Mean flow generation by three-dimensional nonlinear internal wave beams

Abstract

We study the generation of resonantly growing 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. † Email address for correspondence: f.beckebanze@uu.nl arXiv:1805.12356v2 [physics.flu-dyn]

The long view of triadic resonance instability in finite-width internal gravity wave beams

“…If considered analytically (Thorpe & Haines 1986) or numerically (Grisouard et al. 2013) (unfortunately, this manuscript contained a typo in the lead author's name when published and so may appear elsewhere with the lead author (incorrectly) set to ‘Grisouarda’; here we have reverted to the correct spelling of the lead author's name, ‘Grisouard’) in a 2-D setting, only weak horizontal vorticity is generated (with zero Lagrangian mean flow, indicating no mass transport), which is partially suppressed by the background stratification (Beckebanze, Raja & Maas 2019). When considered in a three-dimensional (3-D) setting, however, Grisouard et al.…”

“…This non-propagating disturbance cannot be classed as a wave, but instead should be treated as a forced oscillatory structure that is confined to the interaction region of the primary beam with its reflection. If considered analytically (Thorpe & Haines 1986) or numerically (Grisouard et al 2013) (unfortunately, this manuscript contained a typo in the lead author's name when published and so may appear elsewhere with the lead author (incorrectly) set to 'Grisouarda'; here we have reverted to the correct spelling of the lead author's name, 'Grisouard') in a 2-D setting, only weak horizontal vorticity is generated (with zero Lagrangian mean flow, indicating no mass transport), which is partially suppressed by the background stratification (Beckebanze, Raja & Maas 2019). When considered in a three-dimensional (3-D) setting, however, Grisouard et al (2013) showed, both experimentally and numerically, how a stronger slowly evolving 3-D horizontal mean flow develops from the interaction region of the primary beam with its reflection.…”

Near-field internal wave beams in two dimensions

“…Beckebanze etal. (2019) discussed the appropriate condition at a vertical wave generator and concluded that the boundary layer may be neglected and a free-slip condition used.…”

“…As previously discussed, such a prediction of a boundary layer based on a free-slip boundary condition is questionable. Beckebanze et al (2019) discussed the appropriate condition at a vertical wave generator and concluded that the boundary layer may be neglected and a free-slip condition used. One of the experiments by Mercier et al (2010) aimed at reproducing the self-similar wave beam, propagating downward to the right, generated in a viscous fluid by a point dipole source at (x − = 0, z − = l).…”

“…This can be seen in the inviscid calculations of Oser (1957), Reynolds (1962), Martin & Llewellyn Smith (2011, 2012 b ) and Davis (2012) for a horizontal disc, Hurley (1969) for an inclined plate and Llewellyn Smith & Young (2003) for a vertical plate, or in the viscous calculations of Kistovich & Chashechkin (1999 a , b ) for a two-dimensional inclined plate, Vasil'ev & Chashechkin (2003, 2006 a , b , 2012) for a three-dimensional inclined plate, Tilgner (2000), Bardakov, Vasil'ev & Chashechkin (2007), Davis & Llewellyn Smith (2010), Le Dizès (2015) and Le Dizès & Le Bars (2017) for a horizontal disc, Maurer etal. (2017) and Boury, Peacock & Odier (2019) for a horizontal wave generator and Beckebanze, Raja & Maas (2019) for a vertical wave generator. To some extent this can also be seen in the inviscid calculations of Gabov (1985) for a horizontal plate, Gabov & Pletner (1985) for an inclined plate, Gabov & Krutitskii (1987) for a vertical plate and Gabov & Pletner (1988) for a horizontal disc, although only Gabov & Pletner (1985) considered the determination of the multivalued functions explicitly.…”