Conserved Pseudomomenta in Linear Quasigeostrophic Fluid Flows From Noether's Theorem

TL;DR

This paper derives an infinite set of conserved pseudomomenta in linear quasigeostrophic fluid flows using Noether's theorem, highlighting their separate conservation at each zonal wavenumber.

Contribution

It presents Hamiltonian and Lagrangian formulations revealing an infinite U(1) symmetry and associated conservation laws in linearized quasigeostrophic equations.

Findings

Identifies an infinite U(1) symmetry in linearized equations.

Derives conservation laws (pseudomomenta) via Noether's theorem.

Shows separate conservation of pseudomomenta at each zonal wavenumber.

Abstract

Hamiltonian and Lagrangian formulations for the two-dimensional quasi-geostrophic equations linearized about a zonally-symmetric basic flow are presented. The Lagrangian and Hamiltonian exhibit an infinite U(1) symmetry due to the absence of wave + wave -> wave interactions in the linearized approximation. By Noether's theorem the symmetry has a corresponding infinite set of conservation laws which are the well-known pseudomomenta. There exist separately conserved pseudomomenta at each zonal wavenumber, a point that has sometimes been obscured in past treatments.