Narrowing the solar surface flux transport parameter space through nonlinear feedbacks

Technical Summary: Narrowing the solar surface flux transport parameter space through nonlinear feedbacks

Problem Statement
The Surface Flux Transport (SFT) model is a primary framework for describing the evolution of the Sun's large-scale photospheric magnetic field and predicting future solar cycle amplitudes. While linear transport processes (meridional flow, differential rotation, and turbulent diffusion) are relatively well-constrained, the model's parameter space remains broad and uncertain. Previous optimisation studies (e.g., Petrovay & Talafha 2019) identified "islands" of admissible parameters but often relied on linear assumptions or neglected specific nonlinear feedbacks. Furthermore, the role of a phenomenological flux-decay term in parameterised source formulations has been debated, with some data-assimilative simulations suggesting it may be unnecessary, while others indicate its absence leads to unrealistically persistent dipole fields. The specific impact of nonlinear quenching mechanisms—specifically tilt quenching (TQ) and latitude quenching (LQ)—on the admissible parameter space of meridional flow speed ($u_0$), surface diffusivity ($\eta$), and decay timescale ($\tau$) had not been systematically quantified within a unified optimisation framework.

Methodology
The authors extended the parameter-space optimisation framework established by Petrovay & Talafha (2019) by introducing analytic prescriptions for nonlinear feedbacks into the SFT source term.

• Model Setup: The SFT equation governs the evolution of the radial magnetic field $B(\lambda, t)$, incorporating advection by meridional flow, diffusion, and a decay term ($-B/\tau$). The source term $S(\lambda, t)$ represents the emergence of bipolar magnetic regions (BMRs) based on a statistically averaged solar cycle.
• Nonlinear Feedbacks: Two quenching mechanisms were implemented:
  1. Tilt Quenching (TQ): The latitudinal separation of BMRs (Joy's law) is reduced for stronger cycles, governed by a nonlinearity parameter $b_{joy} = 0.15$.
  2. Latitude Quenching (LQ): The mean emergence latitude increases with cycle amplitude, governed by $b_{lat} = 2.4$.
• Optimisation Strategy: The model was solved numerically over a dense grid of parameters ($u_0$, $\eta$, $\tau$) for 100 solar cycles to reach a steady cyclic state. Admissible solutions were defined by their agreement with three observational constraints:
  1. Polar-field reversal timing ($T_{rev}$).
  2. Polar-field minimum-to-extremum amplitude ratio ($R_{pol}$).
  3. Latitude of the polar cap boundary ($\lambda_{cap}$).
     Additional dipole-based constraints ($D_{rev}$, $D_{min/ext}$) were used for diagnostic purposes.
• Scenarios: The study compared linear cases (no quenching) against cases with TQ only, LQ only, and combined TQ+LQ, across different decay timescales ($\tau = 8$ yr and $\tau = \infty$).

Key Results

• Impact of Quenching: Both TQ and LQ reduce the admissible parameter domain, but LQ exerts a significantly stronger influence. The combined action of TQ and LQ produces a pronounced "ceiling" effect, saturating axial dipole amplification and preventing unrealistically large polar fields in strong cycles.
• Parameter Space Contraction:
  • In the linear case (no quenching) with $\tau = 8$ yr, admissible regions were concentrated around $\eta \approx 600\text{--}650$ km$^2$ s$^{-1}$ and $u_0 \approx 18\text{--}19$ m s$^{-1}$, with a secondary region at lower values.
  • Introducing TQ alone caused only a modest contraction.
  • Introducing LQ alone narrowed the ranges significantly.
  • The combined TQ+LQ case with $\tau = 8$ yr restricted the admissible domain to narrow bands, specifically $u_0 \approx 18\text{--}19$ m s$^{-1}$ and $\eta \approx 600\text{--}650$ km$^2$ s$^{-1}$, effectively eliminating the secondary solution islands found in the linear case.
• Role of Decay Timescale ($\tau$): The inclusion of a finite decay timescale ($\tau \approx 8\text{--}10$ yr) is critical. When $\tau = \infty$ (no decay), the admissible parameter space expands considerably, even with quenching active. However, the authors note that in parameterised models without explicit decay, this can lead to unrealistically persistent dipole fields and delayed reversals. The optimal range of $\tau \approx 8\text{--}10$ yr reflects the timescale required to reproduce observed reversal timing in the presence of nonlinear feedbacks.
• Coupled Balance: Correlations between $u_0$, $\eta$, and $\tau$ reveal a tightly coupled balance between advection, diffusion, and phenomenological flux loss. As $\tau$ decreases (stronger decay), the admissible ranges for $u_0$ and $\eta$ shift and narrow, requiring slower meridional flows or specific diffusivities to maintain observational consistency.

Significance and Claims
The paper claims that incorporating nonlinear quenching mechanisms transforms the SFT optimisation landscape from broad, disconnected "islands" of solutions into narrow, physically consistent "ridges."

• Self-Limiting Regime: The study concludes that nonlinear quenching, combined with a phenomenological flux-loss timescale, defines a self-limiting SFT regime. This regime constrains dynamo saturation by linking surface transport parameters directly to nonlinear feedback in the source term.
• Predictability Limits: The results suggest the solar dynamo operates near a marginally stable saturation threshold. While the mean cycle amplitude is constrained by quenching, the inherent sensitivity to surface transport parameters implies that purely kinematic models have limited predictive horizons due to the stochastic nature of active region emergence.
• Model Dependence: The authors explicitly state that their findings are conditional on the adopted parameterised source formulation. They acknowledge that data-assimilative simulations (e.g., Yeates et al. 2025) which ingest observed active regions directly may not require an explicit decay term, suggesting the necessity of $\tau$ depends on the numerical implementation of the source.
• Physical Interpretation: The "ceiling" effect observed is interpreted as a physical mechanism where additional flux input no longer translates into stronger polar fields once quenching limits dipole build-up efficiency. This provides a unified picture linking surface inflows, quenching, and the deeper dynamo processes.

In summary, the work demonstrates that nonlinear feedbacks are not merely minor corrections but are fundamental in narrowing the viable parameter space of the SFT model, enforcing a self-consistent balance between transport and flux loss that regulates the solar cycle amplitude.