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Mathematicians have proved the universal law of turbulence

Using random processes, three mathematicians have proved the elegant law underlying the chaotic motion of turbulent systems




Imagine a calm river. Now imagine a fast stream of foaming water. What is the difference between them? For mathematicians and physicists, it consists in the fact that a calm river flows in one direction, and a stormy stream flows in several directions at once.

Physical systems with such unsystematic motion are called turbulent . Due to the fact that their motion has so many characteristics at the same time, it is very difficult to study them mathematically. More than one generation of mathematicians will change until researchers learn to describe a turbulent river with exact mathematical expressions.

However new evidencesays that although some turbulent systems seem rebellious, in fact they obey one universal law. This paper provides one of the most rigorous descriptions of turbulence ever given by mathematics. And it appears thanks to a new set of methods that by themselves change the process of researchers studying this hitherto disobedient phenomenon.

“Perhaps this is the most promising approach to turbulence,” said Vladimir Sverak , a mathematician at the University of Minnesota, an expert on turbulence.

The new work provides a way to describe the patterns that arise in moving fluids. They can be clearly seen on the example of sharp fluctuations in temperature at neighboring points of the oceans or mesmerizing pictures obtained by mixing black and white colors. In 1959, the Australian mathematician George Batchelor predicted that these patterns have an accurate and regulated behavior. New evidence confirms the truth of the “Batchelor's Law,” as this prediction was called.

“Batchelor’s law can be seen everywhere,” said Jacob Bedrossian, a mathematician at the University of Maryland at College Park, co-author of the proof with Alex Blumenthal and Samuel Panshon Smith . "By proving this law, we were better able to realize its universality."

Turbulence from top to bottom


And although the new evidence does not describe exactly the same processes that occur in the turbulent course of the river, they are closely related to them and are quite familiar to us. Therefore, let us first imagine them before moving on to the special type of turbulence that mathematicians have analyzed.

Imagine a kitchen sink full of water. Water begins to rotate in the sink almost as a single mass. If we increase the liquid and measure its speed on a smaller scale, we will see the same thing - each microscopic part of the liquid moves in accordance with the others.

“The movement is mostly tied to the scale of the entire conch,” said Blumenthal, a postdoc from the University of Maryland at College Park.


Alex Blumenthal, a postdoc from Maryland University College Park

Now imagine that instead of just letting the water drain by pulling out the cork, you added water jets to the sink, spinning it like in a jacuzzi. With the naked eye you can catch a lot of whirlpools that appear in the water. Choose one of them and increase its scale. If you were a mathematician trying to analyze turbulent shell flows, you could hope that each particle of water in the selected whirlpool moves in the same direction. This would greatly facilitate fluid modeling work.

But, alas, you will find that the whirlpool itself consists of many small whirlpools, each of which moves in a special way. Enlarge its image, and you will again see that it, in turn, consists of various whirlpools, and so on, up to the smallest scale, until the effects of internal friction (or viscosity) of the liquid take up and smooth out the flows.

This is a clear sign of turbulent systems - different behavior of subsystems embedded in one another at different scales. To fully describe the motion of a turbulent system, it is necessary to describe what is happening on all these scales at any given time. None of them can be ignored.

This is a serious requirement - it is similar to modeling the trajectories of the movement of billiard balls, taking into account absolutely everything, from the movement of the Earth through the Galaxy, to the interaction of gas molecules with balls.

“I had to take into account everything at once, which makes this task so incredibly difficult to model,” said Jean-Luc Tiffo of the University of Wisconsin, who studies turbulence.

As a result, mathematicians have been trying for decades to create a description of turbulence that accurately describes what is happening at every point of the turbulent system at any given time. And did not succeed.

“Turbulence is too complex to attack it in the forehead,” Tiffo said. This is true for turbulent rivers and sinks with leaking fluid. This is also true for the special version of turbulence used in the new proof.

Stirring


Shell and river are examples of hydrodynamic turbulence. They are turbulent in the sense that the vectors of fluid velocity — the directions and velocities of particles — vary greatly from point to point. The new work describes other properties of the liquid except the velocity vectors that can be measured at each of its points. To understand what this means, imagine a mix of colors.

Let's start with a can of white paint. We will add black one drop per second, stirring the paint. The first drop will fall into white paint and will stand out like an island. But soon it will begin to dissolve in white paint, stretching into increasingly thin lines. Subsequent drops of black paint will be at different stages of the same transformation: stretch, lengthen, pour into the paint, which gradually turns into gray.

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As the velocity vectors change from point to point in the sink where the water is mixed, so the concentration of black paint in white will change from point to point with mixing: in some in places its concentration will be greater (thicker lines), in some less.

This option is an example of “passive scalar turbulence”. It occurs when one fluid, a “passive scalar” is injected, and milk is added to the other in coffee, black paint in white.

Passive scalar turbulence also describes many natural phenomena - sudden changes in temperature between close points of the ocean. In such an environment, ocean currents “mix” temperatures in the same way that black and white colors mix.

Batchelor’s law predicts the ratio of the number of large-scale phenomena (thick swirls of paint or streams of ocean water of the same temperature) to the number of phenomena on smaller scales (thin lines of paint) when mixing liquids. It is called a law because physicists have been observing this phenomenon in experiments for many years.

“From the point of view of physics, this is enough to call it law,” said Panshon Smith, a mathematician at Brown University. However, before this work there was no mathematical proof of its indispensable performance.


