Regime of Validity of Soundproof Atmospheric Flow Models

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 M ; 1 /30, this amounts to total variations of potential temperature across the troposphere of less than one Kelvin (i.e., to unrealistically weak stratification). Various generalizations of the original anelastic model have been proposed to remedy this issue. Later, Durran proposed the pseudoincompressible model following the same goals, but via a somewhat different route of argumentation. 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 (h sc /u)du/dz , M 2/3 , which corresponds to realistic variations of potential temperature u across the pressure scale height h sc of Duj h sc 0 , 30 K. Specifically, it is shown that (i) for (h sc /u)du/dz , M m with 0 , m , 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 M m ; and (iii) if m , 2 /3, 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.

An unstructured-mesh atmospheric model for nonhydrostatic dynamics

“…The governing equations underlying the local area edge-based model adopt the Lipps-Hemler nonhydrostatic anelastic system [16,17], suitable for simulation of a broad range of atmospheric flows [13,37]. For dry dynamics addressed in this paper, these equations govern the evolution of momentum and entropy, constrained by the incompressible-like mass continuity equation.…”

Scientific Final Report: COLLABORATIVE RESEARCH: CONTINUOUS DYNAMIC GRID ADAPTATION IN A GLOBAL ATMOSPHERIC MODEL: APPLICATION AND REFINEMENT

“…The collaborative work between Rupert Klein (Free University of Berlin), Piotr Smolarkiewicz and colleagues (Klein et al 2010) provided a multiple scale analysis showing that the regime of validity of the two soundproof models (the anelastic LippsHemler and pseudo-incompressible Durran models) covers stratification strengths corresponding to realistic variations of potential temperature < 30 degrees across the pressure scale height. This extends the classical results of Ogura and Phillips (1962) by two orders of magnitude.…”

“…Clearly, EULAG is proven capable of resolving small-scale internal waves in global simulations being driven by planetary wave instability, that is, the results are clearly multiscale. We believe this capability is due to model design (non-oscillatory forward-intime numerics and implicit treatment of potential temperature perturbation) as well as an inherent ability of the anelastic model to successfully capture multiscale effects between synoptic and mesoscale waves (Klein et al 2010).…”

Multi-scale waves in sound-proof global simulations with EULAG

“…Recent steps in this direction have been taken for the anelastic and pseudo-incompressible models. Using the method of distinguished limits, Klein et al (2010) demonstrated that for realistic values of atmospheric stratification (∆θ ∼ 50 K over 15 km depth), both the LH anelastic and pseudo-incompressible models are true multi-scale, sound-proof models in that they feature two asymptotically separated time scales (geostrophic advection and internal gravity waves). The requirement for sound-proof gravity waves to evolve asymptotically closely to compressible gravity waves is H p /H θ < O(M 2/3 ), where H p is the pressure scale height and M is the Mach number.…”