Three-dimensional instability of an elliptic Kirchhoff vortex

The instability of a Kirchhoff vortex [1-3] with respect to three-dimensional perturbations is considered in the linear approximation. The method of successive approximations is applied in the form described in [4-6]. The eccentricity of the core is used as a small parameter. The analysis is restricted to the calculation of the first two approximations. It is shown that exponentially increasing perturbations of the same type as previously predicted and observed in rotating flows in vessels of elliptic cross section [4-9] appear even in the first approximation. As distinct from the case of plane perturbations [1-3], where there is a critical value of the core eccentricity separating the stable and unstable flow regimes, instability is predicted for arbitrarily small eccentricity.