SECT. 3] DYNAMICS OF OCEAN CUKRENTS 345 



let To denote the magnitude of the stress components, the transport function will 

 be of magnitude to/j8 and the curl of the wind stress of magnitude tqIL. Intro- 

 ducing the non-dimensional variables i = xlW, r] = ylL, r'x = rsxlro, r'y = TsylTo 

 and </»' = 0/(to//S), we may write (71) in the non-dimensional form 



H 



pw< 



8^ ^jwy d^ijs' iwy gy 



dij/ _ w 

 d$ ~ L 



L dr'y Bt 



(72) 



By choosing PT = (^ ///^) '/^ ~ 100 km and assuming WjL to be small, we can 

 see that (72) can be approximated by the simpler equation : 



"^ "^ = 0{-] ~ 0. (73) 



8^^ di \L 



We can obtain a solution to (73) of the form 



f =Co + IAne^n?, (74) 



where Co is constant with respect to ^, and the coefficients an satisfy the 

 equation an^= 1. The three roots of the equation are ai= 1, a2= — l/2-f-*\/3/2 

 and a3= — 1/2 — i\/3/2. Thus, there are four independent solutions enabling us 

 to satisfy the required boundary conditions. 



At the western boundary {x — ^ = 0), both i/r' and dip'/d^ must be zero. Further- 

 more, the coefficient A3 is zero because the root ai is positive and would yield 

 a divergent solution otherwise. The two boundary conditions enable us to 

 evaluate A2 and ^3 in terms of Co. The resulting function is 



,, ^ /, a3e"2^ a2e'^^^\ 



ip = Co IH 



\ a2-a3 a2-a3/ 



= Co[l -6-'^'=^^ cos (\/3|/2)- (e-'/^f/V3) sin (\/3|/2)] (75) 



For large values of |, the boundary-layer transport function approaches Co. 

 We can identify Co as the value of the transport function obtained for the 

 interior of the ocean. As the boundary layer is narrow compared with the width 

 of the ocean, we can obtain a reasonable approximation for the value of the 

 interior transport function by neglecting the width of the boundary layer and 

 evaluating ipt at x — O^ Therefore, the transport function for the western 

 region of the ocean is given approximately by 



iP = i/ji{x,y)T{xlW), (76) 



where 



1 As equation (71) is linear, more exact solutions for the boundary layer are readily 

 obtainable in which both the finite width of the layer and curl of wind stress within the 

 layer are taken into account. However, little of interest is gained by this more elaborate 

 procedure. The siinpler procedure outlined here can be applied to non-linear equations in 

 which acceleration terms are included. 



