New ‘Superdiffusion’ Proof Probes the Mysterious Math of Turbulence
26:12 Share

A few weeks ago, I was hiking in the White Mountains in New Hampshire. It's a place that's notorious for its terrible weather, which can shift suddenly and sometimes dangerously in like a heartbeat. After a couple of calm, drizzly hours up in the Alpine zone above the treeline, there was, if you'll excuse the expression, a sudden change in the wind. Just like that, we were being lashed with a cold rain that seemed at its worst moments to be coming from every direction at once.

Overall, the wind was blowing one way, but the rocks all around us, the peaks and valleys were spinning it off into what we officially call turbulence or the seeming chaos that occurs when the flow of a fluid like wind blowing in one direction or water flowing in a stream passes through a complex real world environment.

As you might imagine, understanding such a complicated system mathematically must be insanely complex. So complex, in fact, that figuring it out could be worth a cool million dollars. Welcome to the Quanta podcast, where we explore the frontiers of fundamental science and math. I'm Samir Patel, editor-in-chief of Quanta magazine.

Mathematicians are never the type to shy away from a seemingly impossible problem. And I feel like figuring out how to bring order to something as inherently chaotic as turbulence is one of those. And sure enough, recently there was a major advance in understanding the math of turbulence. And here to discuss it with us is one of Quanta's staff writers, Joe Howlett. Welcome, Joe. Yeah, thanks for having me. Great to be here.

Joe, you're Quanta's math writer, which is not exactly a common position in the journalism world. What's it like writing about math full-time? It's really hard. Where I used to spend my time on calls, even with scientists...

getting good quotes about their experience. Now I spend hours on the phone just trying to understand the actual paper itself because I can't read the first sentence of it usually. That actually is a great transition to talk about a story that you recently wrote for Quanta about turbulence and the math behind it. Before we get into it, what's the big idea? Where are we going with this story? It's the first rigorous mathematical proof that

of a phenomenon about turbulence that we've known since the beginning, or since it started to be studied kind of 100 years ago. That phenomenon, we call it in this context super diffusion, it's the way things get scattered around on the wind if the wind is turbulent. And we've known for a long time that this is something that happens, but it's never been proven mathematically. And this is the first context where it's really been done.

So many of our math stories end up in a very abstract place right away. What's interesting about turbulence as a mathematical concept is it's actually so grounded in the real world, whether it's hiking in the mountains or looking at a stream or driving your car and having the air pass over it. So let's start with

the real world analog of what this is. That's based on the flow of a fluid, right? So what do we mean by a fluid and how does that flow usually look? It's anything that isn't solid, where the molecules can move around each other. And that makes things complicated. How do those molecules interact and move around is like a central question of science that's like flummoxed mathematicians, scientists, engineers for millennia.

Where does turbulence come into play? What is turbulence in a real-world sense? If you imagine, like, very simple flow of water through a pipe, you can kind of imagine, like, maybe everywhere in that pipe, it's more or less moving at the same speed. Yeah, it's interacting a little bit with the edges of the pipe, but the flow of the middle feels relatively consistent if you imagine an idealized version of it. Exactly. That fluid motion is pretty simple. We can write down equations for it.

But when you start speeding it up, adding turns or bends or obstacles, turbulence can start to appear. So turbulence appears whenever the fluid motion becomes fast enough that it's not simple anymore. And you can see this in all kinds of places. Like a classic example is when you're going on a boat and you're like sitting at the back of the boat and you see behind you the wake. Turbulent kind of eddies in all directions.

Another example is if you've ever been kayaking. When you put your oar in the water and you push, you see all these little vortexes go off in different directions.

