Statistical Models of Two-Dimensional and Geostrophic Turbulence

NSF/DMS应用数学布鲁斯·E·图尔金顿马萨诸塞大学数学与统计系，阿默斯特该研究项目涉及湍流运动的统计模型。特别是，这些研究集中在发展的二维或准地转湍流中相干结构的平衡模型。这项工作的主要动机来自于在地球物理流体动力学中的应用。由Gibbs系综定义的统计平衡模型是根据基本流体动力学的守恒量构造的。用概率方法，特别是大偏差理论，分析了这些晶格模型的连续极限。这种方法提供了约束最大熵原理的严格推导，其解是系统最可能的宏观状态。这些宏观状态代表了在急流涡量场中自组织的长期、大规模的流动。例如，它们可以是纬向切变流动或嵌入的涡旋结构。在各种背景下，包括单层正压流动和两层斜压流动，控制这些状态的变分问题通过稳健和精确的方法被数值求解。这样，就可以在不分解所有运动尺度的情况下，对系统在微观尺度上的宏观行为做出定量的预测。研究的另一个方向是将这些平衡理论推广到由集体变量决定的一类新的准平衡理论，这些集体变量不一定是守恒量。这一扩展允许相同的一般方法应用于相干态低变但不稳定的情况。湍流流动仍然是物理科学中尚未解决的谜题之一。更好地从理论上理解湍流将极大地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规模情感--非常难以预测，因为它们的波动剧烈。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备的特殊性质，因此，它们往往在最大的特征中表现出有组织的行为，而在较小的长度尺度上保持无序和随机。在这个项目中进行的研究解决了对这类现象进行建模所涉及的数学和计算问题。具体地说，这项工作试图开发必要的工具来计算近二维流体系统的持久的、主要的状态，而不需要解决其详细行为的全部复杂性。有了这样的工具，地球物理流动的典型或“最可能”状态就可以被表征和计算。除了在流体动力学领域具有基本意义外，这些状态还可以用作预测海洋-大气系统长期趋势的基础，因此它们与天气预报和气候建模有关。

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.