A study of how indicators for 2-D turbulence depend on the driving force in the Navier-Stokes equation

这个项目的重点是发现支持Kraichnan二维完全发展湍流理论的驱动力。在最近的工作中，研究人员和他们的合作者已经确定了用navier - stokes方程（NSE）解的不同范数的平均值表示的临界波数。这些波数为克雷契南理论的成立提供了必要的和几乎充分的条件。该团队还在一定程度上将NSE的全局吸引子定位在由两个规范（一个是能量）跨越的平面上，以帮助理解是哪些驱动力产生了这些条件。所提出的工作将把这种分析与计算优化结合起来，以消除这种力，然后详细研究它们产生的湍流特征。湍流的数学处理在很大程度上是由Kolmogorov、Batchelor和Kraichnan的启发式理论推动的。然而，投射吸引子的方法似乎是全新的。该分析提供的信息将指导计算组件，否则将面临大量可能的驱动力。湍流很容易在三维物理空间中观察到。大多数人想到的是颠簸的飞机（在这种情况下，域是飞机周围的体积）。就像溪流中岩石形成的漩涡一样，飞机周围的空气中也形成了快速变化的图案。湍流理论并不试图预测这些模式的精确发展，而是找到(a)描述能量平均如何转移到较小长度尺度的一致定律，以及(b)这种现象发生变化的临界长度尺度。真正的二维流动在自然界中不那么普遍。最突出的例子是地球大气层，它实际上是一个薄的3-D域，其行为接近于2-D流。能量在不同长度尺度上的命运对于二维流动来说更为复杂，尽管三维流动在某种意义上嵌入其中。虽然二维实验很难在实验室中进行，但它们允许在计算机上进行更精细的研究。在所有二维流中，本项目研究的二维流可以说是最易于分析和有效模拟的。然而，它不仅是二维和接近二维的流动（如大气）的基础，而且由于其普遍性，它也是三维湍流的基础。

This project is focused on the discovery of driving forces that supportthe Kraichnan theory of 2-D fully developed turbulence. In recent workthe investigators and their collaborators have identified critical wavenumbers expressed as averages of different norms of the solution to theNavier-Stokes equations (NSE). These wave numbers provide necessary, andnearly sufficient conditions for the Kraichnan theory to hold. This teamhas also localized to some extent the global attractor of the NSE in aplane spanned by two of these norms (one being the energy), to helpunderstand which driving forces produce these conditions. The proposedwork will combine this analysis with computational optimization to zero inon such forces, and then study in detail the turbulent features theyproduce. The mathematical treatment of turbulence is largely driven bythe heuristic theories of Kolmogorov, Batchelor and Kraichnan. Theapproach taken in projecting the attractor however, seems to be entirelynew. The information provided by this analysis will guide thecomputational component which otherwise would be confronted with a vastlandscape of possible driving forces to consider.Turbulence is readily observed in three-dimensional physical spacedomains. Most people think of a bumpy plane rides (in this case thedomain is the volume around the airplane). Like the swirls generated byrocks in a stream, rapidly changing patterns form in the air around theplane. Turbulence theories do not attempt to predict the precisedevelopment of these patterns, but rather find (a) consistent laws whichdescribe how, on average, energy is transferred to smaller length scales,and (b) critical length scales at which this this phenomenon changes. True 2-D flows in nature are less prevalent. The most prominent example,the earth's atmosphere, is actually a thin 3-D domain, whose behaviorapproaches that of a 2-D flow. The fate of energy over different lengthscales is more complicated for 2-D flow, though that of 3-D flow is insome sense embedded into it. Though 2-D experiments are difficult tocarry out in the laboratory, they allow for much finer study on acomputer. Of all 2-D flows, the one studied in this project is arguablythe most amenable to analysis and efficient simulation. Yet it isfundamental, not only to 2-D and nearly 2-D flows such as the atmosphere,but also to 3-D turbulence due to universality.