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

Marianne Corvellec, Freddy Bouchet|arXiv (Cornell University)|Jul 9, 2012

Fluid Dynamics and Turbulent Flows12 references3 citations

TL;DR

This paper develops a complete analytical theory for low-energy phase diagrams in two-dimensional turbulence using Lyapunov–Schmidt reduction to study bifurcations in energy–Casimir variational problems. It shows that the sign of the quartic coefficient $ a_4 $ in the Casimir functional's Taylor expansion determines whether phase transitions are continuous (second-order) or discontinuous (first-order), with a tricritical point occurring when $ a_4 $ changes sign, revealing non-generic behavior in prior quadratic Casimir approximations.

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).

Motivation & Objective

• To develop a complete analytical theory for phase diagrams of two-dimensional turbulence steady states and statistical equilibria in the low-energy limit.
• To understand the nature of bifurcations in energy–Casimir variational problems that describe stable steady states and statistical equilibria.
• To determine how domain geometry and model parameters—especially the quartic coefficient $ a_4 $—influence the type of phase transitions (continuous vs. discontinuous).
• To demonstrate that prior results based on quadratic Casimir functionals are non-generic and unstable under small perturbations.
• To identify the conditions under which a tricritical point emerges, marking the transition between continuous and discontinuous phase transitions.

Proposed method

• Applies Lyapunov–Schmidt reduction to analyze bifurcations in energy–Casimir variational problems near the low-energy limit.
• Expands the Casimir functional in a Taylor series around its minimum, focusing on the quartic coefficient $ a_4 $ as the key parameter governing nonlinearity.
• Uses symmetry and orthogonality properties of the kernel and range of the linearized operator $ J $ to reduce the infinite-dimensional bifurcation problem to a scalar equation.
• Derives explicit expressions for the bifurcation function $ h(A, eta) $, showing that the cubic nonlinearity in $ A $—driven by $ a_4 $—determines the bifurcation type.
• Applies classical bifurcation theorems to establish the existence and structure of solution branches near critical points.
• Analyzes the dependence of bifurcations on constraints such as energy and circulation, and on domain geometry through eigenfunctions and eigenvalues.

Experimental results

Research questions

• RQ1 How do phase transitions in two-dimensional turbulence equilibria depend on the nonlinearity of the $ q $–$ ψ $ relationship, particularly through the coefficient $ a_4 $?
• RQ2 What determines whether a phase transition in 2D turbulence is continuous (second-order) or discontinuous (first-order)?
• RQ3 Under what conditions does a tricritical point emerge, where the transition type changes from continuous to discontinuous?
• RQ4 Why are previous analytical results based on quadratic Casimir functionals non-generic and unstable under small perturbations?
• RQ5 How does the Lyapunov–Schmidt reduction method enable a complete classification of low-energy phase diagrams in 2D turbulence?

Key findings

• The sign of the quartic coefficient $ a_4 $ in the Taylor expansion of the Casimir functional determines the type of phase transition: $ a_4 < 0 $ leads to a supercritical pitchfork bifurcation and a second-order (continuous) phase transition.
• When $ a_4 > 0 $, the bifurcation is subcritical, resulting in a first-order (discontinuous) phase transition with nontrivial solution branches existing below the critical point.
• A tricritical point occurs when $ a_4 $ changes sign, marking the transition between continuous and discontinuous behavior, and is a generic feature of the bifurcation diagram.
• The bifurcation structure is governed by the cubic nonlinearity in the reduced scalar equation $ h(A, eta) $, with $ ext{sgn}(rac{ ext{d}^3 h}{ ext{d}A^3}(0, eta_c)) = - ext{sgn}(a_4) $, confirming the dependence on $ a_4 $.
• Phase diagrams are highly sensitive to $ a_4 $, and prior results assuming quadratic Casimir functionals are non-generic, as they do not survive small perturbations in the system parameters.
• The theory applies generally to any domain geometry and models analogous to the 2D Euler or barotropic quasi-geostrophic equations, making it broadly relevant to geophysical and laboratory 2D turbulence.