Asymptotic and Statistical Behavior of Hydrodynamic Problems

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This project addresses the asymptotic and/or statistical behavior of two prototype hydrodynamic problems: 1. The asymptotic behavior of the Boussinesq approximation of Rayleigh-Benard convection in the regime of large Prandtl number or small Ekman number. For the large Prandtl number regime, the goal is to derive effective dynamics uniformly valid in time and investigate the validity of such approximation in terms of orbital convergence and convergence in the statistical sense. Another goal here is to investigate asymptotic behavior at large time and large Rayleigh number of the simplified infinite Prandtl number model. For the small Ekman number case, the objective is to derive the leading order of the rate of heat transport in the vertical direction. 2. The long time asymptotic behavior of two-dimensional flows under the bombardment of small-scale vortices. Here the goal is to explain the numerically observed phenomena of emergence of large-scale coherent structures when the flow is forced by random small-scale vortices. The project will be carried out through a combination of asymptotic expansion, rigorous analysis, and numerical computation.Thermal convection, that is, heat driven fluid motion, is one of the most widespread and most important type of fluid motions in the universe. Convection is a major feature of the dynamics of the oceans, the atmosphere, and the interiors of stars and planets. It is also important in many industrial processes. Nevertheless, the understanding of convection is fundamentally incomplete due to the extreme complexity of the problem. Various simplifications are called for in order to make progress. Here we consider situations that are of great importance in geophysical applications, and we derive and study simplified dynamics of heat convection. The emergence of large-scale structures, such as the Great Red Spot on Jupiter, is a ubiquitous feature of geophysical fluid problems. Understanding the mechanism of the emergence and persistence of such structures is one of the most intriguing issues in geophysical fluid dynamics. Here we plan to study the emergence of large-scale structure in a randomly driven environment. The expected successful completion of this project will enhance our understanding of these prototype fluid problems and provide insights into more complex fluid problems that we encounter in industrial manufacturing processes, as well as in the meteorology/climate models that are intimately related to the study of global environment change. The ideas that we develop here will be useful in studying other physical problems as well.