Collaborative Proposal: Study of turbulence in physical systems through complex singularities and determining modes

合作提案：通过复杂奇点和确定模式研究物理系统中的湍流

• 批准号: 1109638
• 负责人: Michael Jolly
• 金额: $ 11.57万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1109638&HistoricalAwards=false
• 关键词: Collaborative; Proposal; Study; turbulence; physical

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.

该项目将两个最近引入的演化方程应用于流体动力湍流的基本问题。 第一个方程用于确定纳维-斯托克斯型微分方程的模式。一组低模决定了，如果对于全局吸引子上的任何解，则所有时间的解的高模可以仅从低模（对于所有时间）产生。将这种关系扩展到从实数到低模的有界函数的 Banach 空间，得到一个确定的形式，即全局 Lipschitz 且耗散的演化方程。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一种替代方法，目前尚不清楚该理论是否适用于纳维-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域及其对流体流动方面（例如间歇性）的影响。大尺度的流体运动，例如在海洋和大气中，表现出不稳定的、看似难以理解的（湍流）在本地短​​时间查看时的功能。然而，随着时间的推移，一致的全球统计模式出现了。这项研究的目的是对这种模式的本质和起源以及其偏差有一个严格的理解，以期实现海洋建模和天气预报等多种应用。例如，天气预报是通过求解一个大型方程组来完成的，该方程组模拟了一段时间内的大气状态。需要大气的当前状态作为输入，以便计算未来的状态。 由于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。系统对此类错误的固有敏感性意味着随着时间的推移，计算出的解决方案会偏离现实。数据同化技术不仅使用系统在单个时刻的有限测量，而且还使用过去较长时间段内的有限测量，以获得更准确的输入。 该项目应用最新的数学发现来开发该技术的新变体。它将产生新的算法，并由研究团队实施和测试。 研究生将在这项合作中发挥重要作用，从而培养下一代 STEM 学科的科学家。