Equilibrium and Nonequilibrium Statistical Theories of Turbulent Geophysical Flows

本课题将现代分析、计算和概率方法应用于地球物理流体动力学中的湍流模拟。具体来说，这些项目研究了湍流中相干结构的统计理论，在模型中，大气或海洋通常以长寿命、大规模喷流和漩涡的形式存在。该方法建立在这种结构的平衡统计理论的最新发展上，该理论现在已经成熟到可以在数学上证明，在计算上实现并成功地应用于地球物理模型。一个典型的例子是最近关于木星大气中纬向喷流和涡旋斑点的模型，它的预测与观测数据非常吻合。鉴于这些发展，该项目的目标是双重的。首先，针对越来越现实的地球物理模型，如多层准地转模型和浅水模型，详细阐述了相干结构的平衡统计理论。通过计算平衡结构族并推导出它们的非线性稳定性定理来研究该理论的物理意义。其次，从相应的非平衡理论出发，提出了一种新的统计闭合方法。与传统的流体湍流闭合方案不同，这种新方法通过调节微观状态的随机路径来推导出一些特定的已解变量的宏观方程。对一些原型问题的非平衡行为进行了理论和计算的结合研究。在具有阻尼和驱动的模型地球物理系统的背景下，这种方法被设想为推导未解析涡流的有效亚网格尺度参数化的一般程序。湍流流体的流动仍然是物理科学的一个未解之谜。需要对湍流有更好的理论认识，作为计算模拟几乎所有自然流体运动的基础。地球物理流体流动——地球海洋和大气的运动——尤其如此，它涉及范围很广的复杂运动，从米到行星大小。每一个用于天气预报或气候预测的现代计算机代码都需要特殊的，但往往是不可靠的，关于小尺度湍流运动如何影响计算出的大尺度行为的假设。在这个项目中进行的研究解决了复杂流体流动建模的一般问题——大气或海洋的数学原型——以这样一种方式，可以可靠地捕获其主要的大规模特征，而无需解决其小规模情绪的全部复杂性。特别是，该工作旨在开发必要的数学和计算工具来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，但在一系列小尺度上表现出无序和随机的运动。为此，该项目利用统计物理学的复杂技术来构建这类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。这种方法最近的一个例子是非常成功地解释了持续的喷流和漩涡，比如巨行星木星大气中的大红斑，这第一次显示了数学理论和美国宇航局航天器观测之间的定量和定性一致。在地球大气和海洋的背景下，这些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.