Mathematical Sciences: Theoretical and Computational Problems in Turbulence and Climatology

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9400615 Temam This project will focus on the study of the equations of fluid mechanics and turbulence as well as on the development of efficient numerical codes for the solution of these equations. Part of the project will involve the study of turbulence, based on a dynamical systems approach; according to dynamical systems theory the permanent regime for a turbulent flow is mathematically represented by an attractor that is a compact set in the phase space. It is hoped that the study of this attractor can improve understanding of the physics of turbulent flows. The approximation of attractors by smooth finite dimensional manifolds called Approximate Inertial Manifolds (AIMs) is part of this study. On the one hand these manifolds are related to the concept of slow manifold used in meteorology for data assimilation. Used from a different point of view in computational fluid dynamics they lead, on the other hand, to new multi-resolution algorithms and to methods of data compression in scientific computing. Thus, it is believed that approximations of this attractor can lead to efficient new algorithms for the numerical simulation of turbulence. Finally, a significant part of the effort will be devoted to problems related to climate and global changes. Various models for climate will be investigated from the mathematical and computational points of view in order to see how they are connected and what are their domains of validity. Such models include the primitive equations of the atmosphere, the primitive equations with vertical viscosity, and the geostrophic and quasi geostrophic equations. Models of this type will be studied for the atmosphere and for the ocean separately and also for the Coupled system of Atmosphere and Ocean (CAO). The understanding of the equations of fluid mechanics and turbulence as well as the development of efficient numerical codes for the solution of these equations are challenging problems of considerable importance in p articular in industry or for studies in meteorology or global climate change. The aims for this project are three fold: (i) to explore what can be learned about these problems from the theoretical and computational viewpoints, using the dynamical systems approach to turbulence; (ii) to develop new efficient computational algorithms taking into account the physics of turbulence and well adapted to the current evolution of large scale computing towards parallel computing; and (iii) to devote special attention to the problems related to the study of climate and global changes.