y-direction: 



9t 



9(pq/h) 



3x 



9(qVh) 

 9y 



-Bh(f . 



8Z 



'-) = 



1 - - 1 ^^ 



+ - (t - t ) - - (■ 

 p ^ sy By'^ p ^ 9x 



2SZ 



3S , 9hT^ 9hT, 

 9y p 9x 9y 



(109) 



where Z is the vertical distance of the bed above an arbitrary datum and 

 always positive (see Fig. 23), and all other terms are as previously defined. 

 Arbitrary beach profiles are readily handled using the gradient of Z, terms. 

 The dependent variables are now p, q, and h as function of x, y, and t. 



c. Other Forms . The equations above are derived by assuming the motion 

 is composed of mean components plus fluctuations at the scale of the short 

 wave motions on the coast. The latter two terms in the motion equations re- 

 sult from the interactions of the mean flow and the wave-induced oscillations, 

 Harris and Bodine (1977)^^ extended this approach by recognizing that two dif- 

 ferent scales of perturbations exist in coastal hydrodynamics: 



(1) the generally organized oscillations due to wind-generated waves 

 (indicated by a tilde) , and 



(2) the generally random fluctuations called turbulence produced by 

 either the mean flow or the waves (indicated by a prime) . 



Each primary variable was divided into three components: 



u(x,y,z,t) = u + ii + u' 



(110) 



v(x,y,z,t) = V + V + V 



w(x,y,z,t) = w + w + w 



p(x,y,z,t) = p + p + p 



These expressions are substituted into the incompressible continuity equation 

 and Navier-Stokes equations (with Coriolis terms) and averaged over some space 

 and time interval. As expected, products of perturbation variables arise that 

 reveal interactions between the large- and small-scale flows. For wave and 

 mean flow interactions, the vertical integrations over the water depth are per- 

 formed first, and then the time-averaging operation is performed. This is 

 identical to the procedures described above. For the random turbulence inter- 

 actions, the time- aver aging process is conducted first followed by vertical 

 depth intergration. Thus the averaging methods employed lack complete mathe- 

 matical exactness but serve to demonstrate the physical interactions of nature 

 (Harris and Bodine, 1977). For example, when the waves are omitted, the 

 Reynolds equations of motion are recovered. 



^^HARRIS, D.L., and BODINE, B.R. , "Comparison of Numerical and Physical 

 Hydraulic Models, Masonboro Inlet, North Carolina," GITI Report 6, U.S. 

 Army, Corps of Engineers, Coastal Engineering Research Center, Fort 

 Belvoir, Va., and U.S. Army Engineer Waterways Experiment Station, 

 Vicksburg, Miss., June 1977 (not in bibliography). 



118 



