In the last few years, we have seen some major natural disasters which have claimed thousands of lives and destroyed millions of dollars worth of infrastructure. Ocean currents and other chaotic phenomena are supposed to be inherently unpredictable. But mathematicians are finding a method to nature’s madness.
In reality, the situation was never so dire—especially on the western coast of Florida. This part of the Gulf Coast was protected for the duration of the oil spill by a persistent, invisible divide. Lying above the continental shelf off of Florida was an unseen line that directed the oil and prevented it from spreading farther east. It was not a solid object, but a wall of water that moved around as ocean currents shifted. Nevertheless, this wall was just as effective as any seawall or containment boom.
Scientists call these invisible walls “transport barriers,” and they are the maritime equivalent of continental divides. They separate water flowing in one direction from water flowing in another. In a chaotic ocean, they provide a road map to tell you where the traffic is going. Although water currents often appear to be almost completely unpredictable, transport barriers restore a measure of order and structure to their chaotic flow.
The study of these structures has blossomed in recent years, and their importance is still not fully appreciated by the scientific community. But already researchers have shown how their study may help explain why the surface oil from the Gulf spill disappeared more rapidly than expected and why none of it escaped through the Strait of Florida into the Atlantic. During future disasters, understanding these flows could make cleanup efforts more efficient. The research could also elucidate how blood flow affects the formation of plaques in arteries and help to predict how allergy-causing spores migrate in the atmosphere.
The study of chaos came of age in the 1970s, when scientists discovered that in certain natural phenomena, even tiny perturbations could lead to profound changes. The proverbial refrain is that the flutter of a butterﬂy’s wing on one side of the globe could make subtle changes in air currents that cascaded, to the point of causing a tornado on the other side weeks later. Flowing ﬂuids—which include gases such as air and liquids such as seawater—are in fact the quintessential example of chaotic systems and one of the most ubiquitous: the dynamics of ﬂuids govern phenomena from the Gulf Stream to the ﬂow of air through a wind turbine to curving penalty kicks in soccer.
The mathematical equations describing ﬂuid ﬂow were unveiled nearly 200 years ago, by Claude-Louis Navier (in 1822) and George Stokes (in 1842). Yet knowing the equations is not the same thing as solving them, and the Navier-Stokes equations remain among the most challenging problems in mathematics. In principle, an exact solution of the Navier-Stokes equations would yield a detailed prediction of the future behavior of a ﬂuid. But the precision of the answer would depend on exact knowledge of the present—or what scientists call the initial conditions.
In reality, you can never know where every molecule of water in the ocean is going, and in a chaotic system any uncertainties—like the effects of a butterﬂy’s motions—grow exponentially over time. Your exact solution to the Navier-Stokes equations will rapidly become moot. And yet “chaotic” does not mean “random” or “unpredictable,” at least in principle. In the past decade or so mathematicians have created a theoretical framework for understanding the persistent structures such as transport barriers that are hidden in chaotic fluids. In 2001 George Haller, a mathematician now at McGill University, gave these structures the rather unwieldy name “Lagrangian coherent structures.”
More poetically, Haller calls the intricate structure of transport barriers “the skeleton of turbulence.” Once you have identiﬁed these structures in a body of ﬂuid, you can make useful short- to medium-term predictions of where the ﬂuid ﬂow will carry an object, for instance, even without a perfect, precise solution of the Navier-Stokes equations. What does a transport barrier look like? You are looking at one every time you see a smoke ring. At its core lies an attracting Lagrangian coherent structure—a curve toward which particles ﬂow as if they were attracted by a magnet. Ordinarily you cannot see such a structure, but if you blow smoke into the air, the smoke particles will concentrate around it and make it visible.
Much harder to visualize are the repelling Lagrangian coherent structures—curves that, if they were visible, would appear as if they were pushing particles away. If you could run time backward, they would be easier to see (because they would attract particles); failing that, the only way to ﬁnd them is to tease them out by computer analysis. Though difficult to observe, repelling structures are particularly important because, as Haller has proved mathematically, they tend to form transport barriers.
An experiment conducted in the summer of 2003 in Monterey Bay off the coast of California showed that Lagrangian coherent structures could be computed in real time and in real bodies of water. Mathematician Shawn C. Shadden of the Illinois Institute of Technology and his collaborators monitored surface currents in the bay using four high-frequency radar stations deployed around the bay. Analyzing the radar data, the researchers discovered that most of the time a long transport barrier snakes across the bay from Point Pinos, at the southern edge, almost all the way to the northern side. Waters to the east of the barrier circulate back into the bay, whereas those on the western side go out to sea. (Occasionally the barrier detaches from Point Pinos and drifts farther out to sea.)
Such information could be vital in case of a pollutant spill. To conﬁrm that the computed structures did actually behave as advertised, Shadden’s team tracked the motion of four drifting buoys they deployed in collaboration with the Monterey Bay Aquarium Research Institute. When they placed drifters on opposite sides of the transport barrier, one drifter would follow the water circulating back into the bay, and the other one would hitch a ride on the currents heading southward along the coast. They also showed that a drifter placed on the recirculating side of the structure would stay in the bay for 16 days— even though they had used only three days of data to compute it. This robustness of their results testiﬁed to the strength and persistence of the transport barrier. For 16 days, it really was like an invisible wall in the water.
- Scientific American Magazine, Dana Mackenzie