Vortex solution in elliptic coordinates

TL;DR

This paper derives transformations between Cartesian and elliptic coordinates to analyze vortex solutions with elliptic streamlines, revealing how constant vorticity flows behave in elliptic coordinates.

Contribution

It introduces a differential geometry approach to relate Cartesian and elliptic coordinates for vortex solutions, providing new insights into vortex behavior in elliptic geometries.

Findings

Constant vorticity flow reduces to =0 and =const in elliptic coordinates.

Transformations between coordinate systems are explicitly derived.

Elliptic vortex solutions are characterized by streamlines matching vortex eccentricity.

Abstract

Vortices (flows with closed elliptic streamlines) are exact nonlinear solutions to the compressible Euler equation. In this contribution, we use differential geometry to derive the transformations between Cartesian and elliptic coordinates, and show that in elliptic coordinates a constant vorticity flow reduces to $\dot{\mu}=0$ and $\dot{\nu}={\rm const}$ along the streamline $\mu_0$ that matches the vortex eccentricity.