Vortex Solution in Elliptic Coordinates

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