Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T18:24:18.473Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiple spatial scales in engineering and atmospheric low Mach number flows

Published online by Cambridge University Press:  15 June 2005

Rupert Klein
Rupert Klein*
Affiliation:
FB Mathematik & Informatik, Freie Universität Berlin, Zuse Institut, Takustr. 7, 14195 Berlin, Germany.
Get access

Abstract

The first part of this paper reviews the single time scale/multiplelength scale low Mach number asymptotic analysis by Klein (1995, 2004). This theory explicitly reveals the interaction of small scale,quasi-incompressible variable density flows with long wave linearacoustic modes through baroclinic vorticity generation and asymptoticaccumulation of large scale energy fluxes. The theory is motivated byexamples from thermoacoustics and combustion. In an almost obvious way specializations of this theory to a singlespatial scale reproduce automatically the zero Mach number variabledensity flow equations for the small scales, and the linear acousticequations with spatially varying speed of sound for the large scales. Following the same line of thought we show how a large number ofwell-known simplified equations of theoretical meteorology canbe derived in a unified fashion directly from the three-dimensionalcompressible flow equations through systematic (low Mach number) asymptotics. Atmospheric flows are, however, characterized by several singularperturbation parameters that appear in addition to the Mach number,and that are defined independently of any particular length or timescale associated with some specific flow phenomenon. These are theratio of the centripetal acceleration due to the earth's rotation vs.the acceleration of gravity, and the ratio of the sound speed vs. therotational velocity of points on the equator. To systematicallyincorporate these parameters in an asymptotic approach, we couple themwith the square root of the Mach number in a particular distinguished sothat we are left with a single small asymptotic expansion parameter,ε. Of course, more familiar parameters, such as the Rossby andFroude numbers may then be expressed in terms of ε as well. Next we consider a very general asymptotic ansatz involvingmultiple horizontal and vertical as well as multiple time scales.Various restrictions of the general ansatz to only one horizontal, onevertical, and one time scale lead directly to the family of simplifiedmodel equations mentioned above. Of course, the main purpose of the general multiple scales ansatz isto provide the means to derive true multiscale models which describeinteractions between the various phenomena described by the members ofthe simplified model family. In this context we will summarize a recentsystematic development of multiscale models for the tropics (with Majda).

