This, and the similar expression for — — , substituted in equations (7a, b) give 



f 



Jz b 



I 



dP ± _d X , dP b P,MP b ) p dzo 

 dx dx dx g dx 



"dP , _d X[ dPb P b a*(Pb) , „ dz b ^ 



7~ dz ~1 — h l h "»7— Ha, b 



z b dy dy by g dy 



Finally, it can be shown that the final terms 



f" 4 d ' 2u i - 



Ay —~^ CiZ — T sx — Tbx , 

 Jz b dz* 



Av—- 2 dz = T S y-T b y, (12a, b) 



Jz b az 



where t sx represents the stress per unit surface area in the x direction and T bx is a similar 

 term for stress at the bottom. By substituting equations (3a, b). (4a, b), (11a, b), and 

 (12a, b) into (la, b), 



du 2 duv dx dP b Pi,ao(Pb) D dz b 



^ + ly- JV ~- -Tx'^x—g- - p bte +A ^ U+Tsx - Tbx 



^f+JV = -f-^^^-P b ^ + A H ^V + r sv -n y . ,13a, b) 



dx dy dy dy g dy 



The bottom pressure and bottom slope terms in (13a, b) can be further simplified as follows: 

 n n , [ Z j dP t> dP* , ["dp dz b 



When z = z b the second term on the right is eliminated so that 



dP b = (dPA _ a d_Zb__ 

 dx \ dx ) z = Zh b h dx 



With this substitution the terms from (13a) become 



PbOoiPb) (dPz\ „ , D . dz b D dz b 

 + PbOtoiPb)pb Pb 



. x )z=z b ox dx 



It is assumed that oto(Ph) = at, and the bottom pressure and slope terms of (13) may be 



represented by the single term ( — -1 



g \dx j z=Zb 



To be useful for the present purposes it is necessary to neglect additional terms in 

 (13a, b). Assume (a) that field acceleration terms are negligible; (b) that the velocities 

 near the bottom are very small so that T b is much smaller than r s , so the vertical eddy vis- 

 cosity terms when integrated over the column are closely approximated by t s : and (c) that 

 in a central region of the ocean horizontal friction terms are negligible (Munk, 1950). With 



21 