All of those motions are technically turbulence, and they're really unpredictable. Like which direction they go in, what size they have, can be really complicated and hard to study. And this is also true of smoke from a cigarette or smoke from a volcano. It rises into the air and then immediately starts coiling in complicated vortices. And that's all turbulence. We can imagine all of the...

real-world ways that these little vortexes and eddies appear. This is also related to the way that a lot of people think of turbulence, which is an airplane, right? Yeah, so what's happening in an airplane is you have this really simple flow of air over the airplane wing. It's really continuous. And

At the back, behind the wing, after the air passes over it, it can do turbulent things, just like in the wake of your boat. The problem is the atmosphere contains tons of turbulence. That turbulent air, when it hits the wing, can break up that constant flow that leaves you comfortable in your seat on an airplane. And that's what makes this kind of bumpiness, right? It's completely random. It's unpredictable. It's chaotic. We mentioned earlier this is a complicated setting. You said that there is a good way to describe a steady flow.

There's a set of equations called the Navier-Stokes equations. It's just the laws of

conservation of energy, conservation of mass, conservation of momentum. There's things that we know about a fluid like it can't break the laws of physics. What those equations do is if you give me an initial state of a fluid, it's moving every point in the fluid has some velocity in some direction. And if I know all of that, I could put it into this equation, turn the crank, and tell you exactly what's going to happen at every point in the future, right? How that fluid's going to continue to move.

And this is an idealized, imagined, mathematical version of that fluid because nothing's ever really that predictable. Nothing's ever really that predictable, but it is deterministic. If you knew everything about that turbulent fluid and how it was moving, you could put it into this equation and predict the future. But because you can never know that, you can never predict the future. And studying these Navier-Stokes equations, this is how people do things like weather prediction. That's where the butterfly effect comes in.

Because we don't know the state of the atmosphere down to every single molecule, we can't put exact numbers into these equations. Small uncertainties balloon over time. And after 10 days, your weather prediction becomes totally unreliable, which is why we can only trust our phone a day or two out. The butterfly effect got turned into a movie. You know, the butterfly flaps its wings in one place. It causes extreme weather somewhere else. Is that true?

Or is that just a useful way to think about a small disturbance at the beginning can result in a big disturbance later? It's both, right? This flies in the face of the way we typically think about science, which is that if you get things approximately right, you can get approximate answers. That's not true in chaotic systems. And the flow of a fluid is what we would call a chaotic system.

It's the chaotic system. Like I said, everything about a fluid is captured in the Navier-Stokes equations. But studying their solutions or even solving them in a particular case can be mathematically impossible. It's considered one of the biggest open problems in all of pure mathematics. It's one of seven of these Millennium Prize problems from the Clay Mathematics Institute. So like you said in the intro, if you were to...

understand the Navier-Stokes equations, you would get a million dollars from this mathematics institute. And the reason it is so difficult mathematically is because of turbulence, right? You have whirls and eddies, which are like these smaller swirling patterns, and vortices, which are the bigger ones. So there are different scales, small and large, and all of this is happening in

The air that's coming out of my mouth and every stream flowing all over the world in the wake of every boat and constantly in the oceans and the atmosphere just all the time. Yeah. So you said scales. That's important to why turbulence is such a problem for mathematicians. If you imagine like a satellite image of a hurricane, that's one big vortex, right?

If you zoomed in to a part of the hurricane, you would see other smaller vortices. And that would be true no matter how much you zoomed in until you get down to the molecular scale where vortices can't form because there's so much noise of molecules hitting each other. At all these different scales, there's these different vortices and they interact with each other. That means like if you want to ask what happens to say a balloon I let loose in the middle of a hurricane, you can't just think about small scale vortices.

you have to think about the whole hurricane and how all those vortices interact and pass down energy that gets transferred to my balloon. Speaking of balloons, one of the origins of the area of mathematics we're about to talk about, so we can get to super diffusion, starts with a balloon race. So in 1906, there was the first iteration of a balloon race that's still going on today. It's like the longest running global balloon race.

