Nonlinear and Computational Problems for Geophysical and Classical Fluid Mechanics

Indiana University, Bloomington IN

Investigators

Abstract

Temam DMS-0906440 This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The investigator and his colleagues study nonlinear problems from meteorology, oceanography, and fluid mechanics, using the mathematical tools offered by analysis, the theory of nonlinear partial differential equations, dynamical systems theory, and computation. Besides improving the understanding of some important problems, an aim of the project is to help improve the numerical simulations of phenomena which bear many societal and industrial applications. The parts of this project related to meteorology and oceanography focus on the issue of open boundary conditions for limited domain simulations, and well-posedness issues for meteorology and oceanography in the absence of viscosity. Other parts of the project relate to the study of singular perturbations, that is, calculations in the presence of small coefficients strongly affecting the equations, and theoretical issues in turbulence. Numerical simulations for weather forecasts are usually done in a limited area around the region of interest; such computations include the usual weather forecasts, or forecasts in extreme situations, like a hurricane, or the study of desertification. Similar geographically local questions arise in oceanography, for example, the study of an estuary. An important issue for limited areas is the choice of the boundary conditions on the nonphysical parts of the boundary that are artificially created for the sake of the computations. This issue becomes more crucial as better computers allow more refined (more precise) calculations. Inappropriate boundary conditions may create incorrect waves (or winds), which may result for instance in incorrect precipitation forecasts. The investigator and his colleagues tackle this problem by theoretical and computational approaches. They work in close collaboration with geoscientists to ensure the practical relevance of the project. The project also includes strong cross-training of students and postdocs in geosciences and mathematics.