On the regime of validity of sound-proof model equations for atmospheric flows

Abstract

This paper summarizes, and slightly extends, the key results of two recent publications (Klein et al., JAS, 2010 and Achatz et al., JFM, 2010). Soundproof atmospheric flow models are considered attractive for two reasons: (i) Since acoustic modes are likely of negligible influence in atmospheric dynamics, sound-proof models have the advantage of a clear-cut focus on the essentials of such flows. (ii) Although several numerical schemes for the compressible atmospheric flow equations are being used in production codes, uncertainties w.r.t. their robustness, accuracy, or flexibility remain. Ogura and Phillips' (1962) derivation of their anelastic model requires the dimensionless stability of the background state to be of the order of the Mach number squared, i.e., (hsc=) d=dz = O("2) as " ! 0. This guarantees the characteristic time scales of advection and internal waves to be of the same order of magnitude in the Mach number, ". Assuming the flow evolution to develop on this time scale only, they asymptotically eliminate the fast sound waves and arrive at the anelastic model featuring internal waves and advection only. For typical values " 1=30, however, the model implies unrealistically weak vertical variation of potential temperature across the troposphere of less than one Kelvin. Later generalizations of the anelastic model (Dutton & Fichtl (1969), Lipps & Hemler (1982)), and Durran's pseudo-incompressible model (Durran 1989) are argued to be valid for stronger stratification, yet their derivations do not account for the fact that one must deal with three separated time scales for sound, internal waves, and advection in this case. Klein et al. (2010) address this issue and show that the Lipps & Hemler anelastic model and Durran's pseudo-incompressible model should be valid up to stratifications (hsc=) d=dz < O("2=3) corresponding to vertical variations of potential temperature of hsc 0 < 30:::50 K. Achatz et al. (2010) study, i.a., the evolution of large amplitude, short wave internal wave packets using WKB theory. They show that the pseudo-incompressible model is asymptotically consistent with the compressible Euler equations even for leading-order stratifications in the considered WKB regime. In contrast, they find this not to be true for the Lipps & Hemler anelastic model, whose asymptotics differs in the first-order terms. We show in the present paper that the anelastic model does correctly represent the amplification of wave packets as they travel upwards in the atmosphere, but that the excitation of higher harmonics and the wave-meanflow-interactions are not consistent with the asymptotics of the compressible Euler and pseudo-incompressible models.