Point Vortex Dynamics on Closed Surfaces

TL;DR

This paper extends point vortex dynamics theory to closed surfaces, providing a comprehensive guide that unifies the theory across different geometries and includes implementation details.

Contribution

It offers a unified framework for point vortex dynamics on closed surfaces with genus zero, expanding the classical theory to new geometrical contexts.

Findings

Unified theory for vortex dynamics on closed surfaces

Implementation details for computational modeling

Connection of surface topology with vortex behavior

Abstract

The theory of point vortex dynamics has existed since Kirchhoff's proposal in 1891 and is still under development with connections to many fields in mathematics. As a strong simplification of the concept of vorticity it excels in computational speed for vorticity based fluid simulations at the cost of accuracy. Recent finding by Stefanella Boatto and Jair Koiller allowed the extension of this theory on to closed surfaces. A comprehensive guide to point vortex dynamics on closed surfaces with genus zero and vanishing total vorticity is presented here. Additionally fundamental knowledge of fluid dynamics and surfaces are explained in a way to unify the theory of point vortex dynamics of the plane, the sphere and closed surfaces together with implementation details and supplement material.