Temam 9705229 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 particular in industry or for studies in meteorology or global climate change. The aims for this project are to explore what can be learned about these problems from the theoretical and computational viewpoints, using the dynamical systems approach to turbulence. More specifically the investigators and their colleagues intend: (i) 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; (ii) to devote special attention to the problems related to the study of the climate and global changes. For (i), new multilevel algorithms related to the concept of attractor and approximate inertial manifolds have been introduced during the past years. The analysis of these algorithms is developed, their performances improved, and their range of application extended, in particular towards problems of practical relevance. During the period of this grant a book will be published on this subject, describing the state of the art on the theoretical side, and including many aspects of their actual implementation on parallel computers. For (ii), a significant part of this project is devoted to the study of models for the atmosphere, the ocean and their coupling. Involved models based on the primitive equations are considered, as well as simpler models such as multilayer or quasi-geostrophic models. The study includes the development of the models and the mathematical and numerical problems that they raise. The interaction of meteorology and mathematics has very long traditions going back to such famous names as Leonard da Vinci, Pierre Simon Laplace or, in this century, after WWII, John Von Neumann. At a more modest level, the investigators pursue a program of research initiated a few years ago and aimed at developing interactions between geosciences and mathematics. These interactions can be mutually beneficial. Meteorology and oceanography raise very challenging mathematical problems very useful for mathematicians and other scientists (for example scientists have learned much from the experience of E. Lorenz in chaos). Conversely, mathematicians might help scientists of the geoscience communities in choosing their models by determining if they are well posed. In the long range they might help also develop new efficient numerical procedures; although the codes (programs) used in meteorology and oceanography are very involved codes written over a long period of time, eventually new codes will be written responding to new needs or new opportunities, and new insights could become useful; it is hoped to contribute to this daunting task.
