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Depth-induced wave breaking

Neither Boussinesq-type wave models nor non-hydrostatic wave-flow models can be directly applied to details of breaking waves, since in both models essential processes such as overturning, air-entrainment and wave generated turbulence, are absent. But, if only the macro-scale effects of wave breaking are of interest, such as the effect on the statistics of wave heights, details of the breaking process can be ignored. By observing that both spilling and plunging breakers eventually evolve into a quasi-steady bore, where the entire front-face of the wave is turbulent, a breaking wave becomes analogous to a hydraulic jump. Consequently, its integral properties (rate of energy dissipation, jump height) are approximately captured by regarding the breaking wave as a discontinuity in the flow variables (free surface, velocities). Proper treatment of such a discontinuity in a non-hydrostatic model (conservation of mass and momentum) can therefore be used to determine the energy dissipation of waves in the surf zone; see Section 5.4.4.


Though a vertical coarse resolution (1-3 layers) is sufficient to describe the wave physics outside the surf zone (e.g. refraction, shoaling, diffraction, nonlinear wave-wave interactions), dissipation due to wave breaking requires a disproportional high vertical resolution ($ \sim$10-20). A coarse resolution will result in an underestimation of the horizontal velocities near the wave crest, and thus an underestimation of the amplitude dispersion. This underestimation implies that at low vertical resolution the influence of the non-hydrostatic pressure gradient is overestimated. Consequently, the stabilizing dispersive effects (i.e. the non-hydrostatic pressures) postpone the transition into the characteristic saw-tooth shape and therefore also the onset of dissipation.


By enforcing a hydrostatic pressure distribution at the front of a wave, we can locally reduce a non-hydrostatic wave-flow model to the nonlinear shallow water equations. The wave then rapidly transitions into the characteristic saw-tooth shape and, consistent with the high resolution approach, dissipation is captured by ensuring momentum conservation over the resulting discontinuity.


The subsequent dissipation is well described by assuming depth uniform velocities and a hydrostatic pressure distribution. In fact, these assumptions often form the basis to derive dissipation formulations to account for depth-induced breaking in energy balance type models, e.g. Battjes and Janssen (1978) among many others. Hence, prescribing a hydrostatic pressure distribution in the model around the discontinuity should result in the correct bulk dissipation.


There is no need to assume a hydrostatic pressure distribution if the vertical resolution is sufficient (i.e. 10 to 20 layers). However, imposing a hydrostatic distribution resolutions at low resolutions (1-3 layers) will ensure that, due the absence of dispersive effects, the front quickly transitions into a bore like shape. Hence, it can be used to initiate the onset of wave breaking, thus allowing for the use of low-vertical resolutions throughout the domain. In practice this means that once a grid point is in the front of a breaking wave, vertical accelerations are no longer resolved, and the non-hydrostatic pressure is set to zero.


A grid point is therefore labelled for hydrostatic computation if the local surface steepness $ \partial$$ \zeta$/$ \partial$x exceeds a predetermined value $ \alpha$. Equivalenty, $ \partial$$ \zeta$/$ \partial$t > $ \alpha$$ \sqrt{{gd}}$. Once labelled, a point only becomes non-hydrostatic again if the crest of the wave has passed. This is assumed to occur when $ \partial$$ \zeta$/$ \partial$t < 0. Furthermore, because grid points only become active again when the crest passes (where w $ \approx$ 0), vertical velocities w are set to zero on the front. To represent persistence of wave breaking, we locally reduce the criterion $ \alpha$ to $ \beta$ if a neighbouring grid point (in x - or y -direction) has been labelled for hydrostatic computation. In this case a point is thus also labelled for hydrostatic computation if $ \partial$$ \zeta$/$ \partial$t > $ \beta$$ \sqrt{{gd}}$, with $ \beta$ < $ \alpha$. In all other grid points, the computations are non-hydrostatic. Based on calibration, the default value for the maximum steepness parameter $ \alpha$ is 0.6, while the persistence parameter $ \beta$ is set to 0.3.


To summarize, in case of a few layers (1-3) we must apply the command BREAK with optionally different values for $ \alpha$ and $ \beta$. In case of a sufficient number of layers (>10) nothing needs to be specified with respect to wave breaking.


next up previous index
Next: Subgrid turbulent mixing Up: Physical parameters Previous: Physical parameters   Index
The SWASH team 2017-04-06