Multiple spatial scales in engineering and atmospheric low Mach number flows.

*(English)*Zbl 1130.76326Summary: 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, \(\varepsilon\). Of course, more familiar parameters, such as the Rossby and Froude numbers may then be expressed in terms of \(\varepsilon\) 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).

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, \(\varepsilon\). Of course, more familiar parameters, such as the Rossby and Froude numbers may then be expressed in terms of \(\varepsilon\) 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).

##### MSC:

76B60 | Atmospheric waves (MSC2010) |

76M45 | Asymptotic methods, singular perturbations applied to problems in fluid mechanics |

76Q05 | Hydro- and aero-acoustics |

86A10 | Meteorology and atmospheric physics |

PDF
BibTeX
XML
Cite

\textit{R. Klein}, ESAIM, Math. Model. Numer. Anal. 39, No. 3, 537--559 (2005; Zbl 1130.76326)

**OpenURL**

##### References:

[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 ). Zbl 1057.86507 · Zbl 1057.86507 |

[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 & Flame 104 ( 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 Geophysics 28 ( 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 . Zbl 0842.76053 · Zbl 0842.76053 |

[13] | R. Klein , Asymptotic analyses for atmospheric flows and the construction of asymptotically adaptive numerical methods . ZAMM 80 ( 2000 ) 765 - 777 . Zbl 1050.76056 · Zbl 1050.76056 |

[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 Mathematics 116 ( 2004 ). MR 2296270 | Zbl pre05263216 · Zbl 1163.86307 |

[15] | R. Klein , Multiple scales asymptotics for atmospheric flows , in Proceedings of the 4th European Conference on Mathematics, Stockholm, Sweden ( 2004 ). Zbl 1137.86308 · Zbl 1137.86308 |

[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 ). · JFM 26.0868.02 |

[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. USA 101 ( 2004 ) 4736 - 4741 . Zbl 1063.86004 · Zbl 1063.86004 |

[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). Zbl 0397.35067 · Zbl 0397.35067 |

[28] | J. Pedlosky , Ed ., Geophysical Fluid Dynamics. Springer, Berlin, Heidelberg, New York, 2nd edition (1987). Zbl 0429.76001 · Zbl 0429.76001 |

[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 Dynamics 16 ( 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 Transfer 35 ( 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. Fluids 144 ( 1998 ) 299 - 324 . Zbl 0920.76061 · Zbl 0920.76061 |

[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 . Zbl 0968.76054 · Zbl 0968.76054 |

[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: M2AN 39 ( 2005 ) 441 - 458 . Numdam | Zbl 1094.35094 · Zbl 1094.35094 |

[35] | V. Smiljanovski , V. Moser and R. Klein , A capturing-tracking hybrid scheme for deflagration discontinuities . J. Combustion Theory Modeling 1 ( 1997 ) 183 - 215 . Zbl 0956.76070 · Zbl 0956.76070 |

[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 Review 100 ( 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 ). MR 1947568 | Zbl 1036.86001 · Zbl 1036.86001 |

[39] | G.B. Whitham , Linear and Non Linear Waves . John Wiley ( 1974 ). Zbl 0373.76001 · Zbl 0373.76001 |

[40] | R.K. Zeytounian , Meteorological Fluid Dynamics . Number m5 in Lecture Notes in Physics. Springer, Heidelberg, Berlin, New York ( 1991 ). MR 1177174 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.