A Hamiltonian description of finite-time singularity in Euler's fluid equations

TL;DR

This paper analyzes a Hamiltonian model of vortex rings to demonstrate finite-time singularity in Euler's fluid equations, revealing integrability and providing explicit solutions that support the singularity formation.

Contribution

It establishes the Hamiltonian structure and integrability of a vortex ring model, linking it to finite-time singularity in Euler's equations, which was not previously demonstrated.

Findings

Hamiltonian structure and invariants identified

Model is integrable with explicit solutions

Finite-time singularity demonstrated through quadrature

Abstract

The recently proposed low degree-of-freedom model of Moffat and Kimura [1,2] for describing the approach to finite-time singularity of the incompressible Euler fluid equations is investigated. The model assumes an initial finite-energy configuration of two vortex rings placed symmetrically on two tilted planes. The Hamiltonian structure of the inviscid limit of the model is obtained. The associated noncanonical Poisson bracket [3] and two invariants, one that serves as the Hamiltonian and the other a Casimir invariant, are discovered. It is shown that the system is integrable with a solution that lies on the intersection for the two invariants, just as for the free rigid body of mechanics whose solution lies on the intersection of the kinetic energy and angular momentum surfaces. Also, a direct quadrature is given and used to demonstrate the Leray form for finite-time singularity in the model. To the extent the Moffat and Kimura model accurately represents Euler's ideal fluid equations of motion, we have shown the existence of finite-time singularity.