Nonlinear Theories — Inertial 117 



to Dr Jule G. Charney of the Institute for Advanced Study at Princeton, and 

 to Dr George W. Morgan of Brown University, and in the spring of 1955 

 these two investigators, independently, developed proper inertial theories 

 of the Stream. Their theories lead to a nonhnear differential equation which 

 can be integrated numerically. 



Let us first examine Morgan's formulation. We consider an ocean bounded 

 by meridional walls at a; = and a; = r, as in Chapter VII. For simphcity we 

 also assume that the wind field over an interval of latitude O^y^s is given 

 by the equation 2\ 



r,= -To^l-^j, r, = 0. (14) 



According to the dynamical equation applying to the central part of the 

 ocean [Chapter VII, equation (42); also Chapter XI, equation (5)] we can 

 solve for the meridional transport per unit width. My, in the interior: 



On the east coast, x = r, the zonal transport M^= —dxjfjdy vanishes; 

 hence we have, from continuity, the following simple results for ijr and M^: 



2'UT 



f=J^.(r-^). (16) 



The advantage of the paraboHc zonal wind-stress law, equation (14), is 

 that it specifies a zonal transport at the western coast, x = 0, independent of 

 the latitude y. Solutions (15), (16), and (17) areindependentof any assump- 

 tion about the vertical density stratification. If we consider a homogeneous 

 ocean, the variation in total depth due to slope of the free surface is negli- 

 gible. If, on the other hand, we consider a two-layer system with density 

 different in each layer, and the bottom layer at rest, we must also take into 

 account the variation of the depth, D, of the top layer. In particular, the 

 pertinent dynamical equation in the interior is the ^/-equation: 



Since, for the purpose of matching the boundary layer to the interior 

 solution, we need only the value of D along the outer edge of the boundary 

 layer (that is, for x = 0), we can integrate equation (18) with respect to y, 

 using My. at a; = from equation (17) and taking into account the variation 

 with latitude of the CorioHs parameter with latitude. We set f=fj^+^^y. 



