Equilibrium and Nonequilibrium Statistical Theories of Turbulent Geophysical Flows

该研究项目应用现代分析、计算和概率方法模拟了在湍流流体动力学中出现的湍流。 具体来说，这些项目研究湍流中相干结构的理论，湍流在模型大气或海洋中通常以长寿命、大尺度射流和涡流的形式出现。 该方法建立在最近的发展，在平衡统计理论的这种结构，现在已经成熟到这一点，它可以证明数学，计算实现和成功地应用于technologicalmodels。 一个主要的例子是最近的木星大气层中纬向喷流和涡旋点的模型，其预测与观测数据非常吻合。 鉴于这些事态发展，该项目的目标是双重的。 首先，阐述了拟序结构的平衡统计理论在日益现实的地球物理模式中的应用，如多层准地转模式和浅水模式。 通过计算平衡结构族和导出它们的非线性稳定性定理，研究了该理论的物理意义。 其次，从相应的非平衡态理论出发，提出了一种新的统计闭合方法。 与传统的流体湍流封闭格式不同，这种新方法通过将微观状态的随机路径作为这些解析变量的集合平均动力学的条件，导出了这些解析变量的宏观方程。 结合理论和计算调查这种近似的非平衡行为进行了一些原型问题。 在具有阻尼和驱动的模型地球物理系统的背景下，这种方法被设想为一个通用的程序，用于导出有效的亚网格尺度参数化的未解决的漩涡。湍流流体流动仍然是物理科学的未解决的难题之一。 对湍流有更好的理论理解是对几乎所有自然流体运动进行计算模拟的基础。 地球物理学的流体流动--地球海洋和大气的运动--尤其如此，它涉及到从米到行星大小的各种尺度的复杂运动。 用于天气预报或气候预测的每一种现代计算机代码都需要特殊的、但往往不可靠的假设，即小尺度湍流运动如何影响计算出的大尺度行为。 该项目的研究解决了模拟复杂流体流动的一般问题--大气或海洋的数学原型--以苏查的方式，可以可靠地捕捉其主要的大尺度特征，而不需要解决其小尺度情感的全部复杂性。 特别是，这项工作旨在开发必要的计算工具，以预测模拟地球物理流体系统的行为，这些系统在大尺度上表现出有组织的特征，但在小尺度范围内表现出无序和随机的运动。为此，该项目利用统计物理学的先进技术来构建这种复杂系统的理论模型，从而提供有效和可靠的方法来计算它们的预期或最可能的行为。 这种方法的一个最近的例子是非常成功地解释了持久的喷流和漩涡，如大红斑，在巨大的木星大气层中，这是第一次展示了南极理论和NASA航天器观测之间的定量和定性一致。 在地球大气层和海洋的背景下，这种理论模型和计算方法可以作为预测海洋-大气系统长期趋势的基础。

The research project applies modern methods in analysis, computationand probability to the modeling turbulent flows arising in geophysicalfluid dynamics. Specifically, the projects investigates statisticaltheories of coherent structures in turbulence, which in a modelatmosphere or ocean usually take the form of long-lived, large-scalejets and vortices. The approach builds on recent developments in theequilibrium statistical theory of such structures, which has nowmatured to the point that it can be justified mathematically,implemented computationally and applied succesfully to geophysicalmodels. A prime example is the recent model of the zonal jets andvortical spots in the atmosphere of Jupiter, whose predictions agreeremarkably well with observational data. In light of thesedevelopments, the goals of the project are two-fold. First, theequilibrium statistical theory of coherent structures is elaboratedfor increasingly realistic geophysical models, such as multi-layerquasi-geostrophic models and shallow-water models. The physicalimplications of the theory are investigated by computing families ofequilibrium structures and deriving nonlinear stability theorems forthem. Second, a novel approach to statistical closure is developedfrom the corresponding nonequilibrium theory. Unlike traditionalclosure schemes for fluid turbulence, this new methodology derivesmacroscopic equations for some specified resolved variables byconditioning random paths of microstates on the ensemble-averageddynamics for those resolved variables. A combined theoretical andcomputational investigation of this approximation to nonequilibriumbehavior is undertaken for some prototype problems. In the context ofmodel geophysical systems with damping and driving, this approach isenvisioned as a general procedure for deriving effective subgrid-scaleparametrizations of unresolved eddies.Turbulent fluid flow remains one of the unsolved puzzles of physicalscience. A better theoretical understanding of turbulence is neededas a basis for the computational simulation of almost all naturalfluid motions. This is especially true of geophysical fluid flows --the motions of the Earth's oceans and atmosphere -- which involvecomplex motions over a wide range of scales, from meters up to theplanetary size. Every modern computer code used in weatherforecasting or climate prediction requires special, but oftenunreliable, assumptions about how the small-scale turbulent motionsaffect the computed large-scale behavior. The research conducted inthis project addresses the general issue of modeling a complex fluidflow -- a mathematical prototype of an atmosphere or ocean -- in sucha way that its predominant large-scale features can be capturedreliably without resolving the full complexity of its small-scalemotions. In particular, the work seeks to develop the mathematicaland computational tools necessary to predict the behavior of modeledgeophysical fluid systems which exhibit organized features on largescales but disordered and random motions on a range of small scales.To do so, the project draws on sophisticated techniques fromstatistical physics to construct theoretical models of complex systemsof this kind, and thereby to provide efficient and reliable methodsfor computing their expected or most probable behavior. A recentexample of this approach is the remarkably successful explanation ofthe persistent jetstreams and vortices, such as the Great Red Spot, inthe atmosphere of the giant planet Jupiter, which for the first timeshows quantitative and qualitative agreement between mathematicaltheory and NASA spacecraft observations. In the context of Earth'satmosphere and oceans, such theoretical models and computationalmethods can used as building blocks in predictions about the long-termtrends in the ocean-atmosphere system.