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

TL;DR

The paper addresses how internal waves in stratified rotating fluids can evade conventional wave attractors by forming Whispering Gallery Modes (WGMs). It develops a ray-dynamics framework with continuous symmetries, reducing 3D dynamics to 2D effective billiards and proving analytically the existence of WGMs in parabolic and trapezoidal channels, with neutral stability and the emergence of Whispering Gallery Beams. A new along-channel, critical-slope attractor is identified, arising from deviations from WGMs, and the work connects these concepts to observed along-canyon energy fluxes and energy accumulation near critical slopes. Together, these results offer a mechanism for attenuation-less energy transport and localized energy growth in natural basins, with potential implications for tidal energy intensification and ocean mixing.

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 2\&3D 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. WGMs neutral stability with respect to different perturbations enable whispering gallery beams, a continuum of rays propagating together coherently. The systems' continuous symmetries also enable projection onto 2D planes that yield effective 2D billiards preserving the original dynamics. By examining rays deviating from these WGMs in parabolic channels we discover a new type of wave attractor which is located along the channel instead of across it as in previous works. This new wave attractor leads to a re-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.

Figures (7)

• Figure 1: (a) Sub-critical internal wave ray reflection. (b) Super-critical internal wave ray reflection. Red arrows indicate wave energy propagation direction. Adapted from thorpe1997interactions. Angles are presented with respect to both up-slope and down-slope although measured with respect to up-slope exclusively.
• Figure 2: (a) WGMs in the right half of a parabolic channel launched from the red dots parallel to the $y$-direction. The critical line is shown in magenta. (b) Two-dimensional projection of (a) onto the $x-z$ plane. vertical lines correspond to rays propagating parallel to the channel.
• Figure 3: (a) Top view of internal wave beam in a parabolic channel of scaled depth $\tau=0.7$ launched inwards with respect to the WGM. Red points indicate launching locations and black points represent reflections from the channel's surface. The magenta critical line is dashed. (b) Side view of two rays focusing after being launched inwards with respect to the WGM. In both figures particle velocities increase, as implied by the focusing of the ray. Color qualitatively highlights diminishing distance between rays in a beam.
• Figure 4: (a) WGMs in a trapezoid channel launched from the red dots parallel to the $y$-direction. All WGMs share $M=1$ (b) Two-dimensional projection of the first half of the channel depicted in (a) onto the $x-z$ plane. Vertical lines correspond to rays propagating parallel to the channel.
• Figure 5: adaptations of fig 6 (left) and fig 7 (right) of ASLAM201820. (a) Depth-integrated along-canyon baroclinic M2 energy flux (blue), HKE (green), and APE (red) with distance along the thalweg. (b) Along-canyon and (c) across-canyon baroclinic M2 energy flux with distance along the thalweg. Positive along-canyon values are toward the head of the canyon limb. Positive acrosscanyon values are to the left when looking up canyon. (f) Along-thalweg slope criticality to the M2 internal tide (blue) and smoothed using a 5-km running mean (black). Near-critical values ($0.8 < \alpha < 1.3$; mcphee2002boundary) are indicated in grey. The dashed grey lines indicate the distance over which bottom intensification occurs.