Nonlinear Theories — Inertial 109 



layers warmer than 16° C. It seems dynamically significant that this 

 vertical shrinking is just about the amount required for the potential 

 vorticity of each isothermal layer to be uniform across the Stream, aU the 

 way from the inshore edge of the Stream, where vertical shrinking is at a 

 maximum, to the Sargasso Sea. At the left-hand edge there is evidently a 

 sharp discontinuity in potential vorticity. This is also true of the T-S 

 correlations of the surface water. 



In order to demonstrate this remarkable uniformity (Stommel, 1955 a) of 

 potential vorticity, let us consider the vertical thickness h of the layer 

 bounded by the 17 and 19° C. isotherms in Worthington's Gulf Stream 

 section shown in fig. 33. We place the ?/-axis in the direction of flow of the 

 stream, and direct the a:-axis toward the right of the current, extending 

 eastward into the Sargasso Sea. 



In the Sargasso Sea, on the right-hand side of the section, the depth of 

 the isothermal layer is constant, and we denote it by Kq. The velocity of the 

 current vanishes there. Moreover, since the current is all in the ^/-direction, 

 the relative vorticity is essentially given by dvjdx. The equation for uniform 

 potential vorticity may then be expressed in the following way: '' 



f+- 



■> (6) 



The velocity v of the water in this isothermal layer may then be computed 

 from the observed values of thickness h by direct integration: 



v{x)=rf(^-\\dx. (7) 



This computed velocity may now be compared with the geostrophically 

 computed velocity. The results of such a comparison are shown in fig. 65. 

 The fact that the velocities computed in these two very different and 

 dynamically independent ways agree is good evidence that the potential 

 vorticity in the Stream is nearly independent of cross-stream position. 



MODEL WITH UNIFORM POTENTIAL VORTICITY 



It is interesting to see how close a representation of the Gulf Stream can be 

 obtained by using nothing but the principle of conservation of potential 

 vorticity and the geostrophic approximation in a simple two-layer model. 

 Taking the axes as before, we assume that a resting layer of uniform depth 

 Dq, and density p^, on top of another resting layer of very great depth, and 

 density yOg) extends indefinitely in the positive a:;-direction. We now suppose 

 that the interface between the two layers comes to the surface along the 

 t/-axis; thus Z) = at a; = 0. Therefore there must be a geostrophic current v 



