Hyper-reflection of inertia-gravity waves by vortices

We examine scattering of waves by a vortex under the f-plane approximation. Two kinds of vortices are considered: eddies (where the angular velocity decreases monotonically with radius) and rings (where the angular velocity peaks at a certain radius before decaying). Both kinds can reflect more energy than the incident wave carries, but over-reflection by eddies is relatively weak, whereas that by rings can be infinitely strong (hyper-reflection). The difference is attributed to stronger or weaker centrifugal force and other curvature-related effects. It is also shown that a hyper-reflecting flow is always marginal to a range of unstable flows, but the converse does not hold: flows that are unstable at frequencies lower than the minimum frequency of inertia-gravity waves cannot radiate; hence, do not hyper-reflect. Finally, we derive an asymptotic model for the coupled dynamics of a vortex and a spectrum of small-amplitude random waves. It is shown that, if the vortex hyper-reflects a certain wavenumber and this wavenumber is present in the spectrum, hyper-reflection almost immediately drains a significant proportion of the vortex's energy.