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Atmospheric dynamics

The fundament of our theoretical studies to charac​terize a variety of scale-depen​dent pheno​mena in the atmos​phere is a unified approach [1,2,3,4], which is based on multiple-scales asymptotics. [5], Important results achieved up to now are


- Model equations for atmospheric motions on planetary spatial scales [6,7,8], as they are relevant for applications in the climate research, [9],
Change in mean air temperature
at ground level in 2100 for a standard scenario of carbon dioxide emissions.
(Picture provided by
S. Rahmsdorf, PIK Potsdam)

- a theory for the temporal evolution of cyclones, which incorpo​rates the inter​action between the internal flow structure, the back​ground flow, and the inclina​tion of the center​line with respect to the vertical. Further​more, it results in the determi​nation of the effec​tive direction of motion and the vortex speed, [10,11],

- two mathematical models for the evolution of deep convective clouds and their interaction with the surrounding atmosphere, [12,13,14,15],

Theory and Simulation of Deep Convective Clouds

Clouds play a decisive role in both the dai­ly wea­ther pat­tern and the long-­term cli­mate varia­tion. They con­sti­tute a moi­sture reser­voir carried by the wind and re­pre­sent the pre­limi­nary stage of pre­ci­pita­tion. By reflec­tion, ab­sorp­tion, and trans­mission of elec​tro​mag-​netic waves in the visi­ble and infra­red spectra they in­fluence directly the atmos­phere's heat budget.

For theory development and computer simu​lation clouds pose a par​ticular chal­lenge since they are deter­mined by the inter­action of a multi​tude of indivi­dual pro­cesses. Some of them take place in the size range of small cloud drop­lets (micrometres), some in the size range of typical turbu­lent flow fluctua­tions (metres), some in the size range of charac­teristical cumulus clouds (one to ten kilo­metres). Large strato­cumulus cloud layers above the oceans even span several thou­sands of kilo­metres. For this rea­son cloud pro­cesses belong to the class of multi scale phe­no­mena inve­stigated inten­sely by natural scien­tists and mathe­mati­cians these days.

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- reduced model equations describing the structure and temporal evolution of different types of atmospheric boundary layers. This includes, in one case, the analysis of model uncertainties, [16,17,17a],

Effects of surface properties on atmospheric boundary layer flows

This project is concerned with a systematic study of effects of the under​­lying surface on atmos-​­pheric flows and to quan­tify the impact of uncer​­tain­ties in sur­face proper­ties on the accu­racy of the boun­dary layer models. The multi­scale asymp­totic method is used to model the boun­dary layer pro­ces­ses captu­ring the spa­tial and tem­poral scales of inte­rests to­gether with the non­linear inter­actions therein. The poly­no­mial chaos method is used in the charac­teri­za­tion of the un­cer­tainty in the flow quan­ti­ties as func­tions of random model inputs such as sur­face rough­ness un­cer­tain­ty.

One of the goals of this in­vesti­gation is to improve on the repre­sen­tation of boun­dary layer pr­ocesses in nume­rical models and also suggest ap­pro­priate coup­ling strate­gies between a boun­dary layer model and the free atmos­phere model. We hope that this will lead to a more accu­rate weather pre­dic­tion and climate fore­casts.

- a critical discussion [18], of well established so called "soundproof" models (where no sound waves are present). This includes different versions of anelastic models [19,20,21,22]and the pseudo-incompressible model[23].

- stochastic models, which efficiently represent certain aspects of weather statistics and planetary fluid mechanics relevant for our climate, [24,25,26,27].

[ 1] Klein, R. (2004) An applied mathematical view of meteorological modeling. Applied Mathematics Entering the 21st century; Invited talks from the ICIAM 2003 Congress., 116 . pp. 177-219.

[ 2] MetStroem lecture notes

[ 3] Klein, R. (2008) An Unified Approach to Meteorological Modelling Based on Multiple-Scales Asymptotics. Advances in Geosciences, 15 . pp. 23-33.

[ 4] Klein, R. (2010) Scale-Dependent Asymptotic Models for Atmospheric Flows. Annual Review of Fluid Mechanics, 42 . (In Press)

[ 5] Schneider, W. (1978) Mathematische Methoden in der Strömungsmechanik, Vieweg

[ 6] Majda, A.J. and Klein, R. (2003) Systematic Multiscale Models for the Tropics. Journal of Atmospheric Sciences, 60 pp. 393-408.

