Mathematical Sciences: Hydrodynamic Stability and Dynamo Theory

数学科学：水动力稳定性和发电机理论

• 批准号: 9301172
• 负责人: Mikhail Vishik
• 金额: $ 7万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-05-15 至 1996-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9301172&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Hydrodynamic; Stability; Dynamo

The PI will investigate the growth rate of the magnetic field in a smooth steady flow of an electrically conducting fluid in the limit of high magnetic Reynolds number. The methods from Geometric Measure Theory and Dynamical Systems provide an estimate from above and possibly an explicit formula for this quantity which is a central problem of "Fast Dynamo" theory. He proposes to study the completeness of a system of Floquet root vectors in time-periodic problems of fluid dynamics. If such a system indeed exists, it would mean that the magnetic field in a time-periodic flow of a conducting fluid cannot decay superexponentially and the same fact is true for small oscillations in a viscous fluid. He will continue the research on the "geometric instability theory" for the Euler equation which would allow treatment of flows of an ideal fluid with singularities. The Dynamo problem concerns the process of a magnetic field generation by the motion of a highly conducting fluid. This process is responsible for the existence of the magnetic field of the Earth, solar magnetic field and magnetic fields in astrophysics. It also can be achieved in laboratory although experimental difficulties are considerable. The suggested mathematical study sheds light on the process of "fast" (most efficient) generation of the magnetic field. This "fast" process takes place on a time scale many orders of magnitude less than the time of Ohmic decay of the magnetic field. Another line of research is the investigation of hydrodynamic instability using ideas from dynamo theory. General conditions for instability of fluid flows are of fundamental importance in the areas of fluid and aerodynamics.

PI 将研究在高磁雷诺数极限下导电流体平稳稳定流动中的磁场增长率。 几何测量理论和动力系统的方法提供了上述估计，并可能为该量提供了一个明确的公式，这是“快速发电机”理论的核心问题。他提议研究流体动力学时间周期问题中 Floquet 根向量系统的完整性。 如果这样的系统确实存在，则意味着导电流体的时间周期流动中的磁场不能超指数衰减，并且对于粘性流体中的小振荡也同样如此。 他将继续研究欧拉方程的“几何不稳定性理论”，该方程可以处理具有奇点的理想流体的流动。 发电机问题涉及高导电流体运动产生磁场的过程。 这个过程导致了地球磁场、太阳磁场和天体物理学中磁场的存在。 尽管实验难度相当大，但它也可以在实验室中实现。 建议的数学研究揭示了磁场“快速”（最有效）产生的过程。 这种“快速”过程发生的时间尺度比磁场欧姆衰减时间小许多数量级。 另一条研究方向是利用发电机理论的思想来研究水动力不稳定性。 流体流动不稳定的一般条件在流体和空气动力学领域至关重要。