# Poisson Structure of the Three-Dimensional Euler Equations in Fourier Space

## TL;DR

This paper derives a Poisson structure for the 3D Euler equations in Fourier space, enabling Hamiltonian analysis and revealing limitations of the Energy-Casimir method for stability studies of shear flows.

## Contribution

It introduces a simple Poisson structure for the 3D Euler equations in Fourier space and explores its properties and implications for stability analysis.

## Key findings

- Poisson structure valid on divergence-free subspace
- Helicity as a Casimir invariant
- Periodic shear flows are equilibrium points of the structure

## Abstract

We derive a simple Poisson structure in the space of Fourier modes for the vorticity formulation of the Euler equations on a three-dimensional periodic domain. This allows us to analyse the structure of the Euler equations using a Hamiltonian framework. The Poisson structure is valid on the divergence free subspace only, and we show that using a projection operator it can be extended to be valid in the full space. We then restrict the simple Poisson structure to the divergence-free subspace on which the dynamics of the Euler equations take place, reducing the size of the system of ODEs by a third. The projected and the restricted Poisson structures are shown to have the helicity as a Casimir invariant. We conclude by showing that periodic shear flows in three dimensions are equilibria that correspond to singular points of the projected Poisson structure, and hence that the usual approach to study their nonlinear stability through the Energy-Casimir method fails.