“They aren’t something you can walk up to and touch,” Jerrold E. Marsden, an engineering and mathematics professor at Caltech, said of the structures. “But they are not purely mathematical constructions, either.”

As an analogy, Dr. Marsden suggests imagining a line that divides a part of a city that has been affected by a disease outbreak from a part that has not. The line is not a fence or a road, but it still marks a physical barrier. And as the outbreak spreads, the line will change.

To find the structures, scientists must track flow, not by watching it go by but from the perspective of the droplets of water or molecules of air moving in it. “It’s like being a surfer,” Dr. Campbell said. “You want to catch the wave and move with the wave.”

Image FLUID MOVEMENT Sensors near Santa Cruz, Calif., take surface current measurements in Monterey Bay. Credit... Francois Lekien/Université Libre de Bruxelles

In the laboratory, researchers shine lasers on tiny particles caught in a flow, capturing their speed and trajectory with fast, high-resolution digital cameras similar to the way tracer rounds from machine guns track the path of bullets. In the ocean or atmosphere, scientists rely on instantaneous data from high-frequency radar, laser detection systems, buoys and satellites. In the human body, phase-contrast magnetic resonance imaging has helped researchers map the complex patterns of blood flow in detail. Computers take in the data from all those sources, applying algorithms that unveil the flow structures.

“We’re just recognizing that these things exist and are playing a role in a variety of scenarios,” said Thomas Peacock, a mechanical engineering professor at M.I.T. who is evaluating how Lagrangian coherent structures affect vehicle performance and efficiency. “The idea is that cars, airplanes and submarines down the line would be fitted with sensors that will help them adapt to these structures.”

Studies of the air flow patterns surrounding Hong Kong International Airport have shown that Lagrangian coherent structures cause unexpected jolts to planes during landing attempts, forcing pilots to waste fuel while they revert to holding patterns. George Haller, an engineering professor at McGill University in Montreal who forged the mathematical criteria for finding such structures in fluid flows, is working with the airport’s officials to design a tool that allows pilots to see and navigate around the structures. It will rely on data from laser scans, analyzed by computers as planes approach the airport.