  For the 2D Euler equations and related models of geophysical flows, minima of
energy--Casimir variational problems are stable steady states of the equations
(Arnol'd theorems). The same variational problems also describe sets of
statistical equilibria of the equations. In this paper, we make use of
Lyapunov--Schmidt reduction in order to study the bifurcation diagrams for
these variational problems, in the limit of small energy or, equivalently, of
small departure from quadratic Casimir functionals. We show a generic
occurrence of phase transitions, either continuous or discontinuous. We derive
the type of phase transitions for any domain geometry and any model analogous
to the 2D Euler equations. The bifurcations depend crucially on a_4, the
quartic coefficient in the Taylor expansion of the Casimir functional around
its minima. Note that a_4 can be related to the fourth moment of the vorticity
in the statistical mechanics framework. A tricritical point (bifurcation from a
continuous to a discontinuous phase transition) often occurs when a_4 changes
sign. The bifurcations depend also on possible constraints on the variational
problems (circulation, energy). These results show that the analytical results
obtained with quadratic Casimir functionals by several authors are non-generic
(not robust to a small change in the parameters).
