N-Vortex Problems: Modeling, Analysis, and Numerics

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We propose to study a wide range of N-vortex problems based on singular or near singular decompositions of the inviscid Euler equations and high Reynolds Navier-Stokes equations written in vorticity form. With such a discretization, the vorticity field equations become particle interaction equations and the models can be viewed as N-body problems with long-range interactions governed by a logarithmic Hamiltonian. The projects focus on three general themes: (i) the N-vortex problem on a rotating sphere with applications to atmospheric flows, (ii) vortex lattice theory with applications to recent experiments in Bose-Einstein condensate systems, (iii) the study of streamline topology, bifurcations, and evolution with the a focus on connecting this with chaotic advection and particle transport. There are seven detailed projects described in the proposal, each is designed to develop new analytical and computational techniques in dynamical systems theory, test current techniques on models that are physically well grounded, and push the models closer towards applications, mainly in oceanographic and atmospheric flows, but also in molecular modeling and the study of Bose-Einstein condensate lattices and vortices in thin film superconductors, where some of the same underlying issues pertain (albeit sometimes with Hamiltonians of a different form). An overriding theme of all of the projects is to develop Hamiltonian particle models in conjunction with probabilistic methods to analyze fluid flows and to develop and apply new techniques in nonlinear dynamical systems theory. The main goals associated with the mathematical models being developed are to model, analyze, and simulate specific atmospheric events, such as the Antarctic pole vortex splitting event of September 2002, which manifested itself in the form of a split ozone hole in the Southern Hemisphere and led to enhanced global particle transport over the full stratospheric layer of the Earth's atmosphere. The specific reasons for this enhancement are not well understood and we will investigate several low dimensional models of this important and unprecedented event. Interestingly, the mathematical models developed in this context are similar to those used in completely different physical settings, such as the study of Abrikosov lattices in superconductivity theory, the recent flurry of experiments done on Bose-Einstein condensate lattices, and the modeling of spherical molecules in chemical physics, such as the famous Buckminsterfullerine molecule. We expect that developments associated with this proposal over the three year funding period will shed light on many physical mechanisms important to these areas as well.