between LQ and TQ, expressed as $R(λ_R) = \mathrm{dev(LQ)}/\mathrm{dev(TQ)}$, transitions near $λ_R \approx 12^\circ$, with LQ dominating at low $λ_R$ and TQ at high $λ_R$; (iii) over $8^\circ \leq λ_R \leq 20^\circ$, the ratio follows a shallow offset power law with exponent $n \approx 0.36 \pm 0.04$, significantly flatter than the $n=2$ scaling assumed in many surface flux--transport (SFT) models; and (iv) symmetric, tilt-asymmetric, and morphology-asymmetric AR prescriptions yield nearly identical $R(λ_R)$ curves, indicating weak sensitivity to AR geometry for fixed transport. These findings demonstrate that nonlinear saturation of the solar cycle can be captured efficiently with algebraic formulations, providing a transparent complement to full SFT simulations. The method highlights that the LQ\--TQ balance is primarily controlled by transport ($λ_R$), not by active-region configuration, and statistically disfavors the SFT-based $1/λ_R^{2}$ dependence.

Figures (10)

• Figure 1: Synthetic butterfly diagram generated using the time-dependent emergence profile from jiang2018predictability, with emergence latitudes declining over the cycle and latitude scatter decreasing with time. Each point represents the latitude and time of an individual active region. The distribution captures the equatorward drift and narrowing latitudinal spread of sunspot emergence throughout the solar cycle.
• Figure 2:
• Figure 3: Final axial dipole moments as a function of normalised cycle amplitude ($A_n/A_0$) and dynamo effectivity range ($\lambda_R$) at the end of synthetic solar cycles. Each panel corresponds to a different nonlinear feedback configuration: Top left — no quenching; Top right — tilt quenching only; Bottom left — latitude quenching only; Bottom right — both tilt and latitude quenching. Colour indicates the strength of the resulting dipole moment (in units of $10^{22}$ Mx).
• Figure 4: Relative importance of LQ vs. TQ in suppressing the solar axial dipole moment, shown as a function of dynamo effectivity range ($\lambda_{R}$) and cycle amplitude ratio ($A_n/A_0$). The colourmap indicates the ratio R($\lambda_R$), with red regions indicating LQ-dominated suppression and blue regions indicating TQ-dominated suppression. Contour lines provide quantitative values of the ratio.
• Figure 5: Final axial dipole moment (in units of $10^{22}\,$Mx) at cycle minimum versus cycle amplitude ratio $A_n/A_0$, at $\lambda_R = 11.72^\circ$. Curves show results with NoQ (black), TQ (red), LQ (blue), and LQTQ (green).