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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.
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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

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