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

本计画应用两个新近引进的演化方程式来探讨流体动力学紊流的基本问题。 第一个方程是Navier-Stokes型微分方程的决定模式。一组低模式是确定的，如果对于全局吸引子上的任何解，该解的所有时间的高模式可以仅从低模式（所有时间）产生。将这种关系扩展到Banach空间的有界函数从实数到低模导致一个确定的形式，一个发展方程，是全球Lipschitz，和耗散。原微分方程的全局吸引子包含在决定型的长时间行为中。 这为惯性形式理论提供了一种替代方法，而惯性形式理论不适用于纳维尔-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解析域的整体吸引子的解决方案和它的影响方面的流体流动，如coincidency.Fluid运动在大尺度上，例如在海洋和大气中，显示不稳定的，似乎难以理解的（湍流）功能时，当地和短时间。然而，随着时间的推移，出现了一致的全球统计模式。这项研究的目的是对这种模式的性质和起源以及偏离这种模式的情况有一个严格的了解，以期实现海洋建模和天气预报等各种应用。例如，天气预报是通过求解一个模拟一段时间内大气状态的大型方程组来完成的，为了计算未来的状态，需要将当前的大气状态作为输入。 由于对系统状态的测量总是有限的，因此该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，计算的解决方案偏离现实。数据同化技术利用系统的有限测量值，不仅是在某一时刻，而是在过去的一段较长时间内，以获得更准确的输入。 该项目应用最新的数学发现来开发这种技术的新变体。它将导致新的算法，将由研究团队实施和测试。 研究生将在这一合作努力中发挥重要作用，从而为培养STEM学科的未来一代科学家服务。

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.