EU Regional School - Klein Seminar
Prof. Klein - Multiscale Dynamics of the Atmosphere-Ocean System and Related Computational Challenges
Freie Universität Berlin
Multiscale Dynamics of the Atmosphere-Ocean System and Related Computational Challenges provides insight into three separate topics that highlight several difficulties Earth Scientists face when modelling geophysical flows.
The first topic is related to fundamental modelling concepts. Textbooks of theoretical meteorology typically address some particular family of typical atmospheric flow phenomena in each of their chapters. These phenomena are theoretically explained by reference to reduced dynamical flow models that are shown to capture the mathematical essence of motions in the related flow regimes. This first part of the lecture elucidates how one may systematically define what is "a regime" through methods of dimensional analysis and asymptotics. It then shows how the majority of reduced models found in textbooks can be derived in a unified fashion from the full three-dimensional compressible flow equations for a moist gas on a rotating and gravitating sphere. This systematization of the "model zoo" then lends itself naturally to investigating interactions across scales by multiple scales methods, when more than one of the just discussed scale-dependent flow processes interact simultaneously. A particular example of such multiscale analyses will be a new theory for the intensification of tropical storms.
The second part of the school addresses conconclusions to be drawn from the multiscale modelling strategy for the construction of numerical methods for weather forecasting and climate research. Since the reduced models mentioned in the previous paragraph emerge from the comprehensive flow equations through singular limit processes, an important question is whether a numerical atmospheric flow solver would be able to robustly reproduce these limit regimes without deterioration of accuracy or performance. For the example of the "soundproof limit" of the compressible flow equations, the entire chain of development from the derivation and analysis of the reduced dynamical model to the construction of a uniformly accurate and robust numerical scheme will be discussed.
The third part of the lecture focuses on the ubiquitous importance of observational data in meteorological and oceanographic modelling. We are not, and will not in the foreseeable future, be able to resolve all flow scales of meteorological interest simultaneously in a computational model. Therefore, numerical weather forecasts and climate simulations suffer from a notorious underdetermination in terms of poorly known initial and boundary data, insufficient knowledge about spatio-temporal scales that cannot be resolved on the computational grid, and the like. In this context, observational data attain outmost importance in that they are the all-decisive final gauge for the quality of simulations. One task of central importance in this context is an abstract characterization of complex high-dimensional time series in terms of dominant features, sudden or smooth transitions between different flow patterns or behaviors, and the like. Over the past 5 to 10 years, Prof. Illia Horenko, Informatics Department, USI Lugano, has developed a comprehensive and unique methodology, "FEM-VARX" and its relatives, that combines ideas of statistics, stochastics, pattern recognition, and information theory and that addresses many of the challenges in this area. The third lecture will motivate and summarize Prof. Horenko's developments.