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.