Mathematical Sciences: Hydrodynamic Stability and Dynamo Theory

PI将调查磁场的增长率， 导电流体的平稳稳定流动， 高磁雷诺数极限。 中的方法 几何测度理论和动力系统提供了一个 从上面估计，可能是一个明确的公式 这是“快速发电机”理论的核心问题。他 提出研究Floquet根系统的完备性 流体动力学的时间周期问题中的向量。 如果这样的 系统确实存在，这意味着磁场在一个 导电流体的时间周期性流动不会衰减 同样的事实也适用于小的 粘性流体中的振荡。 他将继续研究 关于欧拉方程的“几何不稳定性理论” 这将允许处理理想流体的流动， 奇点 发电机问题涉及到磁场的变化过程 通过高传导性流体的运动产生的。 这 磁场的存在是由过程造成的。 地球、太阳磁场和 天体物理学 它也可以在实验室中实现， 实验难度相当大。 建议的 数学研究揭示了“快速”的过程（大多数 有效）产生磁场。 这个“快”的过程 发生在一个时间尺度上， 磁场欧姆衰减的时间。 另一行 研究是利用 发电机理论的观点。 不稳定性的一般条件 流体流动在流体力学领域具有根本的重要性 和空气动力学。

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.