What's this race called? It's called the Gordon Bennett Cup. Gordon Bennett was a rich person who was into aeronautics in England. I learned from our British coworker, Simon France, that he was such a jerk, or maybe his son was such a jerk, that to this day, English people will say, oh, Gordon Bennett, when they're upset about something. It's funny. But yeah, he was just a rich guy who started this cup. Every year, balloonists from different countries would come to one place,

And these are balloons filled with hydrogen or helium, lighter-than-air balloons. And all they have is ballast to let out and a release valve to let out the gas. So they can go up or down, but they can't control anything else. So they all assembled in Paris on September 30th, 1906, and inflated their balloons, drifted off into the sky one by one, with the aim of seeing how far from Paris they could land.

over a certain period of time or just like it or lump it, how far away from Paris are you? How far from Paris can you get? It's not a race because there's no destination because you can't control where you're going. Right. You're totally subject to the winds. All you can do is be like, the winds look better up there and let some ballast out or be like, the winds look better down there and let some gas out. Those are your only controls and you're just scattered to the wind. Okay. And whoever gets the furthest wins.

This sounds like an awful idea. Yeah, it was. Can I just say, because Paris, there's water on several sides of France, you could end up where it's not safe to land, right? Yeah, and that's very much what happened. So it started as a really calm day. They released 5,000 pigeons in celebration of the first event. Everybody lifted off, the balloons dotted the sky, and then drifted off westerly.

But then at night, the winds changed. A very turbulent wind system entered the picture and scattered all these balloons to the northwest.

All of the balloonists immediately realized they were in trouble. They were sailing towards the English Channel or some of them thought the Atlantic Ocean. It's hard to know where you are when you're on one of these balloons above the clouds. Especially in 1906. Exactly. One of them reported they were trying to yell down to people in one of these little French villages they were passing over like, where are we? And it was reported in the press that it was like cries for help, but they didn't know what they were saying. So a lot of the balloonists just landed. But

But some daring balloonists, a few of them were like, I think Britain's over there. I think we can hang on and land in Britain. And a few of them landed in Britain, including the winner. But what was important for the history of science about this balloon race, their landing positions were listed in a 1906 issue of the Aeronautical Journal. So 20 years later, an engineer, meteorologist, interesting kind of maverick scientist named Lewis Fry Richardson said,

was interested in studying turbulence and how it scatters things. He happened upon this journal entry and realized, "I can use the way these balloons were scattered to the winds as an experiment on the turbulence of the atmosphere." He was like, "This is a perfect 1,000 kilometer scale experiment,

I can do a small-scale experiment, too, with just blowing on a dandelion, right? I can look how all the seeds disperse on the wind. I could look at the ash above a volcano and see how far it spreads. I could look at data on hurricane systems and see how they spread things around. So this is about looking at where things land and using that to—

reverse engineer the turbulence that sent them there. Exactly, because a turbulent fluid, it's a total mess. You have to think about every single position of the fluid and where it's moving, right? It's just a simpler question to ask, how does my turbulent fluid scatter things than to be like, how is it moving exactly at every position? What did Richardson see in the records from the 1906 race?

He took the data from that race, put it on the same plot with his dandelion and hurricane data over many, many, many scales. And he saw a really strikingly clear relation. It was just a straight line, a power law, basically, which showed him that all of these scales were somehow interacting to spread these particles faster than random motion. Where the balloons ended up finally landing...

happened more quickly than one would imagine otherwise. So what you're seeing is from all of this little chaotic movement, a larger phenomenon, a larger trend. A really simple order out of the chaos. That was really exciting that we could actually say something about turbulence. And he observed it experimentally with all these different data from aeronautical journals. Really incredible stuff.

