  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.
