SECT. 3] DYNAMICS OF OCEAN CURRENTS 349 



where a„ are the four roots of a„4_A = 0. All four roots are complex for A 

 negative with two roots having negative and two having positive real parts. As 

 the two roots with positive real parts represent divergent solutions, we can set 

 the constants A n multiplying the divergent terms to zero and use the remaining 

 two terms to satisfy the two boundary conditions at 7] = 0. The boundary 

 conditions can be satisfied provided we assume that the interior solution is 

 proportional to e"!-^'^ in the boundary region. Consequently, we must have 



1 ^'i 



The boundary-layer solution is 



0' = j/»'t[l-e-*'' (cos A:T7 + sin A;7j)], (84) 



where yb= (|A|/4)'/i, 



If A is positive, the four roots an are either real or imaginary. Thus, at least 

 one term of the boundary-layer solution is trigonometric and does not diminish 

 in amplitude with distance from the boundary. In this case we cannot separate 

 the ocean into an interior and a boundary region because the frictional terms 

 are not negligible in the interior of the ocean. 



The solution obtained for the zonal boundary layer is of a highly restricted 

 form. In the more general case, the variables cannot be separated and a simple 

 solution is not evident (cf. Rossby, 1937). However, if we assume that the 

 interior transport can be approximated over part of its range by an exponen- 

 tial, or consider {ll^'t) difj'ijd^ to be a slowly varying function so that we may 

 treat it as a constant, we can apply the special solution in a more general way. 

 As i/j'i is zero at the eastern boundary, the ratio {ljip{)8ilj'il8^ will be negative in 

 the zonal boundary region as long as the curl of wind stress does not change 

 sign. The boundary layer will then be narrow compared with the width of the 

 ocean. If the curl of wind stress changes sign in some region along the zonal 

 boundary, a balance of forces cannot be achieved within a finite boundary 

 layer. This implies that a boundary fiow entering such a region would become 

 unstable. 1 We can state these results in an alternative way. The zonal boundary 

 layer is narrow provided westward flow is continuously accelerated and east- 

 ward flow continuously decelerated along the zonal boundary. Eastward 

 intensification leads to instability, 



5. Steady Inertial Circulation 



In spite of the many attractive features of the frictional baroclinic model of 

 ocean circulation, the model has significant shortcomings. Probably the most 



1 An example of such a region may be found along the chain of the Aleutian Islands. A 

 ridge of high atmospheric pressure is occasionally observed to lie across the middle of the 

 chain of islands with large low pressure areas to the east and west. The atmospheric 

 circulation is such that a region of negative curl is formed. Thus, the flow along the island 

 chain could become unstable in this region. 



