Advances in Geosciences www.adv-geosci.net 1680-7340 1680-7359 15 Topics in modern geophysical fluid dynamics 2008 10.5194/adgeo-15-23-2008 http://www.adv-geosci.net/15/23/2008/ http://www.adv-geosci.net/15/23/2008/adgeo-15-23-2008.html http://www.adv-geosci.net/15/23/2008/adgeo-15-23-2008.pdf 23 33 2008-03-18 An unified approach to meteorological modelling based on multiple-scales asymptotics R. Klein rupert.klein@zib.de FB Mathematik & Informatik, Freie Universität Berlin, Arnimallee 2–6, 14195 Berlin, Germany In 2003, the author suggested a mathematical framework for the derivation of reduced meteorological models at a Mathematics conference (5th ICIAM, Sydney, Australia), (Klein, 2004). The framework consists of (i) non-dimensionalization of the 3-D compressible flow equations on the rotating sphere, (ii) identification of universal non-dimensional parameters, (iii) distinguished limits between these and additional problem-specific parameters, and (iv) multiple scales expansions in the remaining small parameter ε. This parameter may be interpreted as the cubic root of the centripetal acceleration due to the Earth's rotation divided by the acceleration of gravity, see also Keller (1951), Eq. (10). For the mojority of reduced models of theoretical meteorology that we have come across, the approach allowed us to generate systematic derivations starting directly from the 3-D compressible flow equations on the rotating sphere. The framework's potential fully shows in multiscale interaction studies such as Klein (2006), in which we incorporated bulk microphysics closures for moist processes and derived scale interaction models for deep convection. Currently, we study the structure, evolution, and motion of Hurricane strength H1/H2 vortices (Mikusky, 2007), large-scale stratocumulus cloud decks, and planetary-synoptic scale interaction models which should be relevant for Earth System Models of Intermediate Complexity (EMICs). Here we summarize the general framework and use the example of quasi-geostrophic theory to demonstrate its application. Biello, J. and Majda, A.: Transformations for temperature flux in multiscale models of the tropics, Theor. Comp. Fluid Dyn., 20, 405–420, 2006. Botta, N., Klein, R., and Almgren, A.: Dry atmosphere asymptotics, PIK-Report 55, Potsdam Institute for Climate Impact Research, 1999. Botta, N., Klein, R., and Almgren, A.: Asymptotic Analysis of a Dry Atmosphere, in: ENUMATH 99 - Proc. 3rd Europ. Conf. on Num. Math., Jyväskylä, Finland, edited by: Neittaanmäki, P. and Tiihonen, T., World Scientific, Singapore, 2000. E, W. and Engquist, B.: The heterogeneous multiscale method, Comm. Math.\ Sci., 1, 87–132, 2003. Grabowski, W.: An improved framework for superparameterization, J. Atmos. Sci., 61, 1940–1952, 2004. Keller, J. and Ting, L.: Approximate equations for large scale atmospheric motions, Internal Report, Inst. for Mathematics & Mechanics (renamed to Courant Institute of Mathematical Sciences in 1962), NYU, (http://www.arxiv.org/abs/physics/0606114), 1951. Klein, R.: Asymptotic Analyses for Atmospheric Flows and the Construction of Asymptotically Adaptive Numerical Methods, ZAMM, 80, 765–777, 2000. Klein, R.: An applied mathematical view of meteorological modelling, in: Applied Mathematics Entering the 21st century, Invited talks from the ICIAM 2003 Congress, Vol. 116, SIAM Proceedings in Applied Mathematics, 177–219, 2004. Klein, R. and Majda, A.: Systematic multiscale models for deep convection on mesoscales, Theor. Comp. Fluid Dyn., 20, 525–551, 2006. Klein, R., Mikusky, E., and Owinoh, A.: Multiple scales asymptotics for atmospheric flows, in: 4th European Conference of Mathematics, Stockholm, Sweden, edited by: Laptev, A., European Mathematical Society Publishing House, 201–220, 2004. Majda, A.: Multiscale Models with Moisture and Systematic Strategies for Superparameterization, J. Atmos. Sci., 64, 2726–2734, 2007a. Majda, A.: New multiscale models and self-similarity in tropical convection, J. Atmos. Sci., 64, 1393–1404, 2007b. Majda, A. and Biello, J.: A multiscale model for tropical intraseasonal oscillations, Proc. Natl. Acad. Sci. USA, 101, 4736–4741, 2004. Majda, A J. and Klein, R.: Systematic Multiscale Models for the Tropics, J. Atmos. Sci., 60, 393–408, 2003. Mikusky, E.: On the structure of concentrated atmospheric vortices in the gradient wind regime and their motion on synoptic scales, Ph.D. thesis, Hamburg University, 2007. Mikusky, E., Owinoh, A., and Klein, R.: On the influence of diabatic effects on the motion of 3D-mesoscale vortices within a baroclinic shear flow, in: Computational Fluid and Solid Mechanics 2005, Elsevier, Amsterdam, 766–768, 2005. Pedlosky, J.: Geophysical Fluid Dynamics, Springer-Verlag, New York, 2. Edn., 1987. Zeytounian, R.: Asymptotic modelling of atmospheric flows, Springer-Verlag, Berlin, Heidelberg, New York, 1990.