Multiple spatial scales in engineering and atmospheric low Mach number flows

Rupert Klein

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 39, Issue: 3, page 537-559
  • ISSN: 0764-583X

Abstract

top
The first part of this paper reviews the single time scale/multiple length 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 linear acoustic modes through baroclinic vorticity generation and asymptotic accumulation of large scale energy fluxes. The theory is motivated by examples from thermoacoustics and combustion. In an almost obvious way specializations of this theory to a single spatial scale reproduce automatically the zero Mach number variable density flow equations for the small scales, and the linear acoustic equations with spatially varying speed of sound for the large scales. Following the same line of thought we show how a large number of well-known simplified equations of theoretical meteorology can be derived in a unified fashion directly from the three-dimensional compressible flow equations through systematic (low Mach number) asymptotics. Atmospheric flows are, however, characterized by several singular perturbation parameters that appear in addition to the Mach number, and that are defined independently of any particular length or time scale associated with some specific flow phenomenon. These are the ratio of the centripetal acceleration due to the earth's rotation vs. the acceleration of gravity, and the ratio of the sound speed vs. the rotational velocity of points on the equator. To systematically incorporate these parameters in an asymptotic approach, we couple them with the square root of the Mach number in a particular distinguished so that we are left with a single small asymptotic expansion parameter, ε. Of course, more familiar parameters, such as the Rossby and Froude numbers may then be expressed in terms of ε as well. Next we consider a very general asymptotic ansatz involving multiple horizontal and vertical as well as multiple time scales. Various restrictions of the general ansatz to only one horizontal, one vertical, and one time scale lead directly to the family of simplified model equations mentioned above. Of course, the main purpose of the general multiple scales ansatz is to provide the means to derive true multiscale models which describe interactions between the various phenomena described by the members of the simplified model family. In this context we will summarize a recent systematic development of multiscale models for the tropics (with Majda).

How to cite

top

Klein, Rupert. "Multiple spatial scales in engineering and atmospheric low Mach number flows." ESAIM: Mathematical Modelling and Numerical Analysis 39.3 (2010): 537-559. <http://eudml.org/doc/194274>.

@article{Klein2010,
abstract = { The first part of this paper reviews the single time scale/multiple length 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 linear acoustic modes through baroclinic vorticity generation and asymptotic accumulation of large scale energy fluxes. The theory is motivated by examples from thermoacoustics and combustion. In an almost obvious way specializations of this theory to a single spatial scale reproduce automatically the zero Mach number variable density flow equations for the small scales, and the linear acoustic equations with spatially varying speed of sound for the large scales. Following the same line of thought we show how a large number of well-known simplified equations of theoretical meteorology can be derived in a unified fashion directly from the three-dimensional compressible flow equations through systematic (low Mach number) asymptotics. Atmospheric flows are, however, characterized by several singular perturbation parameters that appear in addition to the Mach number, and that are defined independently of any particular length or time scale associated with some specific flow phenomenon. These are the ratio of the centripetal acceleration due to the earth's rotation vs. the acceleration of gravity, and the ratio of the sound speed vs. the rotational velocity of points on the equator. To systematically incorporate these parameters in an asymptotic approach, we couple them with the square root of the Mach number in a particular distinguished so that we are left with a single small asymptotic expansion parameter, ε. Of course, more familiar parameters, such as the Rossby and Froude numbers may then be expressed in terms of ε as well. Next we consider a very general asymptotic ansatz involving multiple horizontal and vertical as well as multiple time scales. Various restrictions of the general ansatz to only one horizontal, one vertical, and one time scale lead directly to the family of simplified model equations mentioned above. Of course, the main purpose of the general multiple scales ansatz is to provide the means to derive true multiscale models which describe interactions between the various phenomena described by the members of the simplified model family. In this context we will summarize a recent systematic development of multiscale models for the tropics (with Majda). },
author = {Klein, Rupert},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Low Mach number flows; multiple scales asymptotics; atmospheric flows.},
language = {eng},
month = {3},
number = {3},
pages = {537-559},
publisher = {EDP Sciences},
title = {Multiple spatial scales in engineering and atmospheric low Mach number flows},
url = {http://eudml.org/doc/194274},
volume = {39},
year = {2010},
}

