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} f
ramework involving a constrained Hamiltons 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 Hamiltons principle, we then derive the corresponding 2D vertical slice models for these sound-proof theories.
