SECT. 3] DYNAMICS OF OCEAN CURRENTS 361 



wliich, if we replace x' by {!/mn\ — y)IW, is also the boundary-layer solution for 

 the eastward jet.^ 



The special case, «inin = J, is of particular interest because of its simplicity. 

 It can be seen from (131) that the interface in the interior of the ocean will, in 

 general, be a hyperbola with respect to y if the transport function is linear in y. 

 Hence, the potential vorticity will not be a simple function of i/». For ainin = 2' 

 the potential vorticity is constant. We can verify this feature by assuming 

 filhi to be constant and differentiating it with respect to y. The differentiation 

 yields 



i_A^=l+fj!]^ = (142) 



hi hr Sy hi g'hi^ 



or 



g'hi^^ gV^^ 

 Uihi = ^^- = ^^— (143) 



Multiplying (143) by ymax, we obtain 



|Wi%max| = IfAmaxI = g'ho-^ymaxlfo^ (144) 



But, from (133), |i/fmax| =g'ho^l^ym&x- Hence, the interior potential vorticity is 

 independent of y only if/o = /S?/max, i.e. a=/o//max = |. 



We can gain some insight into the magnitudes of the variables describing 

 the anticyclonic inertial circulation by considering a specific example. Choosing 

 ^0 to be 400 m,/o as 0.5 x 10-^ sec-i and Apjp as 2 x 10"^ and using g= 10^ cm 

 sec~2 {g' = 2 cm sec~2) and |8 = 2 x lO^^^ cm~i sec~i, we obtain: 



/max = fola = 10-4 sec-1 



^max = '2ho = 800 m 



2/max = (/max -fo)!^ = 2500 km 



i^max = g'ho^"l^ymax = 64 X 10^ m^ sec-1 



^0 = ifjm&xlhoym&x = 6.4 cm sec-i 



Vb max = i'Zg'hoY'^ = 400 cm sec-i 



W = (25r'Ao)'V/max = 40 km. 



Contours of the transport function </» and a meridional depth profile are 

 given in Fig. 3. Because of the symmetry of the transport function and depth 

 of the upper layer with respect to the central meridian of the ocean, we do not 

 have to examine the eastern-boundary region explicitly. The western-boundary- 

 layer solution can be readily transformed to apply to the eastern-boundary 

 region. 



1 The boundary -layer solution for the eastward jet can be derived independently. For 

 (hnixi — h the boundary -layer equations become linear and yield the same result as (141) 

 provided the variation of the Coriolis parameter is neglected over the width of the jet. 



