A warm current off the coast of South America shifts weather across the planet every few years. This is El Niño, and its behavior resembles a pendulum that sometimes swings too far. In 1986, meteorologist John Vallis described the process with three equations in which the ocean and atmosphere exchange heat and momentum. The equations turned out to be chaotic: the slightest change in initial conditions leads to a completely different trajectory. But in that model, the ocean remembered only the present moment, as if yesterday's water had no influence on today's.

In the new work, the Vallis equations are rewritten with fractional derivatives. An ordinary derivative shows how fast a quantity changes right now. A fractional derivative adds memory: it accounts for what happened to the system in the past, with recent events mattering more than distant ones. This is like a person making a decision based not only on the present moment but also on their entire experience. The ocean also remembers: warm water that arrived last year continues to affect currents and winds.

The researchers numerically solved the fractional system for different values of the parameter p, which describes the average influence of equatorial winds on the ocean. At α=1, chaos is observed for |p|≤2.45. When α decreases to 0.985, the chaotic range narrows to |p|≤1.5. At α=0.975, chaos remains only in a narrow band near p=0. And at α=0.965, chaos disappears entirely. This means that sufficiently strong ocean memory can quench El Niño's chaotic oscillations.

The critical order below which chaos vanishes was computed exactly. For one set of parameters, it is 0.975, so the total order of the three equations is 2.925. For the incommensurate system, where different equations have different derivative orders, the critical total order turned out to be even lower: 2.71 with α1=0.71 and α2=α3=1. This shows that memory in just one equation can be decisive.

It was previously known that the integer-order Vallis model is chaotic and linked to El Niño, but the influence of fractional order on chaos remained unexplored. Now it is established that decreasing the fractional order narrows the chaotic region in parameter p, and below a critical value chaos disappears entirely. Exact critical orders were found for both commensurate and incommensurate cases. This result was obtained through bifurcation analysis and computation of Lyapunov exponents for the fractional system across all values of parameter p and various derivative orders.

Ocean memory turns out to be not just a detail but a tool that can suppress or amplify El Niño. If memory in the real ocean is strong enough, climate model forecasts must account for fractional order. It is not yet clear how to measure this memory in nature, but the work provides quantitative benchmarks. Perhaps in the future, climatologists will be able to tune models to better predict when the warm current will again alter the planet's weather.