Mathematical Problems of Geophysical Fluid Mechanics: Uncertainties, Modeling, Theory, and Computing

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This research project addresses problems of great significance in ocean and atmosphere sciences, related to weather or climate predictions. The studies have also potential applications in other areas of sciences but this project only addresses the geophysical problems. The two classes of problems considered are the study of boundary conditions for predictions made in a limited domain, and the study of moist convection, which is related to the understanding of the behavior of clouds. It is generally considered that our limited understanding of the physics of clouds is a major cause of uncertainty in current weather and climate predictions. Developing methods that improve weather predictions and climate modeling is an objective of important human, societal, and economical interest. The training of graduate students and postdoctoral students at the interface of mathematics, geophysical and industrial fluid mechanics, and scientific computing is another broader impact of the project. A major problem in developing computational models of geophysical flows, when the flows extend over a large area but the region of particular interest is smaller, is to develop suitable boundary conditions for limited-area models when the physics does not provide natural boundary conditions. This old problem, already mentioned by von Neumann and Charney, is expected to become very significant. Indeed, when new computers will allow the use of fine spatial resolutions, it is predicted that inappropriate boundary conditions will introduce spurious modes. The problem arises also in other contemporary developments like the use of multilevel numerical methods or modeling. The investigator has obtained a number of relevant results for linear and nonlinear two-dimensional equations. Here he aims to continue building a sound mathematical theory for the inviscid primitive equations and other related equations. Concerning the outstanding problem of moist convection, precipitation and clouds, the the problems to be addressed relate to the theory, computations, and modeling. On the theoretical side, the tools of convex analysis and variational inequalities are used to address questions of existence and uniqueness of solutions for the equations of the humid atmosphere with saturation, which are nonlinear discontinuous partial differential equations (PDEs) not directly connected to any existing PDE theory. On the modeling side, using again tools of convex analysis, he models the equations of the humid atmosphere corresponding to water clouds below freezing levels by considering a more refined description of the micro physics at saturation, which would introduce additional state parameters. On the computational side, he studies numerically the superparameterization of the very complex convection phenomena occurring near the equator in the presence of topography, using finite volumes and multilevel methods. The importance of these phenomena, which involve tropical rains and affect the climate far into the mid-latitude regions, is well known. Finally, certain questions of probability are addressed in connection with the modeling of the many uncertainties occurring in the phenomena described above. Graduate students and postdoctoral students are engaged in the work of the project.