AMM

Abstract

The design regime of established sound-proof flow models (anelastic, pseudo-incompressible) involves tropospheric flows at length scales of 10-100 km and associated advective time scales. Most derivations of such models require "weak background stratification" in some sense. In this lecture I demonstrate through arguments of multiple scales asymptotics that this constraint can be quantified: The anelastic and pseudo-incompressible approximations should be valid up to dimensionless potential temperature stratifications of the order of the Mach number to the power 2/3.

Building on the morning lecture "Weakly compressible flows I: regime of validity of sound-proof appoximaitons", I will discuss regimes of length scales different from the orginal design regime of sound-proof models, which involves tropospheric flows at horizontal scales of 10-100 km and associated advective scales. In particular, I will show that short wave internal wave packets in the stratosphere can be described by the pseudo-incompressible approximation, but that the anelastic model looses its formal validity in this case.

At very small scales the pseudo-incompressible model reduces to the zero Mach number variable density equations, whereas the anelastic model reduces to the more restrictive Boussinesq approximation.

Sound-proofing appoximations affect the thermodynamic relationships. I will introduce a definition for the notion of "thermodynamic consistency" of sound-proof approximations, show how complying anelastic and pseudo-incompressible models can be formulated, and discuss their regimes of validity.

References:

Klein R., Achatz U., Bresch D., Knio O.M., and Smolarkiewicz P.K., Regime of Validity of Sound-Proof Atmospheric Flow Models, JAS vol. 67, pp 3226--3237 (2010)

Klein R., Scale-Dependent Asymptotic Models for Atmospheric Flows, Ann. Rev. Fluid Mech., 42, 249-274 (2010)