Random Models for Turbulent Fluid Systems

这项研究的总体目标是对湍流流体中的非线性过程有一些基本的了解，特别注意在粗粒度水平上描述复杂系统动力学的有效方程的发展。 两个特殊的物理现象进行检查：混合和运输的物质浸没在湍流流体和弱非线性波之间的相互作用。调查的常见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包含了大尺度的平均流，它可以依赖于空间和时间，和一个小尺度的速度场的波动分量。 这使得研究如何大规模和小规模的湍流流体相互作用，在确定的被动标量密度的有效演变。 工作将针对扩展一些严格的均匀化定理和调查的一些物理现象所产生的新功能的模型。 第二个研究领域涉及的简化方程的发展来描述弱非线性的影响下的波传播。 一个著名的弱湍流理论已经在许多情况下成功地处理了这样的系统，但它的基础和局限性仍在积极的研究中。 弱湍流理论的一些特定方面将在费米-帕斯塔-乌拉姆模型上进行审查和说明。 分析的结果将被用来建议弱湍流理论的标准动力学方程的修改，这可能会提高其精度和推广其域的application.Both这些研究都是针对实现更好地理解如何湍流系统可以有效地描述通过简化的方程。 这一研究主题的主要应用是为气候和天气预测而设计的大气-海洋模型。 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湍流。

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.