*riiere C3 is the vertical comptonent of the relative vorticity of the current 

 "V, and f is the coriolis parameter. This form of the vorticity equation 

 does not consider the twisting effect of differential vertical advection of 

 vorticity or the baroclinic (solenoidal) generation of relative vorticity, 

 but does include the generation of relative vorticity by horizontal diver- 

 gence in the current. Flow over a sloping bottom will cause both horizontal 

 divergence and vertical motion that enter into equation (1); so long as this 

 divergaice and vertical motion are small, it is safe to assume that the cur- 

 rent is essentially horizontal and non-divergent. 



Since the fluid is assumed incompressible, the continuity equation can be 

 written 



ff^y-V = 0, (2) 



where w is the vertical componait of the flow. The assumption that the 

 flow is vertically constant in direction is expressed by 



V('x,y,E,t) = A(2) V(x,y,t), (3) 



>^ere the bar operator, vhen applied to the flow, is defined by 



'il. 



V = tl ydi. (4) 



The depth field h(x, y) is measured positive; thus the directional sense of 

 the z-axis is reversed from the normal right-handed Cartesian coordinate sys- 

 tem. The vertical profile parameter A is in reality also a function of x, 

 y , and even t , but it is permissible to limit A to variation only in the 

 z direction. The vorticity of equation (3) is given by 



^ = /c.V7xv =k'V>i(Ay) =/4('A'17^v) =/45. (5) 



Equations (3) and (5) are now substituted in the vorticity equation (1), 

 and the result is integrated vertically with the bar operator, equation (4): 



or 



and finally. 



If ^ y'\7(A'3 -tf) + (A'd ^ f ) 7-y = 0, (6) 



as 



ilAd.-l, ilA'cl^-A\ 



(7) 



136 



