stresses are usually described in terms of an eddy viscosity coefficient due 

 to the weak understanding of the surf zone. Care is required to ensure that 

 the results are independent of the coordinate system chosen and consistent 

 with surf zone turbulence. For example, Vreugdenhil (1980) has shown that 

 the contribution of the pressure fluctuations to the normal stresses is rela- 

 tively small so that the following expressions result for the effective 

 lateral stresses: 



^Lxx = ^Lx^l^ - P (101a) 



- - ,3u 8vx 



^Lxy = ^Lxy % + ^^ (101b) 



.9v 9u^ . „, . 



^Lyy = ^Ly ^^ " ^> (101c) 



where the eddy viscosity coefficients are direction dependent. For homo- 

 geneous, horizontal, time-averaged wave scale turbulence 



^L = ^Lx = hy = i^Lxy (102) 



and only in this case do the cross gradient terms 3/8xy disappear. Finally, 



it is also common for the mean water depth h, in the effective, lateral stress 



gradient terms to be mistakenly removed from the differentials. Since h is 

 a function of x and y it must also be differentiated. 'If the water depths 



are canceled, forms often found in the literature (e.g., Bowen, 1967, 1969b) 

 result, i.e., 



x-direction: 



a: 



3^u, 



(^^ + ^^) (103a) 



9x2 gy2 



y-direction: 



... -V, (•^ + ^) (103b) 



^ 9x2 9y2 



that can only be considered as approximations of the true expressions for 

 the effective lateral stress terms. Here v = y /p is the kinematic eddy 

 viscosity, and only the last terms of the full motion equations are shown. 



If the motion equations are dimensionless, it can be seen that the re- 

 lative importance of the effective lateral stress terms compared with the 

 convective acceleration terms is given by a time-averaged lateral Reynolds 

 ntimber 



116 



