The result is fascinating. Any form of self-similarity across scales means that it is a fractal with a specific dimension. The dimension describes how thoroughly it is mixed.
This should be somewhat self-evident to anyone familiar with fractals who has spent time watching turbulence in clouds, fluids, etc. (particularly if under the influence of LSD).
It's fractal down to the Kolmogorov limit, was my understanding - beyond a certain size it's not meaningfully a "fluid" anymore.
There are lots of things that appear to be true and aren't, and there is a long way between observing something that appears to be true and proving it mathematically.
Your comment is misleading because it implies we should all have been assuming this based on the evidence you state -- and we certainly should not have.
That's not even remotely true. Most things in nature people talk about being fractal aren't, in the mathematical sense, at all. They show statistical variation between areas and only have a few levels of self-similarity. A few minutes of Googling with confirm that.
You're missing the forest for the trees - perhaps literally.
Trees are a perfect example. Fractal in form, down to a certain level; but also an example of something that builds over time like a cellular automaton and is being influenced by the environment. Compromises are made, the perfect form is impossible to achieve, but the gestalt is still there.
The fact that the most realistic simulated / computer-generated trees we can render are made primarily of simple fractals is a great indicator.
We see the same thing in terms of self-similarity in mountain ranges, lightning, rivers, lungs, and now clouds and water. At this point if you want to deny it, all I can surmise is that you either never go outside, or simply don't know what to look for.
There is an important distinction you're missing between how you're using fractal and how this article is using it. It's saying something fundamentally different, though related, about turbulence, than you're saying about trees etc., even if you don't realize it.
I live in the mountains and I'm outside in a forest setting multiple times a week. I see exactly what you're talking about; it's just not the same as what the article is saying, even if you use the same word to describe it.
This result applies to turbulent flows. Not all flows are turbulent, and the Navier-Stokes equations should describe laminar, transitional, and turbulent flows (3 major regimes of fluid flow). Transitional flows are not yet fully turbulent and can behave differently than a fully turbulent flow behaves. My background is in fluid dynamics, so I can't speak directly about the mathematical difficulties in proving the existence and uniqueness of solutions of the Navier-Stokes equations. However, I can say that any general solution to the Navier-Stokes equations must apply to laminar, transitional, and turbulent flows, so finding a particular result for turbulent flows alone is insufficient. Any such result would have to transcend its regime in some way. Surely I think more proofs like this couldn't hurt, but I do not see any direct connection between this result and the Navier-Stokes existence/uniqueness problem (coming from a fluid dynamics background).
If what you mean by "general solution for Navier-Stokes" is something like the Millennium Prize problem (proving general existence & uniqueness of solutions for NS), then no, at least not as far as I can see. This work is about a different problem than NS: they are mainly interested in the advection of passive tracers, e.g., a blob of ink, in a chaotically-evolving fluid field. To do this, they assume a number of things: their fluid equations are subjected to stochastic forcing, which makes available a bigger bag of mathematical tools than deterministic chaotic systems (like high Reynolds number NS without stochastic forcing). They also assume periodic boundary conditions, which enables explicit calculations using Fourier series representations. (Physical boundaries play an important role in many kinds of turbulent flows but make the mathematical analysis harder.) And, in 3d, a hyperviscosity term (4th derivatives in space instead of the usual 2nd derivatives associated with viscosity), which makes solutions more smooth.
That being said, proving new theorems in this area is hard. I think it's very nice work.
Thanks! (I had the benefit of sitting down with one of the authors and having some of this explained to me.) And thanks for the clarification about the Millennium Prize.
I think it is relevant to note this proof is based on the incompressible Navier--Stokes equations. Real fluids are compressible. From the point of view of kinetic theory, even the compressible Navier--Stokes equations are only a first-order approximation to the Boltzmann equation. Further, many authors (including myself) have questioned whether the Chapman--Enskog approximation is justifiable for high Reynolds number flows. Certainly it is known that NS does not correctly describe shock flows, which are also high Reynolds number flows. I would say these papers represent an interesting mathematical exercise whose relevance to the physics of turbulence remains to be assessed.
This is not a bad question (turbulent transport is very important for magnetic confinement fusion), but the answer is unfortunately "very little". This work considered only hydrodynamics, which does not describe the inclusion of magnetic fields. Turbulence in magnetohydrodynamics is much more complicated and it is not clear how the present result would be generalized to cover that case as well. Also we do have results for neo-classical transport in MHD turbulence, even we don't have mathematically rigorous proofs.
Also, there are almost no implications to real-world applications from this proof because the result has been assumed to be true for a long time. I don't mean to diminish the value of the proof, which is an impressive bit of work; it's just that the value here is more academic and in the theory realm than in the practical or experimental where the theorem has been treated as true without a formal proof.
Oh hey I’m sitting in an office down the hall from a neoclassically optimized stellarator. :)
I’m not a scientist, but from my layman view there seems to be a lot of empirical work being done to characterize turbulence beyond what MHD models tell us. Afaict, hydrodynamics is a set of empirically derived equations that describe fluid macro scale behavior. Neoclassical MHD models apply Maxwell’s equations to these, but that isn’t sufficient for most plasma regimes. Gyrokinetic approximations of particle simulations are the lead that most people are following, but they aren’t able to agree with real world measurements very well yet. I have the impression that particle level simulation works but we are several orders of magnitude away from that in terms of computational capacity.
