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

TL;DR

The paper establishes that linear quasigeostrophic flows linearized about a zonal base state possess an infinite set of conserved pseudomomenta, rooted in a continuous $U(1)$ symmetry. By formulating both Hamiltonian and Lagrangian descriptions, it shows that each zonal wavenumber $k$ carries a conserved charge ${\cal M}(k)$, derivable as a Noether charge and also commuting with the linear Hamiltonian via a noncanonical bracket. The Lagrangian treatment makes the symmetry explicit and yields ${\cal M}(k) = \int \frac{\omega'(k,y,t) \omega'^*(k,y,t)}{\partial_y \bar{\omega} + \beta} \, dy$ as the conserved quantity, clarifying why per-$k$ pseudomomenta are sometimes obscured in prior work. The discussion highlights limitations in the nonlinear (quasilinear) regime and points to future work on extending the variational structure to capture potential conserved quantities there and their predictive value.

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.