TY - JOUR
AU - Klein, Rupert
TI - Multiple spatial scales in engineering and atmospheric low Mach number flows
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 39
IS - 3
SP - 537
EP - 559
AB - The first part of this paper reviews the single time scale/multiple length 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 linear acoustic modes through baroclinic vorticity generation and asymptotic accumulation of large scale energy fluxes. The theory is motivated by examples from thermoacoustics and combustion. In an almost obvious way specializations of this theory to a single spatial scale reproduce automatically the zero Mach number variable density flow equations for the small scales, and the linear acoustic equations with spatially varying speed of sound for the large scales. Following the same line of thought we show how a large number of well-known simplified equations of theoretical meteorology can be derived in a unified fashion directly from the three-dimensional compressible flow equations through systematic (low Mach number) asymptotics. Atmospheric flows are, however, characterized by several singular perturbation parameters that appear in addition to the Mach number, and that are defined independently of any particular length or time scale associated with some specific flow phenomenon. These are the ratio of the centripetal acceleration due to the earth's rotation vs. the acceleration of gravity, and the ratio of the sound speed vs. the rotational velocity of points on the equator. To systematically incorporate these parameters in an asymptotic approach, we couple them with the square root of the Mach number in a particular distinguished so that we are left with a single small asymptotic expansion parameter, ε. Of course, more familiar parameters, such as the Rossby and Froude numbers may then be expressed in terms of ε as well. Next we consider a very general asymptotic ansatz involving multiple horizontal and vertical as well as multiple time scales. Various restrictions of the general ansatz to only one horizontal, one vertical, and one time scale lead directly to the family of simplified model equations mentioned above. Of course, the main purpose of the general multiple scales ansatz is to provide the means to derive true multiscale models which describe interactions between the various phenomena described by the members of the simplified model family. In this context we will summarize a recent systematic development of multiscale models for the tropics (with Majda).
LA - eng
KW - Low Mach number flows; multiple scales asymptotics; atmospheric flows.
UR - http://eudml.org/doc/194274
ER -

