Abstract:

Point vortices are singularities in the vorticity field of a perfect fluid. Physically, they show up as points in the fluid around which the fluid rotates, much like the fluid rotating around the core of a hurricane in the atmosphere.  From a fluid-dynamical point of view, the dynamics of point vortices forms a (finite-dimensional) Hamiltonian system, the solutions of which are weak solutions of Euler's equation, and there is consequently a great deal of interest in modeling and simulating the dynamics of point vortices.

Despite the fact that the equations of motion are Hamiltonian, however, this system has a number of special features which make direct numerical simulation hard: the vector field is not Lipschitz continuous, the symplectic form preserved is not the standard one, and there exists no Lagrangian formulation. The latter two problems especially make the construction of a symplectic integrator for this system a non-trivial task.

In this talk I will use the Hopf fibration to develop a new formulation of the point vortex equations on the sphere S^2. More specifically, I will show that there exists a nonlinear-Schrodinger-like equation on the Lie group SU(2), or alternatively the space of unit quaternions, whose solutions project down onto the trajectories of point vortices in S^2. Using the connection one-form of the Hopf fibration, I will also construct a Lagrangian for these equations, and we will investigate some symplectic integrators based on this formulation. We will illustrate the long-term qualitative features of these integrators with a number of examples from the recent literature, such as the spherical von Karman vortex street.