SECT. 3] DYNAMICS OF OCEAN CURKENTS 359 



where W = v ( Uol^) is a characteristic width and Fr is the internal Froude 

 ninnber at the southern boundary. From (130) we can see that the meridional 

 extent of the circulation is limited. In particular, if ?/max is less than the distance 

 between the northern and southern boundaries of the ocean, the surface layer 

 cannot cover the entire ocean and lower-layer water will be exposed to the 

 surface north of ?/max- Consequently, in the two-layer ocean, we can have a 

 separation of the western-boundary current from the boundary at an interior 

 point. For a given depth ho at the southern boundary, the northern limit of 

 the upper layer varies inversely as the square root of the speed of westward 

 drift. The stronger the flow, the more southerly will be the point of separation 

 of the flow from the western boundary. 



The depth of the upper layer in the interior of the ocean is a function of ipt. 

 Substituting (128) into (124), we obtain 



.2 , . , 2foUoho , ^Uoho 



(131) 



g g Uoho 



At a given latitude in the boundary region, the layer depth is related to the 

 transport function by (121). As each transport line ipb in the boundary region 

 must be connected to the interior transport line ipt having the same value, we 

 can evaluate hi in terms of ht,. Eliminating the transport function from (131) 

 by substituting from (121), we obtain 



hi^ = ho^-i^{hbo^-h^) + l-^&-^{h,o^-h^)^ 



J PUoho ^J32) 



= ho^ - a{ho^ - hb^) + la^b^ho^ - hb^)^ 



where a=folf< 1 and h = ^g' jJo^Uoho. As hb must approach hi for large values of 

 X, a solution of (132) for hb — hi must exist. The condition for a real solution to 

 exist is 



(l-a)2/a252 ^ ho^-hbo^ 

 or 



101 = Uohoy ^ g'{ho^-hbo^)l^y. (133) 



But from (126) we see that the equality must hold if ipi is a linear function of 

 y. Hence 



hi^ = ho^-a{hbo^-hb^)+ ^.l^\~''l\A hbo^-hb^)^. (134) 



We can obtain the variation of depth of the upper layer in the boundary 

 region from (116) and (119) by neglecting Ub in comparison to Vb. As the flow is 



