The distance from shore / x / was used in all models that also used u„ : 



as reference. Battjes (1975) again departed from the norm and used the water 1 



depth as the characteristic size of the eddies, i.e. the mixing length. In j 



effect, this brought in the beach slope as an additional variable. J 



No single model for the lateral turbulent eddy viscosity has emerged as 

 superior to the others. The tendency is to use separate formulations inside 

 and outside the breaking zone. This recognizes the large differences present 

 in mixing processes present in each. In almost all cases some type of closure 

 coefficient is present that can be determined by calibration with field and i 

 laboratory data. Further discussion can be found imder the Surf Zone Empiricism 

 later in this section. j 



d. Modified Theory of Kraus and Sasaki (1979) . It is instructive to ' 

 summarize here the latest and perhaps most complete modified analytic theory. I 

 Kraus and Sasaki (1979) extended the weak current-large wave angle theory to 

 include lateral mixing. For plane, dissipative-type prototype beaches, it 

 is possibly the best closed- form theory available at this writing. The many 

 modifications incorporated into the theory when compared with the original 

 theory of Longuet-Higgins (1970) are summarized in Table 4. Partial wave 

 setup, refraction, angle-dependent bottom friction stress, and different lat- i 

 eral mixing formulations within and beyond the breakers are the primary changes. 

 Near the breaker line, from the results of Longuet-Higgins (1970) (Fig. 24) 

 and the results of Liu and Dalrymple (1978) (Fig. 26), the effects of both 

 lateral mixing and large incident wave angle are comparable. Both effects ^ 

 have been isolated and studied independently as discussed below. New labora- 1 

 tory and field data presented to confirm the theory will be reviewed in j 

 Chapter 4. | 



le the breaker zone, the driving stress di>-^/ 

 (74) including refraction and partial wave setup (normal incidence) . It is 

 set to zero outside the breaker line. The time-averaged bottom shear stress 

 is calculated from the expression 



Y(l+sin^a^ T— ) inside breaker line 



\ = ^C^v(gh)- 



(91) 



T-(l+sin^a, ■; — ) outside breaker line 

 h b h, 



D 



This is essentially that derived by Liu and Daljmiple (1978) for weak currents 

 but large angles. Note that outside the breakers, the bottom stress is not 

 assumed negligible. Also, the wave height outside the breakers is approxi- 

 mated from long wave theory to be 



n = (rf) R^ (92) 



and not assumed to increase linearly as implied in the Longuet-Higgins (1970) 

 model. For lateral turbulent mixing stress dTj^/dx, they used the eddy vis- 

 cosity model given by equation (55) with kinematic viscosity as specified in 



102 



