Problems of Geophysical Fluid Mechanics: Modeling, Theory and Computing

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This project will address problems in mathematical ocean and atmosphere sciences, related to weather and climate prediction. Two classes of problems will be investigated. The first one is that of the boundary conditions for limited area weather models for which physical models do not provide natural boundary conditions. This problem is as old as numerical weather prediction itself, and it is expected that inappropriate boundary conditions will introduce spurious modes, in particular as improved computing capabilities allow better spatial and temporal resolutions. The aim is to derive boundary conditions that lead to mathematically correct problems, and that are computationally satisfactory in the sense that they let the waves move freely inside and outside the domain of computation. The PI will continue his investigations in this direction. The second class of problems is that of moist convection, precipitations and clouds. Our limited understanding of the physics of clouds is one of the primary causes of uncertainty in the current weather and climate predictions. The difficulties are in part due to the large differences in scales, ranging from cloud sizes of hundreds of kilometers to particles (droplets of water, ice or aerosols) of a few microns. There is a similar range of time scales. The project will study this physical problem using mathematical tools from convex analysis and variational inequalities. Numerical multilevel methods will be used as well to study the interaction of clouds and topography near the equator; and finally the tools of convex analysis will also be used for the very modeling of the changes of phase in the clouds. These studies will be conducted in close collaboration with physicists. The project will involve graduate students and post-doctoral associates who will be trained in this important interdisciplinary area. This research project will study fundamental problems in the numerical prediction of weather and climate, as well as problems in the mathematical modeling of clouds and evaporation. The project component related to numerical weather prediction addresses the problem of using information from coarse featured global weather data for fine grained local and regional predictions. If the coarse data are not coupled correctly to the regional model that has higher resolution, then spurious systematic errors (e.g. "ghost" weather fronts) may arise. The project will investigate the mathematical foundations of this problem. The project component on the modeling of clouds and evaporation will use advanced mathematical tools to contribute to the understanding of a set of basic open problems in weather and climate research: Do clouds help cool the planet (because they reflect sunlight) or do they contribute to warming (because they can hold thermal energy)? Weather prediction and climate modeling are objectives with important human, social and economic interest. The project will also train young scientists at the interface of mathematics, geophysical fluid mechanics, and scientific computing.