THERE’S something lurking in the waters of Monterey Bay, off the California coast. It attaches itself to the southern point of the bay, wends its way almost 40 kilometres northward towards Santa Cruz, then meanders south-east before disappearing into the depths of an underwater canyon at the centre of the bay. It’s not a living creature and it’s not even visible to the eye, yet it can mean life or death to the fish and other wildlife that inhabit this marine sanctuary.
So what is it? It’s a moving wall of water that effectively controls which currents will flow out to the open sea, and which will loop back into the bay. For marine biologists, this divide is all-important because it controls the flow of pollutants as well as water. Near the middle of the bay, a power plant discharges warm waste water that can raise the average temperature of the bay enough to kill some native fish. Depending which side of the effluent pipe the divide lies on, the waste water will either flow out to sea – where its effect will be negligible – or recirculate into the bay, heating it up.
Mathematicians and physicists describe this wall of water as a “Lagrangian coherent structure”. This insight allows them to model the complex flow of water in the bay and suggest a strategy for minimising the damaging effects of the effluent from the power station. It is a striking example of how researchers are gaining a practical handle on turbulence, one of the most intractable phenomena in physics.
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Turbulent flow governs many everyday processes – the way your milk and coffee mix and which direction smoke blows, for instance. It can also cause serious problems, such as when an aircraft hits a rough patch of air. Until recently, however, no one had managed to create a predictable model of turbulence in the real world. Now coherent structures are allowing researchers to solve a raft of problems in oceanography, aviation, cardiology and other areas.
For the better part of the 20th century, turbulence was thought to be essentially random and, by definition, unpredictable. There was no problem modelling smooth, “laminar” flow – in which adjacent particles in a fluid always go in roughly the same direction – but getting to grips with fast, turbulent flow was a different story. In a rapidly moving stream, for instance, some particles will zip along in the fastest channel while others, even if they start as neighbouring particles, may get caught in an eddy or swept into a lazier part of the stream. “Turbulence was hopelessly complex,” says George Haller, a mathematician at the Massachusetts Institute of Technology.
That picture is now changing, thanks to Lagrangian coherent structures. Predicting where a particle of fluid will go is actually easy if you know which side of the coherent structure it’s on in the first place. The tricky bit is pinpointing the structure itself – generally it is a convoluted object – and tracking it in real time.
Stretch it like taffy
An added complication is that there are two kinds of these coherent structures. One type – the most familiar example being a smoke ring – attracts nearby particles. The structure itself is invisible, but it becomes visible to the naked eye when smoke particles congregate around it. The second type is much more elusive: a curve that repels nearby particles. Obviously, the smoke-ring trick will not work for visualising these structures. Yet they seem to be just as important as attracting structures such as rings and eddies. “These [repelling] structures do all the stirring up,” Haller says. They apportion the fluid to one attracting region or another, while the attractors suck the fluid up and then stretch it out like taffy.
Although coherent structures were shown to exist in theory in the late 1960s, nobody had actually demonstrated their physical existence. So Haller made it his mission to show that coherent structures were real – to unveil what he calls “the skeleton of turbulence”. Tracking the motion of particles in 3D would be too difficult, though, so he first had to find a case of 2D turbulence. That’s hard to do; if you agitate water, it likes to flow in 3D.
In 2005, Haller found his dream experiment. Harry Swinney, a physicist at the University of Texas, Austin, had built a half-metre-tall cylindrical tank that rotates about as fast as a DJ’s record deck, with holes in the bottom through which water can be injected and drained. Although the flow at the bottom of the tank is complicated and 3D, at the top it is as smooth as a stack of pancakes. Within each pancake, however, the water flows in complex patterns: its velocity points in all directions and changes over time in a way that reveals no pattern. For all practical purposes, this is .
Haller and Swinney were in business. They injected fluorescent polystyrene balls, each about the size of a fine grain of sand, into the water. Using a laser, they could track the position of each particle over time, and so measure the velocities of individual particles. Haller then fed the measured positions into a fluid-dynamics computer model of the flow, which interpolated between the data points and allowed him to add as many virtual particles as he wanted.
This turned out to be very convenient, because of a remarkable property of coherent structures: if you reverse time, a repelling structure becomes attracting, and that means you can find it using the smoke-ring trick. So Haller injected virtual particles at the end of the simulation, ran the equations backwards for 8 seconds – and out popped a beautiful coherent structure, winding around like a strand of spaghetti. In fact, the team produced images of both attracting and repelling structures, looping around and interweaving (). Their approach is “nothing short of revolutionary”, says Jerrold Marsden, a dynamical systems theorist at the California Institute of Technology in Pasadena.
The experiment was the first clear demonstration of the underlying structure of turbulence, raising the possibility of coherent structures being used to make predictions in the real world. A drawback, though, was that Haller’s set-up took an hour of computing time just to process 8 seconds of data. What’s more, an ocean is nothing like an idealised 2D fluid: in addition to water flowing horizontally across the surface, there are also vertical upwellings and downwellings.