You can derive hydrodynamical (and magnetohydrodynamical) equations in a rigorous (as opposed to empirical) way. BUT that gets you an infinite hierarchy of equations where the equation for the density contains the flows, the equations for the flows contains the pressure, the equations for the pressure contains the heat flux and so on. So in praxis we end this hierarchy at some point and "close" the system of equations by imposing e.g. a empirical description of the pressure based on density, flow and temperature (this would be an equation of state. you could also set the heatflux based on lower order equations and close there).
That said, yes there is deviation from what even MHD turbulence predicts for a stellerator. Gyrokinetic simulations are more complicated (you keep lot and lot of the individual particles that make up the fluid, but ignore at what angle along their gyro orbit they are, basically describing them as charged little rings), consequently much more computationally costly, but closer to real life. A full particle simulation (retaining pointlike particles with a correct gyro phase) with something like a PiC code would be even better but is indeed order of magnitude out of reach at the moment.
tl;dr: you have acquired a good high-level view via diffusion from the people around you.
For anyone looking for a simple writeup, the Wikipedia topic title is "Batchelor vortexes", and it has not been updated to reflect the papers mentioned here.
> In their first paper, the mathematicians focused on what happens during the mixing process to two points of black paint that begin the process right next to each other. They proved that the points follow chaotic paths and go off in their own directions. In other words, the nearby points can’t ever get stuck in a vortex that will keep them close forever.
> “The particles move together initially,” Blumenthal said, “but eventually they split apart and go in completely different directions.”
> In the second and third papers, they took a broader look at the mixing process. They proved that in a chaotic fluid, generally speaking, the black and white paint mixes as quickly as possible. This further established that the turbulent fluid doesn’t form the kinds of local imperfections (vortices) that would prevent the elegant global picture described by Batchelor’s law from being true.
> In these first three papers, the authors did the hard mathematics required to prove that the paint mixes in a thorough, chaotic fashion. In the fourth, they showed that in a fluid with those mixing properties, Batchelor’s law follows as a consequence.
So no, they are not "proving something by not being able to disprove it." A better way of phrasing their strategy is, "proving something by proving that disproving it is impossible."
In Computer Science, there is a similar concept for proving asymptotic bounds of algorithms called an "adversarial proof." The idea is, given some query that your algorithm performs (e.g. in a graph algorithm, a query could be "are two vertices connected") come up with a worst-case adversary that answers queries in the absolute worst way possible, that would necessitate even more queries to complete the problem. In this way, you can prove a universal lower bound for the cost of solving some problem. See [1].
In this case, the adversary is trying to come up with the worst-case initial conditions for this particular brand of turbulence. Basically they are saying, no matter what, you couldn't come up with an initial condition that challenges Batchelor's law more.
> proving something by proving that disproving it is impossible.
No, I take issue with this phrasing as well. There are things that can neither be proven or disproven (by godel's theorem), proving that disproving it is impossible would not have been sufficient.
Without having read beyond what is in the article, I imagine what they must have shown is that
1. For all systems x, if x does not obey Batchelor's law than neither would the thing they are talking about in the 4th paper.
2. The system they are talking about in the 4th paper obey's Batchelor's law.
The immediate corollary is all systems obey Batchelor's law, otherwise you would have a contradiction (the 4th system both would and would not).
> No, I take issue with this phrasing as well. There are things that can neither be proven or disproven (by godel's theorem), proving that disproving it is impossible would not have been sufficient.
Yes, but this is only a trivial mis-speaking in what is obviously meant to be a description of proof by contradiction: proving something by showing that its opposite is impossible (not that disproving it is impossible).
> but to me this sounds like the "proved" something by not being able to disprove Batchelor's law?
Not quite. The statement is more like, if Batchelor's law were to fail, then it has to be in one of the following specific ways. Then you show that these specific ways can't happen and get the result.
This is a common approach and needs a few ingredients:
- How could things go wrong?
- Show that these are all possibilities and that otherwise things work (hard problem).
- Isolate each scenario from the first step and show that things don't go wrong (hard problem again).
A classical example would be something like global existence for the two-dimensional Euler equations. If the solution were to fail after a finite time, then necessarily some quantity has to go to infinity, because otherwise we could find a solution for a small additional time (Beale-Kato-Majda criterion). We then show that this quantity does not go to infinity and we are done.
I see, very helpful, thanks. But doesn't this leave room for what you potentially have not accounted for? I am just trying to wrap my head around how one can write a proof by saying something doesn't occur
Good question. That is part of the second point. You have to show that there is no room left.
For example say you want to show that a real valued solution stays bounded. Then you have to show that a solution always exists, starts at some small value and "it never occurs that the absolute value of the solution is bigger than 1000". Because you ruled out other scenarios this then implies that the solution is always bounded by 1000.
https://news.ycombinator.com/item?id=21771684