Nonlinear Theories — Inertial 113 



mates the total derivative of the potential vorticity, (d/dt) [iC+f)/D], by 

 fiv/D. Thus C. is regarded as small, and no account is taken of variations 

 in D. The left-hand member of the potential-vorticity equation, formula (5) 

 of this chapter, must balance against something, and hence all that remain 

 are the various possible types of frictional terms. 



In the high- velocity regions of the Gulf Stream, the relative vorticity ^is 

 negative and is not at all neghgible as compared to /; inspection of the 

 surface velocity profiles in fig. 32 wUl show the truth of this statement. 

 Therefore it is quite conceivable that as a fluid column moves into the Gulf 

 Stream, the changes in C, and D can occur in such a way as to keep the 

 potential vorticity constant and the conservation equation satisfied without 

 the use of the frictional terms. Thus the theoretical model of a frictionless, 

 nonhnear stream may describe the real Gulf Stream, the essential balance 

 of terms within an isopy cnal layer being of the foUomng form : 



U+f)dD d^ 



D dx dx 



+ v 



U+f)dD_dC_^ 



D dy dy 



= 0. (13) 



A very interesting series of free, steady solutions of the motion m a 

 frictionless, homogeneous ocean of constant depth has recently been 

 obtained by FofonofiF (1954). The conclusion drawn from his work is that 

 eastward-flowing currents cannot be broad, slow streams, but must be 

 narrow and have high velocity and high relative vorticity. As yet, Fofonoff 

 has not been able to obtain similar solutions for a two-layer ocean in which 

 horizontal divergence is appreciable, but he has indicated that, in the North 

 Atlantic, this will produce a westward, as well as a northward, intensifi- 

 cation. When these solutions have been obtained we should expect the 

 profiles of the strong currents to be similar to those obtained earlier in this 

 chapter, in the consideration of conservation of potential vorticity. 

 Fofonoff's study is very similar to the uniform potential-vorticity model 

 described above, and the models by Charney and Morgan described below. 

 His paper (1954) was somewhat formal, however, and its applicabihty to the 

 Gulf Stream becomes obvious only in the fight of these and other studies. 



In formulating the physical model for a nonhnear steady-state theory of 

 ocean currents we shall try to preserve the following characteristics. 



i. The ocean will be density-stratified. 



ii. The solution for central ocean transport will be the same as Sverdrup's 

 and Munk's. 



iu. The narrow western stream (Gulf Stream) will be frictionless and 

 convergent, and there will be a tendency toward development of a Fofonoff- 

 type eastward flow in the region of the North Atlantic Drift. 



iv. All dissipation of the energy in the Stream will occur in a limited 

 region toward the end, where meanders break up and multiple streams form. 



