Three-dimensional stability of elliptical vortex columns in external strain flows

The Kirchhoff-Kida family of elliptical vortex columns in flows with uniform strain and rotation displays a rich variety of dynamical behaviours, even in a purely two-dimensional setting. In this paper, we address the stability of these columns with respect to three-dimensional perturbations via the geometrical optics method. In the case when the external strain is equal to zero, the analysis reduces to the stability of a steady elliptical vortex in a rotating frame. When the external strain is non-zero, the stability analysis reduces to the theory of a Schrödinger equation with quasi-periodic potential. We present stability results for a variety of different Kirchhoff-Kida flows. The vortex columns are typically unstable except when the interior vorticity is approximately the negative of the background vorticity, so that the flow in the inertial frame is nearly a potential flow.