Local Stability of Two-Dimensional Steady Irrotational Solenoidal Flows with Closed Streamlines

We investigate the linear stability of regions of two-dimensional steady irrotational flows in which streamlines are closed, with respect to three-dimensional perturbations of short wavelengths. The fluid is assumed to be inviscid and incompressible. The geometrical optics equations derived by Hameiri and Lifschitz are employed. It is demonstrated that, at large times, both short-wave perturbation velocity and vorticity grow in magnitude at most linearly in time. It implies that there are no exponentially growing short-wave instabilities in such a flow domain. The same result holds if the flow field is steady relative to a suitably chosen steadily rotating frame and the net vorticity relative to the inertial frame is zero.