Mixing and coherent structures in 2D viscous flows

We introduce a dynamical description based on a probability density ϕ ( σ , x , y , t ) of the vorticity σ in two-dimensional viscous flows such that the average vorticity evolves according to the Navier–Stokes equations. A time-dependent mixing index is defined and the class of probability densities that maximizes this index is studied. The time dependence of the Lagrange multipliers can be chosen in such a way that the masses m ( σ , t ) ≔ ∫ d x d y ϕ ( σ , x , y , t ) associated with each vorticity value σ are conserved. When the masses m ( σ , t ) are conserved then (1) the mixing index satisfies an H-theorem and (2) the mixing index is the time-dependent analogue of the entropy employed in the statistical mechanical theory of inviscid 2D flows. In the context of our class of probability densities we also discuss the reconstruction of the probability density of the quasi-stationary coherent structures from the experimentally determined vorticity-stream function relations.