On the anelastic approximation for a compressible atmosphere

The equations of motion for a compressible atmosphere under the influence of gravity are reexamined to determine the necessary conditions for which the anelastic approximation holds. These conditions are that (i) the buoyancy force has an O(1) effect in the vertical momentum equation, (ii) the characteristic vertical displacement of an air parcel is comparable to the density scale height, and (iii) the horizontal variations of the thermodynamic state variables at any height are small compared to the static reference value at that height. It is shown that, as a consequence of these assumptions, two additional conditions hold for adiabatic flow. These ancillary conditions are that (iv) the spatial variation of the base-state entropy is small, and (v) the Lagrangian time scale of the motions must be larger than the inverse of the buoyancy frequency of the base state. It is argued that condition (iii) is more fundamental than (iv) and that a flow can be anelastic even if condition (iv) is violated, provided diabatic processes help keep a parcel's entropy close to the base-state entropy at the height of the parcel. The resulting anelastic set of equations is new but represents a hybrid form of the equations of Dutton and Fichtl and of Lipps and Hemler for deep convection. The advantageous properties of the set include the conservation of energy, available energy, potential vorticity, and angular momentum as well as the accurate incorporation of the acoustic hydrostatic adjustment problem. A moist version of the equations is developed that conserves energy.