Classification of self-organized vortices in two-dimensional turbulence: the case of a bounded domain

Abstract

We calculate steady solutions of the Euler equations for any given value of energy and circulation (and angular momentum in the case of a circular domain). A linear relationship between vorticity and stream function is assumed. These solutions correspond to the predicted self-organization into a maximum-entropy state, in the limit of strong mixing. Vorticity mixing is then only weakly restricted by the constraint of energy conservation. While maximum-entropy solutions depend in general on the whole probability distribution of vorticity levels, these linearized results depend only on a single control parameter, yet keeping much of the general structure of the problem. A convenient classification of the maximum-entropy states is thus provided. We show furthermore how to extend these linearized results as an expansion in energy, involving successive moments of the vorticity probability distribution. They are applied to a rectangular domain and compared with existing numerical and laboratory results. We predict that the flow organizes into a single vortex in the square domain, but into a two-vortex dipolar state in a rectangle with aspect ratio greater than 1.12. The case of a circular domain is also explicitly solved, taking into account the conservation of the angular momentum.