terms on the right side of equations (14a, b) as follows: 



pf(v b + v g ) = d £, 



and integrating 



fi dP 



fV b +fV g =\ —dz. 



Substituting for V b from (18b), 



g \dx J 2=Zb )z b dx 



Then using (11a) and the arguments following (13a, b) regarding bottom pressure terms, 



g {dxJ^J^* dx + g \dx)^ b (19) 



Identical pressure terms can be eliminated from the two sides of this expression. This, 

 together with a similar development for U g yields 



trj =- d -X fv = §X. 



jUg dy fyg dx (20a, b) 



The components of the Ekman transport are defined by the equations 



fU E = T sy , fV E =-T sx . (21a, b) 



Although the sum of the divergences of the three modes must be zero (15), the divergences 

 of each mode taken separately need not be zero. Expressions for the individual diver- 

 gences can be derived by cross differentiating the equations that define the barotropic, 

 baroclinic, and Ekman transports — (18a, b), (20a, b), and (21a, b). Cross differentiating 

 leads to 



/(f + £) + «--"(»£ + -£)- 



\ dx dy ) dx dy 



In equation (22) p' is closely approximated by the mean density of the column. 



The right side of equation (22) can be neglected when the gradient of the bottom is 

 sufficiently small or when flow along the bottom is nearly normal to the slope of the bottom. 

 (Applicability of this approximation is examined in section 2.2.) With this approximation, 

 on adding equations (22), (23), and (24) 



A d(U b + u g+ u E) + d(v b + v g+ v E) i [v v Vb]= b^_b^ 



L 5x dy J dx dy 



23 



