Restrictions on the geometry of the periodic vorticity equation

TL;DR

This paper demonstrates that certain fluid dynamics equations, including quasi-geostrophic and axisymmetric Euler flows, cannot be represented as metric Euler equations on the diffeomorphism group of the circle.

Contribution

It establishes non-realizability results for several important fluid models as metric Euler equations on the circle's diffeomorphism group.

Findings

Quasi-geostrophic model cannot be realized as a metric Euler equation.

Axisymmetric Euler flow in higher dimensions cannot be realized as a metric Euler equation.

De Gregorio's vorticity model cannot be realized as a metric Euler equation.

Abstract

We prove that several evolution equations arising as mathematical models for fluid motion cannot be realized as metric Euler equations on the Lie group of all smooth and orientation-preserving diffeomorphisms on the circle. These include the quasi-geostrophic model equation, the axisymmetric Euler flow in higher space dimensions, and De Gregorio's vorticity model equation.