Batchelor’s law predicts the ratio of the number of large-scale phenomena (thick swirls of paint or streams of ocean water of the same temperature) to the number of phenomena on smaller scales (thin lines of paint) when mixing liquids. This ratio remains unchanged when zoomed out, as small nesting dolls keep the proportions large.

To realize the idea of ​​Batchelor, back to the paint. Imagine that you continue this experiment for a while, adding drops of black paint and stirring. Now stop the time. You will see thick strips of black paint (it was kneaded the least), thinner strips (they were kneaded longer), and even thinner (they were kneaded even longer).

Batchelor’s law predicts that the number of thick strips, thinner and very thin strips obeys the exact proportion - something like dolls obey the same proportions.

“Strips of different scales are visible in a given fragment of liquid, because part of the droplets have only just begun to mix, and some have been mixed for some time,” said Blumenthal. “Batchelor’s law describes the size distribution of strips of black paint.” It is difficult to describe the exact proportion in a nutshell, but more thin strips are obtained than thick ones, and a certain number of times.

The law predicts that the proportion is maintained even if you look at the fragment of fluid with increasing. Strips of various thicknesses, both in a small area of ​​liquid and in the entire bank, will have exactly the same ratio in quantity; and zooming out, we will see the same ratio. The pattern is the same on all scales, as in hydrodynamic turbulence, where in each whirlpool there are small whirlpools.

A rather bold prediction, which, moreover, is difficult to model mathematically. The complex nesting of phenomena at different scales makes it impossible to accurately describe the appearance of Batchelor's law in a single fluid flow.

But the authors of the work figured out how to get around this complexity and prove it.

Random approach


Bedrossian, Blumenthal and Punchon Smith have adopted an approach that considers the average behavior of fluids in all turbulent systems. Mathematicians have tried this strategy before, but no one has successfully implemented it.

This approach works because randomness sometimes allows accurate predictions of system behavior. Imagine a vertical board studded with nails. Drop a coin along it from above, and it will bounce off the nails until it hits one of the slots below. It is difficult to predict where a particular coin will fall - too many factors affect where it will bounce after each collision.


Samuel Punchon Smith

Instead, you can consider the system as random - and that for every nail there is a chance that the coin will bounce right and left. If the probabilities are correctly calculated, it will be possible to make accurate predictions about the behavior of the system as a whole. For example, you may find that coins are more likely to fall into specific slots.

“What's good with randomness is the ability to do averaging,” Tiffo said. “Averaging is a very reliable idea, in the sense that many small details don’t touch it.”

What does this mean for turbulence and color mixing? Since exact and deterministic statements are beyond the scope of mathematics, it would be more useful to imagine that certain random forces act on the paint - sometimes interfering with it here, sometimes there, without any regularity. This approach is called random, or stochastic. It allows mathematicians to use high-level statistical calculations and study what is happening in the systems as a whole, without burying themselves in the specifics of every detail.

“A little bit of coincidence allows us to overcome difficulties,” said Punchon Smith.

This, finally, allowed three mathematicians to prove Batchelor's law.

Understanding mix


One way to prove a physical law is to imagine the conditions that would invalidate it. If it can be proved that such conditions do not arise, it will prove that the law always works. The team realized that in order to avoid the laws predicted by Batchelor’s law, kneading must have very specific characteristics.

The proof of the law is divided into four works published online between September 2018 and November 2019. The first three focused on understanding certain movements of the mixed paint that would not allow Batchelor’s law to work out and excluding such movements. They proved that even if you took a fluid specially formulated to defeat Batchelor’s law, the pattern would still appear in it.

“The main thing you need to understand is that the liquid cannot conceive anything against you,” Bedrossian said.


Jacob Bedrossian

For example, Batchelor’s law would not work if the mixing process resulted in persistent whirlpools, or funnels, in the paint. Such funnels would hold some drops of black paint in one place - like debris at the edge of the stream - and the paint would not mix.

“In such a whirlpool, the particle paths will not be chaotic; they do not separate quickly, but spin all together, ”said Bedrossian. “If your system doesn’t mix paint at the right speed, Batchelor’s law will not manifest.”

In the first work, mathematicians focused on what happens during the mixing process with two dots of black ink that were originally next to each other. They proved that points follow random paths and diverge in different directions. In other words, closely spaced points cannot get stuck in a whirlpool that would keep them together all the time.

“Initially, particles move together,” Blumenthal said, “but in the end they separate and diverge in completely different directions.”

In the second and third works, they looked broader at the mixing process. They proved that in a chaotic liquid, in the general case, black and white paint mix as fast as possible. Then they determined that local imperfections (whirlpools) would not form in the turbulent fluid, which could interfere with the appearance of an elegant global picture described by Batchelor's law.

In the first three works, the authors performed the complex mathematical calculations necessary to prove that the paint mixes thoroughly and randomly. In the fourth, they showed that in a fluid with such mixing properties, Batchelor’s law arises as a necessary consequence.

This is one of the strongest mathematically rigorous statements regarding turbulent systems. More importantly, it provides us with opportunities for a new stream of mathematical ideas. Turbulence is a chaotic phenomenon, almost random in its movement. Three mathematicians figured out how to deal with randomness using randomness. Other specialists in this field will almost certainly follow them.

“Their biggest contribution is to provide us with a platform on which to build evidence,” Tiffo said. “I think chance is one of the few ways to build a turbulence model that we can mathematically understand.”

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