Waves on a vortex: rays, rings and resonances

Abstract

We study the scattering of surface water waves with irrotational draining vortices. At small depth, this system is a mathematical analogue of a rotating black hole and can be used to mimic some of its peculiar phenomenon. Using ray-tracing methods, we exhibit the existence of unstable orbits around vortices at arbitrary depth. These orbits are the analogue of the light rings of a black hole. We show that these orbits come in pairs, one co-rotating and one counter-rotating, at a critical radius that varies with the frequency. We derived an explicit formula for this radius in the deep water regime. Our method is validated by comparison with recent experimental data from a wavetank experiment. We finally argue that these rings will generate a discrete set of damped resonances that we characterize and that could possibly be observed in future experiments.

21 pages, 6 figures

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• Classical Hydrodynamics for Analogue Spacetimes: Open Channel Flows and Thin Films
• Mimicking inflation with 2-fluid systems in a strong gradient magnetic field
• The sound-ring radiation of expanding vortex clusters
• Optical analogue of the Schwarzschild-Planck metric
• Estimate of the superradiance spectrum in dispersive media
• Analogue gravitational lensing in optical Bose-Einstein condensates
• A deep-water closure model for surface waves on axisymmetric swirling flows