Model Reduction and Statistical Closure of Turbulent Dynamics

湍流动力学的模型简化和统计收敛

• 批准号: 1312576
• 负责人: Bruce Turkington
• 金额: $ 31.88万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1312576&HistoricalAwards=false
• 关键词: Model; Reduction; Statistical; Closure; Turbulent

This project aims to develop and apply a new methodology of model reduction and statistical closure for deterministic, nonlinear dynamical systems. The overall objective of this work is to derive coarse-grained equations that approximate the effect of interactions between resolved and unresolved scales of motion from given, fine-grained, physical equations of motion. For a typical complex system, say a Hamiltonian system with many degrees of freedom, the method of reduction proceeds as follows: a vector of resolved variables is selected to describe the coherent, slow behavior of the system, and a parametric statistical model is canonically associated with this resolved vector; the predicted, or estimated, evolution of the mean resolved vector is defined by that path in the statistical parameter space which optimally fits the Liouville equation for the given dynamics. A cost functional is introduced to quantify the lack-of-fit of parameter paths; it is an information-theoretic metric on the Liouville residual containing weights that determine the adjustable constants in the ensuing closure. The desired equation for the mean resolved vector is derived by applying Hamilton-Jacobi theory to the optimization principle defining the best-fit closure. Recently the best-fit theory has been validated on a prototypical turbulent dynamics (truncated Burgers-Hopf equation), for which it produces a coarse-grained dynamics having novel dissipative effects and modified nonlinear interactions. Effective coarse-grained equations will be derived for barotropic and baroclinic quasi-geostrophic flows, and these closures will be tested against benchmark numerical simulations of fully-resolved ensembles. The best-fit approach will also be applied to represent approximate nonequilibrium spectra for these models, after first investigating these questions on some simpler turbulent wave dynamics. Mathematical models of many physical phenomena such as geophysical fluid dynamics, which are used to predict the motions of the Earth's atmosphere and oceans, typically display very complex, turbulent, behavior. It is therefore important to design appropriate reduced models of such complex systems that capture, at least approximately, the coherent behavior of the full system in many fewer variables. The P.I.'s research is directed towards developing general mathematical and statistical tools for constructing reduced models of this kind. Specifically, the research endeavors to simplify nonlinear dynamical models of the atmosphere and oceans by devising effective computational schemes in which unresolved turbulence is parameterized by statistical models that best fit the full equations of motion. A reduction method of this kind would furnish a mathematically justified technique for improving the performance of numerical models of the large-scale, low-frequency dynamics of weather and climate. Specifically, the work seeks to build up an understanding of model reduction and closure through a hierarchy of simplified, prototype problems examined using a systematic and robust methodology that draws on physical concepts from nonequilibrium statistical mechanics and thermodynamics, mathematical techniques from optimization theory and dynamical systems, as well as tools from statistics and information theory. A postdoctoral research associate will be hired to collaborate with the P.I., and the training of the postdoc in these interrelated topics will be an important outcome of the project.

本计画的目的是发展及应用一种新的方法论，用于确定性、非线性动态系统的模型简化与统计封闭。这项工作的总体目标是从给定的细粒度物理运动方程中推导出粗粒度方程，该粗粒度方程近似于已解决的和未解决的运动尺度之间的相互作用的效果。对于一个典型的复杂系统，例如多自由度的Hamilton系统，约化方法是：选择一个分解变量向量来描述系统的相干慢行为，并将参数统计模型正则地与这个分解向量相关联;预测的或估计的平均分辨矢量的演变由统计参数空间中最佳地拟合给定动态的刘维尔方程的路径定义。引入一个成本函数来量化参数路径的拟合不足;它是一个信息论度量，包含确定随后闭合中的可调常数的权重的刘维尔残差。通过将Hamilton-Jacobi理论应用于定义最佳拟合闭合的优化原理，推导出平均解析向量的所需方程。最近的最佳拟合理论已被验证的原型湍流动力学（截断Burgers-Hopf方程），它产生了一个粗粒度的动力学具有新颖的耗散效应和修改的非线性相互作用。有效的粗粒度方程将推导出正压和斜压准地转流，这些关闭将测试基准数值模拟的完全解决合奏。最佳拟合的方法也将适用于代表近似非平衡谱这些模型，首先调查这些问题后，一些简单的湍流波动力学。 许多物理现象的数学模型，如用于预测地球大气和海洋运动的地球物理流体动力学，通常显示出非常复杂的湍流行为。因此，重要的是要设计适当的简化模型，这样的复杂系统，捕捉，至少近似地，整个系统的一致性行为在许多更少的变量。私家侦探的研究是针对发展一般的数学和统计工具，用于构建这种简化模型。具体而言，研究致力于简化大气和海洋的非线性动力学模型，设计有效的计算方案，其中未解决的湍流参数化的统计模型，最适合完整的运动方程。这种简化方法将为改进天气和气候的大尺度低频动力学数值模式的性能提供一种数学上合理的技术。具体来说，这项工作旨在建立一个简化的层次结构，原型问题的研究，使用系统和强大的方法，借鉴非平衡统计力学和热力学的物理概念，数学技术从优化理论和动力系统，以及统计和信息理论的工具，通过模型简化和关闭的理解。将聘请一名博士后研究助理与私人侦探合作，对博士后进行这些相关主题的培训将是该项目的一个重要成果。