Constructing efficient score functions for rare event simulation in high-dimensional ocean-climate models

Abstract

Calculating transition probabilities between different states of multistable climate tipping systems is computationally challenging in high-dimensional models. Targeted algorithms, such as the Trajectory-Adaptive Multilevel Splitting (TAMS) method, require an adequate score function to be successful, i.e., to provide an estimate of a transition probability with an acceptable variance when only a relatively small ensemble of model trajectories can be computed. Here, we present a data-driven method to derive a score function based on projecting the model dynamics in a reduced state space. Using a spatially two-dimensional partial differential equation model of the Atlantic Meridional Overturning Circulation, we show that this score function performs better than currently available ones. Using the new score function, transition probabilities can be determined with low variance, even in the case of small noise amplitudes. Besides purely noise-induced transitions, we also consider the scenario of combined stochastic and time-dependent deterministic forcing, presenting a strategy to efficiently simulate AMOC tipping events in global ocean and climate models subject to transient climate change.

Figures (18)

• Figure 1: (a) Time evolution of the score function during a noise-induced transition from the AMOC-on state to the AMOC-off state. Red dots indicate the time points of the depicted instantaneous fields; (b) stream function $\psi$ and (c) salinity field at four instants during the transition.
• Figure 2: Time series of the normalised AMOC strength $\xi_1^0$ (see Eq. (\ref{['eq:sf_naive']}) below) for trajectories under deterministic forcing with $\alpha_0 \in [0.6,0.63]$. The grey area highlights the control period.
• Figure 3: Sequential steps in constructing the score function $\xi^0_{3}$ from a TAMS run: 1) Assemble the data matrix $\mathbf{Q}$ from the ensemble of trajectories obtained after a TAMS run, 2) perform POD to construct the latent space $\mathcal{V}_{POD}$, 3) project $\mathbf{Q}$ in $\mathcal{V}_{POD}$ and construct the ensemble transition path $\mathcal{P}_{tr}$, 4) map the full latent space with a score function $\xi^0_{3}$ using the arc-length along $\mathcal{P}_{tr}$.
• Figure 4: POD modes $\mathbf{U}_{1-4}$ from left to right, for the stream function $\psi$ (top row) and salinity $S$ (bottom row).
• Figure 5: Mean transition paths (MTPs) (full lines) obtained during the score function improvement iterations, projected in three pairs of POD modes. The FW instanton (dashed line) and the saddle state (red star) are added for reference.