Forecasting climate events such as El Niño, heat waves, or droughts lies in a gray zone between deterministic weather predictability (about two weeks) and climatology, where initial conditions are forgotten. For phenomena on timescales from months to decades, it is useful to know the probability of an event conditioned on the current state of the system. This work sits alongside studies of rare events in the quantum world and classical probability, but focuses on a concrete climate example, introducing the mathematical machinery of the committor to describe this intermediate region.
The authors use the low-dimensional stochastic El Niño model of Jin and Timmermann, which describes the evolution of three variables: sea surface temperature in the western and eastern equatorial Pacific and thermocline depth anomaly. Gaussian white noise is added to mimic random wind stress fluctuations. To compute the committor, they employ two approaches: direct ensemble simulation of trajectories from each point in phase space, and estimation from long observed trajectories using the ergodic theorem. Transitions between a strange attractor and a limit cycle, corresponding to regimes without and with strong El Niño, are studied through statistical analysis of first passage times.
The main result is the identification of four regions in phase space regarding the predictability of strong El Niño events: two regions of deterministic predictability (probability 0 or 1), a region of probabilistic predictability (probability between 0 and 1, weakly dependent on initial conditions), and a region of probabilistic unpredictability (strong sensitivity of probability to initial conditions). At intermediate noise amplitude (σ = 5×10⁻⁵), these regions coexist, with the probabilistically predictable region arising from local instability associated with positive finite-time Lyapunov exponents. Furthermore, the mean transition time between attractors does not follow the Arrhenius law but grows as a power law σ⁻¹ over a wide range of small noise, contradicting expectations from large deviation theory.
This work introduces a distinction between probabilistic predictability and unpredictability that may be common to many climate forecasting problems at the edge of predictability. Unlike related studies that focus on other systems or general theorems, here a concrete example with a climate model is provided, demonstrating the practical importance of the committor. The violation of the Arrhenius law in the Jin-Timmermann model points to a possible counterexample to the expected universal behavior of transitions in chaotic systems with weak noise, necessitating a revision of theoretical foundations.
The topic could develop toward applying the committor to more realistic climate models and real data. The authors already announce the use of machine learning methods to compute the committor in the CESM2.1 model for forecasting extreme heat waves. Also promising is the investigation of the mathematical nature of the Arrhenius law violation, possibly related to the fractal structure of basin boundaries.
The results will influence the methodology for forecasting rare climate events, the assessment of predictability in nonlinear systems, and the development of transition theory in stochastic dynamical systems.
Next steps include developing efficient machine learning algorithms for computing the committor in high-dimensional systems and providing a theoretical explanation for the observed violation of the Arrhenius law.
This work directly addresses the fundamental problem of predictability in chaotic systems: how long and with what accuracy can rare events be forecast in the presence of noise. It also touches on the question of the universality of the Arrhenius law for transitions between complex attractors, which remains open in dynamical systems theory.
The mean transition time between regimes in the model can reach 500,000 dimensionless time units, corresponding to millennia, and the distribution of transition times is exponential, like radioactive decay.