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

A complete theory of low-energy phase diagrams for two-dimensional turbulence steady states and equilibria