Investigating Nonlinear Quenching Effects on Polar Field Buildup in the Sun Using Physics-Informed Neural Networks

Abstract

The solar dynamo relies on the regeneration of the poloidal magnetic field through processes strongly modulated by nonlinear feedbacks such as tilt quenching (TQ) and latitude quenching (LQ). These mechanisms play a decisive role in regulating the buildup of the Sun's polar field and, in turn, the amplitude of future solar cycles. In this work, we employ Physics-Informed Neural Networks (PINN) to solve the surface flux transport (SFT) equation, embedding physical constraints directly into the neural network framework. By systematically varying transport parameters, we isolate the relative contributions of TQ and LQ to polar dipole buildup. We use the residual dipole moment as a diagnostic for cycle-to-cycle amplification and show that TQ suppression strengthens with increasing diffusivity, while LQ dominates in advection-dominated regimes. The ratio $ΔD_{\mathrm{LQ}}/ΔD_{\mathrm{TQ}}$ exhibits a smooth inverse-square dependence on the dynamo effectivity range, refining previous empirical fits with improved accuracy and reduced scatter. The results further reveal that the need for a decay term is not essential for PINN set-up due to the training process. Compared with the traditional 1D SFT model, the PINN framework achieves significantly lower error metrics and more robust recovery of nonlinear trends. Our results suggest that the nonlinear interplay between LQ and TQ can naturally produce alternations between weak and strong cycles, providing a physical explanation for the observed even-odd cycle modulation. These findings demonstrate the potential of PINN as an accurate, efficient, and physically consistent tool for solar cycle prediction.

Figures (6)

• Figure 1: Simulation results for SFT parameters $u_0 = 9\,\mathrm{m/s}$, $\tau = 8\,\mathrm{years}$, and $\eta = 350\,\mathrm{km^2/s}$. (a) Dipole moment distribution in Gauss across latitudes for each time step over 15 cycles. (b) Time evolution of the residual dipole moment,$D_{\mathrm{res}}(P=11, \tau = 8)$ (c) Change in dipole moment of each cycle from the previous cycle for different processes (TQ, LQ, LQ+TQ, and linear). Deviations are quantified at twice the mean of the maximum source.
• Figure 2: Parameter sensitivity of the nonlinear quenching effects across the SFT parameter space. Panels (a), (c), (e) and (b), (d), (f) show the sensitivity maps of $\Delta D_{\mathrm{LQ}}$, $\Delta D_{\mathrm{TQ}}$, and their ratio across the $(u_{0}, \eta)$ parameter space for $\tau=8$ and $\tau=\infty$, respectively.
• Figure 3: Relative importance of nonlinear quenching mechanisms. Ratio $\Delta D_{\mathrm{LQ}}/\Delta D_{\mathrm{TQ}}$ is plotted as a function of the dynamo effectivity range $\lambda_{R}$ for finite $\tau=8$ (a) and $\tau=\infty$ (b). The teal symbols show the PINN results, the dashed black curve is the best-fit inverse-square law ($C_{1}+C_{2}/\lambda_{R}^{2}$), and the crimson points are values from paper2. Best-fit parameters: $C_1 = -0.3847,\, C_2 = 136.86$ ($\tau = 8$); $C_1 = -0.3679,\, C_2 = 158.19$ ($\tau = \infty$). The shaded regions highlight regimes dominated by LQ (green) and TQ (magenta), showing that LQ dominates at low $\lambda_{R}$ (advection-dominated regimes) and TQ becomes increasingly effective at higher $\lambda_{R}$ (diffusion-dominated regimes).
• Figure 4: Comparison of dipole moment deviations $\Delta D_{\mathrm{LQ}}$ and $\Delta D_{\mathrm{TQ}}$ for different values of $u_{0}$ and $\eta$. The ratio quantifies the relative importance of the two quenching mechanisms for finite $\tau = 8$ and $\tau \to \infty$. Transition values ($\Delta D_{\mathrm{LQ}}/\Delta D_{\mathrm{TQ}} \approx 1$) are shown in bold.
• Figure 5: Schematic illustration of the roles of latitude quenching (LQ) and tilt quenching (TQ) in regulating solar cycle strength. Panels (a–c): strong-cycle scenario. (a) Due to TQ, the mean tilt angle of emerging BMRs decreases from $\alpha_{w}$ (weak cycle) to $\alpha_{s}$ (strong cycle), enhancing flux cancellation between trailing and leading polarities. (b) LQ simultaneously shifts the mean emergence latitude poleward from $\bar{\lambda}_{0w}$ to $\bar{\lambda}_{0s}$, reducing cross-equatorial cancellation. (c) The combined effect decreases poleward transport of flux, leaving a weaker polar field and reducing the amplitude of the subsequent cycle (d). Panels (e–f): weak-cycle scenario. (e) With weaker TQ, BMRs retain larger Joy’s-law tilts, while LQ shifts the mean emergence latitude equatorward, enhancing cross-equatorial cancellation. (f) This facilitates stronger poleward transport of trailing flux, yielding an enhanced polar field that seeds a stronger following cycle (a). Together, these processes illustrate how the interplay of LQ and TQ can produce cycle-to-cycle modulation consistent with the Gnevyshev–Ohl rule and the observed even–odd alternation in solar activity.