Random Models for Turbulent Fluid Systems

这项研究的总体目标是发展对湍流流体中的非线性过程的一些基本理解，特别是关注在粗粒度水平上描述复杂系统的动力学的有效方程的发展。研究了两种特殊的物理现象：浸没在湍流中的物质的混合和输运以及弱非线性波之间的相互作用。常见的调查模式是对简化的随机模型进行精确分析。湍流输运模型的主要特点是它既包含了大尺度的平均流，又包括了速度场的小尺度脉动分量。这使得我们能够研究湍流的大小尺度如何相互作用来确定被动标量密度的有效演化。这项工作旨在推广一些严格的齐化定理，并研究由该模型的新特征引起的一些物理现象。第二个研究领域是建立描述弱非线性影响下的波传播的简化方程。一个著名的弱湍流理论已经在许多情况下发现了处理这类系统的一些成功，但其基础和局限性仍在积极的研究中。弱湍流理论的一些特殊方面将在Fermi-Pasta-Ulam模型上仔细研究和说明。这一分析的结果将被用来建议对弱湍流理论的标准动力学方程进行修改，以提高其精度并推广其适用范围。这两项研究的目的都是为了更好地理解如何通过简化的方程来有效地描述湍流系统。这一研究主题的一个主要应用是为气候和天气预报设计的大气-海洋模型。现有数据和超级计算资源的限制使得完全详细的模拟在可预见的未来是不可能的，湍流的影响必须由一些可管理的参数集来表示。上述研究将有助于更好地理解如何在大气-海洋科学模式中以更合理和有效的方式对湍流进行参数化。

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.