The Tropospheric Circulation 



eii 



of the physical sea level and —d and p_a are the depth and the corresponding density 

 of the layer of no motion, then from (XIX. 16) when m_<j = and 



1 



it follows 



Vr = 



P = 



7 



l + d 



If dpjcx is exactly zero then 



Vr = 



- P-d ^ I ^ ^ 

 p dx p 8x 



g P — P~d ^ 

 f p dx 



(XIX. 17) 



(XIX. 18) 



which corresponds to a strictly equivalent-barotropic field where the boundary surface 

 lies at a depth —d and the current layer has a density p while the motionless layer 

 beneath has a density p-^. Since in the transverse section through the Gulf Stream 

 dpjcx is very small, the effect of the first term in (XIX. 17) will predominate and the 

 actual distribution will approximate closely that of an equivalent-barotropic model. 



A model of this type has been worked out by Charney (1955). Earlier investigations 

 by Stommel (1953) and Charney (1955) have clarified the question how a boundary 

 current possibly might be influenced by a consideration of the inertial terms in the 

 total equations of motion and by stratification of the water masses. 



This model assumes the j'-axis along the edge of the continental shelf with the :v-axis 

 at right angles ; with a slight approximation it can be assumed that the j-axis points 

 northward and the .v-axis eastward. Ignoring the unimportant kinematic effects of the 

 earth's curvature it follows from the theorem of the Constance of potential vorticity 

 (p. 336) that for a steady current in a water layer h above a motionless lower layer 



8x 



h 



+ v 



8y 



t+f 







(XIX. 19) 



Introducing from the equation of continuity the volume transport stream function i/< 

 which is defined by 



hu 



hv = ^ 

 ox 



(XIX.20) 



allows (XIX. 19) to be rewritten in the form 



[dx \h oxj ^ dy \h dyj 



+ f 



/-(</.) 



(XIX.21) 



where F is a function of ip to be determined. A second equation relating iJj and h is 

 the Bernoulli-equation. This gives 



1 r/1 dipy _ /I d,p\ 



2 [[hd^j ^ [hdj'j 



g*h = G(0), 



(XIX.22) 



where ^* = (ph-- p)l Ph-S and pn is the density of the lower motionless layer and p is 

 the density of the upper moving layer. G(^) is another function of i/*, which has to be 

 determined. 