References

top
  1. P.R. Bannon, On the anelastic approximation for a compressible atmosphere. J. Atmosphere Sci.53 (1996) 3618–3628.  
  2. G.K. Batchelor, The conditions for dynamical similarity of motions of a frictionless perfect gas atmosphere. Quart. J. Roy. Meteorol. Soc.79 (1953) 224–235.  
  3. J.R. Biello and A.J. Majda, A new multiscale model for the madden julian oscillation. J. Atmosphere Sci.62 (2005) in press.  
  4. N. Botta, R. Klein and A. Almgren, Asymptotic analysis of a dry atmosphere. ENUMATH, Jyväskylä, Finland (1999).  
  5. C. Bretherton and A. Sobel. The gill model and the weak temperature gradient (wtg) approximation. J. Atmosphere Sci.60 (2003) 451–460.  
  6. J.G. Charney, A note on large-scale motions in the tropics. J. Atmosphere Sci.20 (1963) 607–609.  
  7. S.B. Dorofeev, V.P. Sidorov, A.E. Dvoinishnikov and W. Breitung, Deflagration to detonation transition in large confined volume of lean hydrogen-air mixtures. Combustion & Flame104 (1996) 95–110.  
  8. D.R. Durran, Improving the anelastic approximation. J. Atmosphere Sci.46 (1989) 1453–1461.  
  9. A.E. Gill, Some simple solutions for heat-induced tropical circulation. Quart. J. Roy. Meteorol. Soc.87 (1980) 447–462.  
  10. I.M. Held and B.J. Hoskins, Large-scale eddies and the general circulation of the troposphere. Adv. in Geophysics28 (1985) 3–31.  
  11. J.C.R. Hunt, K.J. Richards and P.W.M. Brighton, Stably stratified shear flow over low hills. Quart. J. Roy. Meteorol. Soc.114 (1988) 859–886.  
  12. R. Klein, 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.  
  13. R. Klein, Asymptotic analyses for atmospheric flows and the construction of asymptotically adaptive numerical methods. ZAMM80 (2000) 765–777.  
  14. 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 Mathematics116 (2004).  
  15. R. Klein, Multiple scales asymptotics for atmospheric flows, in Proceedings of the 4th European Conference on Mathematics, Stockholm, Sweden (2004).  
  16. R. Klein and S. Vater, Mathematical modelling in climate research. Technical report, Freie Universität, Berlin, Germany (2005).  
  17. H. Lamb, Hydrodynamics. Dover Publishers, New York (1981).  
  18. F. Lipps and R. Hemler, A scale analysis of deep moist convection and some related numerical calculations. J. Atmosphere Sci.39 (1982) 2192–2210.  
  19. F. Lipps and R. Hemler, Another look at the scale analysis of deep moist convection. J. Atmosphere Sci.42 (1985) 1960–1964.  
  20. R. Madden and P. Julian, Description of global cell circulation cells in the tropics with a 40-50 day period. J. Atmosphere Sci.29 (1972) 1109–1123.  
  21. A.J. Majda and J.A. Biello, A multiscale model for tropical intraseasonal oscillations. Proc. Natl. Acad. Sci. USA101 (2004) 4736–4741.  
  22. A. Majda and R. Klein, Systematic multi-scale models for the tropics. J. Atmosphere Sci.60 (2003) 393–408.  
  23. A. Majda and J. Sethian, The derivation and numerical solution of the equations for zero Mach number combustion. Combust. Sci. Tech.42 (1985) 185–205.  
  24. T. Matsuno, Quasi-geostrophic motions in the equatorial area. J. Meteorol. Soc. Jap.44 (1966) 25–43.  
  25. T.M.J. Newley, H.J. Pearson and J.C.R. Hunt, Stably stratified rotating flow through a group of obstacles. Geophys. Astrophys. Fluid Dynam.58 (1991) 147–171.  
  26. Y. Ogura and N.A. Phillips, Scale analysis of deep moist convection and some related numerical calculations. J. Atmosphere Sci.19 (1962) 173–179.  
  27. 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).  
  28. J. Pedlosky, Ed., Geophysical Fluid Dynamics. Springer, Berlin, Heidelberg, New York, 2nd edition (1987).  
  29. V. Petoukhov, A. Ganopolski, V. Brovkin, M. Claussen, A. Eliseev, C. Kubatzki and S. Rahmstorf, Climber-2: A climate system model of intermediate complexity. Part I: Model description and performance for the present climate. Climate Dynamics16 (2000) 1–17.  
  30. A.S. Worlikar and O.M. Knio, Numerical study of oscillatory flow and heat transfer in a loaded thermoacoustic stack. Numer. Heat Transfer35 (1999) 49–65.  
  31. A.S. Worlikar, O.M. Knio and R. Klein, Numerical simulation of a thermoacoustic refrigerator. II: stratified flow around the stack. J. Comput. Fluids144 (1998) 299–324.  
  32. Th. Schneider, N. Botta, R. Klein and 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.  
  33. 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).  
  34. S. Schochet, The mathematical theory of low mach number flows. ESAIM: M2AN39 (2005) 441–458.  
  35. V. Smiljanovski, V. Moser and R. Klein, A capturing-tracking hybrid scheme for deflagration discontinuities. J. Combustion Theory Modeling1 (1997) 183–215.  
  36. A.A. Sobel, J. Nilsson and L. Polvani, The weak temperature gradient approximation and balanced tropical moisture waves. J. Atmosphere Sci.58 (2001) 3650–3665.  
  37. P.J. Webster, Response of the tropical atmosphere to local steady forcing. Monthly Weather Review100 (1972) 518–541.  
  38. 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).  
  39. G.B. Whitham, Linear and Non Linear Waves. John Wiley (1974).  
  40. R.K. Zeytounian, Meteorological Fluid Dynamics. Number m5 in Lecture Notes in Physics. Springer, Heidelberg, Berlin, New York (1991).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.