Damping of 3D internal wave attractors by lateral walls

The reflection of internal gravity waves at sloping boundaries leads to focusing or defo- cusing. In closed domains, focusing dominates and projects the wave energy onto ’wave attractors’. Previous theoretical and experimental work on 2D steady state wave attrac- tors has demonstrated that geometric focusing by wave reflection can be balanced either by viscous dissipation at high wave numbers (Hazewinkel et al., 2008), or by nonlinear wave-wave interactions (Scolan et al., 2013). The present study considers a weakly nonlinear 3D internal wave beam under steady state conditions in a semi-infinite domain between two vertical walls. We analyze the effect of the Stokes boundary layers at these two vertical side walls on the interior velocity field. With a perturbation approach, we find that the two lateral Stokes boundary layers generate a fully three-dimensional interior velocity field component, proportional to ν1/2, with ν the dynamical viscosity. This velocity field dampens the wave beam at high wave numbers, thereby providing a new mechanism to establish an energetic balance for steady state wave attractors. The analytical results agree well with the 3D numerical wave attractor simulation by Brouzet et al. (2016).