Nonlinear aspects of focusing internal waves

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)epsilon(-1/2)f(theta), with oscillation amplitude A, f(theta) a function for the variation of the wave amplitude with wave angle theta, and epsilon(1/2) = root 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 >= 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 approximate to 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.