Fluid Dynamics with Incompressible Schrödinger Flow

This thesis introduces a new way of looking at incompressible fluid dynamics. Specifically, we formulate and simulate classical fluids using a \({\Bbb C}^2\)-valued Schrödinger equation subject to an incompressibility constraint. We call such a fluid flow an incompressible Schrödinger flow (ISF). The approach is motivated by Madelung's hydrodynamical form of quantum mechanics, and we show that it can simulate classical fluids with particular advantage in its simplicity and its ability of capturing thin vortex dynamics. The effective dynamics under an ISF is shown to be an Euler equation modified with a Landau-Lifshitz term. We show that the modifying term not only enhances the dynamics of vortex filaments, but also regularizes the potentially singular behavior of incompressible flows. Another contribution of this thesis is the elucidation of a general, geometric notion of Clebsch variables. A geometric Clebsch variable is useful for analyzing the dynamics of ISF, as well as representing vortical structures in a general flow field. We also develop an algorithm of approximating a ``spherical'' Clebsch map for an arbitrarily given flow field, which leads to a new tool for visualizing, analyzing, and processing the vortex structure in a fluid data.