Waves on a Stuart vortex

Abstract

We consider the linear stability of Stuart vortices, an exact periodic solution of the two-dimensional inviscid Euler equations of hydrodynamics consisting of an infinite row of co-rotating vortices, to wavelike disturbances when the vortex amplitude is small but still much larger than the disturbances. An unsteady critical layer analysis is used, with outer and inner expansions away from and near the vortex cores which we match to obtain disturbance growth rates. Disturbance-vortex interactions occur inside the critical layer and are neglected elsewhere. We consider α+-1∼O(1) and α+≈1 separately, where α+ is ratio of the disturbance wavenumber to that of the periodic vortex row. We consider primarily three-dimensional (oblique) disturbances but touch upon two-dimensional disturbances in the vortex plane. There are resonance mechanisms for oblique disturbances when k+⩽1, with k+ the ratio of the streamwise disturbance wavenumber to that of the periodic vortex row. When k+=1, this is fundamental mode or Benney–Lin instability, thought to be connected to streamwise streaks seen in experiments, and when k+=1/2, this is subharmonic instability or helical pairing, thought to be connected to vortex pairing.