Asymptotic and Statistical Behavior of Hydrodynamic Problems

本项目研究两个原型流体力学问题的渐近和/或统计行为：1.瑞利-贝纳德对流的Boussinesq近似在大Prandtl数或小Ekman数区域的渐近行为。对于大Prandtl数区，我们的目标是得到在时间上一致有效的有效动力学，并从轨道收敛和统计意义上的收敛来考察这种近似的有效性。本文的另一个目的是研究简化的无限Prandtl数模型在大时间和大Rayleigh数下的渐近行为。对于小Ekman数的情况，目标是推导出垂直方向上热传递速率的先导顺序。2.小尺度涡轰击下二维流动的长时间渐近行为。这里的目的是解释数值观察到的当流动被随机的小尺度旋涡强迫时出现大尺度相干结构的现象。该项目将通过渐近展开、严格分析和数值计算相结合的方式进行。热对流，即热驱动流体运动，是宇宙中最广泛和最重要的流体运动类型之一。对流是海洋、大气以及恒星和行星内部动力学的一个主要特征。它在许多工业过程中也很重要。然而，由于问题的极端复杂性，对对流的理解从根本上来说是不完整的。为了取得进展，需要进行各种简化。在这里，我们考虑了在地球物理应用中非常重要的情况，并推导和研究了热对流的简化动力学。大尺度结构的出现，如木星上的大红斑，是地球物理流体问题的一个普遍特征。了解这种构造产生和持续的机制是地球物理流体动力学中最有趣的问题之一。在这里，我们计划研究随机驱动环境中大尺度结构的出现。预计该项目的成功完成将增强我们对这些原型流体问题的理解，并为我们在工业制造过程中以及与全球环境变化研究密切相关的气象/气候模型中遇到更复杂的流体问题提供见解。我们在这里提出的想法也将对研究其他物理问题有用。

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.