Mathematical Sciences: Hydrodynamic Stability and Dynamo Theory

PI将研究在高磁雷诺数极限下导电流体的平稳流动中磁场的增长速度。几何测量理论和动力系统的方法提供了上述估计，并可能提供了一个明确的公式，这是“快速发电机”理论的中心问题。他提出研究流体动力学时间周期问题中Floquet根向量系统的完备性。如果这样的系统确实存在，这将意味着磁场在导电流体的时间周期流动中不会呈超指数衰减，同样的事实也适用于粘性流体中的小振荡。他将继续研究欧拉方程的“几何不稳定性理论”，该理论将允许处理具有奇点的理想流体的流动。发电机问题涉及由高导电性流体的运动产生磁场的过程。这个过程是地球磁场、太阳磁场和天体物理学中磁场存在的原因。虽然实验难度相当大，但也可以在实验室中实现。这个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”过程发生在比磁场欧姆衰减时间短许多数量级的时间尺度上。另一个研究方向是利用发电机理论的思想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具有根本的重要性。

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.