Singular Casimir Elements of the Euler Equation and Equilibrium Points

TL;DR

This paper investigates the structure of Casimir elements in the Hamiltonian formulation of the Euler equation for incompressible fluids, revealing singularities and their impact on equilibrium points and energy-Casimir extremals.

Contribution

It introduces a generalized functional derivative approach to identify singular Casimir elements caused by the inhomogeneity of the symplectic operator in infinite-dimensional systems.

Findings

Analysis of the kernel of the symplectic operator in 2D flows

Identification of singular Casimir elements at operator nullity changes

Formulation of a solvable system for 2D Euler flows

Abstract

The problem of the nonequivalence of the sets of equilibrium points and energy-Casimir extremal points, which occurs in the noncanonical Hamiltonian formulation of equations describing ideal fluid and plasma dynamics, is addressed in the context of the Euler equation for an incompressible inviscid fluid. The problem is traced to a Casimir deficit, where Casimir elements constitute the center of the Lie-Poisson algebra underlying the Hamiltonian formulation, and this leads to a study of the symplectic operator defining the Poisson bracket. The kernel of the symplectic operator, for this typical example of an infinite-dimensional Hamiltonian system for media in terms of Eulerian variables, is analyzed. For two-dimensional flows, a rigorously solvable system is formulated. The nonlinearity of the Euler equation makes the symplectic operator inhomogeneous on phase space (the function space of the state variable), and it is seen that this creates a singularity where the nullity of the symplectic operator (the "dimension" of the center) changes. Singular Casimir elements stemming from this singularity are unearthed using a generalization of the functional derivative that occurs in the Poisson bracket.