# Mathematicians Prove Universal Law of Turbulence, Hacker News

Mixing liquids and other turbulent systems have long been observed to follow a universal rule known as Batchelor’s law. Researchers have finally proved it mathematically.

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.

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.

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 2019, 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 co-author 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.”

In the same way that the velocity varies from point to point in the churning sink, the concentration of black paint will vary from point to point within the mixing paint: more concentrated in some places (the thicker sinews) and less in others.

This variation is an example of “passive scalar turbulence.” You can think of it as what happens when you mix one fluid, considered the “passive scalar,” into another – milk into coffee, say, or black paint into white.

Passive scalar turbulence also characterizes many phenomena in the natural world, like the dramatic temperature variations between nearby points in the ocean. In that environment, the ocean currents “mix” temperatures the way stirring mixes black paint into white.

Batchelor’s law is a prediction about the ratio of large-scale phenomena (thick tendrils of paint, or thick bands of ocean water at the same temperature) to phenomena at smaller scales (thinner tendrils) when a fluid is mixed into another. It’s referred to as a law because physicists have observed it in experiments for years.

“From a physics standpoint that’s good enough to call it a law,” said Punshon-Smith, a mathematician at Brown University. But prior to this work there’s been no mathematical confirmation that it holds absolutely.

To get a sense of what Batchelor had in mind, return to the paint. Imagine you’ve run the process for a while, adding drops of black paint as you stir. Now freeze the picture. You’ll see thick tendrils of black paint (paint that’s been stirred for the least amount of time), along with thinner tendrils (paint that’s been stirred longer) and even thinner tendrils (paint that’s been stirred even longer).

Batchelor’s law predicts that the number of thick tendrils, thinner tendrils and thinnest tendrils conforms to an exact ratio – similar to the way the nested figurines that comprise a Russian doll follow an exact ratio (in that case, one figurine per length scale).

“In a given patch of fluid, I’ll see stripes of different scales because some droplets have barely begun to mix, while others have been mixing for a while,” Blumenthal said. “Batchelor’s law tells you the distribution of the sizes of those stripes of black paint.” The exact ratio it predicts is complicated to describe, but thinner tendrils will be more numerous than thicker tendrils in an exact proportion.

The law predicts that the ratio holds even as you zoom in on a patch of fluid. You’ll see exactly the same relationship between tendrils of different sizes in the paint can and in a small patch of paint; if you zoom in more, on an even smaller patch, you’ll still see it. The pattern looks the same at every scale, just as it does in hydrodynamic turbulence, where each vortex contains other vortices.

It’s a powerful prediction that’s also very hard to model mathematically. The complicated nesting of phenomena at different length scales makes it impossible to exactly describe the emergence of Batchelor’s law in a single fluid flow.

But the authors of the new work figured out how to get around this difficulty and prove the law anyway.