The fundament of our theoretical studies to characterize a variety of scale-dependent phenomena in the atmosphere is a unified approach [1,2,3,4], which is based on multiple-scales asymptotics. [5], Important results achieved up to now are

Theory:

- 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 incorporates the interaction between the internal flow structure, the background flow, and the inclination of the centerline with respect to the vertical. Furthermore, it results in the determination of the effective 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 electromag-netic waves in the visible and infrared spectra they influence directly the atmosphere's heat budget.

For theory development and computer simulation clouds pose a particular challenge since they are determined by the interaction of a multitude 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],

- 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].## Effects of surface properties on atmospheric boundary layer flows

This project is concerned with a systematic study of effects of the underlying surface on atmos-pheric flows and to quantify the impact of uncertainties 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.

- 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)

[ 2] MetStroem lecture notes

[ 3] Klein, R. (2008)

[ 4] Klein, R. (2010)

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

[ 6] Majda, A.J. and Klein, R. (2003)

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

[ 8] Dolaptchiev, S. and Klein, R. (2009)

[ 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)

[12] Carqué, G. (2009)

[13] Carque et al. ZIB/Reports; Carqué, G. and Schmidt, H. and Stevens, B. and Klein, R. (2008)

Carqué, G. and Owinoh, A.Z. and Klein, R. and Majda, A. J. (2008)

[14] Majda, A. J. and Klein, R. (2006)

[15] Ruprecht, D. and Klein, R. and Majda, A. J. (2009)

[16] Owinoh, A.Z. and Hunt, J. and Orr, A. and Clark, P. and Klein, R. and Fernando, H. and Nieuwstadt, F. (2005)

[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)

[18] Klein, R. (2009)

[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)

[25] Horenko, I. and Dolaptchiev, S. and Eliseev, A. and Mokhov, I. and Klein, R. (2008)

[26] Horenko, I. and Klein, R. and Dolaptchiev, S. and Schütte, Ch. (2008)

[27] Franzke, Ch. and Horenko, I. and Majda, A. J. and Klein, R. (2009)