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Internal and inertia-gravity wave focusing at large Stokes numbers

Natalia Shmakova1,2, Bruno Voisin1, Joel Sommeria1, and Jan-Bert Flór1,*

  • 1Laboratoire des Écoulements Géophysiques et Industriels (LEGI), CNRS, Université Grenoble Alpes, Grenoble INP, 38000 Grenoble, France
  • 2Lavrentyev Institute of Hydrodynamics, Siberian Branch of the Russian Academy of Sciences, Prospekt Lavrentyev 15, Novosibirsk 630090, Russia

  • *jan-bert.flor@legi.cnrs.fr
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Phys. Rev. Fluids 6, 114804 – Published 16 November, 2021

DOI: https://doi.org/10.1103/PhysRevFluids.6.114804

Abstract

In this experimental study we consider the geometric focusing of internal waves by the horizontal oscillation of a torus in a stratified or rotating stratified fluid. In order to minimize viscous dissipation the experiments have been conducted at the 13−m-large Coriolis platform and a large torus was used so that a Stokes number of approximately 6000 was reached. For such large Stokes numbers the waves are bimodal, i.e., they are emitted at the critical locations where the rays are tangential to the torus boundary such that each section of the torus generates for each direction of propagation two wave rays. In contrast to the unimodal case for which these waves diffuse into a single wave beam and there is a single focusing region, there are now four focal regions where the waves emitted from the inner and outer sides of the sections meet. The focusing of the waves on the axis gives rise to strongly nonlinear processes. A linear theory for internal waves emitted by an oscillating torus is presented. For small oscillation amplitudes, we obtain a reasonable agreement of the bimodal wave structure with this linear theory. Resonant wave triads are observed in the stratified (nonrotating) case, similar to the triads of unimodal waves previously observed at moderate Stokes numbers. In the presence of background rotation, no triad is present since the oscillation frequency falls outside the range for which triads can exist. For higher oscillation amplitudes, the waves become highly nonlinear. This results in a spreading of the fundamental wave beams which have relatively less energy than for small oscillation amplitudes, while higher harmonics in the focal region and in the boundary layer of the torus have slightly more energy. A similar tendency is observed for inertia-gravity waves. A large spectrum of waves in resonant interactions is thus observed. This behavior strongly differs from its moderate-Stokes-number counterpart for which higher oscillation amplitudes result in an increase of the energy of the fundamental wave beams. In contrast to observations for small Stokes numbers that showed a decrease in Richardson number with the oscillation amplitude, the Richardson number near the torus and in the focusing region remains constant and close to 1, and overturning wave motions are absent. The results are discussed in the context of the ocean.

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Phys. Rev. Fluids 6, 114804– Published 16 November, 2021
  • Received 28 June 2021
  • Accepted 22 October 2021
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