Thomas Peacock

THE connection between an 18th-century savant called Joseph-Louis Lagrange and the problem of landing safely at Hong Kong International Airport may not, at first, be obvious. But there is one. Hong Kong airport is notorious for rocky and sometimes aborted landings caused by the disturbed air flow from nearby mountains. Though laser technology is deployed alongside its runways to monitor changes in wind speed and thus forewarn pilots, that is often not enough. What is needed is a better understanding of the theory of the winds themselves.

And this is where Lagrange comes in. He was a pioneer of the study of moving fluids (among many other things), but his ideas outran the computational tools of his day. Only now, with supercomputers available to help with the calculations, is it possible to explore those ideas completely. What is emerging is a picture of fluid dynamics more subtle and more complex than anything dreamed of even a decade ago. The atmosphere and the ocean are, it seems, dominated by invisible barriers that have come to be known as Lagrangian coherent structures. They govern the movement of everything from the trajectories of aircraft to the distribution of pollution, the migration of jellyfish and the tracks taken by hurricanes. They are, as it were, the skeletons of the sea and the air.

To understand what a Lagrangian coherent structure is, it helps to imagine a crowd at a railway terminus, says Thomas Peacock of the Massachusetts Institute of Technology, who studies the structures. Some people will be arriving. Some will be leaving. And, whichever they are doing, they will be going to and from numerous different platforms. The result is chaos, but structured chaos. What emerges is a shifting pattern of borders between groups of people with different goals. These borders are Lagrangian coherent structures. They are intangible, immaterial and would be undetectable if the passengers stopped moving. But they are also real enough to be treated mathematically.

Crucially, they form a barrier to exchange between groups, as anyone who has been caught up in a crowd and pulled inexorably in the wrong direction will know. Substitute unreasoning air or water molecules for sentient humans and you have a chaotic fluid. And it turns out to be easier to study the behaviour of such fluids by looking at the barriers than at the bodies of fluid which those barriers separate.

A coherent approach

The theory of Lagrangian coherent structures was started about ten years ago by George Haller, who was then at Brown University, in Rhode Island. He had been working in the fashionable mathematical field of chaos theory—an explanation in search of a problem—when he came across a suitable problem in the form of mountains of data accumulated by various ocean and atmospheric scientists of his acquaintance.

At the moment Dr Haller, who is now at McGill University in Montreal, is collaborating with Wenbo Tang of Arizona State University and Pak Wai Chan of the Hong Kong Observatory to apply Lagrangian theory to the problem of aircraft taking off and landing in difficult circumstances. The lasers by the runways act a bit like radars. They cannot see air molecules directly, but they are able to track aerosol particles that are tossed around by the currents. The Doppler effect (familiar from the changing pitch of a passing police siren) gives away the velocities of these particles and Dr Haller's computers are able to crunch the resulting data in the light of Lagrange's theories and work out the network of coherent structures that is causing their movements—and the pattern of updraughts and downdraughts that should result.

The initial predictions from the crunching compare well with the actual movement up or down of planes coming in to land at Hong Kong—and, crucially, the system highlights areas of turbulence that traditional methods miss. If all continues to go well, the results will be plugged into the airport's air-traffic control system for real within a few months.

Lagrangian coherent structures should also prove useful at sea. François Lekien of the Free University of Brussels, in Belgium, and his colleagues have used radar measurements to do for the ocean what Dr Haller's lasers do for the air. In this case, the purpose is to predict the fate of pollutants.

Dr Lekien's radars are pointed at the waters of Monterey Bay, in California. Again, the Doppler shift is the telltale. It allows him (or, rather, his computer) to see the ever-shifting Lagrangian pattern in the bay. That, in turn, predicts at any given moment which regions of the bay will wash pollutants out to sea and which will confine them to the coast. This information should help the authorities work out when it is safe to permit operations that risk causing leakage, such as repairs to shore-based oil tanks.

Similar projects are also being carried out near Fort Lauderdale in Florida and close to Brest on the west coast of France—a stretch of coastline that is notoriously dangerous for oil tankers. Lagrangian analysis might allow the behaviour of spills to be predicted more easily. Analysing the sea in this way might prove useful, too, in rescuing people whose boats have broken down or in locating the wreckage of aircraft that have crashed in mid-ocean.

It should also please zoologists. John Dabiri of the California Institute of Technology, for example, has used the technique to study the hunting behaviour of jellyfish. He has shown how parts of the ocean are temporarily protected from the depredations of these creatures because they cannot cross the invisible barriers imposed by Lagrange. It sounds esoteric, but America's Office of Naval Research is using the results to help design better unmanned submarines.

And that is not all. Lagrangian coherent structures can appear at all sorts of scales. What goes for an airport or a bay can be scaled up to an ocean, or the air above it, and down to the flow off an aircraft's wingtip, or a ship's hull. As a consequence Jerry Marsden, one of Dr Dabiri's colleagues at Caltech, thinks the coherent-structure approach might also be used to help predictions of the passage of hurricanes; they, too, are constrained by the invisible barriers that Lagrange's theory describes. Dr Marsden reckons the technique might even improve understanding of how water from melting ice caps will affect ocean currents, including the Gulf Stream. Dr Peacock, meanwhile, concentrates on what it can say about aerodynamic drag on cars and lift under aircraft wings.

The upshot is that Dr Haller and his disciples have created a powerful new way of looking at the world. They have shown that the boundaries between things are often as important as the things themselves. And they have issued a reminder that even the most abstruse and arcane branches of mathematics can sometimes be deeply pragmatic, too.