Poisson geometry and first integrals of geostrophic equations

TL;DR

This paper explores the Poisson geometric structure underlying geostrophic equations, revealing their first integrals and similarities to fluid dynamics invariants through Hamiltonian reduction.

Contribution

It provides a geometric framework for understanding first integrals of geostrophic equations and connects them to Poisson structures derived from Hamiltonian reduction.

Findings

Identification of first integrals similar to enstrophy invariants

Description of Poisson structure on the space of densities

Connection between symplectic reduction and geostrophic equations

Abstract

We describe first integrals of geostrophic equations, which are similar to the enstrophy invariants of the Euler equation for an ideal incompressible fluid. We explain the geometry behind this similarity, give several equivalent definitions of the Poisson structure on the space of smooth densities on a symplectic manifold, and show how it can be obtained via the Hamiltonian reduction from a symplectic structure on the diffeomorphism group.