Variational formulations of sound-proof models

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-Poincaré 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.

1st version, comments welcome!. arXiv admin note: text overlap with arXiv:1211.5827

References in corpus (1)

• On Noether's theorem for the Euler-Poincaré equation on the diffeomorphism group with advected quantities

Cited by in corpus (4)

• Oscillatory convection and limitations of the Boussinesq approximation
• Thermodynamically consistent semi-compressible fluids: a variational perspective
• Hamilton's Principle with Phase Changes and Conservation Principles for Moist Potential Vorticity
• Conservation laws for potential vorticity in a salty ocean or cloudy atmosphere