SECT. 3] DYNAMICS OF OCEAN CXIRRENTS 357 



If the boundary flow is northward {fb>fi), an increase in velocity yields an 

 increase in the magnitude of relative vorticity ^b. Conversely, if the boundary 

 flow is towards the equator (/&</<), an increase in velocity yields a decrease in 

 the magnitude of relative vorticity. In poleward flow the intensification is 

 limited by (116) as hi, cannot become negative. Thus, the velocity along each 

 transport line cannot exceed y'(2gr'A,f). Moreover, the relative vorticity is 

 negative so that the maximum poleward velocity must occur at the western 

 boundary. Hence, we obtain the stronger condition that the velocity cannot 

 exceed Vmax, where 



i^max = Vi^g'ho) (118) 



and ^0 is the interior depth for the transport line «/» = 0. 



If the flow is towards the equator, the intensification in the western-boundary 

 region is accompanied by a decrease in the depth of the layer. The decrease in 

 depth compensates the decrease of Coriolis parameter so that the potential 

 vorticity can remain constant along a transport line with lower magnitudes of 

 relative vorticity than that of the poleward flow. Moreover, the potential 

 energy and, hence, depth in the boundary current has to be greater than that 

 of the interior flow at the same latitude in order for flow toward the equator 

 to exist. Thus, the velocities cannot be as intense as in the poleward flow. From 

 (117) we can see that the relative vorticity will become negative if the velocity 

 at the western boundary exceeds [2g''Ao(/o— /&)//o]''^^, where ^o,/o are the interior 

 values for = 0. Should the relative vorticity become negative, the maximum 

 velocity could no longer occur at the boundary. i 



In order to determine the characteristics of the western-boundary current 

 in more detail, we shall use the geostrophic approximation to obtain velocities 

 and (113) and (114) to connect the boundary solutions to the interior solution. 

 The velocity across a given latitude circle in the boundary region is approxi- 

 mated by 



M-y^ (119) 



and the transport by 



/-*' = -4 = t- (^^«) 



Integration of (120) yields 



fh = Xf>o-Xb = lg'{ho^-h^), (121) 



where ^^o is the depth of the upper layer at the western boundary {x = 0). 



1 If the transport function in the interior is assumed to be a linear function of y, it is 

 possible to show that the relative vorticity is always positive for cyclonic circulation, i.e. 

 hb>hifb/fi- However, the relative vorticity might become negative for more complex 

 interior flows. 



