When does decaying 2D turbulence reach its statistical-mechanics equilibrium?
In plain words
Freely decaying flat turbulence organizes itself into a few large vortices or jets, and a theory from statistical mechanics predicts the final pattern by maximizing a mixing entropy. The prediction works in some cases and fails in others, and no rule says when.
Precise statement
For 2D Euler or high-Re Navier-Stokes flow from given initial vorticity in a bounded or periodic domain, the Miller-Robert-Sommeria theory predicts the late-time coarse-grained vorticity as the maximum-entropy state constrained by energy and by all integrals of functions of vorticity (Casimirs). Determine the conditions on initial vorticity distribution, domain geometry and Re under which the observed late-time state matches the prediction, and characterize the incomplete relaxation where it does not. Answer: a criterion validated by DNS and experiment.
What would settle it
Ensembles of high-resolution DNS and soap-film or thin-layer experiments compared with computed equilibria, with a criterion that predicts agreement or failure in advance.
Status in the literature
Cases of agreement and of incomplete relaxation are reviewed in Bouchet and Venaille, Physics Reports 2012.