Mathematical Sciences: Hydrodynamic Stability and Dynamo Theory

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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.