The N-Vortex Problem on a Symmetric Ellipsoid: A Perturbation Approach

TL;DR

This paper develops a perturbation method to analyze the N-vortex problem on a symmetric ellipsoid, deriving explicit first-order equations and demonstrating non-integrability for three vortices.

Contribution

It introduces a conformal transformation approach to simplify vortex equations on an ellipsoid and derives explicit first-order equations in eccentricity.

Findings

First-order equations explicitly derived

Numerical evidence of non-integrability for three vortices

Method applicable to symmetric ellipsoids

Abstract

We consider the N-vortex problem on a ellipsoid of revolution. Applying standard techniques of classical perturbation theory we construct a sequence of conformal transformations from the ellipsoid into the complex plane. Using these transformations the equations of motion for the N-vortex problem on the ellipsoid are written as a formal series on the eccentricity of the ellipsoid's generating ellipse. First order equations are obtained explicitly. We show numerically that the truncated first order system for the three-vortices system on the symmetric ellipsoid is non-integrable.