The mixing efficiency of interfacial waves breaking at a ridge. Part I: Overall mixing efficiency

The overall mixing efficiency of periodic, interfacial waves breaking at a Gaussian ridge is investigated through laboratory experiments. Cumulative measurements are used to investigate the fraction of the wave energy lost in the breaking event that contributes to irreversible mixing of the background density gradient. Using the tank as a control volume, the distribution of energy into reflected waves, transmitted waves, and dissipation and irreversible mixing from the breaking event is determined. The overall fraction of wave energy lost in the breaking event that is converted irreversibly to mixing is found to be 3–8%, which is low compared with typical values of around 20% for steady, parallel, stratified shear instabilities. Spatial variability in the mixing event may contribute to the relatively low overall efficiency of the event.

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 Ri_g is calculated. The transition to breaking occurred when 0.2 ≤ Ri_g ≤ 0.4. In these wave-ridge breaking events, the destabilizing effects of waves steepening in shallow layers may be responsible for breaking at higher Ri_g 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 Ri_g > 0.25 is not sufficient to predict the stability of interfacial waves. A simple model is developed to estimate Ri_g 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 Ri_g 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.

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.