What open questions did he leave that we're able to continue to examine today? Richardson didn't say or know is why the hell is this happening? He could conjecture about it, but he didn't know why. So that sat for decades. We had this empirical law and some kind of heuristic understanding of why it could happen, but no real mathematical understanding. And then in the 70s, some physicists came along and realized that

That a new technique from particle physics might be useful for the study of turbulence, including understanding this fast spread of particles. And what was that technique? It's called renormalization. It's a famously...

hand wavy technique. Richard Feynman called it mathematical hocus pocus. You're kind of cheating. In mathematics, you're not allowed to divide by zero. There's a bunch of other rules like that too. And physicists were breaking all of them. But they were answering really important questions in the right way. Like they were able to predict how two electrons interact and collide really, really precisely. And crucial to renormalization is uniting many different scales. So relating all these different scales is something renormalization was able to do.

So some of these fluid dynamics physicists were like, wait, if this thing can be applied to problems with many different scales, maybe it can help with turbulence. So they started applying it to this problem, and they were able to explain Richardson's thing, this simple law of how particles or rubber ducks or balloons get scattered on the wind. They were able to get exactly the right answer for how this happens.

This was mainly confined to simplified scenarios. So instead of taking a real fluid with the full Navier-Stokes equations, you take a kind of simplified fluid where you still see this super diffusion, right? That's what the physicists do. Boil it down to its essential essence. And in that context, they were able to use renormalization to explain it and get an answer.

But to a mathematician, is that answer right? I don't know. Mathematicians don't trust renormalization. It's not a rigorous mathematical tool. They've tried over the years to make renormalization rigorous in all kinds of contexts. It's a multi-decade effort. And only in a few cases has it been able to work out.

So Richardson makes his super diffusion observations in the 1920s. And then by the 1980s, there's this first mathematical step to explain it. But people don't really trust that. What happens next? Another few decades pass and mathematicians were trying to use their own rigorous tools to say something about these systems. They were able to say things.

But instead of saying, this is the exact way these particles will scatter, they'll say they'll scatter on average more quickly than this if you do it a million times or if you imagine every possible motion of the fluid. So they were able to make much weaker statements. So you either use a broken mathematical technique or you use really good rigorous ones, but you can't say anything as grand or firm as you want. Exactly. The dichotomy is you can either use loose mathematics and say something strong.

Or you can use rigorous mathematics and say very little or make some assumptions that simplify everything to make some kind of statement. The mathematicians we're talking about today insisted there was a way to make renormalization rigorous in this context and really attack the problem head on and say something that's mathematically rigorous and powerful.

So these mathematicians who are from NYU's Courant Institute and the University of Helsinki, did they have a particular technique to get to that? Yeah, so their idea was a mathematical tool called homogenization. It's something people had been doing for a while where you look at things from different scales. You zoom in very close and then you zoom very far out and you kind of relate the system at this disordered close scale to this zoomed out simple scale mathematically. Okay.

These mathematicians thought, okay, if we do it step by step, instead of zooming out a lot, we zoom out a little bit and draw a grid over our fluid. So you imagine in every grid square, some of them have little eddies, some of them have fluid flowing right through them. And in every square, they ask, what is the motion of the fluid? And how different is it from square to square?

So they're like pixelating a fluid in a way. I'm thinking of like a Chuck Close painting, trying to aggregate what's in one individual square at a time. It's like a digital photo, right? They're just trying to simplify each square to its one number. Like a low resolution fluid. Yes. And then they zoom out more and draw a coarser grid. They reduce the resolution. Okay. And they relate this coarser grid to the finer grid. And they show that it becomes less heterogeneous. Everything kind of becomes simpler.

And they thought this was a way to take that renormalization method and do it mathematically, right? Eventually you get this simple picture and you know exactly how you got there and you've proven it. And their proof mathematically nails that down and shows how the chaos becomes order and tells you what the number is, how the particles get spread around. So now they're able to tell you exactly how balloons will get scattered on the wind and

at that large scale. And they're able to do it rigorously. And the answer they get is the same one the physicists got in the 1970s with their renormalization. We take the real-world observation of superdiffusion. Renormalization gives us a partial explanation or one we don't feel like we can trust. But renormalization combined with homogenization in what I assume is an elaborate and complex mathematical proof is

is telling us why super diffusion happens, why the balloons got scattered further and faster than anyone thought in the 1906 Bennett Cup race. Yeah, exactly. Using this thing homogenization, this mathematical tool, they made the physicist thing renormalization mathematically solid in this one context of turbulence.

