Boundary streaming by internal waves

Abstract

Damped internal wave beams in stratified fluids have long been known to generate strong mean flows through a mechanism analogous to acoustic streaming. While the role of viscous boundary layers in acoustic streaming has thoroughly been addressed, it remains largely unexplored in the case of internal waves. Here we compute the mean flow generated close to an undulating wall that emits internal waves in a viscous, linearly stratified twodimensional Boussinesq fluid. Using a quasi-linear approach, we demonstrate that the form of the boundary conditions dramatically impacts the generated boundary streaming. In the no-slip scenario, the early time Reynolds stress divergence within the viscous boundary layer is much stronger than within the bulk while also driving flow in the opposite direction. Whatever the boundary condition, boundary streaming is however dominated by bulk streaming at large time. Using a WKB approach, we investigate the consequences of adding boundary streaming effects to an idealised model of wave-mean flow interactions known to reproduce the salient features of the quasi-biennial oscillation. The presence of wave boundary layers has a quantitative impact on the flow reversals.

Particle transport induced by internal wave beam streaming in lateral boundary layers

“…We speculate that there exists a continuous transition from our streaming mechanism at lateral (vertical) walls to the streaming mechanism over a flat bottom, recently investigated by Renaud & Venaille (2018). The viscous boundary layer description by Kistovich & Chashechkin (1995b,a) for supercritical internal waves reflecting at inclined boundaries may be a good starting point to extend the lateral-wall streaming analysis to oblique wave beam reflections at inclined boundaries.…”

“…Strong mean flow generation is known to occur due to horizontal cross-beam variation (Bordes et al 2012;Kataoka & Akylas 2015;Semin et al 2016;Beckebanze et al 2018b), with important modifications by planetary rotation (Grisouard & Bühler 2012;Fan et al 2018), and upon reflection where incident and reflected beams interact (Thorpe 1997;Grisouard et al 2013;Zhou & Diamessis 2015;Raja 2018). Renaud & Venaille (2018) recently also found strong mean flow generation in a flat bottom boundary layer.…”

Near-field internal wave beams in two dimensions

“…When the same vertical velocity is imposed on both sides of the boundary, corresponding to the oscillations of a rigid plate, the present analysis for n = 1 yields a boundary layer of the same order as the waves; the same conclusion has been reached by Hurley & Hood (2001) using a free-slip boundary condition. When a given vertical velocity profile is imposed on part of an otherwise fixed boundary, as for the wave generator in § 8.3, and the image of the profile through this boundary is added, the present analysis for n = 0 predicts that the boundary layer is negligible compared with the waves; for an undulating horizontal wall, the boundary layer has been found by Renaud & Venaille (2019) to be negligible compared with the waves when a free-slip boundary condition is used, and of the same order as them when a no-slip condition is used.…”

“…The particular case of a two-dimensional horizontal boundary has been considered by Hurley & Hood (2001) and Renaud & Venaille (2019). When the same vertical velocity is imposed on both sides of the boundary, corresponding to the oscillations of a rigid plate, the present analysis for yields a boundary layer of the same order as the waves; the same conclusion has been reached by Hurley & Hood (2001) using a free-slip boundary condition.…”

Energy budget in internal wave attractor experiments

“…It has been also recently shown that viscous boundary layers can induce strong mean flow due to streaming (Horne et al. 2019; Renaud & Venaille 2019).…”