Calculating Wave Breaking on a Current 



Dissipation rates 



The motivation for these laboratory experiments was to find a wave dissipation 

 formulation for wave breaking on a current that can be applied in numerical 

 wave-transformation models. Two likely parameters to correlate with dissipation 

 are wave steepness (as in deepwater breaking and whitecapping relationships) and 

 wave height (as in bore dissipation models). Also, existing dissipation formula- 

 tions discussed in the introduction are evaluated with dissipation calculated from 

 the measurements. 



Dissipation was calculated from the laboratory measurements by applying the 

 action balance equation, Equation 6, modified to include energy dissipation: 



a (r^(C^r-U)] 



dx 



E 



CO. 



f 



The action balance equation was applied between two wave gauges to solve for the 

 dissipation D. Figures 9 and 10 show the calculated dissipation as a function of 

 steepness and wave height, respectively. Figure 9 shows that tiie steepness 

 parameterization segregates the data by peak wave period. Although dissipation 

 increases with steepness, for a given steepness, dissipation is higher for longer 

 peak periods. This result foreshadows that the whitecapping dissipation formula- 

 tions, which are sfl-ongly a function of wave steepness, will not provide good 

 estimates of dissipation for this data set. The calculated dissipation is highly 

 correlated with wave height, and wave period does not seem to be a controlling 

 parameter (Figure 10). This wave-height dependence implies that bore-type 

 dissipation formulations, which are functions of wave height, are good candidates 

 for estimating dissipation for this data set. Figure 1 1 shows the Miche linut 

 (Equation 13) in terms of maximum wave steepness as a function of relative depth 

 (solid line). The measurements, some of which correspond to breaking and some 

 do not, also are plotted. The Miche criterion serves as a conservative upper limit 

 to the data. The conservatism is not surprising because the formulation is for 

 regular waves, and the data correspond to irregular waves. 



Three dissipation formulations were evaluated with the laboratory data, those of 

 Komen et al. (Equation 15), Resio (Equation 16), and Battjes and Janssen (Equation 

 14). Although current does not enter explicitly in any of these formulations, current 

 has been included in the calculation of wavelength and wave number using conser- 

 vation of waves and linear theory (see, e.g., Jonsson (1990)). The Resio formula- 

 tion is given in Figure 12 as the solid line. This formulation sUghtly overpredicts 



20 Chapter 4 Results 



