Equation (6) is the vorticity equation of the vertically-averaged velocity 

 V . By being prescribed in a special manner, the vertical variability of the 

 cvirrent has_been integrated out of the vorticity equation leaving only the 

 parameter A* . An ejqDression for the divergence V'V of the mean flow must 

 be derived to acccmodate the effect of bottom topography. 



By expanding th« ea^ression for V*V > we have 



vy = i7y^/'W= -^*-^^V</.A ivh-v, ^-kfv-vj,. 



or 



\/'\^-7rV'Vh -^j;V,'Vh i- j;£V'Vdi 



(8) 



The last term on the right side of equation (8) can be evaluated by applying 

 the bar operator, equation (4), to the continuity equation (2): 



ilVvJ. = ;r/r-tM =-k(K-^.). (9) 



The vertical velocities w. , at the top of the fluid, and Wj^ , at the bot- 

 tcoi of the fluid, must satisfy the kinanatic boundary condition that there be 

 no cross-boundary flow; therefore: 



W^ = O , Wm = "l;*7/l . (10) 



When equations (8), (9), and (10) are combined, we have the following: 



7-v = —kV'Vh =-V'Vlo^h, (11) 



Substitution of equation (11) into the integrated vorticity equation (6) 

 yields the equation 



or 



dt 



i-V-V(A'^+f-ilc^h) =0, (12) 



where fg is an average value of the coriolis parameter for the fluid. Equa- 

 tion (12) is the mathematical relationship liiich describes the equivalent 

 barotropic model for the ocean,_ This equation differs from that for a simple 

 barotropic model in the terms A*" , vihich is a measure of the vertical vari- 

 ability of the current, and -fg log h , which results from the sloping bot- 

 tom topography. In the application of equation (12), a stream function 

 ^(x, y, t) is introduced, from \iiich the vertically-averaged current and vor^ 

 ticity can be derived. The resulting model equation is then 



V"^^Tj(f,K^V'ftf-fJo^H) =0, (13) 



137 



