modified to conform to the formulations normally used in wave-induced current 

 models. A velocity, as opposed to discharge, version of WIFM that included 

 nonlinear advective terms was used. 

 Equations of motion 



26. The hydrodynamic equations used in the model for wave-induced cur- 

 rents and setup may be derived from the Navier-Stokes equations (Phillips 

 1969). It is assumed in the derivation that the fluid is homogeneous and 

 incompressible, and the vertical accelerations are negligible so that the 

 pressure distribution is hydrostatic. By integrating the three-dimensional 

 form of the equations in the vertical direction and applying appropriate 

 boundary conditions, the depth-averaged two-dimensional form of the equations 

 of motion and continuity are obtained. These equations are derived by time- 

 averaging over the wave period. The momentum equations (Figure 6) are 



3U ., 3U ,, 3U 9n 1 



t— +U-7— +V-— + e — - + — - t + — - l — - — + - J I — ^- =0 ( 26 ) 



9t 3x 3y B 3x pd bx pd \ 3x 3y / p 3y 



3V „ 3V ,, 3V 3n 1 1 



1E +U ^ + V ^ + 8^y- + ^d T by + ^d l 



The continuity equation is 



I? + -I- (Ud) + -jL (Vd) = (28) 



3t 9x 3y 



Here U and V are the depth-averaged horizontal velocity components at time 

 t in the x and y directions, respectively; n is the mean free surface dis- 

 placement; p is the mass density of sea water; d = n + h is the total 



water depth where h is the local still-water depth; t, and t, are the 



r bx by 



bottom friction stresses in the x and y directions, respectively; S xx , S , 

 and S are the radiation stresses which arise because of the excess momen- 

 tum flux due to waves; and t is the lateral shear stress due to turbulent 



xy _ 



mixing. Note that when the still-water level is zero, the condition r\ > 

 is called "setup" and n < is called "setdown." 



27. The numerical model uses a linear formulation for friction (Longuet- 

 Higgins 1970). Thus, 



T bx= pc <u orb> U (29) 



21 



