Abstract
Ogura and Phillips derived the original anelastic model through systematic formal asymptotics using the flow Mach number as the expansion parameter. To arrive at a reduced model that would simultaneously represent internal gravity waves and the effects of advection on the same time scale, they had to adopt a distinguished limit requiring that the dimensionless stability of the background state be on the order of the Mach number squared. For typical flow Mach numbers of 
Specifically, it is shown that (i) for (hsc/θ)dθ/dz < Mμ with 0 < μ < 2, the atmosphere features three asymptotically distinct time scales, namely, those of advection, internal gravity waves, and sound waves; (ii) within this range of stratifications, the structures and frequencies of the linearized internal wave modes of the compressible, anelastic, and pseudoincompressible models agree up to the order of Mμ; and (iii) if μ < ⅔, the accumulated phase differences of internal waves remain asymptotically small even over the long advective time scale. The argument is completed by observing that the three models agree with respect to the advective nonlinearities and that all other nonlinear terms are of higher order in M.
Corresponding author address: Rupert Klein, FB Mathematik und Informatik, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany. Email: rupert.klein@math.fu-berlin.de
