Boundary-layer-separation-driven vortex shedding beneath internal solitary waves of depression

Abstract

We investigate global instability and vortex shedding in the separated laminar boundary layer beneath internal solitary waves (ISWs) of depression in a two-layer stratified fluid by performing high-resolution two-dimensional direct numerical simulations. The simulations were conducted with waves propagating over a flat bottom and shoaling over relatively mild $(S= 0. 05)$ and steep $(S= 0. 1)$ slopes. Over a flat bottom, the potential for vortex shedding is shown to be directly dependent on wave amplitude, for a particular stratification, owing to increase of the adverse pressure gradient ($\mathrm{d} P/ \mathrm{d} x\gt 0$ for leftward propagating waves) beneath the trailing edge of larger amplitude waves. The generated eddies can ascend from the bottom boundary to as high as 33 % of the total depth in two-dimensional simulations. Over sloping boundaries, global instability occurs beneath all waves as they steepen. For the slopes considered, vortex shedding begins before wave breaking and the vortices, shed from the bottom boundary, can reach the pycnocline, modifying the wave breaking mechanism. Combining the results over flat and sloping boundaries, a unified criterion for vortex shedding in arbitrary two-layer continuous stratifications is proposed, which depends on the momentum-thickness Reynolds number and the non-dimensionalized ISW-induced pressure gradient at the point of separation. The criterion is generalized to a form that may be readily computed from field data and compared to published laboratory experiments and field observations. During vortex shedding events, the bed shear stress, vertical velocity and near-bed Reynolds stress were elevated, in agreement with laboratory observations during re-suspension events, indicating that boundary layer instability is an important mechanism leading to sediment re-suspension.

Numerical simulations of shoaling internal solitary waves of elevation

“…The figure shows that the most significant differences from the laboratory-scale simulations are the vortices generated near the surface behind the main wave. This phenomenon is very similar to the vortex shedding discussed in Aghsaee et al (2012), and is due to the no-slip and no-flux conditions imposed at the top boundary. A free-slip surface boundary condition will remove the near-surface instability, and altering the numerical model to allow a no-slip bottom but a free-slip top could be another future research direction as well.…”

“…While the majority of past work has examined waves of depression (e.g. Lamb, 2002Lamb, , 2003Aghsaee et al, 2012;Carr et al, 2012) as they are commonly observed in oceans and deep lakes, waves of elevation, which occur when the pycnocline is below the mid-depth, are more typical of shallow waters such as near-coastal regions (e.g. Klymak and Moum, 2003;Scotti and Pineda, 2004).…”

“…In the laboratory experiments performed by Boegman et al (2005), high-frequency internal wave breaking was observed along sloping topography, which led to energy transfer from the wind-forced basin-scale motions to the turbulent motions. Aghsaee et al (2012) showed, using twodimensional numerical simulations, that for waves of depression, boundary layer instability in the form of separation bubble bursting that leads to vortex shedding can be generated during shoaling. The instability can eventually reach the pycnocline, modifying the wave breaking mechanism, though it is unclear to what extent three-dimensional effects will modify this observation.…”

Topographically generated internal waves and boundary layer instabilities

“…This has been suggested by multiple authors in field, experimental, and numerical work. 1,7,12,17 Recent numerical work has confirmed that sediment resuspension occurs in a coupled hydrodynamic-sediment model. 6 More broadly, a wide variety of field studies have pointed to a link between increased near-bed sediment concentrations and shoaling internal waves on a slope.…”

“…Particularly, it is important for simulations to include realistic nonlinear internal waves, either with the Dubreil-Jacotin-Long (DJL) equation 11 or direct simulation. Further, reports of stability regimes in two-dimensional simulations of fully nonlinear waves 12 and weakly nonlinear waves 4 both show inconsistencies with laboratory experiments 10 suggesting that three-dimensional effects are very important. The mechanisms that lead to the generation of these large internal waves have been under investigation for some time, with stratified flow over topography a primary example.…”