Variational formulations of sound-proof models

TL;DR

This paper derives a unified family of ideal sound-proof fluid models, including AA and PIA, using a Hamiltonian framework, and extends them to 2D vertical slice models, highlighting their conserved properties.

Contribution

It introduces a Hamiltonian-based derivation of a family of sound-proof models, unifying AA and PIA, and extends these models to 2D vertical slices with preserved physical invariants.

Findings

Models include Kelvin-Noether circulation theorem

Conservation of potential vorticity on fluid parcels

Existence of a Lie-Poisson Hamiltonian formulation

Abstract

We derive a family of ideal (nondissipative) 3D sound-proof fluid models that includes both the Lipps-Hemler anelastic approximation (AA) and the Durran pseudo-incompressible approximation (PIA). This family of models arises in the Euler-Poincar\'{e} framework involving a constrained Hamilton's principle expressed in the Eulerian fluid description. The derivation in this framework establishes the following properties of each member of the entire family: the Kelvin-Noether circulation theorem, conservation of potential vorticity on fluid parcels, a Lie-Poisson Hamiltonian formulation possessing conserved Casimirs, a conserved domain integrated energy and an associated variational principle satisfied by the equilibrium solutions. \smallskip Having set the stage with the derivations of 3D models using the constrained Hamilton's principle, we then derive the corresponding 2D vertical slice models for these sound-proof theories.