The mixing efficiency of interfacial waves breaking at a ridge. Part II: Local Energetics

The efficiency with which internal wave energy is converted irreversibly to diapycnal mixing in bathymetry-induced mixing events is of great importance to larger-scale ocean modeling efforts. High-resolution laboratory measurements are used to investigate the spatial and temporal variability of interfacial wave breaking events at a submerged bathymetric ridge. From high spatial resolution measurements, it appears the local efficiency can vary significantly. Parameterizations based on the turbulent Reynolds number and Froude number suggest that the local mixing efficiency within the overturning patch at the interface is 10–17% but the local efficiency is near zero within the relatively homogenous layers. When the local mixing efficiency is integrated over the entire event, the resulting overall efficiency is consistent with the result from Part 1 that the overall event efficiency is 3–8%. This spatial variation in the mixing efficiency reinforces the importance of dynamic mixing efficiency parameterizations based on local stratified turbulence parameters.

The breaking of interfacial waves at a submerged bathymetric ridge

The breaking of periodic progressive two-layer interfacial waves at a Gaussian ridge is investigated through laboratory experiments. Length scales of the incident wave and topography are used to parameterize when and how breaking occurs. Qualitative observations suggest both shear and convection play a role in the instability of waves breaking at the ridge. Simultaneous particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) measurements are used to calculate high resolution, two-dimensional velocity and density fields from which the local gradient Richardson number Rig is calculated. The transition to breaking occurred when 0.2 ≤ Rig ≤ 0.4. In these wave-ridge breaking events, the destabilizing effects of waves steepening in shallow layers may be responsible for breaking at higher Rig than for similar waves breaking through shear instability in deep water (Troy & Koseff, J. Fluid Mech., vol. 543, 2005b, p. 107). Due to the effects of unsteadiness, nonlinear shoaling and flow separation, the canonical Rig > 0.25 is not sufficient to predict the stability of interfacial waves. A simple model is developed to estimate Rig in waves between finite depth layers using scales of the incident wave scale and topography. The observed breaking transition corresponds with a constant estimated value of Rig from the model, suggesting that interfacial shear plays an important role in initial wave instability. For wave amplitudes above the initial breaking transition, convective breaking events are also observed.

The viscous decay of progressive interfacial waves

The viscous damping of progressive, two-layer interfacial waves is examined theoretically and experimentally. Traditional water wave theory is modified to derive the damping rates associated with interfacial wave propagation in a rectangular channel. The individual wave damping contributions are considered from the bottom, side, and interfacial boundary layers, as well as the damping associated with the wave-induced velocities within the homogenous fluid layers. These results show that for most laboratory-scale experiments, sidewall friction plays the dominant role in wave damping. Laboratory experiments are conducted to verify the damping rates for progressive two-layer internal waves in a rectangular channel. Experiments are conducted on both monochromatic and polychromatic wave trains. The results of these experiments are in good agreement with the derived damping rates, but show poorer agreement for large-amplitude waves when the sidewall boundary layers become turbulent. More work is necessary to quantify the damping associated with nonlinear internal waves in order to allow for accurate interpretation of the results from laboratory experiments.

The instability and breaking of long internal waves

Laboratory experiments are carried out to determine the nature of internal wave breaking and the limiting wave steepness for progressive, periodic, lowest-mode internal waves in a two-layer, miscible density stratification. Shoaling effects are not considered. The waves investigated here are long relative to the thickness of the density interface separating the two fluid layers. Planar laser-induced fluoresence (PLIF) flow visualization shows that wave breaking most closely resembles a Kelvin–Helmholtz shear instability originating in the high-shear wave crest and trough regions. However, this instability is strongly temporally and spatially modified by the oscillations of the driving wave shear. Unlike a steady stratified shear layer, the wave instability discussed here is not governed by the canonical $it Ri{=}1/4$ stability limit. Instead, the wave time scale (the time scale of the destabilizing shear) imposes an additional constraint on instability, lowering the critical Richardson number below 1/4. Experiments were carried out to quantify this instability threshold, and show that, for the range of wavenumbers considered in this study, the critical wave steepness at which the wave breaking occurs is wavenumber-dependent (unlike surface waves). The corresponding critical wave Richardson numbers at incipient wave breaking are well below 1/4, in consonance with a modified instability analysis based on results from stratified shear flow instability theory.

The generation and quantitative visualization of breaking internal waves

New techniques for the generation and quantitative visualization of breaking progressive internal waves are presented. Laboratory techniques applicable to general stratified flow experiments are also demonstrated. The planar laser-induced fluorescence (PLIF) technique is used to produce calibrated images of the wave breaking process, and the details of the PLIF measurements are described in terms of the necessary corrections and considerations for the application of PLIF to stratified flows. Results of the flow visualization and wave generation techniques are presented, which show that the nature of internal wave breaking is strongly dependent on the type of breaking internal wave considered.