An extended self-organisation principle for modelling and calculating the dissipation of 2D confined vortices

Steady Euler flows in a periodic square that, for positive vorticity distributions, minimise the entropy at given values of the energy and the circulations are non-confined vortices for optimal values of the circulation, and are confined vortices for certain non-optimal values. An extension of the self-organisation principle for the 2D Navier-Stokes equations is presented which models the dissipative evolution of such confined vortices by changing the values of the constraints in time in accordance with the effect of dissipation. An efficient numerical procedure for calculating these confined states is described and calculations of the dissipative evolution are presented.