This is where Monterey Bay comes in. It is one of the most thoroughly studied bodies of water in the world and is very rich in sea life, as there is a 2-kilometre-deep canyon in the centre of the bay which creates a source for upwellings of cold water that bring nutrients to the surface. Four high-frequency radar stations located around the bay – one at the northern end in Santa Cruz, one in Moss Landing near the power plant, one at the Naval Postgraduate School in Monterey, and one at Pinos Point at the southern end of the bay – enable near real-time estimates of the water velocity to be made.
The set-up was the perfect opportunity for Haller. In 2003, researchers from Princeton University had used radar to plot the first rough images of Lagrangian coherent structures in the bay, including the repelling structure that controls the flow of pollutants (see Diagram). “That structure marks the boundary between recirculation and quick escape to the ocean,” says physicist Francois Lekien, who was part of the team at Princeton. “It gives us an indicator of which way [particles] are going to go beforehand.”
As the coherent structure meanders around the bay, it occasionally passes from one side of the power plant’s effluent pipe to the other, sometimes leaving the pipe on the inland side, and at other times on the seaward side. Using the radar data taken in 2003, Lekien and Haller’s group found that the dividing line between outward-flowing and recirculating water typically spends about four days on each side of the pipe before crossing over. This means the ideal pollution-control strategy would involve holding the hot effluent in a storage tank until the structure crosses over to the desired side, when the waste can be released and flow out to sea.
Torpedo bots
For this, the turbulent backbone would need to be tracked in real time. Lekien had the chance to try this in a more ambitious follow-up study. In August, a consortium of researchers led by oceanographer Steven Ramp of the Naval Postgraduate School in Monterey and mechanical engineer Naomi Leonard of Princeton released a fleet of torpedo-shaped robotic vehicles into the bay. The robots provided data on where particles in different regions of the water would go. The idea was to use a model of ocean currents to figure out the best place to send the robots so as to track the coherent structures accurately, says Lekien, now at the Free University of Brussels (ULB) in Belgium.
To find the repelling structure using Haller’s approach, however, you would need information from the future, so that you can run the model backwards in time. That is impossible, of course. Instead, Lekien did the next best thing. He fed the current data into the model, ran it backwards and figured out where the structure was 8 hours earlier. That gave him a rough idea of where it would currently be.
This method is not perfectly accurate, but according to the team’s estimates it is good enough for a timed-release pollution-control strategy to work. Such a strategy could potentially halve the peak concentration of waste water in the bay, says Lekien. “That would save a lot of species.”
Although there are no plans yet to implement the timed-release approach for pollution control – it would require quite a sophisticated set-up – the experiment in Monterey Bay shows , and what might be possible in the future.
“The experiment in Monterey Bay shows what kinds of effects of ocean turbulence can be predicted”
Meanwhile, researchers are also working on detecting Lagrangian coherent structures in other settings. Haller has a grant from the US air force to study whether aircraft can use LIDAR – a type of radar that employs light instead of radio waves – to scan the skies for clear-air turbulence. These are spontaneous and potentially damaging jets of air that cannot be detected on normal radar because they aren’t associated with storm systems. “Boundaries of clear air turbulent regions are delineated by very strong Lagrangian coherent structures,” says Haller. The main obstacle to using on-board LIDAR is that it can measure only one component of the air’s velocity – towards or away from the aircraft. In practice, velocity measurements in 3D will be needed to reveal the coherent structure.
Another field that could benefit greatly is biomedicine. Shawn Shadden, a postdoctoral researcher at Stanford University in California, is helping cardiologist Charles Taylor map out the 3D blood flow in human arteries. Shadden is looking for places where a Lagrangian coherent structure separates from the blood vessel wall. He has focused on the carotid bifurcation, where the carotid artery splits into two – one branch going to the brain, the other to the face and neck. The internal carotid artery, which supplies the brain, often develops a vortex of recirculating blood. This region of stagnant flow is believed to be a likely spot for atherosclerotic plaques to form, and perhaps even to be the trigger for their formation.
Imaging techniques in 3D now make it possible to model blood flow through the carotid bifurcation, though the procedure isn’t yet ready for clinical use. Cardiologists previously believed that the pulsation of the blood flow in the carotid artery would break up the stagnation zone. Shadden and Taylor have shown that it doesn’t; the coherent structure remains intact through an entire heartbeat.
Lagrangian coherent structures have shed light on the nature of turbulence, allowing researchers to predict where particles in a turbulent fluid will flow in some practical cases. Marsden seems impressed by the scope of the work. “Which particles will recirculate in Monterey Bay, and which will flow on by and go south? Which blood cells will recirculate in a blood vessel and lead to cardiac disease, and which will flow smoothly to their intended destination?” he asks. “Lagrangian structures let one determine the answers to such clearly important questions.”
