Random Models for Turbulent Fluid Systems

Rensselaer Polytechnic Institute, Troy NY

Investigators

Abstract

This research is directed toward the general goal of developing some fundamental understanding of nonlinear processes in turbulent fluids, with particular attention to the development of effective equations which describe the dynamics of complex systems on a coarse-grained level. Two particular physical phenomena are examined: the mixing and transport of substances immersed in a turbulent fluid and the interaction between weakly nonlinear waves. The common mode of investigation is a precise analysis of simplified stochastic models. The main feature of the turbulent transport model is its inclusion of both a large-scale mean flow, which can depend on both space and time, and a small-scale fluctuating component of the velocity field. This allows a study of how the large and small scales of a turbulent fluid interact in determining the effective evolution of the passive scalar density. The work will be directed toward extending some rigorous homogenization theorems and investigating some physical phenomena which arise from the new features of the model. The second research area concerns the development of simplified equations to describe wave propagation under the influence of weak nonlinearity. A well-known weak turbulence theory has found some success in treating such systems in a number of contexts, but its foundations and limitations are still under active investigation. Some particular aspects of the weak turbulence theory will be scrutinized and illustrated on the Fermi-Pasta-Ulam model. The outcomes of this analysis will be used to suggest modifications of the standard kinetic equations of weak turbulence theory which may improve their accuracy and generalize their domain of applicability. Both of these studies are directed toward achieving a better understanding of how turbulent systems can be effectively described through simplified equations. A chief application of this research theme is in atmosphere-ocean models designed for climate and weather prediction. Limitations on both available data and supercomputing resources make fully detailed simulations impossible for the forseeable future, and the effects of turbulence must be represented by some managable set of parameters. The research described above will contribute toward a better understanding of how turbulence can be parameterized in atmosphere-ocean science models in a more rational and effective manner.