Nonlinear Theories — Inertial 111 



where g' = giP2—Pi)/P2- The configuration we have conjured up thus far does 

 specify the total transport T of the stream, 



T=j\Ddx=?j^, (9) 



but does not specify the shape of the stream profile, the width of the stream, 

 or whether counter currents, and so forth, exist. The stream might be wide 

 or narrow. Therefore some further physical statement must be introduced, 

 to determine the stream completely. Suppose that we now require that the 

 potential vorticity of the upper layer be the same for all x: 



8v 



Combining this requirement with the geostrophic equation, we are led to 

 the following equation in D: 



|^=^,P-i>.), (11) 



where A^ = g'DJp. Therefore the cross-stream profile of depth and velocity 

 is completely determined : 



(12) 



If we now introduce numerical values Dq = 800 m . , and (pg — y^i ) /P2 = ^ x 1 0"^, 

 the model gives a reahstic transport, T = Q4x 10®m.^/sec. The maximum 

 velocity of the Stream is 4 m./sec. The width of the Stream, taken as the 

 distance between the point of maximum velocity and that at which the 

 velocity is reduced to 1/e of the maximum, is 40 km. The length. A, has been 

 called by Rossby (1936a) the radius of deformation. Rossby introduced it 

 in the presentation of his wake -stream theory, to explain counter currents. 

 Today it would seem that this term may actually be suitable for describing 

 the Stream itself. 



Since this very crude model of uniform potential vorticity is independent 

 oiy, that is, of latitude, there is no provision for the growth of the stream, 

 such as is observed in the real Gulf Stream between the Florida Straits and 

 Cape Hatteras. In the actual Gulf Stream the water present at any latitude 

 y' is mostly drawn from lower latitudes, and hence one might reasonably 

 ask why there should be any reason to expect the potential vorticity in the 

 Stream at y' to correspond to that outside the Stream at y'. The justification 

 for applying the simple model described rests on the observed fact that in 

 the central Atlantic between 10 and 35° N. latitude, the potential vorticity 



