In the Pacific Ocean, every few years the water off the coast of Peru warms up, and fishermen call it El Niño. Sometimes the warming is mild, sometimes so strong that it changes weather across the entire planet. Forecasting a strong El Niño six months to a year ahead is a task at the edge of possibility: weather is forgotten within two weeks, while climate remembers for decades. Between these time scales lies a gray zone where the question is not "will the event happen" but "with what probability will it occur given the current state of the ocean."

The scientists took a simple El Niño model that describes water temperature in the western and eastern equatorial Pacific and the depth of the warm layer. They added random wind bursts as noise that constantly nudges the ocean. Then, for each point in phase space, that is, for every possible ocean state, they computed the committor: the probability of a strong El Niño in the coming years. The result was as varied as a map with different zones.

In some regions, the probability is zero or one, where everything is determined, like in a spring-driven clock. In others, the probability changes smoothly from state to state, and forecasts can be made with confidence, just as a meteorologist predicts rain. But there are regions where the slightest shift in the initial state changes the probability abruptly: today 10 percent, and a bit to the right, already 90. This is probabilistic unpredictability, and it is worse than ordinary chaos because even perfect knowledge of initial conditions does not save the forecast.

Transitions between regimes without and with strong El Niño resemble jumps between two attractors. An attractor is a state toward which the system tends, like a ball rolling into a pit. The model has a strange attractor, which is chaotic, and a limit cycle, which is periodic, like a pendulum. The average transition time between them under moderate noise is about a thousand dimensionless units, corresponding to centuries. But the key point is not the number itself, but that this time does not follow the familiar Arrhenius law. This means that the theory of large deviations, which climatologists have relied on, may fail in chaotic systems.

A compass that shows not direction but the probability of a storm can now be built for real climate models. The authors are already applying machine learning to compute the committor in a large model and predict extreme heat waves. Previously, it was unclear how to assess the predictability of event probabilities on timescales from months to decades. Now it is clear: in phase space, there are regions where the probability depends weakly on the initial state, and regions where it is extremely sensitive. This distinction determines where forecasting is possible and where we are left with a compass whose needle trembles at the slightest breeze.