Damping of quasi-2D internal wave attractors by rigid-wall friction

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 wave numbers by geometric focusing can be balanced by viscous dissipation at high wave numbers. 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-2D 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 3D wave attractors in equilibrium closes the gap between observations and theory by Hazewinkel et al. (2008)

Mean flow generation by three-dimensional non-linear internal wave beams

We study the generation of strong mean flow by weakly non-linear internal wave beams. With a perturbational expansion, we construct analytic solutions for 3D internal wave beams, exact up to first order accuracy in the viscosity parameter. We specifically focus on the subtleties of wave beam generation by oscillating boundaries, such as wave makers in laboratory set-ups. The exact solutions to the linearized equations allow us to derive an analytic expression for the mean vertical vorticity production term, which induces a horizontal mean flow. Whereas mean flow generation associated with viscous beam attenuation - known as streaming - has been described before, we are the first to also include a peculiar inviscid mean flow generation in the vicinity of the oscillating wall, resulting from line vortices at the lateral edges of the oscillating boundary. Our theoretical expression for the mean vertical vorticity production is in good agreement with earlier laboratory experiments, for which the previously unrecognized inviscid mean flow generation mechanism turns out to be significant

On functional equations leading to exact solutions for standing internal waves

The Dirichlet problem for the wave equation is a classical example of a problem which is ill-posed. Nevertheless, it has been used to model internal waves oscillating harmonically in time, in various situations, standing internal waves amongst them. We consider internal waves in two-dimensional domains bounded above by the plane z=0 and below by z=−d(x) for depth functions d. This paper draws attention to the Abel and Schröder functional equations which arise in this problem and use them as a convenient way of organising analytical solutions. Exact internal wave solutions are constructed for a selected number of simple depth functions d

On functional equations leading to exact solutions for standing internal waves

The Dirichlet problem for the wave equation is a classical example of a problem which is ill-posed. Nevertheless, it has been used to model internal waves oscillating harmonically in time, in various situations, standing internal waves amongst them. We consider internal waves in two-dimensional domains bounded above by the plane z= 0 and below by z= -d(x) for depth functions d. This paper draws attention to the Abel and Schröder functional equations which arise in this problem and use them as a convenient way of organising analytical solutions. Exact internal wave solutions are constructed for a selected number of simple depth functions d

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)

Mass transport generated by stratified internal wave boundary layers

Internal waves, generated by tidal oscillations over rough bottom topography at the margins of shallow seas, are known to be important for the mixing budget of the ocean. One of the open questions in the dynamics of the ocean is related to mechanisms by which energy is transferred to smaller scales, where mixing takes place. Using small-amplitude expansions, we investigate the mass transport generated by monochromatic internal wave beams between two lateral boundaries in the laminar regime. We find that the peculiar 3D structure of the lateral viscous boundary layers results in effective Reynolds stresses near the lateral walls, which generates a horizontal circulation in the interior. This induced circulation increase linearly over time, and as such, it may lead to the onset of wave-mean flow interactions and to turbulent mixing. Surprisingly, even very thin boundary layers (∼ 1% of wall to wall distance) have a significant impact on the mass transport in the interior. The theory is verified by laboratory experiments on particle transport induced by quasi-2D internal wave beams

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

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)