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的2-D充分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确定了临界波数，用不同范数的N-S方程(NSE)解的平均值表示。这些波数为Kraichnan理论的成立提供了必要且近乎充分的条件。这个团队还在某种程度上将NSE的全球吸引者定位在一个平面上，这两个规范(一个是能量)跨越了这个平面，以帮助理解是什么驱动力产生了这些条件。这项工作将把这种分析与计算优化相结合，对这种作用力进行零化，然后详细地研究它们产生的湍流特性。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。这一分析提供的信息将指导计算部分，否则将面临巨大的可能驱动力的景观。湍流很容易在三维物理空间中观察到。大多数人会想到颠簸的飞机飞行(在这种情况下，范围是飞机周围的体积)。就像溪流中的岩石产生的漩涡一样，飞机周围的空气中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)描述能量如何平均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。在自然界中，真正的二维流动并不那么普遍。最突出的例子是地球大气，实际上是一个很薄的三维域，其行为接近于二维流。对于二维流来说，不同长度尺度上的能量的命运要复杂得多，尽管三维流的命运在某种意义上是嵌入其中的。尽管2-D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。在所有的二维流动中，本项目研究的流动可以说是最易于分析和有效模拟的。然而，它是基本的，不仅对2-D和近2-D流动如大气，而且由于普遍性而对3-D湍流也是如此。

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.