5t ^iy ay H I ^i^ax y H 



tiydy H 





^y ay 



Hdf 



'sy 



■ ''' 'L 



;H.D.)y da 



JLiiL + D, 



^y sy y 



(2.38) 



where the vertically-integrated velocities are defined as 



(U. V) 





H (u,v) do 



(2.39) 



and (H.D.)j^ and (H.D.)y represent the horizontal diffusion terms shown in 

 Equations (2.25) and (2.26), respectively. Notice that simplifying 

 approximations have been made in deriving the nonlinear terms shown in (2.37) 

 and (2.38). A more general representation of these terms should include a 

 unity-order multiplication factor which depends on the vertical shapes of 

 horizontal velocities (Sheng et al., 1978). Due to difficulty in estimating 

 the shape parameter which is usually a function of time and space, most 

 vertically integrated models use the approximate nonlinear terms and hence 

 underestimate the nonlinear effects. Chen (1981) attempted to account for 

 this effect of a non-uniform velocity profile in a vertically-integrated model 

 by introducing an internal stress term. In the present three-dimensional 

 model, the nonlinear inertia terms can be accurately computed from the 

 three-dimensional velocity field and hence no approximations are required of 

 the vertical shapes of horizontal velocities. 



38 



