A wave travels westward, reflects off the coast, and returns six months later to alter temperatures along the eastern Pacific. This journey of Rossby and Kelvin waves Mori-Zwanzig formalism turns into an equation with memory. Previously, climatologists added delays to El Niño models by intuition, relying on physical insight. Now the delay has been derived rigorously, step by step, from the ocean's equations of motion.
The authors took a two-strip ocean model and projected its dynamics onto a single quantity: surface temperature. The result is an equation where the rate of temperature change depends not only on the current state but also on what happened about 200 days earlier. The main surprise is an additional cubic term absent from the classic Suarez-Schopf model. It slows the oscillations: the period grows from 2-3 years to 3.5-4.5 years, closer to the observed 4-7 years.
The exact method of variation of parameters allowed comparison of approximate calculations with a benchmark. It turned out that the popular pseudo-orthogonal dynamics approximation introduces a noticeable error. This means that more careful methods will be needed for more complex models.
The echo in the ocean is no longer a metaphor but a precise mathematical object. Waves returning with a delay control temperature, and this can now be derived rather than postulated. This approach applies not only to El Niño but also to other climate phenomena where distant waves carry information about the past.
The delay in the El Niño model corresponds to the time it takes for Rossby waves to cross the Pacific basin, reflect off the western boundary, and return as Kelvin waves. This journey takes about half a year.