Generalizing the Boussinesq approximation to stratified compressible flow

Dale R. Durran; Akio Arakawa

doi:10.1016/j.crme.2007.08.010

Boussinesq approximation, geophysical flows

Generalizing the Boussinesq approximation to stratified compressible flow
[Généralisation de l'approximation de Boussinesq à un flux compressible stratifié]

Dale R. Durran ¹ ; Akio Arakawa ²

¹ Department of Atmospheric Sciences, University of Washington, Seattle, WA 98195, USA
² Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, CA 90095, USA

Comptes Rendus. Mécanique, Joseph Boussinesq, a Scientist of bygone days and present times, Volume 335 (2007) no. 9-10, pp. 655-664.

Résumé
Abstract

Les simplifications requises pour l'application de l'approximation de Boussinesq à un fluide compressible sont comparées avec celles d'un fluide incompressible. Le niveau plus élevé d'approximation requis pour décrire la conservation de la masse dans un fluide compressible stratifié au moyen de l'équation de continuité de Boussinesq conduit au développement de différents systèmes d'équations « anélastiques » qui peuvent être considérés comme des généralisations de l'approximation originale de Boussinesq. Ces systèmes anélastiques filtrent les ondes sonores tout en permettant une représentation des perturbations non-acoustiques plus précise que celle qui est obtenue avec le système de Boussinesq. Nous comparons les propriétés de plusieurs systèmes anélastiques quant à la conservation de l'énergie, sous l'hypothèse que les perturbations des variables thermodynamiques à partir d'un état hydrodynamique d'équilibre de référence sont faibles. Nous montrons que le système « pseudo-incompressible » conserve l'énergie cinétique totale et l'énergie anélastique statique sèche sans qu'il soit nécessaire de modifier aucune des équations principales, sauf l'équation de continuité. Au contraire, d'autres systèmes anélastiques conservant l'énergie requièrent des approximations supplémentaires dans d'autres équations principales. Le système « pseudo-incompressible » inclut les effets des changements de température sur la densité dans l'équation de conservation de la masse, alors que cet effet est omis dans d'autres systèmes anélastiques. Nous présentons une généralisation de l'équation pseudo-incompressible et la comparons avec l'équation diagnostique de continuité pour un flux quasi-hydrostatique dans un système de coordonnées transformé dans lequel la coordonnée verticale est fonction de la seule pression.

The simplifications required to apply the Boussinesq approximation to compressible flow are compared with those in an incompressible fluid. The larger degree of approximation required to describe mass conservation in a stratified compressible fluid using the Boussinesq continuity equation has led to the development of several different sets of ‘anelastic’ equations that may be regarded as generalizations of the original Boussinesq approximation. These anelastic systems filter sound waves while allowing a more accurate representation of non-acoustic perturbations in compressible flows than can be obtained using the Boussinesq system. The energy conservation properties of several anelastic systems are compared under the assumption that the perturbations of the thermodynamic variables about a hydrostatically balanced reference state are small. The ‘pseudo-incompressible’ system is shown to conserve total kinetic and anelastic dry static energy without requiring modification to any governing equation except the mass continuity equation. In contrast, other energy conservative anelastic systems also require additional approximations in other governing equations. The pseudo-incompressible system includes the effects of temperature changes on the density in the mass conservation equation, whereas this effect is neglected in other anelastic systems. A generalization of the pseudo-incompressible equation is presented and compared with the diagnostic continuity equation for quasi-hydrostatic flow in a transformed coordinate system in which the vertical coordinate is solely a function of pressure.

