Damping of quasi-two-dimensional internal wave attractors by rigid-wall friction

Abstract

The reflection of internal gravity waves at sloping boundaries leads to focusing or defocusing. In closed domains, focusing typically dominates and projects the wave energy onto ‘wave attractors’. For small-amplitude internal waves, the projection of energy onto higher wavenumbers by geometric focusing can be balanced by viscous dissipation at high wavenumbers. Contrary to what was previously suggested, viscous dissipation in interior shear layers may not be sufficient to explain the experiments on wave attractors in the classical quasi-two-dimensional trapezoidal laboratory set-ups. Applying standard boundary layer theory, we provide an elaborate description of the viscous dissipation in the interior shear layer, as well as at the rigid boundaries. Our analysis shows that even if the thin lateral Stokes boundary layers consist of no more than 1 % of the wall-to-wall distance, dissipation by lateral walls dominates at intermediate wave numbers. Our extended model for the spectrum of three-dimensional wave attractors in equilibrium closes the gap between observations and theory by Hazewinkel et al. (J. Fluid Mech., vol. 598, 2008, pp. 373–382).