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

该项目的重点是发现支持二维完全发展湍流的 Kraichnan 理论的驱动力。 在最近的工作中，研究人员及其合作者已经确定了临界波数，以纳维-斯托克斯方程（NSE）解的不同范数的平均值表示。 这些波数为克莱奇南理论的成立提供了必要且几乎充分的条件。 该团队还在某种程度上将 NSE 的全局吸引子定位在由其中两个规范（其中一个是能量）跨越的平面上，以帮助理解哪些驱动力产生了这些条件。 拟议的工作将把这种分析与计算优化结合起来，使这些力为零，然后详细研究它们产生的湍流特征。 湍流的数学处理很大程度上是由柯尔莫哥洛夫、巴彻勒和克莱奇南的启发式理论推动的。 然而，投影吸引子​​所采用的方法似乎是全新的。 该分析提供的信息将指导计算组件，否则计算组件将面临大量可能的驱动力需要考虑。在三维物理空间域中很容易观察到湍流。 大多数人会想到颠簸的飞机之旅（在这种情况下，域是飞机周围的体积）。 就像溪流中的岩石产生的漩涡一样，飞机周围的空气中也会形成快速变化的图案。 湍流理论并不试图预测这些模式的精确发展，而是找到（a）一致的定律，描述能量平均如何转移到较小的长度尺度，以及（b）这种现象发生变化的临界长度尺度。自然界中真正的二维流动并不常见。 最突出的例子是地球大气层，它实际上是一个薄的 3D 域，其行为接近 2D 流。 对于 2-D 流，不同长度尺度上的能量命运更为复杂，尽管 3-D 流在某种意义上已嵌入其中。 尽管二维实验很难在实验室中进行，但它们可以在计算机上进行更精细的研究。 在所有二维流中，本项目研究的流可以说是最适合分析和高效模拟的。 然而，它不仅对于大气等二维和近二维流动至关重要，而且由于普遍性而对于 3-D 湍流也至关重要。

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.