assumption in the surf zone, but the data were used an3rway. An explicit ex- 

 pression for the combined, nonlinear friction factor was obtained that pro- 

 perly reflected the weak and strong current theories. It is not clear how 

 the effect of wave approach angle is included by this calibration approach. 

 The magnitudes of the bottom shear-stress coefficients required in all 

 the above theories are discussed in Chapter 4. 



c. Modified Lateral Turbulent Mixing Shear Stresses . Theoretical know- 

 ledge is very weak regarding the horizontal transfer of momentum due to tur- 

 bulent mixing processes . Turbulence length scales represented here are on the 

 order of the water depths, wave particle excursions, or surf zone widths, and 

 generated by wave breaking shears, bore-bore interactions, and swash zone 

 mixing. These processes in the surf zone are not sufficiently understood to 

 permit detailed models of the effective stresses that result. Consequently, 

 recourse has been universally made to some type of eddy viscosity model to 

 linearly relate the time-averaged lateral mixing shear stress, Tt to the long- 

 shore current gradient dv/dx. If the velocity fluctuations (u, v) in the x- 

 y directions are considered respectively due to wave orbital motions and inter- 

 actions, then for one- dimensional motion, as in equation (55) with u^ the time- 

 averaged lateral eddy viscosity 



- dv dv 



Tj^ = - pav = Mj^ ^ = pv^ -^ (86) 



The tilde symbol here means velocity fluctuations on the scale of wave motion. 

 The prime sjnnbol is reserved for true, random velocity fluctuations due to 

 turbulence, however generated. The kinematic eddy viscosity is v = u /p. 

 Prandtl's mixing length hypothesis 



;..£ (87) 



could also have been employed giving 



^L = -^i (88) 



so that the eddy viscosity is simply 



U^ = pv^ = p|gI-I (89) 



The eddy viscosity is related to the reference velocity u and length scale 

 a in some time- averaged sense. A rigorous and physically defensible deriva- 

 tion of equation (89) can be found in Battjes (1975). 



Longuet-Higgins (1970, 1972a) chose to take the characteristic velocity 

 proportional to the maximum bottom wave orbital velocity (u„ ) and the refer- 

 ence length scale proportional to distance from shore (x) . The closure coef- 

 ficient N in his equation (eq. 55) incorporated both proportionality factors 

 plus assumed that turbulence velocities u were about 10 percent of mean velo- 

 cities, as in normal turbulence (Longuet-Higgins, 1972a) . 



99 



