Committor Functions for Climate Phenomena at the Predictability Margin: The Example of El Niño–Southern Oscillation in the Jin and Timmermann Model

Abstract

Abstract Many atmosphere and climate phenomena lie in the gray zone between weather and climate: they are not amenable to deterministic forecast, but they still depend on the initial condition. A natural example is medium-range forecasting, which is inherently probabilistic because it lies beyond the deterministic predictability time of the atmosphere, but for which statistically significant prediction can be made, which depends on the current state of the system. Similarly, one may ask the probability of occurrence of an El Niño event several months ahead of time. We introduce a quantity that corresponds precisely to this type of prediction problem: the committor function is the probability that an event takes place within a given time window, as a function of the initial condition. We compute it in the case of a low-dimensional stochastic model for El Niño, the Jin and Timmermann model. In this context, we show that the ability to predict the probability of occurrence of the event of interest may differ strongly depending on the initial state. The main result is the new distinction between probabilistic predictability (when the committor function is smooth and probability can be computed, which does not depend sensitively on the initial condition) and probabilistic unpredictability (when the committor function depends sensitively on the initial condition). We also demonstrate that the Jin and Timmermann model might be the first example of a stochastic differential equation with weak noise for which transition between attractors does not follow the Arrhenius law, which is expected based on large deviation theory and generic hypothesis. Significance Statement A key problem for atmospheric and climate phenomena is to predict events beyond the time scale over which deterministic weather forecast is possible. In a simple model of El Niño, we demonstrate the existence of two regimes, depending on initial conditions. For initial conditions in the “probabilistic predictability” regime, the system is unpredictable deterministically because of chaos, but the probability of occurrence of the event can still be predicted because it depends only weakly on the initial condition. In the “probabilistic unpredictability” regime, even predicting probabilities is difficult, because the probability depends strongly on initial conditions. These new concepts of probabilistic predictability and unpredictability should be key in understanding the predictability potential for rare events in climate problems, as well as in other complex dynamics.