The Hidden Math of Ocean Waves

"The best perk of Alberto Maspero 's job, he says, is the view from his window. Situated on a hill above the ancient port city of Trieste, Italy, his office at the International School for Advanced Studies overlooks a broad bay at the northern tip of the Adriatic Sea. "It's very inspiring," the mathematician said. "For sure the most beautiful view I've ever had.""

"Italians call Trieste la città della bora, after its famed "bora" wind, which blows erratically down off the Alps and over the city. When the bora is strong enough, it drives the waves into reverse. Instead of breaking against the docks, they stream away from the city, back toward the open sea. But they never actually get there. Watching from his window on these gusty days, Maspero can see the retreating waves slowly disperse as they exit the port, eventually giving way to a calm, still surface."

"The equations that mathematicians use to study the flow of water and other fluids-which Leonhard Euler first wrote down nearly 300 years ago-look simple enough. If you know the location and velocity of each droplet of water, and simplify the math by assuming there's no internal friction, or viscosity, then solving Euler's equations will allow you to predict how the water will evolve over any time period. The rich menagerie of phenomena we see in the world's oceans-tsunamis, whirlpools, riptides-are all solutions to Euler's equations."

A team of Italian mathematicians made significant advances in understanding even the simplest ocean waves. Observations in Trieste illustrate how the bora wind can reverse waves, which then disperse and leave a calm surface. Euler's equations, formulated nearly 300 years ago, formally govern fluid flow but are usually impossible to solve in practice. Knowing droplet positions and velocities and neglecting viscosity would allow exact prediction in principle, yet steady trains of gently rolling waves remain mathematically challenging to extract from those equations. These challenges motivate deeper analytical work and recent mathematical breakthroughs.

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