Reçu le : 2007-01-15
Accepté le : 2007-05-14
Publié le : 2007-09-01

DOI : 10.1016/j.crme.2007.08.010

Keywords: Fluid mechanics, Boussinesq approximation, Anelastic, Pseudo-incompressible, Pressure coordinates, Pseudo-height coordinates, Low Mach number
Mots-clés : Mécanique des fluides, Approximation de Boussinesq, Anélastique, Pseudo-incompressible, Coordonnées de pression, Coordonnées pseudo-hauteur, Nombre de Mach faible

Affiliations des auteurs :

Dale R. Durran ¹ ; Akio Arakawa ²

¹ Department of Atmospheric Sciences, University of Washington, Seattle, WA 98195, USA
² Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, CA 90095, USA

@article{CRMECA_2007__335_9-10_655_0,
     author = {Dale R. Durran and Akio Arakawa},
     title = {Generalizing the {Boussinesq} approximation to stratified compressible flow},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {655--664},
     publisher = {Elsevier},
     volume = {335},
     number = {9-10},
     year = {2007},
     doi = {10.1016/j.crme.2007.08.010},
     language = {en},
}

TY  - JOUR
AU  - Dale R. Durran
AU  - Akio Arakawa
TI  - Generalizing the Boussinesq approximation to stratified compressible flow
JO  - Comptes Rendus. Mécanique
PY  - 2007
SP  - 655
EP  - 664
VL  - 335
IS  - 9-10
PB  - Elsevier
DO  - 10.1016/j.crme.2007.08.010
LA  - en
ID  - CRMECA_2007__335_9-10_655_0
ER  -

%0 Journal Article
%A Dale R. Durran
%A Akio Arakawa
%T Generalizing the Boussinesq approximation to stratified compressible flow
%J Comptes Rendus. Mécanique
%D 2007
%P 655-664
%V 335
%N 9-10
%I Elsevier
%R 10.1016/j.crme.2007.08.010
%G en
%F CRMECA_2007__335_9-10_655_0

Dale R. Durran; Akio Arakawa. Generalizing the Boussinesq approximation to stratified compressible flow. Comptes Rendus. Mécanique, Joseph Boussinesq, a Scientist of bygone days and present times, Volume 335 (2007) no. 9-10, pp. 655-664. doi : 10.1016/j.crme.2007.08.010. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.08.010/

Bibliographie
Cité par

[1] G.K. Batchelor An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967 (615 p)

[2] E.A. Spiegel; G. Veronis On the Boussinesq approximation for a compressible fluid, Astrophys. J., Volume 131 (1960), pp. 442-447

[3] Y. Ogura; N. Phillips Scale analysis for deep and shallow convection in the atmosphere, J. Atmos. Sci., Volume 19 (1962), pp. 173-179

[4] F. Lipps; R. Hemler A scale analysis of deep moist convection and some related numerical calculations, J. Atmos. Sci., Volume 29 (1982), pp. 2192-2210

[5] F. Lipps On the anelastic approximation for deep convection, J. Atmos. Sci., Volume 47 (1990), pp. 1794-1798

[6] D.R. Durran Improving the anelastic approximation, J. Atmos. Sci., Volume 46 (1989), pp. 1453-1461

[7] J.R. Holton An Introduction to Dynamic Meteorology, Elsevier, Amsterdam, 2004 (535 p)

[8] B.J. Hoskins; F.P. Bretherton Atmospheric frontogenesis models: Mathematical formulation and solution, J. Atmos. Sci., Volume 29 (1972), pp. 11-37

[9] L.B. Nance; D. Durran A comparison of three anelastic systems and the pseudo-incompressible system, J. Atmos. Sci., Volume 51 (1994), pp. 3549-3565

[10] J. Boussinesq Théorie Analytique de la Chaleur. Vol. II, Gauthier–Villars, Paris, 1903 (625 p)

[11] P.-A. Bois A unified asymptotic theory of the anelastic approximation in geophysical gases and liquids, Mech. Res. Comm., Volume 33 (2006), pp. 628-635

[12] P.M. Morse Vibration and Sound, McGraw–Hill, New York, 1948 (468 p)

[13] R. Wilhelmson; Y. Ogura The pressure perturbation and the numerical modeling of a cloud, J. Atmos. Sci., Volume 29 (1972), pp. 1295-1307

[14] P.R. Bannon On the anelastic approximation for a compressible atmosphere, J. Atmos. Sci., Volume 53 (1996), pp. 3618-3628

