On internal wave whispering gallery modes in channels and critical-slope wave attractors

Abstract

Internal waves are an important feature of stratified fluids, both in oceanic and lake basins and in other settings. Many works have been published on the generic feature of internal wave trapping onto planar wave attractors and super-attractors in two and three dimensions and the exceptional class of standing global internal wave modes. However, most of these works did not deal with waves that escape trapping. By using continuous symmetries, we analytically prove the existence of internal wave whispering gallery modes (WGMs), internal waves that propagate continuously without getting trapped by attractors. The WGM’s neutral stability with respect to different perturbations enables whispering gallery beams, a continuum of rays propagating together coherently. The systems’ continuous symmetries also enable projection onto two-dimensional planes that yield effective two-dimensional billiards preserving the original dynamics. By examining rays deviating from these WGMs in parabolic channels, we discover a new type of wave attractor that is located along the channel’s critical depth – the depth at which the bottom slope is identical to the ray slope, instead of cross-channel, as in previous works. This new critical-slope wave attractor leads to a new understanding of WGMs as sitting at the border between the two basins of attraction. Finally, both critical-slope wave attractors and whispering gallery beams are used to propose explanations for along-channel energy fluxes in submarine canyons and tidal energy intensification near critical slopes.