Statistical equilibrium states for two-dimensional flows

We explain the emergence of organized structures in two-dimensional turbulent flows by a theory of equilibrium statistical mechanics. This theory takes into account all the known constants of the motion for the Euler equations. The microscopic states are all the possible vorticity fields, while a macroscopic state is defined as a probability distribution of vorticity at each point of the domain, which describes in a statistical sense the fine-scale vorticity fluctuations. The organized structure appears as a state of maximal entropy, with the constraints of all the constants of the motion. The vorticity field obtained as the local average of this optimal macrostate is a steady solution of the Euler equation. The variational problem provides an explicit relationship between stream function and vorticity, which characterizes this steady state. Inertial structures in geophysical fluid dynamics can be predicted, using a generalization of the theory to potential vorticity.