Mathematical Sciences: Equilibria Instabilities and Waves in Fluids and Plasmas

数学科学：流体和等离子体中的平衡不稳定性和波动

• 批准号: 9623033
• 负责人: Alexander Lipton
• 金额: $ 5.73万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623033&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Equilibria; Instabilities; Waves

9623033 Lipton The Principal Investigator (PI) will continue to study equilibria, instabilities and waves in fluids and plasmas with the emphasis on the comparative aspect of the problem. The PI is particularly interested in the following topics: (a) stationary states of magnetized plasmas with flow and their group-theoretical properties; (b) the stability and evolution of elliptical self-gravitating fluid masses (the Riemann ellipsoids) with applications to astrophysics; (c) the linear and nonlinear dynamics of fixed-boundary ellipsoids containing fluid. The PI is going to use various analytical techniques including the constrained minimization, `exact' Galerkin, group-theoretical, and geometrical optics methods in conjunction with numerical techniques such as the three-dimensional vortex method, large scale symbolic manipulations, and visualization, in order to achieve these goals. Further directions of attack include self-similar, time-dependent vortex rings, short wavelength instabilities of unmagnetized and magnetized accretion discs, global instabilities of collisionless gravitating systems,and spectral problems of hydrodynamics and magnetohydrodynamics. This research will advance current knowledge of the behavior of both laboratory and astrophysical fluids and plasmas. In particular, this research will contribute towards a better understanding of the properties of nonlinear PDEs of mixed elliptic/hyperbolic type describing symmetric stationary states of plasmas with flow; the stability of Riemann ellipsoids and fluid- and plasma-filled bodies; the evolution of nonaxisymmetric stars, and the validity of the fission theory of binary stars. %%% The dynamics of fluids and plasmas is of great interest to mathematicians and physicists alike. There are at least two reasons for this interest. First, fluid and plasma motions play a fundamental role in nature and technology with applications stretching from astrophysics, to oceanography, to weather predicti on, to controlled thermonuclear fusion. Second, a very rich and sophisticated mathematical apparatus is required in order to give an adequate description of these motions. The equilibrium theory studies the structure and properties of steady solutions of the nonlinear equations of fluid and plasma dynamics. These equilibrium solutions represent special states which do not change in time when left undisturbed. Usually it is rather difficult to describe equilibrium solutions, however, it is still much easier then to describe the general solutions. In addition, equilibrium solutions are the most interesting ones from a practical point of view. The stability theory studies the impact of initially small perturbations on a given steady fluid or plasma equilibrium. An equilibrium is called stable and can occur in nature if perturbations do not have a profound effect on its properties. An equilibrium is called unstable if under the influence of perturbations it either evolves into a different equilibrium, or loses its steady character altogether. Over a period of years many classical stability problems were solved, however, several important stability problems are still open. The wave theory studies the behavior of small perturbations in the vicinity of stable plasma equilibria. The main issues include the analysis of the oscillation frequencies, the structure of the corresponding eigenfunctions, and the asymptotic behavior of perturbations in time. The present research is aimed at advancing current understanding of equilibria, instabilities and waves. It blends theoretical and symbolic methods with high-performance computing and graphics. The outcome of this research will have both theoretical and practical value. Among other things, it will help to understand the dynamics of plasmas in the so-called tokamaks (installations which are used for controlled thermonuclear fusion), and the stability of stars. ***

小行星9623033 主要研究者（PI）将继续研究平衡， 流体和等离子体中的不稳定性和波动，重点放在问题的比较方面。PI对以下主题特别感兴趣：（a）流动的磁化等离子体的定态及其群论性质;（B）椭圆自引力流体质量的稳定性和演化（黎曼 椭球）与天体物理学的应用;（c）包含流体的固定边界椭球的线性和非线性动力学。PI将使用各种分析技术，包括约束最小化、“精确”Galerkin、群论和几何光学方法，并结合数值技术，如三维涡旋方法、大尺度符号方法和几何光学方法。 操纵和可视化，以实现这些目标。 进一步的研究方向包括自相似的含时涡环，非磁化和磁化吸积盘的短波长不稳定性，无碰撞的整体不稳定性， 引力系统，以及流体力学的光谱问题， 磁流体力学这项研究将推进实验室和天体物理流体和等离子体行为的现有知识。 特别是，本研究将有助于更好地理解混合椭圆/双曲型非线性偏微分方程的性质 描述对称稳态的等离子体与流动;稳定的黎曼椭球体和流体和等离子体填充机构;非轴对称恒星的演变，和有效性的裂变理论的双星。 流体和等离子体的动力学对数学家和物理学家都有很大的兴趣。这种兴趣至少有两个原因。首先，流体和等离子体运动在自然界和技术中起着基础作用，其应用范围从天体物理学到海洋学，到天气预报，再到受控热核聚变。第二，需要一个非常丰富和复杂的数学工具，以便给出 对这些运动的充分描述。 平衡理论研究流体和等离子体动力学的非线性方程定常解的结构和性质。这些平衡解代表了当不受干扰时不随时间变化的特殊状态。通常，描述平衡解是相当困难的，然而，描述一般解仍然容易得多。此外，从实用的角度来看，平衡解是最有趣的。稳定性理论研究了初始小扰动对给定定常系统的影响， 流体或等离子体平衡。 一个平衡被称为稳定的，如果扰动对其没有深刻的影响，它就可以在自然界中发生。 特性. 如果在扰动的影响下，一个平衡态要么演化成另一个平衡态，要么完全失去其稳定性，那么这个平衡态就被称为不稳定的。 多年来，许多 经典的稳定性问题已经解决，然而，一些重要的稳定性问题仍然是开放的。波动理论研究稳定等离子体平衡附近的小扰动行为。主要问题包括对振荡的分析 频率，相应的本征函数的结构，以及扰动在时间上的渐近行为。本研究旨在推进目前对平衡、不稳定性和波动的理解。它融合了理论和象征的方法， 高性能计算和图形。本文的研究成果具有一定的理论意义和实用价值。除其他外，它将有助于了解所谓的托卡马克（用于受控的装置）中等离子体的动力学。 热核聚变）和恒星的稳定性。 ***