Motion 



x-direction: 



9u , - 3u , - 9v _ , 1 /- - x 

 9l+"93^ + ^37 ~ ^ ^'sx - "Bx^ 



(99) 



9S 9S , 9hT^ 9hT^ 

 1 , XX ^. xy x 1^ , Lxx Lxy .. 



ph ^ 9x 9y '^ ph ^ 9x 9y 



y-direction: 



9v , - 9v , - 9v , 1 ,- - V, 



1- u -r 1- V -r— = H r- (t - T ) 



9t 9x 9y ph sy sy 



(100) 



9S 9S -, 9hTT 9hT^ 



_L ( yy + xy . _ J_ , LxZ + L^Z) 



ph ^ 9y 9x -^ ph ^ 9x 9y 



where u, v = the depth-averaged and time-averaged velocity components 

 T^^^ = the time-averaged surface wind shear-stress components 

 = the time-averaged bottom shear-stress components 



sy 



Bx' By 

 S , etc. = the radiation stress components 



Lxx' 



etc.= the effective lateral stress components. 



In this Eulerian fonn, the mass equation neglects the small contribution of 

 mass transport due to finite-amplitude wave orbital motion. The motion 

 equations include a possible surface wind shear stress for completeness. For 

 steady, uniform motion, with no wind and no coupling with the y-direction 

 momentum terms, equation (99) reduces to the simple momentum balance for wave 

 setdown and setup derivations (eq. 28). Similarly, equation (100) reduces 

 to the y-direction momentum balance (eq . 42) used to derive the longshore 

 current profile. 



The motion equations (99,100) are equivalent to those for nearly hori- 

 zontal free-surface flows if the radiation stress gradients are neglected. 

 Then, all variables and the stresses are no longer time-averaged quantities 

 and the pressure distribution is assumed hydrostatic. The radiation stress 

 gradients, by definition, account for the deviation from hydrostatic pressure 

 resulting from the streamline curvature present in short waveforms. Conse- 

 quently, the pressure distribution is nonhydrostatic in the motion equations. 

 The radiation stresses effectively arise from time-averaging the mean velo- 

 city plus the wave orbital velocity fluctuation in analogy with Reynolds 

 stresses . 



Horizontal momentum flux is due to the convective acceleration terms, 

 radiation stress gradient terms, and the effective lateral stress terms. 

 The latter effective stresses combine momentum fluxes due to both horizontal 

 mixing of wave scale motions and deviations of the local velocity from its 

 depth-averaged value (Vreugdenhil, 1980). As before, the effective lateral 



15 



