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

合作研究：通过复杂奇点和确定模式研究物理系统中的湍流

• 批准号: 1425877
• 负责人: Animikh Biswas
• 金额: $ 3.22万
• 依托单位: University of Maryland Baltimore County
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-11-18 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1425877&HistoricalAwards=false
• 关键词: Collaborative; Research; 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.

该项目将最近引入的两个进化方程式应用于流体动力湍流的基本问题。 第一个方程是用于确定Navier-Stokes类型的微分方程的模式。一组低模式正在确定，如果对于全局吸引子上的任何解决方案，则可以一直以低模式（始终）产生解决方案的高模式。将这种关系扩展到有界函数的Banach空间从真实的模式到低模式，导致了确定的形式，一个全球lipchitz和耗散的进化方程。原始微分方程的全局吸引子包含在确定形式的长时间行为中。 这为惯性形式的理论提供了替代方法，该理论不适用于Navier-Stokes方程。另一个进化方程是用于空间变量中的分析性半径。 我们将使用该方程来研究全球吸引子解决方案的分析域及其对流体流动方面的影响，例如间歇性。大规模的流体运动，例如在海洋和大气中，在局部和短时间内均显示出不稳定的，看似不可理解的（湍流）特征。但是，随着时间的流逝，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进行严格的理解，并与之偏离，以期对海洋建模和天气预报等各种应用。例如，天气预报是通过解决一段时间内对大气状态进行建模的大型方程式来完成的。需要将大气的当前状态作为输入，以便将来计算状态。 由于系统状态的测量始终有限，因此此输入不可避免地包含一些错误。系统对这种错误的固有敏感性意味着随着时间的流逝，计算的解决方案与现实不同。数据同化的技术不仅在一次时间时刻使用了系统的有限测量值，而且在过去的延长时间段上使用了更准确的输入。 该项目应用了最新的数学发现，以开发有关此技术的新变体。它将导致研究团队将实施和测试的新算法。 研究生将在这项合作努力中发挥至关重要的作用，从而培训STEM学科中未来的科学家。