Nonlinear and Computational Problems for Geophysical and Classical Fluid Mechanics

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TemamDMS-0906440 This award is funded under the American Recovery andReinvestment Act of 2009 (Public Law 111-5). The investigator and his colleagues study nonlinear problemsfrom meteorology, oceanography, and fluid mechanics, using themathematical tools offered by analysis, the theory of nonlinearpartial differential equations, dynamical systems theory, andcomputation. Besides improving the understanding of someimportant problems, an aim of the project is to help improve thenumerical simulations of phenomena which bear many societal andindustrial applications. The parts of this project related tometeorology and oceanography focus on the issue of open boundaryconditions for limited domain simulations, and well-posednessissues for meteorology and oceanography in the absence ofviscosity. Other parts of the project relate to the study ofsingular perturbations, that is, calculations in the presence ofsmall coefficients strongly affecting the equations, andtheoretical issues in turbulence. Numerical simulations for weather forecasts are usually donein a limited area around the region of interest; suchcomputations include the usual weather forecasts, or forecasts inextreme situations, like a hurricane, or the study ofdesertification. Similar geographically local questions arise inoceanography, for example, the study of an estuary. An importantissue for limited areas is the choice of the boundary conditionson the nonphysical parts of the boundary that are artificiallycreated for the sake of the computations. This issue becomesmore crucial as better computers allow more refined (moreprecise) calculations. Inappropriate boundary conditions maycreate incorrect waves (or winds), which may result for instancein incorrect precipitation forecasts. The investigator and hiscolleagues tackle this problem by theoretical and computationalapproaches. They work in close collaboration with geoscientiststo ensure the practical relevance of the project. The projectalso includes strong cross-training of students and postdocs ingeosciences and mathematics.