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In this work we investigate the statistical mechanics of a family of twodimensional (2D) fluid flows, described by the generalized Euler equations, or$\alpha$-models. These models describe both nonlocal and local dynamics, withone example of the latter given by the Surface Quasi Geostrophy (SQG) model forwhich the existence of singularities is still under discussion. Furthermore,SQG is relevant both for atmosphere and ocean dynamics, and in particular, itis proposed to understand the oceanic submesoscale structures, front andfilaments, associated with horizontal gradients of buoyancy. We discuss underwhich conditions the statistical theory suggests a principle of selective decayfor the whole family of turbulent models, and then we explore the selectivedecay principle numerically, the transition to equilibrium and the formation ofsingularities, starting from initial conditions (i.c.s) corresponding to ahyperbolic saddle.  We study the topological transitions in the flow configuration induced byfilaments breaking. Furthermore we compare the theoretical equilibrium states,and the functional relation between the generalized vorticity $q$ and itscorrespondent stream function $\psi$, with the results of the simulations. Forthe particular i.c.s investigated, and domain used for the simulations, we havenot noticed transition from tanh-like to sinh-like $\psi-q$ functionalrelation, which would be expected by the emergence of coherent structures thatare, however, filtered by the time averages used to compute the mean fields.

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