Phase mixing in the fluid mechanics and kinetic theory

流体力学和动力学理论中的相混合

• 批准号: 1462029
• 负责人: Jacob Bedrossian
• 金额: $ 12.03万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-21 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1462029&HistoricalAwards=false
• 关键词: Phase; mixing; fluid; mechanics; kinetic

Nonlinear mixing is a ubiquitous behavior observed in fluids and plasmas which plays an important role in understanding certain aspects of atmosphere and ocean sciences, turbulence, astrophysics and controlled fusion. For example, it is thought to be an important organizing mechanism in atmospheric dynamics that helps to explain why hurricanes retain their characteristic vortex shape for extended periods of time. Despite being a fundamental physical mechanism, the theoretical and mathematical analysis of nonlinear phase mixing has remained elusive. Moreover, many of these effects are difficult to isolate in computational or physical experiments and here mathematical analysis can provide important insights. This project specifically aims to expand our mathematical understanding by studying the effect in a variety of fundamental 'case studies' found in fluid mechanics and plasma physics.The purpose of this program is to further the mathematical analysis of nonlinear phase mixing, generally regarded as difficult due to the appearance of highly non-normal linear evolutions and unusual regularity issues intertwined with subtle nonlinear resonances. The PI will work to develop new tools necessary to approach these problems with a focus on working towards four specific settings: the vanishing viscosity limit of the incompressible Navier-Stokes equations (NSE) near the 2D Couette flow, the nonlinear instability of 3D shear flows, inviscid 'vortex axisymmetrization' in 2D Euler and Landau damping in Vlasov-Poisson (or Vlasov-Maxwell) in presence of external magnetic fields. Many related physically relevant problems exist, and may serve as intermediate steps or additional lines of research. The work in fluid mechanics will expand further on ideas employed by the PI and collaborator Nader Masmoudi in the recent work on the stability of 2D Couette flow [arXiv:1306.5028], especially the use of adaptive coordinate systems and the construction of norms which account for weakly nonlinear resonances and the non-normal transient linear behavior. The work in kinetic theory will further develop techniques used in the work of the PI and collaborators Nader Masmoudi and Clement Mouhot on Landau damping [arXiv:1311.2870], especially the efficient treatment of the plasma echoes. Most likely, in order to expand the range of kinetic theory applications, some ideas from the work in fluid mechanics may need to be adapted to the kinetic setting.

非线性混合是在流体和等离子体中观察到的一种普遍现象，它在理解大气和海洋科学、湍流、天体物理和受控聚变的某些方面发挥着重要作用。例如，它被认为是大气动力学中的一个重要组织机制，有助于解释为什么飓风在很长一段时间内保持其特有的涡旋形状。尽管这是一种基本的物理机制，但对非线性相位混合的理论和数学分析仍然难以捉摸。此外，这些影响中的许多很难在计算或物理实验中分离出来，在这里，数学分析可以提供重要的见解。这个项目的具体目的是通过研究流体力学和等离子体物理中发现的各种基本案例研究中的影响来扩大我们的数学理解。这个项目的目的是进一步非线性相混合的数学分析，由于高度非正态的线性演化和与微妙的非线性共振交织在一起的不寻常的规律性问题的出现，通常被认为是困难的。PI将致力于开发解决这些问题所需的新工具，重点放在四个具体的设置上：2D Couette流附近不可压缩Navier-Stokes方程(NSE)的粘性极限消失，3D剪切流的非线性不稳定性，2D Euler中的无粘轴对称涡旋轴对称，以及存在外部磁场时Vlasov-Poisson(或Vlasov-Maxwell)中的朗道阻尼。存在许多相关的物理相关问题，可以作为中间步骤或额外的研究方向。流体力学的工作将进一步扩展PI和合作者Nader Masmudi在最近关于2D Couette流稳定性的工作中所采用的思想[arxiv：1306.5028]，特别是自适应坐标系的使用和范数的构建，这些范数解释了弱非线性共振和非正常的瞬时线性行为。运动论方面的工作将进一步发展PI及其合作者纳德·马斯穆迪和克莱门特·穆霍关于朗道衰减的工作中使用的技术[arxiv：1311.2870]，特别是有效地处理等离子体回波。最有可能的是，为了扩大动力学理论的应用范围，流体力学中的一些思想可能需要适应动力学环境。