0.8 - 



2 0.2 



EXAMPLE VELOCITY PROFILES 

 NO. SYMBOL REFERENCE 



O 



D 



O BATTJES (1974) 



• LONGUET HIGGINS(I972) 



■ BATTJES (1974) 



DISTANCE FROM SHORE 



Figure 27.- Effects of nonlinear bed shear stress on theoretical longshore cur- 

 rent velocity profile neglecting lateral mixing (from Bijker and 

 v.d. Graaff, 197826). 



deviate from an exact triangular shape where the bed friction factor is as- 

 sumed constant in the surf zone (eq. 61). Profile v^ used the full nonlinear 

 equation (80) for bed shear and full expression for dS^y/dx with no shallow- 

 water approximations. The results differ up to 20 percent near the breakpoint. 

 Use of V21, as a reference velocity in the dimensionless equation with lateral 

 mixing would then reflect this difference in the magnitude of the longshore 

 currents calculated. 



(4) Empirical Formulation . Finally, in addition to the above 

 theoretical modifications of the original bed-stress formulation by Longuet- 

 Higgins (1970), the empirical, curve fit approach of Madsen, Ostendorf, and 

 Reynolds (1978) must be included (see also Ostendorf and Madsen, 1979). To 

 remove the weak current and small-angle assumptions, they first postulated 

 that the form of the longshore current profile (with lateral mixing) is to 

 remain that given by equations (65), (66), and (67) from the original model 

 theory. A scaling factor is introduced between the characteristic reference 

 velocity v? (with setup) used by Longuet-Higgins (1970) and that for full 

 nonlinear bottom stress proposed. This scaling factor is also proportional 

 to the ratio v^/ug . Curve-fitting procedures are used to obtain an expres- 

 sion for the scaling factor from a surf zone force balance. The modified 

 model also includes new formulations for lateral turbulent mixing and the 

 breaking criteria, y. The resulting modified model requires appropriate values 

 to be selected for the bottom stress coefficient and mixing parameter, as 

 usual. Madsen, Ostendorf, and Reynolds (1978) used the laboratory data of 

 Galvin and Eagleson (1965) to calibrate the model. It was recognized that 

 plunging breakers present in these experiments did not match the y = constant 



98 



