A novel formulation of point vortex dynamics on the sphere: geometrical and numerical aspects

TL;DR

This paper introduces a new Lagrangian formulation for point vortices on the sphere by leveraging the Hopf fibration and Lie group structures, leading to a symplectic integrator that outperforms classical methods.

Contribution

A novel Lagrangian formulation on the sphere using the Hopf fibration and Lie groups, resulting in an efficient symplectic integrator for vortex dynamics.

Findings

The integrator is symplectic and second-order.

It preserves the unit-length constraint of vortices.

It outperforms classical Runge-Kutta and midpoint methods.

Abstract

In this paper, we present a novel Lagrangian formulation of the equations of motion for point vortices on the unit 2-sphere. We show first that no linear Lagrangian formulation exists directly on the 2-sphere but that a Lagrangian may be constructed by pulling back the dynamics to the 3-sphere by means of the Hopf fibration. We then use the isomorphism of the 3-sphere with the Lie group SU(2) to derive a variational Lie group integrator for point vortices which is symplectic, second-order, and preserves the unit-length constraint. At the end of the paper, we compare our integrator with classical fourth-order Runge--Kutta, the second-order midpoint method, and a standard Lie group Munthe-Kaas method.