Following the ocean swirls

The mathematics of dynamical systems reveals ocean dynamics, an understanding that could improve the monitoring of ocean processes.

Climate is an intimate dance between ocean and atmosphere. The oceans are a dark partner in this dance, though, much less understood than the air moving above them. Mathematical techniques from the study of dynamical systems are helping illuminate its movements.

Eddies swirl around in the Southern Ocean, between the tip of South America and Antarctica. These currents are key to the mixing of air and water.

The big question is just how much carbon dioxide the ocean absorbs. Though the precise value is unknown, what is known is that the ocean absorbs a lot, perhaps as much carbon as plants on land absorb. Climate scientists especially need to know whether that amount is going up, going down or staying the same.

Oceanographers understand the big picture pretty well: Great currents driven by wind and ocean mixing suck deep water from the North Atlantic southward toward Antarctica, where it rises to the surface and encounters open air. While in contact with the air, it exchanges carbon dioxide with the air before it is pumped back below the surface to return north.

The hard question is just how much carbon the water absorbs or gives off before plunging back to the depths, and that depends on how much it’s stirred up while at the surface. It’s just like when a sugar cube dissolves in coffee: It happens much faster when stirred. So the more the eddies in the water whirl and dance, the more heat and carbon gets distributed evenly throughout the ocean.

Furthermore, the mixing itself influences the strength of the current, helping to determine how fast the water moves toward the Antarctic and then back up to the Equator.

Oceanographers simply don’t have the data they need to get that kind of relatively fine-scale view of the oceans. Ships take some measurements of the ocean, and satellites can measure crude ocean statistics like sea-surface height. About 1,250 floating weather stations drift around the oceans, reporting their data to satellites. But that leaves huge portions of the ocean largely unobserved, and there aren’t the resources to stud the oceans with weather stations every few miles.

But recently, mathematicians have figured out how to make a small number of floating weather stations generate far more information. Oceanographers have tended to spread floats evenly around the ocean. “After all, if you place a bunch nearby each other and they always stay near each other, you might as well have placed just one,” says Chris Jones, a mathematician at the University of North Carolina at Chapel Hill. “But the smart thing to do is to place them where the flow field will distribute them, so they start close together and then disperse quite far.”

That will happen, Jones and his colleague Kayo Ide of the University of California, Los Angeles have realized, if the floaters are tossed into the ocean at the “hyperbolic points,” the spots that are like saddle points between the eddies. Then the floats will drift apart, spreading all around the edge of the eddy. The collection of floats will then show the full regions they can cover — what dynamical systems experts call the “unstable manifold.”

Satellite readings of the height of the sea surface give a rough picture of the eddies in the ocean and allow oceanographers to identify about where the hyperbolic points are at a particular moment, guiding the placement of the floaters. The eddies move around the ocean over time, causing the floaters placed at the hyperbolic points to trace out the crazy zigzag patterns of the unstable manifolds. The mathematicians have developed tools allowing oceanographers to compute the degree of mixing directly from those shapes.

“The geometric patterns formed by the eddies show you the mixing structure,” says Emily Shuckburgh of the British Antarctic Survey. Shuckburgh and her colleagues are now preparing for a trip to test out the technique.

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