Committor Functions for Climate Phenomena at the Predictability Margin: The example of El Ni\~no Southern Oscillation in the Jin and Timmerman model

Many phenomena in the climate system 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 depend on the current state of the system. Similarly, one may ask the probability of occurrence of an El Ni\~no event several months ahead of time. In this paper, we introduce a quantity which 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 explain the main mathematical properties of this probabilistic concept, and compute it in the case of a low-dimensional stochastic model for El-Ni\~no, the Jin and Timmerman 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 intrinsic probabilistic predictability (when the committor function is smooth and probability can be computed which does not depend sensitively on the initial condition) and intrinsic probabilistic unpredictability (when the committor function depends sensitively on the initial condition). We also demonstrate that the Jin and Timmerman model might be the first example of a stochastic differential equation with weak noise for which transition between attractors do not follow the Arrhenius law, which is expected based on large deviation theory and generic hypothesis.