Statistical Measures and Selective Decay Principle for Generalized Euler Dynamics: Formulation and Application to the Formation of Strong Fronts

In this work we investigate the statistical mechanics of a family of two dimensional (2D) fluid flows, described by the generalized Euler equations, or $$\alpha $$-models. These models describe both nonlocal and local dynamics, with one example of the latter given by the surface quasi geostrophy (SQG) model for which the existence of singularities is still under discussion. Furthermore, SQG is relevant both for atmosphere and ocean dynamics, and in particular, it is proposed to understand the oceanic submesoscale structures, front and filaments, associated with horizontal gradients of buoyancy. We discuss under which conditions the statistical theory suggests a principle of selective decay for the whole family of turbulent models, and then we explore the selective decay principle numerically, the transition to equilibrium and the formation of singularities, starting from initial conditions (i.c.s) corresponding to a hyperbolic saddle. We study the topological transitions in the flow configuration induced by filaments breaking. Furthermore we compare the theoretical equilibrium states, and the functional relation between the generalized vorticity q and its correspondent stream function $$\psi $$, with the results of the simulations. For the particular i.c.s investigated, and domain used for the simulations, we have not noticed transition from tanh-like to sinh-like $$\psi -q$$ functional relation, which would be expected by the emergence of coherent structures that are, however, filtered by the time averages used to compute the mean fields.