Pseudomomenta in Linear and Quasi-Linear Approximations to Fluids

We present Hamiltonian and Lagrangian formulations for the two-dimensional fluid equations linearized about a zonally-symmetric basic flow. The Lagrangian and Hamiltonians 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. We emphasize that there exist separately conserved pseudomomenta at each zonal wavenumber, a point that has sometimes been obscured in past treatments. We highlight the relationship between the infinite U(1) symmetry and possible conserved quantities in quasi-linear systems.