Regime of validity of soundproof atmospheric flow models

The present paper provides a scale analysis showing that the regime of validity of two of these extended models covers stratification strengths on the order of (hsc/θ)dθ/dz < M2/3, which corresponds to realistic variations of potential temperature θ across the pressure scale height hsc 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 < ⠔, 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.