The time-dependent boundary layer induced by a weakly nonlinear solitary internal wave in shallow water is examined through direct numerical simulation. Waves of depression and elevation are both considered. The mean density field corresponds to that typical of the coastal ocean and lakes where the lower fraction of the water column is subject to the stabilizing effect of a diffuse stratification. Sufficient resolution of the “inviscid” dynamics of the boundary layer is ensured through use of a Legendre spectral multidomain discretization scheme in the vertical direction. At higher Reynolds numbers, where the simulations become underresolved, because of restrictions in available computational resources, spectral accuracy and numerical stability at the scales of physical interest are preserved through use of a penalty scheme in the vertical and explicit spectral filtering. Thus, a highly accurate description of the qualitative dynamics of the wave-induced global instability is possible and finescale physical mechanisms critical to the appearance of this instability are not smeared out by the high artificial dissipation inherent in lower-order finite-difference schemes. Results indicate that, for a wave amplitude exceeding a critical value, the global instability occurs in regions near the bottom boundary where the wave induces an adverse pressure gradient. The structure of the associated separation bubble is modified through the establishment of coherent and synchronous dynamics, characterized by elevated levels of bottom shear stress and a periodic shedding of coherent vortex structures. Although details of the vortex shedding depend on the particular wave forcing involved, these vortical structures always ascend high into the water column. All findings suggest that this global instability is a potent mechanism for benthic turbulence, mixing, and possible sediment resuspension in shallow waters, presumably even more intense than the nominal turbulent boundary layer.

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The mean velocity profile scaling and the vorticity structure of a stably stratified, initially turbulent wake of a towed sphere are studied numerically using a high-accuracy spectral multi-domain penalty method model. A detailed initialization procedure allows a smooth, minimum-transient transition into the non-equilibrium (NEQ) regime of wake evolution. A broad range of Reynolds numbers,Re= UD/ν ∈ [5 × 103, 105] and internal Froude numbers,Fr= 2U/(ND) ∈ [4, 64] (U,Dare characteristic velocity and length scales, andNis the buoyancy frequency) is examined. The maximum value ofReand the range ofFrvalues considered allow extrapolation of the results to geophysical and naval applications.At higherRe, the NEQ regime, where three-dimensional turbulence adjusts towards a quasi-two-dimensional, buoyancy-dominated flow, lasts significantly longer than at lowerRe. AtRe= 5 × 103, vertical fluid motions are rapidly suppressed, but atRe= 105, secondary Kelvin–Helmholtz instabilities and ensuing turbulence are clearly observed up toNt≈ 100. The secondary motions intensify with increasing stratification strength and have significant vertical kinetic energy.These results agree with existing scaling of buoyancy-driven shear onRe/Fr2and suggest that, in the field, the NEQ regime may last up toNt≈ 1000. At a given highRevalue, during the NEQ regime, the scale separation between Ozmidov and Kolmogorov scale is independent ofFr. This first systematic numerical investigation of stratified turbulence (as defined by Lilly,J. Atmos. Sci.vol. 40, 1983, p. 749), in a controlled localized flow with turbulent initial conditions suggests that a reconsideration of the commonly perceived life cycle of a stratified turbulent event may be in order for the correct turbulence parametrizations of such flows in both geophysical and operational contexts.

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.

The internal gravity wave (IGW) field emitted by a stably stratified, initially turbulent, wake of a towed sphere in a linearly stratified fluid is studied using fully nonlinear numerical simulations. A wide range of Reynolds numbers, $\mathit{Re}= UD/ \nu \in [5\times 1{0}^{3} , 1{0}^{5} ] $ and internal Froude numbers, $\mathit{Fr}= 2U/ (ND)\in [4, 16, 64] $ ($U$, $D$ are characteristic body velocity and length scales, and $N$ is the buoyancy frequency) is examined. At the higher $\mathit{Re}$ examined, secondary Kelvin–Helmholtz instabilities and the resulting turbulent events, directly linked to a prolonged non-equilibrium (NEQ) regime in wake evolution, are responsible for IGW emission that persists up to $Nt\approx 100$. In contrast, IGW emission at the lower $\mathit{Re}$ investigated does not continue beyond $Nt\approx 50$ for the three $\mathit{Fr}$ values considered. The horizontal wavelengths of the most energetic IGWs, obtained by continuous wavelet transforms, increase with $\mathit{Fr}$ and appear to be smaller at the higher $\mathit{Re}$, especially at late times. The initial value of these wavelengths is set by the wake height at the beginning of the NEQ regime. At the lower $\mathit{Re}$, consistent with a recently proposed model, the waves propagate over a narrow range of angles that minimize viscous decay along their path. At the higher $\mathit{Re}$, wave motion is much less affected by viscosity, at least initially, and early-time wave propagation angles extend over a broader range of values which are linked to increased efficiency in momentum extraction from the turbulent wake source.