Executive Summary : | The point vortex model model is the simplest model for studying vortex interactions in a two-dimensional ideal flow. In point vortex the vorticity profile tends to be a delta distribution. The point vortex equations have a Hamiltonian form and are well suited for statistical hydrodynamics. Like the N -body problem in celestial mechanics, the motion of a system of mutually interacting N -point vortices is called an N -vortex problem. The solutions to the one- and two-vortex problems are somewhat trivial. While it is known that the 3-vortex system is integrable, it does not permit closed-form solutions for large N in general. The interaction of more than three vortices leads to chaotic vortex trajectories. Thus the analysis of larger vortex systems is complicated from both numerical and analytical points of view. The proposal aims to develop a generic numerical framework for understanding vortex motion far from the integrable regime, where the number of vortices is large. In particular, the proposal will study point vortex solutions associated with vortex patterns that move without a change of shape and size-referred to as the relative equilibrium (RE) configuration. In a RE configuration, vortices move as a rigid body preserving the initial geometrical shape and size. Such RE configurations are also found as long-lived vortex patterns, for example, in geophysical flows and in various two-dimensional flow situations such as super-fluids, superconductors, Bose-Einstein condensates. The RE configurations resulting from the interaction of 3-point vortices in a two-dimensional ideal fluid is well-known. In particular it is known that initial vortex positions in the form of an equilateral triangle is the only strictly planar arrangement of vortices that yield a RE configuration. On contrary, the understanding of point vortex systems for N large is quite limited except for a very few special cases. The only known facts about the 4-vortex system are that the number of possible RE configurations is finite for generic circulations and that there are infinitely many strictly planar geometries possible for RE configurations. Note that the four vortex system is the smallest vortex system in which infinitely many strictly planar geometries possible for RE configurations. To the best of PI's knowledge, a systematic study addressing the infinite collection of all possible quadrilateral geometries in RE configurations has never been carried out. Such studies would undoubtedly shine some light on the attempts to understand more complex larger point vortex systems. Objectives of the project are: 1) to establish numerical methods for finding RE configuration four vortex, 2) to describe the distribution of possible geometries for the 4-vortex RE configurations, 3) to classify the RE configurations, 4) To describe some interesting quadrilateral RE configurations with symmetry, 5) To develop a numerical framework based on optimization for larger N. |