Collaborative Research: Study of Turbulence in Physical Systems Through Complex Singularities and Determining Modes

本项目将两个最近引入的演化方程应用于流体动力学湍流的基本问题。第一个方程用于确定Navier-Stokes型微分方程的模态。一组低模态决定了，如果对于全局吸引子上的任何解，解的所有时间的高模态，只能由（所有时间的）低模态产生。将这种关系扩展到有界函数的Banach空间，从实数到低模态，可以得到一个决定形式，一个全局Lipschitz的演化方程和耗散方程。原微分方程的全局吸引子包含在确定形式的长时间行为中。这为惯性形式理论提供了另一种选择，而惯性形式理论并不适用于纳维-斯托克斯方程。另一个演化方程是空间变量的解析半径。我们将使用这个方程来研究全局吸引子解的解析性域及其对流体流动的影响，如间歇性。大尺度的流体运动，例如海洋和大气，在局部和短时间内表现出不稳定的、似乎难以理解的（湍流）特征。然而，随着时间的推移，一致的全球统计模式出现了。这项研究的目的是发展对这种模式的性质和起源的严格理解，以及对其偏差的理解，以期实现各种应用，如海洋建模和天气预报。例如，天气预报是通过求解一个大的方程组来完成的，这些方程组模拟了一段时间内的大气状态。为了计算未来的状态，需要将大气的当前状态作为输入。由于对系统状态的测量总是有限的，因此这种输入不可避免地包含一些误差。系统对这种误差的固有敏感性意味着，随着时间的推移，计算出的解会偏离现实。数据同化技术不仅利用系统在某一时刻的有限测量值，而且利用过去一段较长时间的测量值，以获得更准确的输入。这个项目应用了最近的数学发现来开发这种技术的新变体。这将导致新的算法，将由研究小组实施和测试。研究生将在这一合作努力中发挥重要作用，从而有助于培养STEM学科的下一代科学家。

This project applies two recently introduced evolution equations to fundamental issues in hydrodynamic turbulence. The first equation is for the determining modes of a differential equation of Navier-Stokes type. A set of low modes is determining, if for any solution on the global attractor, the high modes of the solution for all time, can be produced from only the low modes (for all time). Extending this relationship to the Banach space of bounded functions from the reals into the low modes leads to a determining form, an evolution equation that is globally Lipschitz, and dissipative. The global attractor of the original differential equation is contained in the long time behavior of the determining form. This provides an alternative to the theory of inertial forms which is not known to be applicable to the Navier-Stokes equations. The other evolution equation is for the radius of analyticity in the spatial variable. We will use this equation to study the domain of analyticity for solutions on the global attractor and its effect on aspects of fluid flow such as intermittency.Fluid motions at large scales, for instance in ocean and atmosphere, display erratic, seemingly incomprehensible (turbulent) features when viewed locally and for short times. However, over time, consistent global statistical patterns emerge. The purpose of this research is to develop a rigorous understanding of the nature and origin of such patterns, and of deviations from it, with a view towards diverse applications such as ocean modeling and weather forecasting. For instance, weather forecasting is done by solving a large system of equations which model the state of the atmosphere over a period of time.The current state of the atmosphere is needed as input in order to compute the state in the future. As there are always limited measurements of the state of the system, this input inevitably contains some error. The inherent sensitivity of the system to such an error means that as time goes on the computed solution diverges from reality. The technique of data assimilation uses the limited measurements of the system at not just a single moment of time, but rather over an extended time period in the past, to arrive at a more accurate input. This project applies recent mathematical discoveries to develop new variations on this technique. It will lead to new algorithms that will be implemented and tested by the research team. Graduate students will play an essential role in this collaborative effort, thus serving to train future generation of scientists in the STEM disciplines.