Mathematical Sciences: Nonlinear Phenomena in Fluid Dynamicsand Related P.D.E.'s with Applications to Atmosphere-Ocean Science

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9301094 Majda The investigator continues his research on the partial differential equations of fluid dynamics, involving a combination of modelling, mathematical theory, asymptotic methods, and scientific computing. The present project centers around various theoretical issues regarding turbulence in incompressible flows, with a new direction involving potential applications to atmosphere-ocean science. This proposal contains specific research projects in the following areas: (a) Eddy diffusivity, renormalization, and Monte Carlo simulation in turbulent transport; (b) The generation of small scale three-dimensional structures in incompressible flow; (c) The statistical interaction of internal gravity waves and vortical structure in stably stratified flow fields. The range of physical scales necessary for understanding fluid flow in the atmosphere or ocean is enormous. The processes associated with molecular diffusion involve scales smaller than centimeters while the large scale patterns of significance for weather prediction and other environmental issues involve scales of hundreds to thousands of kilometers. Even the largest supercomputers process observations over the globe at a mesh spacing with each side no smaller than 500 kilometers. How can one assess the mean impact of the many scales, which cannot be described in detail on the large-scale processes of environmental significance? The proposal takes up this question. The work bears on current issues in atmospherics and oceanics. ***