Klein, R. (2009) Asymptotics, structure, and integration of soundproof atmospheric flow equations. Theoretical and Computational Fluid Dynamics, 23 (3). pp. 161195. ISSN 09354964 (Print) 14322250 (Online)

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Official URL: http://www.springerlink.com/content/09354964
Abstract
Relative to the full compressible flow equations, soundproof models filter acoustic waves while maintaining advection and internal waves. Two wellknown soundproof models, an anelastic model by Bannon and Durran's pseudoincompressible model, are shown here to be structurally very close to the full compressible flow equations. Essentially, the anelastic model is obtained by suppressing @t in the mass continuity equation and slightly modifying the gravity term, whereas the pseudoincompressible model results from dropping @tp from the pressure equation. For length scales small compared to the density and pressure scale heights, the anelastic model reduces to the Boussinesq approximation, while the pseudoincompressible model approaches the zero Mach number, variable density flow equations. Thus, for small scales, both models are asymptotically consistent with the full compressible flow equations, yet the pseudoincompressible model is more general in that it remains valid in the presence of large density variations. For the relatively small density variations found in typical atmosphereocean flows, both models are found to yield very similar results, with deviations between models much smaller than deviations obtained when using different numerical schemes for the same model. This in agreement with Smolarkiewicz and Dörnbrack (2007). Despite these useful properties, neither model can be derived by a lowMach number asymptotic expansion for length scales comparable to the pressure scale height, i.e., for the regime they were originally designed for. Derivations of these models via scale analysis ignore an asymptotic time scale separation between advection and internal waves. In fact, only the classical Ogura & Phillips model, which assumes weak stratication of the order of the Mach number squared, can be obtained as a leadingorder model from systematic low Mach number asymptotic analysis. Issues of formal asymptotics notwithstanding, the close structural similarity of the anelastic and pseudoincompressible models to the full compressible flow equations makes them useful limit systems in building computational models for atmospheric flows. In the second part of the paper we propose a secondorder finitevolume projection method for the anelastic and pseudoincompressible models that observes these structural similarities. The method is applied to test problems involving free convection in a neutral atmosphere, the breaking of orographic waves at high altitudes, and the descent of a cold air bubble in the smallscale limit. The scheme is meant to serve as a starting point for the development of a robust compressible atmospheric flow solver in future work.
Item Type:  Article 

Subjects:  Mathematical and Computer Sciences > Mathematics Mathematical and Computer Sciences > Mathematics > Applied Mathematics 
Divisions:  Department of Mathematics and Computer Science > Institute of Mathematics Department of Mathematics and Computer Science > Institute of Mathematics > Geophysical Fluid Dynamics Group 
ID Code:  467 
Deposited By:  Ulrike Eickers 
Deposited On:  18 Jun 2009 13:53 
Last Modified:  03 Mar 2017 14:40 
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