
Mathematicians prove universal law of turbulence - theafh
https://www.quantamagazine.org/mathematicians-prove-batchelors-law-of-turbulence-20200204/
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semi-extrinsic
Previous discussion of this same result from 2 months ago, 141 points and 16
comments:

[https://news.ycombinator.com/item?id=21771684](https://news.ycombinator.com/item?id=21771684)

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btilly
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.

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core-questions
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.

~~~
agf
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.

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core-questions
Everywhere else in nature that we have seen things that appear fractal, it's
been borne out by analysis, as far as I've ever seen.

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agf
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.

~~~
core-questions
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.

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agf
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.

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HenryKissinger
To any fluid physicist or mathematician here: Does this move us closer to a
general solution for Navier-Stokes?

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atrettel
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).

Hacker News has already discussed this result previously in an earlier thread:
[https://news.ycombinator.com/item?id=21771684](https://news.ycombinator.com/item?id=21771684)

~~~
mar77i
Damn it, you beat me to bringing up laminar flow!

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kfmu
Wonder what the implications are for magnetic confinement fusion.

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petschge
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.

~~~
willis936
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.

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petschge
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.

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fernly
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.

[https://en.wikipedia.org/wiki/Batchelor_vortex](https://en.wikipedia.org/wiki/Batchelor_vortex)

~~~
semi-extrinsic
No, that's a different thing, what you link to is an analytical vortex that
can approximate a boundary layer.

The result posted here is that the power density of the turbulent energy
spectrum scales with the inverse of wavenumber.

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whatitdobooboo
Not too well on mathematical proofs, but to me this sounds like the "proved"
something by not being able to disprove Batchelor's law?

Also, the part about randomness makes sense in theory but the jump to an
actual proof seems a little wide to me.

Are many mathematical laws proved in this way?

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flafla2
From the article:

> 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.

[1]
[https://www.cs.cmu.edu/afs/cs/academic/class/15451-s20/www/l...](https://www.cs.cmu.edu/afs/cs/academic/class/15451-s20/www/lectures/lec2.pdf)
Section 3.2

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gpm
> 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).

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JadeNB
> 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).

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esch89
My not very mathematically-inclined mind: the universe is patterns.

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Ameo
That may be true, but it might just be a side effect of the fact that most of
our perception is based off of layered pattern recognition.

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wvenable
Our perception might be based off of pattern recognition because the universe
is patterns.

~~~
Ameo
Hahaha that's totally true. I love the... pattern in that logic.

