Chaotic Kramers' Law: Hasselmann's Program and AMOC Tipping

TL;DR

The paper addresses tipping in bistable systems driven by chaotic forcing by generalizing Kramers' law through homogenization and large-deviation theory. It derives chaotic Kramers' law under a regime where the forcing speed $\varepsilon$ is much smaller than the noise strength $\delta$, and validates the theory with a chaotically forced 3-box AMOC model coupled to Lorenz-63 systems. The results show exponential scaling of mean transition times with the effective barrier, extending to non-Gaussian chaotic forcing and illustrating practical numerical strategies via ensemble methods. The study provides a potential mechanism for abrupt AMOC responses in climate models and highlights the limits of the Hasselmann program when forcing is fast but not infinitely fast, suggesting directions for multiplicative forcing and higher-order corrections in future work.

Abstract

In bistable dynamical systems driven by Wiener processes, the widely used Kramers' law relates the strength of the noise forcing to the average time it takes to see a noise-induced transition from one attractor to the other. We extend this law to bistable systems forced by fast chaotic dynamics, which we argue is in some cases a more realistic modeling approach than unbounded noise forcing. Transitions similar to the noise-driven case can only occur if the amplitude of the chaotic forcing is large enough. If this is the case, in our numerical example - a reduced-order model of the Atlantic Meridional Overturning Circulation (AMOC) - we observe the chaotic Kramers' law to hold even when the chaotic forcing is far from the stochastic limit. We discuss the limitations of the chaotic Kramers' law, how to address the numerical issues associated with the timescale separation, and give a possible explanation for the dynamics of recently found AMOC collapses and recoveries in complex climate models.

Figures (7)

• Figure 1: Sample timeseries of the chaotically forced 3-box AMOC model with $\epsilon = 1$ and $\delta = 0.5$ for different hosing values. For all three values of $H$, the deterministic unforced system is bistable.
• Figure 2: Weak invariance principle-based estimates of $\sigma_L^2$, obtained from integrating the first component of the Lorenz-63 system. Estimated over $N = 500$ trajectories with timestep $t=1$ and lengths up to $K = T = 10000$.
• Figure 3: Plots of observed tipping times in the AMOC 3-box model calibrated to match HadGEM3-MM, starting in the on-state for selected values of $\delta$. The black dots show the logarithmic waiting time until the first transition to the off-state, and their average over $50$ samples is shown by the blue dots. The red line is fitted to these averages (fit with $200$ sampled transition times per value of $\delta$) for the SDE case, but the exact same line is still shown in panels (b) to (d) for comparison to the chaotic ODE case. Note that the SDE case can be viewed as the ODE's asymptotic behavior for $\delta \rightarrow 0$. In green, the relative bounds of the error in the prefactor $\kappa^{-1} \in [e^{-2}, e^1]$. We use the noise matrix $A = 10A_{MM}$ from Chapman_2024 that was obtained via Maximum Likelihood estimation to match the HadGEM3-MM fluctuations.
• Figure 4: Plots of the increments in the AMOC 3-box model calibrated to match HadGEM3-MM, after reaching the on-state for $\delta = 1$ in comparison to the SDE limit. The increments are defined by $X(t)-X(t-\Delta t)$ with $\Delta t = 0.1$ for $t \in \left[500, 100000\right] \cap 0.1 \mathbb{N}$.
• Figure 5: Mean squared deviation of the mean transition time in the chaotically forced 3-box AMOC model from Kramers' law for different values of $\epsilon$

Theorems & Definitions (10)

• Lemma 2.1.1
• Definition 2.1.2: Weak invariance principle
• Theorem 2.2.1: Kramers' law
• Definition 2.2.2: Quasipotential
• Theorem 2.2.3: The irreversible Kramers formula
• Lemma 2.2.4: chaotic Kramers' law
• Lemma 2.2.5
• proof
• Theorem 2.2.6: Chaotic LDP
• Theorem B.1.1: Anisotropic Freidlin-Wentzell Theorem