Mathematical Sciences: Equilibria Instabilities and Waves in Fluids and Plasmas

9623033立顿首席研究员(PI)将继续研究流体和等离子体中的平衡、不稳定性和波动，重点放在问题的比较方面。PI对下列主题特别感兴趣：(A)流动磁化等离子体的定态及其群论性质；(B)椭圆自重流体质量(黎曼椭球)的稳定性和演化及其在天体物理学中的应用；(C)含有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，以及三维涡旋方法、大规模符号处理和可视化等数值技术，以实现这些目标。进一步的攻击方向包括自相似的、依赖于时间的涡环、未磁化和磁化吸积盘的短波不稳定性、无碰撞引力系统的全局不稳定性以及流体动力学和磁流体动力学的光谱问题。这项研究将促进对实验室和天体物理流体和等离子体行为的当前了解。特别是，这项研究将有助于更好地理解描述流动等离子体对称定态的椭圆/双曲型混合非线性偏微分方程组的性质；黎曼椭球和充满流体和等离子体的物体的稳定性；非轴对称恒星的演化，以及双星裂变理论的有效性。流体和等离子体的动力学是数学家和物理学家都非常感兴趣的。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发挥着基础性的作用，应用范围从天体物理学、海洋学、天气预报到受控的热核聚变。其次，为了对这些运动进行充分的描述，需要非常丰富和复杂的数学工具。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。这些平衡解代表了特殊的状态，当不受干扰时，这些状态不会随时间变化。通常，描述均衡解相当困难，然而，描述一般解仍然容易得多。此外，从实践的角度来看，均衡解是最有趣的。稳定性理论研究初始微小扰动对给定的稳定流体或等离子体平衡的影响。平衡被称为稳定的，如果扰动不会对其性质产生深刻的影响，那么在自然界中就可能发生平衡。如果一个平衡点在扰动的影响下演化成另一个不同的平衡点，或者完全失去了它的稳定特性，那么这个平衡点就叫做不稳定。多年来，许多经典的稳定性问题已经得到解决，然而，一些重要的稳定性问题仍然是悬而未决的。波动理论研究了稳定等离子体平衡附近的微小扰动行为。主要问题包括振荡频率的分析，相应的特征函数的结构，以及时间扰动的渐近行为。本研究的目的是促进当前对平衡、不稳定和波动的理解。它将理论和符号方法与高性能计算和图形相结合。本研究成果具有一定的理论意义和实用价值。在其他方面，它将有助于理解所谓托卡马克(用于受控热核聚变的装置)中等离子体的动力学，以及恒星的稳定性。***

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. ***