We consider a modified version of the two-dimensional NS equation introduced in ,  preserving energy and momentum of inertia. The physical motivation of this approach is the occurrence of di¤erent dissipation time scales. The constraint we consider is related to the gradient flow structure of the 2-D Navier-Stokes equation with respect to the Wasserstein metric. In this paper we justify the choice of this metric from the point of view of the stochastic vortex theory.
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Mario Pulvirenti, E. Rossi, The 2- constrained equation and stochastic vortex theory. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 23 (2012), no. 1, pp. 1–27DOI 10.4171/RLM/614