Model Reduction and Statistical Closure of Turbulent Dynamics

该项目旨在开发和应用一种新的方法，用于确定性、非线性动力系统的模型简化和统计闭合。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，该方程近似于已解析和未解析运动标度之间的相互作用的影响。对于一个典型的复杂系统，例如一个具有多个自由度的哈密顿系统，简化的方法如下进行：选择一个可分辨变量的矢量来描述系统的相干、缓慢行为，并且参数统计模型典型地与该可分辨矢量相关联；平均可分辨矢量的预测或估计的演变由统计参数空间中的该路径定义，对于给定的动力学，该路径最适合于Liouville方程。引入了代价泛函来量化参数路径的不适配性；它是关于Liouville残差的信息理论度量，其中包含确定随后闭包中的可调整常数的权重。将哈密顿-雅可比理论应用于定义最佳闭包的最优化原理，得到了平均分辨向量的期望方程。最近，最佳拟合理论已经在一个典型的湍流动力学(截断的Burgers-Hopf方程)上得到了验证，它产生了一个具有新的耗散效应和修正的非线性相互作用的粗粒动力学。对于正压和斜压准地转流，将推导出有效的粗粒度方程，这些闭合将与全分辨集合的基准数值模拟进行比较。在首先研究了一些较简单的湍流波动动力学问题之后，还将使用最佳拟合方法来表示这些模型的近似非平衡谱。许多物理现象的数学模型，如地球物理流体动力学，用于预测地球大气和海洋的运动，通常表现出非常复杂的湍流行为。因此，重要的是设计这样的复杂系统的适当的简化模型，至少近似地用更少的变量来捕捉整个系统的一致行为。P.I.S的研究旨在开发通用的数学和统计工具，以构建这类简化模型。具体地说，这项研究致力于通过设计有效的计算方案来简化大气和海洋的非线性动力学模型，其中未分解的湍流由最适合完整运动方程的统计模型来参数化。这种折减方法将为改进大尺度、低频天气和气候动力学数值模式的性能提供一种数学上合理的技术。具体地说，这项工作试图通过使用系统和稳健的方法审查简化的原型问题来建立对模型简化和闭合的理解，该方法借鉴了非平衡统计力学和热力学的物理概念，来自最优化理论和动力系统的数学技术，以及来自统计学和信息论的工具。将聘请一名博士后研究助理与P.I.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。

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.