Does applying homogenization to renormalization mean that this intellectual leap could have wider application?

The authors think it might, and so do some others in the field. You know, we mentioned those seven clay millennium prize problems. Another one has to do with exactly this question in the context of particle physics. Can you make renormalization rigorous in particle physics? And this seems like it could be a new approach to formulate renormalization in a more rigorous way.

No one's won any of these Millennium Prizes on the basis of this work, but it does provide a potentially promising approach to answering a bunch of questions about chaotic systems in multiple fields that could potentially down the road lead to more mathematical solutions that people would be really excited about. Yeah, and the author is

They're evangelists for this stuff. They really believe it's enormously powerful. They've been singing its praises for decades. And for a while, people didn't really believe it could make renormalization rigorous. They see this as the first step in a long process of using this technique on all kinds of problems. It's also a rare foothold on...

on turbulence, right? Turbulent fluids have been beyond our understanding for millennia. And just even understanding how it spreads particles about is a rare thing and a special thing. Thank you for explaining this story to us, Joe. If you want to read more about it, you can find it on quantummagazine.org. Joe's story is called New Super Diffusion Proof Probes the Mysterious Math of Turbulence. Every episode, we like to ask our guests for a recommendation. Joe, what's exciting your imagination this week?

Okay, I have two very different ones. Great. You can choose one. I do both. Okay. One on topic and one off topic. The on topic one is writing this story finally got me to read...

something on my bucket list for a while, James Glick's book called Chaos. It's a classic of science communication. It's incredible. And it tells the story of this 1970s scientific revolution in chaos theory. It's cool because the people who work on fluid dynamics are really eclectic weirdos who come from all different parts of science. And a lot of times it tanks their career because it's kind of like a weird silo within the scientific world. So it's just a great book and I absolutely recommend it.

My totally off topic one, I think it was several happy hours ago, you and I connected over Pat Benatar. Yes. In particular, her 1979 album In the Heat of the Night and how it opens with banger after banger. I feel like this is a no skips album. No duds. It's really, really good. And Pat Benatar is completely underrated. Yeah. I want people to listen to it.

to In the Heat of the Night. In particular, the song We Live for Love, which was like a hit at the time, but I feel like has gotten lost to current generations. It's unbelievable. Agreed. And I wholly endorse that particular recommendation.

Also on Quanta this week, we get to know fermions and bosons, the two most fundamental particles in physics. And we have a conversation with a scientist who is exploring a possible feedback loop between machine intelligence and human cognition. We're going to leave you today with a bit of what turbulence sounds like. This is the wind recorded by the Mount Washington Observatory in New Hampshire at the top of the tallest peak in the Northeast.

Mount Washington is notorious for having some of the strongest winds and worst weather in the world. Notably, this recording was actually made just hours before my hiking party and I reached the summit in June 2025. It was much less windy when we got there.

The Quanta Podcast is a podcast from Quanta Magazine, an editorially independent publication supported by the Simons Foundation. I'm Quanta's Editor-in-Chief, Samir Patel.

Funding decisions by the Simons Foundation have no influence on the selection of topics, guests, or other editorial decisions in this podcast or in Quanta Magazine. The Quanta podcast is produced in partnership with PRX Productions. The production team is Ali Budner, Deborah J. Balthazar, Genevieve Sponsler, and Tommy Bazarian. The executive producer of PRX Productions is Jocelyn Gonzalez.

From Quanta Magazine, Simon France and myself provide editorial guidance, with support from Matt Karlstrom, Samuel Velasco, Simone Barr, and Michael Kenyongolo. Our theme music is from APM Music. If you have any questions or comments for us, please email us at quanta at simonsfoundation.org. Thanks for listening. From PR.