The Universal Law of Turbulence Isn’t So Unruly After All

Picture a calm river. Now picture a torrent of white water. What is the difference between the two? To mathematicians and physicists it’s this: The smooth river flows in one direction, while the torrent flows in many different directions at once.

Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research develop­ments and trends in mathe­matics and the physical and life sciences.

Physical systems with this kind of haphazard motion are called turbulent. The fact that their motion unfolds in so many different ways at once makes them difficult to study mathematically. Generations of mathematicians will likely come and go before researchers are able to describe a roaring river in exact mathematical statements.

But a new proof finds that while certain turbulent systems appear unruly, they actually conform to a simple universal law. The work is one of the most rigorous descriptions of turbulence ever to emerge from mathematics. And it arises from a novel set of methods that are themselves changing how researchers study this heretofore untamable phenomenon.

“It may well be the most promising approach to turbulence,” said Vladimir Sverak, a mathematician at the University of Minnesota and an expert in the study of turbulence.

The new work provides a way of describing patterns in moving liquids. These patterns are evident in the rapid temperature variations between nearby points in the ocean and the frenetic, stylized way that white and black paint mix together. In 1959, an Australian mathematician named George Batchelor predicted that these patterns follow an exact, regimented order. The new proof validates the truth of “Batchelor’s law,” as the prediction came to be known.

“We see Batchelor’s law all over the place,” said Jacob Bedrossian, a mathematician at the University of Maryland, College Park and coauthor of the proof with Alex Blumenthal and Samuel Punshon-Smith. “By proving this law, we get a better understanding of just how universal it is.”

Turbulence All the Way Down

While the white waters of a choppy river aren’t the exact kind of turbulence at issue in the new proof, they are closely related and more familiar. So it’s worth thinking about them for a moment before turning to the specific kind of turbulence the mathematicians analyzed.

Picture a kitchen sink full of water. Open the drain. The water in the sink will start to rotate nearly as a single body. If you zoomed in on the fluid and measured its velocity at finer scales, you’d still observe the same thing — each microscopic portion of the fluid moves in lockstep with the others.

“The motion is predominantly at the scale of the sink itself,” said Blumenthal, a postdoctoral fellow also at the University of Maryland, College Park.

Now imagine that instead of merely draining the water, you pulled the plug while also adding water jets to the sink, churning it like a jacuzzi. With the naked eye, you might observe a handful of different vortices rotating in the water. Choose one of the vortices and zoom in on it. If you were a mathematician trying to analyze the flow of the turbulent sink, you might hope that every particle of water within that chosen vortex was moving in the same direction. This would make the task of modeling the fluid easier.

But alas, you’d find instead that the vortex is itself made up of many different vortices, each moving its own way. Zoom in on one of those and you’ll see that it, too, is made up of many different vortices, and so on all the way down, until the effects of internal friction (or viscosity) within the fluid take over and the flow smooths out.

This is a hallmark of turbulent systems — they feature distinct behaviors nested within each other at different scales. In order to fully describe the motion of a turbulent system, you need a picture of what’s going on at all of these scales at each moment in time. You can’t ignore any of them.

That’s a tall order, akin to modeling the trajectory of billiard balls using everything from Earth’s motion through the galaxy down to the interactions between gas molecules around the balls.

“I have to take it all at once, which is what makes it incredibly difficult to model,” said Jean-Luc Thiffeault of the University of Wisconsin, who studies turbulence.



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