Dynamics Near an Unstable Kirchhoff Ellipse

We describe the dynamics of the Kirchhoff ellipse by formulating a nonlinear equation for the boundary of a perturbed vortex patch in elliptical coordinates. We demonstrate that in the regime for which the linearized equation of motion is unstable, the nonlinear dynamics of a rather general initial perturbation of the Kirchhoff ellipse are determined by the fastest growing mode for the corresponding linearized equation, on a time scale when the nonlinear instability occurs. In particular, we resolve a question suggested by Love's results and prove that such elliptical patches are indeed unstable in the full nonlinear sense.