Nonlinear reflection of a two-dimensional finite-width internal gravity wave onto a slope

Matthieu Leclair (1, 2) , Keshav Raja (1, 3) , Chantal Staquet (1)

1 LEGI - Laboratoire des Écoulements Géophysiques et Industriels [Grenoble]
2 ETH Zürich - Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich]
3 USM - University of Southern Mississippi

The nonlinear reflection of a finite-width plane internal gravity wave incident onto a uniform slope is addressed, relying on the inviscid theory of Thorpe (1987, 1997) for pure plane waves. The aim of this theory is to determine the conditions under which the incident and the reflected waves form a resonant triad with the second harmonic wave resulting from their interaction. Thorpe's theory leads to an indeterminacy of the second-order amplitude at resonance. In waiving this indeterminacy, we show that the second-order amplitude has a finite behavior at resonance, increasing linearly from the slope. We investigate the influence of background rotation and find similar results with a weaker growth rate. We then adapt the theory to the case of an incident plane wave of finite-width. In this case, nonlinear interactions are confined to the area where the incident and reflected finite-width waves superpose implying that the amplitude of the second-order wave is bounded at resonance. We find good agreement with the results of numerical simulations in a vertical plane as long as the dissipated power of the incident and reflected waves remain smaller than the power transferred to the second harmonic. This is the case for small slope angles. As the slope angle increases, the focusing of the reflected wave enhances viscous effects and dissipation eventually dominates over nonlinear transfer. We finally discuss the relevance of laboratory experiments to assess the validity of these theoretical results.

Domaines: Mécanique des fluides [physics.class-ph]

Matthieu Leclair, Keshav Raja, Chantal Staquet. Nonlinear reflection of a two-dimensional finite-width internal gravity wave onto a slope. 2019. ⟨hal-02126145⟩