Welsh 



where d is the depth of the ocean, Tg is the surface temperature, 

 and n is a parameter of the model. Note that this prescribes a 

 temperature gradient which has the same direction at all depths and, 

 together with its n - 1 derivatives, vanishes at the ocean floor, 

 for n > . The equation for the vertical shear of the geostrophic 

 flow V can be written 



|-v(.) 



dZ 



g 



a J k X VT(z) 



(2) 



where f is the coriolis parameter and a is the linear coeffi- 

 cient of thermal expansion for sea water. Now Eq. (1) can be sub- 

 stituted into Eq. (2) and the result integrated in the vertical to 

 yield 



>in+l 



V(z) 



agd 



f(n+l) 



-f) 



k X VT 



(3) 



where the assumption of zero bottom flow has been used. From Eq. 

 (3) , derive the further parameters 



1, 

 d 



V(z)dz 



agd 



f(n+l)(n+2) 



^ X VT 



(4) 



A(z) 



V_(_z) 

 f 



(n + 2) 



i-f 



>|n+l 



(5) 



1^ 

 d 



[A(z)]2 dz 



(n + 2)- 

 2n + 3 



(6) 



and note that 



A(z = 0) 



n + 2 . 



(7) 



Furthermore, it follows from Eq. X4) that the stream function i|) 

 of the vertically averaged flow ^ is given by 



agh 







f,(n+l)(n+2) s 



kT , 

 s ' 



(8) 



where 



agb. 



fp(n+l)(n+2) 



(9) 



is the conversion factor, and hg and fg are suitable mean values 



for d and f , respectively. 



185 



