Mathematical Sciences: Topics in Hydrodynamics and Magnetohydrodynamics

这位研究者将与芝加哥大学的M. M. Vishik博士合作，继续研究流体动力学方面的课题。她使用逆散射技术来获得欧拉方程的解，欧拉方程控制着不可压缩、无粘性流体的流动。这项工作是从研究者和Vishik导出的欧拉方程的Lax对公式开始的。她还建议应用发电机理论技术来扩展她和Vishik最近获得的线性化流体方程的不稳定性准则的适用性。研究者将继续她的研究，通过使用哈密顿技术，三维轴对称磁流体动力平衡的稳定性/不稳定性。本研究的主要主题涉及不稳定性存在的标准（即，当稳定结构中的小变化随着时间变得越来越大时）。流体不稳定性是决定许多重要物理系统行为的关键因素。例如，所谓的斜压不稳定性控制着大气的大尺度环流；对流不稳定性在海气相互作用中发挥重要作用，对可能的全球变暖具有影响；磁流体动力学的不稳定性是等离子体聚变现象的基础；地球磁场和太阳磁场的存在取决于发电机的不稳定性。研究者对流体运动方程的数学分析得出了控制某些类型不稳定性存在的具体标准，她提出的研究将继续推进我们对地球物理和天体物理流体行为的理解。

The investigator, in collaboration with Dr. M. M. Vishik of University of Chicago, will continue to investigate topics in fluid dynamics. She uses inverse scattering techniques to obtain solutions to the Euler equations that govern the flow of an incompressible, inviscid fluid. This work follows from a Lax pair formulation for the Euler equations derived by the investigator and Vishik. She also proposes to apply dynamo theory techniques to extend the applicability of the instability criteria that she and Vishik have recently obtained for the linearized fluid equations. The investigator will continue her examination, via the use of Hamiltonian techniques, of the stability/instability of three dimensional axisymmetric magnetohydrodynamic equilibria. The primary subject of this research concerns criteria for the existence of instabilities (i.e., when a small change in a steady configuration becomes larger and larger with time.) Fluid instabilities are crucial factors in determining the behavior of many important physical systems. For example, the so called baroclinic instability governs the large scale circulation of the atmosphere; convective instabilities play an important role in air-sea interaction with implications for possible global warming; magnetohydrodynamic instabilities underlie the phenomenon of plasma fusion; the existence of terrestrial and solar magnetic fields is dependent on dynamo instabilities. The investigator's mathematical analysis of the equations of fluid motion has led to specific criteria governing the existence of certain types of instabilities and her proposed research will continue to advance our understanding of the behavior of geophysical and astrophysical fluids.