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

TL;DR

This paper analyzes the bifurcation diagrams of variational problems related to 2D turbulence steady states, revealing generic phase transitions and the influence of higher-order Casimir terms on stability and bifurcation types.

Contribution

It introduces a Lyapunov--Schmidt reduction approach to study phase transitions in 2D turbulence models, highlighting the role of quartic Casimir coefficients and domain constraints.

Findings

Phase transitions are generically observed in the bifurcation diagrams.

The type of phase transition depends on the quartic coefficient a_4.

Analytical results with quadratic Casimirs are non-generic and sensitive to parameters.

Abstract

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).