[ 7] Dolaptchiev, S. (2008) Asymptotic models for planetary scale atmospheric motions (phdthesis) Freie Universität Berlin

[ 8] Dolaptchiev, S. and Klein, R. (2009) Planetary geostrophic equations for the atmosphere with evolution of the barotropic flow. Dynamics of Atmospheres and Oceans Volume, 46 (1-4). pp. 46-61

[ 9] Petukhov, V. Et Al (2000)
CLIMBER-2: A climate system model of intermediate complexity. Part I: Model description and performance for the present climate, Journal Climate Dynamics, Volume 16, pp 1-17

[10] Mikusky Dissertation

[11] Mikusky, E. and Owinoh, A.Z. and Klein, R. (2005) On the influence of diabatic effects on the motion of 3D-mesoscale vortices within a baroclinic shear flow. In: Third MIT Conference on Computational Fluid and Solid Mechanics, June 14-17, 2005.

[12] Carqué, G. (2009) Derivation and Validation of an Asymptotic Column Model for Deep Convective Precipitating Clouds. PhD thesis, Freie Universität Berlin.

[13] Carque et al. ZIB/Reports; Carqué, G. and Schmidt, H. and Stevens, B. and Klein, R. (2008) Plausibility Check of an Asymptotic Column Model for Deep Convective Clouds. ZIB-Report, 08 (44)

Carqué, G. and Owinoh, A.Z. and Klein, R. and Majda, A. J. (2008) Asymptotic Scale Analysis of Precipitating Clouds. ZIB-Report , 08-03 . ISSN 1438-0064

[14] Majda, A. J. and Klein, R. (2006) Systematic Multiscale Models for Deep Convection on Mesoscales. Theoretical and Computational Fluid Dynamics, 20

[15] Ruprecht, D. and Klein, R. and Majda, A. J. (2009) Moisture - Gravity Wave Interactions in a Multiscale Environment. Journal of Atmospheric Sciences . (Submitted)

[16] Owinoh, A.Z. and Hunt, J. and Orr, A. and Clark, P. and Klein, R. and Fernando, H. and Nieuwstadt, F. (2005) Effects Of Changing Surface Heat Flux On Atmospheric Boundary-Layer Flow Over Flat Terrain. Boundary Layer Meterology, 116 (2). pp. 331-361.

[17] Klein R., Mikusky E., Owinoh A. (2005) Multiple Scales Asymptotics for Atmospheric Flows,
in: 4th European Conference of Mathematics, Stockholm, Sweden, 2004, Ari Laptev (ed.), 201--220; European Mathematical Society Publishing House,

[17a] Schmidt, H. and Oevermann, M. and Bastiaans, R.J.M. and Kerstein, A.R. (2009) A priori Tabulation of Turbulent Flame Speeds via a Combination of a Stochastic Mixing Model and Flamelet Generated Manifolds, Extended to Incorporate Strain Effects. ZIB Report, 09-09 . ISSN 1438-0064

[18] Klein, R. (2009) Asymptotics, structure, and integration of sound-proof atmospheric flow equations. Theoretical and Computational Fluid Dynamics, 23 (3). pp. 161-195. ISSN 0935-4964 (Print) 1432-2250 (Online)

[22] Bannon, P.R. (1996) On the anelastic approximation for a compressible Atmosphere (article)
Journal J.Atmosph.Sci., 53, pp 3618--3628

[23] Durran, D.R. (1989) Improving the anelastic approximation (article)
Journal of Atmosphere Sciences, 46, pp 1453--1461

[24] Petoukhov, V. and Eliseev, A. and Klein, R. and Oesterle, H. (2008) On the statistics of the free-troposphere synoptic component. Part I: An evaluation of skewnesses and estimation of the third-order moments contribution to the synoptic-scale dynamics and meridional fluxes of heat and humidity. Tellus, 60 (1). pp. 11-31.

[25] Horenko, I. and Dolaptchiev, S. and Eliseev, A. and Mokhov, I. and Klein, R. (2008) Metastable Decomposition of High-Dimensional Meteorological Data with Gaps. Journal of the Atmospheric Sciences, 65 (11). pp. 3479-3496.

[26] Horenko, I. and Klein, R. and Dolaptchiev, S. and Schütte, Ch. (2008) Automated Generation of Reduced Stochastic Weather Models I: Simultaneous Dimension and Model reduction for Time Series Analysis. Mult. Mod. Sim., 6 (4). pp. 1125-1145.

[27] Franzke, Ch. and Horenko, I. and Majda, A. J. and Klein, R. (2009) Systematic Metastable Atmospheric Regime Identification in an AGCM. Journal for Atmospheric Sciences, 66 . pp. 1997-2012.

Complete list of publications