Every few years the eastern equatorial Pacific warms by a degree or more, and rainfall shifts across half the planet. The cause is a conversation between wind and water: a feedback loop that amplifies any nudge, ocean waves that take months to cross the basin, and a delay that turns the loop into an oscillation.
This is a slice along the equator from the western Pacific to South America. The trade winds blow west and pile warm surface water against Indonesia. That tilts the thermocline, the boundary between warm surface water and cold deep water. In the east it sits close to the surface, so upwelling brings cold water up. Weaken the trades and the tilt relaxes. This is an idealized toy model with round-number constants, built to show the mechanism.
How to read this: a side-on slice along the equator, with air on top and ocean below (deeper is lower). Colors show water temperature, using the bar above. Amber arrows are the trade winds blowing west. The green loop is air rising over warm water and sinking over cool water.
Try this: click El Niño, then La Niña, and compare the dashed line's tilt. Then click Neutral and press "Let the loop run" to let the ocean and wind drive each other.
Warm water heats the air above it, which rises and lowers surface pressure. Cold water does the opposite. So the east–west temperature gap sets the pressure difference that drives the trades. Every link pushes the same way, which makes the loop a positive feedback. Jacob Bjerknes described it in 1969. Run it in either direction from neutral and it races to an extreme and stays there. Something else has to turn it around (lab 2).
Wind stress τ pushes the warm layer west until the slope of the thermocline pushes back. With ρa = 1.2 kg/m³, CD = 1.3×10⁻³, ρ = 1025 kg/m³, a density step of 4 kg/m³ (g′ = 0.038 m/s²), a warm layer and a 15,600 km basin, a trade wind gives a tilt of about .
Toy rules for the rest: sea surface temperature mixes toward the temperature of the water just below, which is colder when the thermocline is shallow, and mixing grows with wind stress. Trades respond to the east–west temperature gap, capped at . Niño-3.4 averages 170°W–120°W.
The ocean can't respond everywhere at once. A burst of westerly wind in the central Pacific pushes warm water east as a bump in the thermocline: an equatorial Kelvin wave. At the same time it sends a shallowing west as a Rossby wave, three times slower. The Rossby wave reflects off Asia as a Kelvin wave that lifts the thermocline and cools the east, months later.
How to read this: the top is a map of the ocean near the equator. The bottom is a side view of the floor of the warm layer (bumps are exaggerated; lower means deeper warm water). Watch the orange (warming) bump race east and the blue (cooling) signal follow months later.
Try this: press New wind burst and watch the day counter in the tag.
A surface wave on the ocean feels full gravity and runs at √(gH), about in deep water. The thermocline separates water that differs in density by only about 0.4%, so the restoring force is that much weaker. The bump crawls along at a couple of meters per second and takes months to cross the Pacific.
Coriolis deflection flips sign across the equator, and that traps the wave within a few hundred kilometers of it. That width is √(c/β), with β = 2Ω/R = 2.29×10⁻¹¹ m⁻¹s⁻¹.
Equatorial Kelvin waves only travel east. The reason is that Coriolis deflection changes sign at the equator, so the equator acts as a guide that works in one direction. In 2017 Delplace, Marston and Venaille showed in Science that this one-way behavior follows from topology, the same mathematics used for edge states in topological insulators. Their argument predicts exactly two eastward-only modes: the Kelvin wave and the Yanai wave.
At the coast of South America part of the Kelvin wave reflects as Rossby waves and part runs poleward along the coast. This toy leaves those out.
Combine the two labs. Bjerknes feedback makes the eastern Pacific temperature T grow. The waves bring back a signal of the opposite sign, but only after a delay δ. Suarez and Schopf (1988) wrote this as one equation, the delayed oscillator. The page integrates it as you watch. Everything here is nondimensional, so T is an index and time is in model units.
How to read this: top, the green line is how warm the east Pacific is, scrolling left as time passes. The gold dot marks the value one delay ago, and that old value is what pulls the line back the other way. Bottom, how long one full cycle takes for each delay.
Try this: drag Delay δ below about 1.5 and watch the swings stop.
T is the Bjerknes feedback: warmth makes more warmth. −T³ caps the growth, standing in for limits like the thermocline reaching its flattest. −α·T(t−δ) is the wave signal from lab 2. It reports what T was a time δ ago, with the opposite sign.
For the real index, NOAA calls an El Niño when the three-month average Niño-3.4 sea surface temperature anomaly is at least +0.5 °C for five overlapping seasons, and La Niña at −0.5 °C or below. The toy T has no units, so its ±0.5 bands only borrow that convention.
Real events recur every 2 to 7 years. Wave delays of a few months give periods that are too short, so the toy captures the mechanism and misses the timing. Noise stands in for weather such as westerly wind bursts, which makes each cycle different.