ON THE ANELASTIC APPROXIMATION FOR A COMPRESSIBLE ATMOSPHERE

The equations of motion for a compressible atmosphere under the influe nce of gravity are reexamined to determine the necessary conditions fo r which the anelastic approximation holds. These conditions are that ( i) the buoyancy force has an O(1) effect in the vertical momentum equa tion, (ii) the characteristic vertical displacement of an air parcel i s comparable to the density scale height, and (iii) the horizontal var iations of the thermodynamic state variables at any height are small c ompared to the static reference value at that height. It is shown that , as a consequence of these assumptions, two additional conditions hol d for adiabatic flow. These ancillary conditions are that (iv) the spa tial variation of the base-state entropy is small, and (v) the Lagrang ian time scale of the motions must be larger than the inverse of the b uoyancy frequency of the base state. It is argued that condition (iii) is more fundamental than (iv) and that a flow can be anelastic even i f 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 inc lude the conservation of energy, available energy, potential vorticity , and angular momentum as well as the accurate incorporation of the ac oustic hydrostatic adjustment problem. A moist version of the equation s is developed that conserves energy.