Trapped Free Surface Waves for a Lamb-Oseen Vortex Flow

TL;DR

This study numerically investigates how free-surface gravity waves interact with a Lamb-Oseen vortex, revealing conditions under which waves become trapped and how their propagation direction depends on vortex rotation speed.

Contribution

It introduces a spectrally accurate numerical method with a novel buffer region to analyze trapped surface waves around a vortex, highlighting the transition from decay to trapping as rotation rate increases.

Findings

Trapped modes occur at higher vortex rotation rates.

Wave propagation direction reverses at low rotation speeds.

Modes become nearly neutrally stable with increased rotation.

Abstract

Trapped surface waves have been observed in a swimming pool trapped by, and rotating around, the cores of vortices. To investigate this effect, we have numerically studied the free-surface response of a Lamb--Oseen vortex to small perturbations. The fluid has finite depth but is laterally unbounded. The numerical method used is spectrally accurate, and uses a novel non-reflecting buffer region to simulate a laterally unbounded fluid. While a variety of linear waves can arise in this flow, we focus here on surface gravity waves. We investigate the linear modes of the vortex as a function of the perturbation azimuthal mode number and the vortex rotation rate. We find that at low rotation rates, linear modes decay by radiating energy to the far field, while at higher rotation rates modes become nearly neutrally stable and trapped in the vicinity of the vortex. While trapped modes have previously been seen in shallow water surface waves due to small perturbations of a bathtub vortex, the situation considered here is qualitatively different owing to the lack of an inward flow and the dispersive nature of non-shallow-water waves. We also find that for slow vortex rotation rates, trapped waves propagate in the opposite direction to the vortex rotation, whereas, above a threshold rotation rate, waves co-rotate with the flow.