Reflection of an internal gravity wave beam off a horizontal free-slip surface

Abstract

The reflection of a planar finite-amplitude internal gravity wave beam off a free-slip flat horizontal surface is investigated numerically in a uniformly stratified Boussinesq fluid. Nonlinear effects such as mean currents and harmonics are observed in the wave reflection zone. Mean currents form a stationary, vertically oscillatory, layered structure under the free-slip reflecting surface. The vertical wavelength of the mean-flow layers equals half of the vertical wavelength of the reflecting wave. An empirical predictive model for the steady-state mean flow strength, based on the degree of wave nonlinearity and hydrostaticity, is proposed and subsequently compared to the weakly nonlinear theory by Tabaei et al. [J. Fluid Mech. 526, 217–243 (2005)10.1017/S0022112004002769]. Very strong agreement between simulation results and theory is observed for all waves considered, suggesting although weakly nonlinear in its formulation, the Tabaei et al. theory is valid for the full range of finite amplitudes for which a wave remains stable. Both propagating and evanescent superharmonics are observed, and for waves with steepnesses of O(5%), parametric subharmonic instabilities can occur in the later stages of the reflection process. When a subsurface mixed layer is incorporated into the simulations, the mean currents at the middle of the underlying pycnocline are similar in structure and magnitude to their uniformly-stratified counterparts.