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 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.