[15] D.R. Durran, Generalizing the Boussinesq continuity equation to low-Mach-number compressible stratified flow, J. Fluid Mech. (2007), in press

[16] D.R. Durran Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, Springer-Verlag, New York, 1999 (465 p)

M. Perrot; F. Lemarié; T. Dubos Energetically Consistent Eddy‐Diffusivity Mass‐Flux Convective Schemes: 1. Theory and Models, Journal of Advances in Modeling Earth Systems, Volume 17 (2025) no. 1 | DOI:10.1029/2024ms004273
Zhe Wu; Changhao Wang; Ran Lang; Yanbo Li; Bingyuan Hong The distribution of components in hydrogen-blended pipelines under different gas stream injection pattern, Fuel, Volume 375 (2024), p. 132577 | DOI:10.1016/j.fuel.2024.132577
Weihong Qian; Jun Du Anomaly Format of Atmospheric Governing Equations with Climate as a Reference Atmosphere, Meteorology, Volume 1 (2022) no. 2, p. 127 | DOI:10.3390/meteorology1020008
Yasser Abbassi; Hossein Ahmadikia; Ehsan Baniasadi Prediction of pollution dispersion under urban heat island circulation for different atmospheric stratification, Building and Environment, Volume 168 (2020), p. 106374 | DOI:10.1016/j.buildenv.2019.106374
Jacob Koshy Mulamootil; Sukanta Kumar Dash Augmentation and diminution of non-Boussinesq effects due to non-Newtonian power-law behavior in natural convection, International Journal of Thermal Sciences, Volume 151 (2020), p. 106263 | DOI:10.1016/j.ijthermalsci.2020.106263
Nikolaos D. Katopodes Geophysical Effects, Free-Surface Flow (2019), p. 710 | DOI:10.1016/b978-0-12-815489-2.00010-1
Yuan Lian; Roger V. Yelle Damping of gravity waves by kinetic processes in Jupiter's thermosphere, Icarus, Volume 329 (2019), p. 222 | DOI:10.1016/j.icarus.2019.04.001
David G. Dritschel; Steven J. Böing; Douglas J. Parker; Alan M. Blyth The moist parcel‐in‐cell method for modelling moist convection, Quarterly Journal of the Royal Meteorological Society, Volume 144 (2018) no. 715, p. 1695 | DOI:10.1002/qj.3319
Roger Grimshaw; Dave Broutman; Brian Laughman; Stephen D. Eckermann Solitary Waves and Undular Bores in a Mesosphere Duct, Journal of the Atmospheric Sciences, Volume 72 (2015) no. 11, p. 4412 | DOI:10.1175/jas-d-14-0351.1
Felix Rieper; Stefan Hickel; Ulrich Achatz A Conservative Integration of the Pseudo-Incompressible Equations with Implicit Turbulence Parameterization, Monthly Weather Review, Volume 141 (2013) no. 3, p. 861 | DOI:10.1175/mwr-d-12-00026.1
Jahrul M. Alam Toward a Multiscale Approach for Computational Atmospheric Modeling, Monthly Weather Review, Volume 139 (2011) no. 12, p. 3906 | DOI:10.1175/2011mwr3533.1
Dale R. Durran Stabilizing Fast Waves, Numerical Techniques for Global Atmospheric Models, Volume 80 (2011), p. 105 | DOI:10.1007/978-3-642-11640-7_6
Ulrich Achatz; R. Klein; F. Senf Gravity waves, scale asymptotics and the pseudo-incompressible equations, Journal of Fluid Mechanics, Volume 663 (2010), pp. 120-147 | DOI:10.1017/s0022112010003411 | Zbl:1205.76071
Akio Arakawa; Celal S. Konor Unification of the Anelastic and Quasi-Hydrostatic Systems of Equations, Monthly Weather Review, Volume 137 (2009) no. 2, p. 710 | DOI:10.1175/2008mwr2520.1

Cité par 14 documents. Sources : Crossref, zbMATH

Commentaires - Politique