Keywords

Low Mach number flowsmultiple scales asymptoticsatmospheric flows.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 3: Special issue on Low Mach Number Flows Conference , May 2005 , pp. 537 - 559
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bannon, P.R., On the anelastic approximation for a compressible atmosphere. J. Atmosphere Sci. 53 (1996) 3618–3628. 2.0.CO;2>CrossRef
Batchelor, G.K., The conditions for dynamical similarity of motions of a frictionless perfect gas atmosphere. Quart. J. Roy. Meteorol. Soc. 79 (1953) 224–235.
J.R. Biello and A.J. Majda, A new multiscale model for the madden julian oscillation. J. Atmosphere Sci. 62 (2005) in press.
N. Botta, R. Klein and A. Almgren, Asymptotic analysis of a dry atmosphere. ENUMATH, Jyväskylä, Finland (1999).
C. Bretherton and A. Sobel. The gill model and the weak temperature gradient (wtg) approximation. J. Atmosphere Sci. 60 (2003) 451–460.
Charney, J.G., A note on large-scale motions in the tropics. J. Atmosphere Sci. 20 (1963) 607–609. 2.0.CO;2>CrossRef
Dorofeev, S.B., Sidorov, V.P., Dvoinishnikov, A.E. and Breitung, W., Deflagration to detonation transition in large confined volume of lean hydrogen-air mixtures. Combustion & Flame 104 (1996) 95–110. CrossRef
Durran, D.R., Improving the anelastic approximation. J. Atmosphere Sci. 46 (1989) 1453–1461. 2.0.CO;2>CrossRef
Gill, A.E., Some simple solutions for heat-induced tropical circulation. Quart. J. Roy. Meteorol. Soc. 87 (1980) 447–462.
Held, I.M. and Hoskins, B.J., Large-scale eddies and the general circulation of the troposphere. Adv. in Geophysics 28 (1985) 3–31. CrossRef
Hunt, J.C.R., Richards, K.J. and Brighton, P.W.M., Stably stratified shear flow over low hills. Quart. J. Roy. Meteorol. Soc. 114 (1988) 859–886.
Klein, R., Semi-implicit extension of a godunov-type scheme based on low mach number asymptotics I: One-dimensional flow. J. Comput. Phys. 121 (1995) 213–237. CrossRef
Klein, R., Asymptotic analyses for atmospheric flows and the construction of asymptotically adaptive numerical methods. ZAMM 80 (2000) 765–777. 3.0.CO;2-1>CrossRef
R. Klein, An applied mathematical view of theoretical meteorology, in Applied Mathematics Entering the 21st Century; Invited talks from the ICIAM 2003 Congress, SIAM Proceedings in Applied Mathematics 116 (2004).
R. Klein, Multiple scales asymptotics for atmospheric flows, in Proceedings of the 4th European Conference on Mathematics, Stockholm, Sweden (2004).
R. Klein and S. Vater, Mathematical modelling in climate research. Technical report, Freie Universität, Berlin, Germany (2005).
H. Lamb, Hydrodynamics. Dover Publishers, New York (1981).
Lipps, F. and Hemler, R., A scale analysis of deep moist convection and some related numerical calculations. J. Atmosphere Sci. 39 (1982) 2192–2210. 2.0.CO;2>CrossRef
Lipps, F. and Hemler, R., Another look at the scale analysis of deep moist convection. J. Atmosphere Sci. 42 (1985) 1960–1964. 2.0.CO;2>CrossRef
Madden, R. and Julian, P., Description of global cell circulation cells in the tropics with a 40-50 day period. J. Atmosphere Sci. 29 (1972) 1109–1123. 2.0.CO;2>CrossRef
Majda, A.J. and Biello, J.A., A multiscale model for tropical intraseasonal oscillations. Proc. Natl. Acad. Sci. USA 101 (2004) 4736–4741. CrossRef
Majda, A. and Klein, R., Systematic multi-scale models for the tropics. J. Atmosphere Sci. 60 (2003) 393–408. 2.0.CO;2>CrossRef
Majda, A. and Sethian, J., The derivation and numerical solution of the equations for zero Mach number combustion. Combust. Sci. Tech. 42 (1985) 185–205. CrossRef
Matsuno, T., Quasi-geostrophic motions in the equatorial area. J. Meteorol. Soc. Jap. 44 (1966) 25–43.
Newley, T.M.J., Pearson, H.J. and Hunt, J.C.R., Stably stratified rotating flow through a group of obstacles. Geophys. Astrophys. Fluid Dynam. 58 (1991) 147–171. CrossRef
Ogura, Y. and Phillips, N.A., Scale analysis of deep moist convection and some related numerical calculations. J. Atmosphere Sci. 19 (1962) 173–179. 2.0.CO;2>CrossRef
J. Oliger and A. Sundstroem, Theoretical and practical aspects of some initial boundary value problems in fluid dynamics. NASA STI/Recon Technical Report N 77 25465 (November 1976).
J. Pedlosky, Ed., Geophysical Fluid Dynamics. Springer, Berlin, Heidelberg, New York, 2nd edition (1987).
Petoukhov, V., Ganopolski, A., Brovkin, V., Claussen, M., Eliseev, A., Kubatzki, C. and Rahmstorf, S., Climber-2: A climate system model of intermediate complexity. Part I: Model description and performance for the present climate. Climate Dynamics 16 (2000) 1–17. CrossRef
Worlikar, A.S. and Knio, O.M., Numerical study of oscillatory flow and heat transfer in a loaded thermoacoustic stack. Numer. Heat Transfer 35 (1999) 49–65. CrossRef
Worlikar, A.S., Knio, O.M. and Klein, R., Numerical simulation of a thermoacoustic refrigerator. II: stratified flow around the stack. J. Comput. Fluids 144 (1998) 299–324.
Th. Schneider, N. Botta, R. Klein, K.J. Geratz, Extension of finite volume compressible flow solvers to multi-dimensional, variable density zero Mach number flow. J. Comput. Phys. 155 (1999) 248–286. CrossRef
Th. Schneider, R. Klein, E. Besnoin and O. Knio, Computational analysis of a thermoacoustic refrigerator, in Proceedings of the joint EAA/ASA meeting (March 1999).
Schochet, S., The mathematical theory of low mach number flows. ESAIM: M2AN 39 (2005) 441–458. CrossRef
Smiljanovski, V., Moser, V. and Klein, R., A capturing-tracking hybrid scheme for deflagration discontinuities. J. Combustion Theory Modeling 1 (1997) 183–215. CrossRef
Sobel, A.A., Nilsson, J. and Polvani, L., The weak temperature gradient approximation and balanced tropical moisture waves. J. Atmosphere Sci. 58 (2001) 3650–3665. 2.0.CO;2>CrossRef
Webster, P.J., Response of the tropical atmosphere to local steady forcing. Monthly Weather Review 100 (1972) 518–541. 2.3.CO;2>CrossRef
A.A. White, A view of the equations of meteorological dynamics and various approximations, in Large Scale Atmosphere-Ocean Dynamics I: Analytical Methods and Numerical Models. J. Norbury and I. Roulstone, Eds., Cambridge University Press (2002).
G.B. Whitham, Linear and Non Linear Waves. John Wiley (1974).
R.K. Zeytounian, Meteorological Fluid Dynamics. Number m5 in Lecture Notes in Physics. Springer, Heidelberg, Berlin, New York (1991).