Waves in a coherent two-dimensional flow

A condensate in two-dimensional turbulence confined to a finite domain was predicted by Kraichnan. The most common spatial form of the condensate is a coherent vortex of radius comparable with the domain size, which is statistically steady over times much longer than its turnover time. The vortices were successively studied during the last four decades, and their time-averaged properties were recently and extensively considered in theory and measured in experiments. Here, we consider weak perturbations of the coherent flow in the vortices. Our interest lies in slow perturbations, which implies that they are homogeneous along the streamlines of the coherent flow. We show that such kind of perturbations can be considered as waves of condensate propagating in the radial direction with some dispersion law. In the present work, the dispersion law and the propagation length of the waves are found as a function of the radial position inside a vortex flow. Cases of condensate saturated due to bottom friction and of viscous condensate are different. In the first case, there are waves with low damping. In the second case, all waves are of a similar kind as the shear waves in an unsteady laminar boundary layer.