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)
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.