Energy cascade in internal wave attractors

Energy cascade in internal wave attractors Thierry Dauxois 1 , C. Brouzet 1 , E. Ermanyuk 1,2 , S. Joubaud 1 , H. Scolan 1 , I. Sibgatullin 1,3 1. Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon, France 2. Lavrentyev Institute of Hydrodynamics, Novosibirsk, Russia 3. Institute of Mechanics, Moscow State University, Russia Thierry.Dauxois@ens-lyon.fr Abstract One of the pivotal questions in the dynamics of the oceans is related to the cascade of mechanical energy in the abyss and its contribution to mixing. Here, we propose a unique self-consistent experimental and numerical set up that models a cascade of triadic interactions transferring energy from large-scale monochromatic input to multi- scale internal wave motion. We also provide, for the first time, explicit evidence of a wave turbulence framework for internal waves. Finally, we show how beyond this regime, we have a clear transition to a cascade of small-scale overturning events which induce mixing. Introduction The continuous energy input to the ocean interior comes from the interaction of global tides with the bottom topography yielding a global rate of energy conversion to internal tides of the order of 1TW. The subsequent mechanical energy cascade to small-scale internal-wave motion and mixing is a subject of active debate in view of the important role played by abyssal mixing in existing models of ocean dynamics. The oceanographic data support the important role of internal waves in mixing, at least locally: increased rates of diapycnal mixing are reported in the bulk of abyssal regions over rough topography in contrast to regions with smooth bottom topography. A question remains: how does energy injected through internal waves at large vertical scales induce the mixing of the fluid? Let us consider a stratified fluid with an initially constant buoyancy frequency N = [(−g/¯ ρ )(dρ/dz)] 1/2 , where ρ(z) is the density distribution over the vertical coordinate z, and g the gravity acceleration. The dispersion relation is θ = arcsin(Ω). Here θ is the slope of the wave beam to the horizontal, and Ω (resp. ω = ΩN ) is the non-dimensional (resp. dimensional) frequency of oscillations. The anisotropic dispersion relation requires preservation of the slope of the internal wave beam upon reflection at a rigid boundary. In the case of a sloping boundary, this property gives a purely geometric reason for a strong variation of the width of internal wave beams (focusing or defocusing) upon reflection. Internal wave focusing provides a necessary condition for large shear and overturning, as well as shear and bottom layer instabilities at slopes. In a confined fluid domain, focusing usually prevails, leading to a concentration of wave energy on a closed loop, the internal wave attractor [1]. At the level of linear mecha- nisms, the width of the attractor branches is set by the competition between geometric focusing and viscous broadening. High concentration of energy at attractors make them prone to triadic resonance instability which sets in as the energy injected into the sys- tem increases [2]. Note that the particular case for which both unstable secondary waves have a frequency equal to half of the forcing frequency is of particular interest in the VIII th Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016