Random Models for Turbulent Fluid Systems

本研究的总体目标是发展对湍流非线性过程的一些基本理解，特别关注在粗粒度水平上描述复杂系统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。常用的调查模式是对简化的随机模型进行精确分析。紊流输运模型的主要特征是既包含了大尺度平均流（可以依赖于空间和时间），又包含了速度场的小尺度波动分量。这允许研究湍流的大小尺度如何相互作用，以确定被动标量密度的有效演化。这项工作将致力于扩展一些严格的均匀化定理，并研究一些由模型的新特征引起的物理现象。第二个研究领域涉及发展简化方程来描述弱非线性影响下的波传播。一个著名的弱湍流理论在许多情况下成功地处理了这类系统，但其基础和局限性仍在积极研究中。弱湍流理论的某些特定方面将在Fermi-Pasta-Ulam模型上进行仔细研究和说明。该分析结果将用于建议对弱湍流理论的标准动力学方程进行修正，以提高其准确性并扩大其适用范围。这两项研究都是为了更好地理解如何通过简化方程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而设计的大气-海洋模式。可用数据和超级计算资源的限制使得在可预见的未来不可能进行完全详细的模拟，并且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湍流如何以更合理和有效的方式在大气-海洋科学模式中参数化。

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.