Statistical Models of Two-Dimensional and Geostrophic Turbulence

二维和地转湍流的统计模型

• 批准号: 9971204
• 负责人: Bruce Turkington
• 金额: $ 10.5万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-15 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971204&HistoricalAwards=false
• 关键词: Statistical; Models; Two; Dimensional; Geostrophic

ABSTRACT for NSF/DMS Applied MathematicsBruce E. TurkingtonDepartment of Mathematics and StatisticsUniversity of Massachusetts, AmherstThe research project concerns statistical models of turbulent fluidmotions. In particular, these investigations focus on equilibriummodels of coherent structures in developed turbulent flows that areeither two-dimensional or quasi-geostrophic. The principal motivationfor this work comes from applications in geophysical fluid dynamics.Statistical equilibrium models defined by Gibbs ensembles areconstructed from the conserved quantities for the underlying fluiddynamics. The continuum limit of these lattice models is analyzedusing probabilistic techniques, especially the theory of largedeviations. This approach furnishes rigorous derivations of theconstrained maximum entropy principles whose solutions are the mostprobable macroscopic states of the systems. These macrostatesrepresent long-lived, large-scale flows that self-organize in theturbulent vorticity field. For instance, they can be zonal shearflows or embedded vortex structures. In a variety of contexts,including one-layer barotropic flows and two-layer baroclinic flows,the variational problems governing these states are solved numericallyby means of a robust and accurate method. In this way, quantitativepredictions about the macroscopic behavior of the system, which isturbulent on the microscopic scales, can be made without resolving allthe scales of motion. Another direction of the research is to extendthese equilibrium theories to a new class of quasi-equilibriumtheories determined by collective variables that are not necessarilyconserved quantities. This extension allows the same general approachto be applied in situations where the coherent states areslow-varying, but unsteady.Turbulent fluid flow remains one of the unsolved puzzles of physicalscience. A better theoretical understanding of turbulence wouldgreatly improve our ability to compute the behavior of natural fluidmotions. In particular, geophysical fluid flows -- the large-scalemotions of the Earth's oceans and atmosphere -- are very difficult topredict because of their turbulent fluctuations. Nevertheless, theseflow systems, whose motions are mainly horizontal, have specialproperties not shared by general fluid motions in three dimensions.Consequently, they tend to exhibit organized behavior in their largestfeatures while they remain disordered and random on a range of smallerlength scales. The research conducted in this project addresses themathematical and computational issues involved in modeling phenomenaof this type. Specifically, the work seeks to develop the toolsnecessary to calculate the persistent, predominant states of nearlytwo-dimensional fluid systems without resolving the full complexity oftheir detailed behavior. With such tools in hand, the typical or"most probable" states of geophysical flows can be characterized andcomputed. Besides being of fundamental interest in the field of fluiddynamics, these states can used as building blocks in predictionsabout the long-term trends in the ocean-atmosphere system, and hencethey are relevant to weather forecasting and climate modeling.

摘要 美国国家科学基金会/DMS应用数学Bruce E.马萨诸塞州大学数学与统计系，阿默斯特研究项目涉及湍流流体运动的统计模型。 特别是，这些调查集中在发达的湍流，无论是二维或准地转相干结构的平衡模型。 本文的主要工作来源于地球物理流体动力学的应用，由Gibbs系综定义的统计平衡模型是从流体动力学的守恒量出发建立的。 利用概率方法，特别是大偏差理论，分析了这些格点模型的连续极限。 这种方法严格推导了约束最大熵原理，其解是系统的最可几宏观态。 这些宏观状态代表了在湍流涡度场中自组织的长寿命的大规模流动。 例如，它们可以是纬向剪切流或嵌入涡结构。 在各种情况下，包括一层正压流和两层斜压流，这些状态的变分问题的数值求解通过一个强大的和准确的方法。 用这种方法，可以对系统的宏观行为进行定量预测，而不需要解决所有的运动尺度。 研究的另一个方向是将这些平衡理论扩展到一类新的由集体变量决定的准平衡理论，这些集体变量不一定是守恒量。 这一扩展使得相同的一般方法可以应用于相干态变化小但不稳定的情况。湍流流体流动仍然是物理科学中未解决的难题之一。 对湍流有更好的理论理解将大大提高我们计算自然流体运动行为的能力。 特别是，地球物理流体流动--地球海洋和大气的大尺度流动--由于它们的湍流波动而非常难以预测。 然而，这些流动系统的运动主要是水平的，具有一般三维流体运动所不具有的特殊性质。因此，它们往往在最大尺度上表现出有组织的行为，而在较小尺度上则保持无序和随机。 在这个项目中进行的研究解决了这类现象建模中涉及的数学和计算问题。 具体来说，这项工作旨在开发必要的工具来计算近二维流体系统的持续，主导状态，而不解决其详细行为的全部复杂性。 有了这些工具，地球物理流的典型或“最可能”状态就可以被描述和计算出来。 除了在流体动力学领域的基本利益，这些状态可以被用作预测海洋-大气系统的长期趋势的构建块，因此它们与天气预报和气候建模有关。