Nonlinear aspects of focusing internal waves

Abstract

When a torus oscillates horizontally in a linearly stratified fluid, the wave rays form a double cone, one upward and one downward, with two focal points where the wave amplitude has a maximum due to wave focusing. Following a former study on linear aspects of wave focusing (Ermanyuk et al. , J. Fluid Mech. , vol. 813, 2017, pp. 695–715), we here consider experimental results on the nonlinear aspects that occur in the focal region below the torus for higher-amplitude forcing. A new non-dimensional number that is based on heuristic arguments for the wave amplitude in the focal area is presented. This focusing number is defined as $Fo=(A/a)\\unicode[STIX]{x1D716}^{-1/2}f(\\unicode[STIX]{x1D703})$ , with oscillation amplitude $A$ , $f(\\unicode[STIX]{x1D703})$ a function for the variation of the wave amplitude with wave angle $\\unicode[STIX]{x1D703}$ , and $\\unicode[STIX]{x1D716}^{1/2}=\\sqrt{b/a}$ the increase in amplitude due to the focusing, with $a$ and $b$ , respectively, the minor and major radius of the torus. Nonlinear effects occur for $Fo\\geqslant 0.1$ , with the shear stress giving rise to a mean flow which results in the focal region in a central upward motion partially surrounded by a downward motion. With increasing $Fo$ , the Richardson number $Ri$ measured from the wave steepness monotonically decreases. Wave breaking occurs at $Fo\\approx 0.23$ , corresponding to $Ri=0.25$ . In this regime, the focal region is unstable due to triadic wave resonance. For the different tori sizes under consideration, the triadic resonant instability in these three-dimensional flows resembles closely the resonance observed by Bourget et al. ( J. Fluid Mech. , vol. 723, 2013, pp. 1–20) for a two-dimensional flow, with only minor differences. Application to internal tidal waves in the ocean are discussed.