## Friday, October 19, 2012

### Turbulence in a 2-dimensional Box: Pretty

Physicists like systems with fewer than the three spatial dimensions that we are used to. Not so much because that's easier, but because it often brings in qualitatively new features. For example, in two dimensions vortices in a fluid fulfill a conservation law that does not hold in three dimensions.

The vorticity of a fluid is a local quantity that measures, roughly, the spinning around each point of a fluid. In a two dimensional system, the only spinning that can happen is around the axis perpendicular to the two dimensions of the system. That is, if you have fluid in a plane, the vorticity is a vector that is always perpendicular to the plane, so the only thing that matters is the length and direction of this vector. In two dimensions now, the integral of the vorticity is a conserved quantity, called the enstrophy.

Pictorially this means if you create a vortex - a point that is itself at rest but around which the fluid spins - you can only do that in pairs that spin in opposite direction.

This neat paper:
Dynamics of Saturated Energy Condensation in Two-Dimensional Turbulence
Chi-kwan Chan, Dhrubaditya Mitra, Axel Brandenburg
Phys. Rev. E 85, 036315 (2012)
arXiv:1109.6937 [physics.flu-dyn]
studies what happens if you put a 2-dimensional fluid in a box with periodic boundary conditions, and disturb it by a force that is random in direction but at a distinct frequency. Due to dispersion the energy that enters the system at the frequency of the driving force cascades down to longer wavelengths. However, in a box of finite size there's a longest wavelength that will fit in. So the energy "condenses" into this longest possible wavelength. At the same time, the random force creates turbulence that leads to the formation of two oppositely rotating vortices.

Below is a plot of the vorticity of the fluid in the box. The two white/red and white/blue swirls are the vortices.
 Fig 1 from arXiv:1109.6937. Pseudocolor plot of vorticity of fluid in 2-dimensional box, showing condensation into long wavelength modes.
My mom likes to say "symmetry is the art of the stupid", and she's right in that symmetry all by itself is usually too strict to be interesting. Add a little chaos to symmetry however and you get a good recipe for beauty.

1. Hi Bee,

Not to disagree, but I would say symmetry is what has the world to be beautiful as it can even bring order to chaos. Now before I get into any more trouble with mom I would like to thank her for the part she played in having her offspring come to appreciate the role of both as to have them explored:-)

”Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”

- Benoit Mandelbrot,” The Fractal Geometry of Nature”, page 1, (1982)

Best,

Phil

2. You got a quote for every occasion :o)

3. Hi Bee,

Just my way of attempting to find some beauty in chaos :-)

Best,

Phil

4. Again, already when reading the title I was sure Axel was an author. :-)

5. Best clue to Navier-Stokes I've ever seen.

Add a little CDT and ... quantum gravity's core nugget.

6. The image you use from article is as if taken from the brush strokes of Van Gogh Also see: Van Gogh Knew Turbulence

Now might one consider such beauty as "symmetry breaking" or a moving away from "a state of calmness?"

In Van Gogh's case, it was obvious, yet the summation of the work in either state revealed a certain creativeness and expression of. Can we call it a "Organized Chaos?"

So like using Benoit, you wonder about the signature, as a sign of the global perspective and what arises as from the pattern.

Best,

7. Just as a matter of interest....A bubble is a minimal-energy surface

Best,

8. Hi Bee,
This is kind of off topic, but I hope you don't mind me posting this here. I am doing introductory courses in Quantum Field Theory and General Relativity and today the professor had some vague side remarks. He told us that the cosmological constant as measured is very small and that QFT predicts a very very large cosmological constant. This would be a result of our failure to combine gravity and quantum mechanics. Do you know any good review articles about this topic. I ask this because I know you are a quantum gravity theorist :) Thanks in advance!
Erik

9. Hi Erik,

maybe you check out this post ;-)

10. This comment has been removed by the author.

11. Hi Stefan,
If I tried to post a terrific question in the most unthreatening manor to people who have already made up their minds about it I could not have done it as well as Erik did. If he didn't exist one would almost have to invent him. Great question and great answer, even if it was a bit off topic. Who says the universe isn't slightly magical. :-)

Best,
Eric

Btw, I am not Erik writing as a pseudonym.

12. It's just me - or the quantum field theorists are gradually recognizing the dense aether model...?

13. It's just you.

14. LOL:-)

BTW Bee I'm in Germany (Aachen) for bizz and I was wondering where exactly you live in Germany.

15. I looked through the paper briefly but must've missed it: does the number of vortices have to be two? Can a "higher excited state" be created, with say 4 of them.
Also, it would be interesting to see how the picture would change gradually if transversal direction is allowed to be non-zero. Probably, depending on viscosity,
the vortices will survive until certain size, but then some sort of instability might set in.

16. Hi Giotis,

In Germany, we live in the Heidelberg area. Alas, I'm in Canada this week. Best,

Sabine

I don't know, it's a good question. I guess in principle it can happen, but not sure it would be stable, ie they would tend to combine to just two. Best,

B.

18. Are these vortices analogous to the audio nodal patterns revealed by Chladni plates? I gather these vortices arise within the medium at some remove from any real physical boundary and any constraints to movement are internal and integral to that medium.

Is it legitimate to say that the vortices are nature’s way of containing more energy within a given space?

There is a YouTube video, “Enstrophy amplification events in three-dimensional turbulence” by JÃ¶rg Schumacher, Maik Boltes, Herwig Zilken, Marc-AndrÃ© Hermanns, Bruno Eckhardt, and Charles R. Doering. It is interesting, but it lacks the Van Gogh extravagance of your image.

Thanks.

19. Thanks Stefan! Very interesting to read and very thoroughly written!