SECT. 3] DYNAMICS OF OCEAN CUBRENTS 373 



For the vertical velocity to be independent of x, we must choose x of the form 

 A{z) + B{z)x. The dependence on y will enter parametrically into the functions 

 A{z) and B{z). A simple particular solution of the linear form in x is 



X = 4pokvfHxo-x)l^z, (175) 



where xo is an arbitrary constant. As the zonal component of velocity is zero 

 at x = xo, we can interpret xq as the position of a meridional boundary. By 

 differentiating (175), we obtain the vertical velocity 



-=/4|=-^^ (176, 



and the density anomaly 



Pb-R 



1 d^x _ 8pokvfHxo-x)^ ^^^^ 



g dz^ g^z^ 



As z is negative in the interior of the ocean, a stable solution, p^ > p, occurs 

 only for x > xq. Hence, xq must represent a western boundary. 

 The horizontal velocities represented by (175) are 



The meridional flow is uniform zonally and increases with latitude. The zonal 

 flow is westward and independent of y. Stream-lines at each level form a 

 family of hyperbolae in the x-y plane with the flow curving in the anticyclonic 

 sense. Although this simple solution is not of a sufficiently general form to be 

 applied to a real ocean, it has important theoretical implications. In particular, 

 no solutions of (174) exist in which the flow is towards the equator. Even in the 

 unstable case, x < xo, the flow is poleward. We conclude, therefore, that flow 

 towards the equator cannot exist as a purely baroclinic mode. 



The vertical velocity (176) is independent of x and y as assumed. It is pro- 

 portional to the eddy diflfusivity, kv, and increases towards the surface where 

 the Ekman transport must be divergent {we > 0) and the surface source, Q* 

 positive. We can eliminate ky between (176) and (177) to obtain 



Substitution of the representative values: w = 4:X 10"^^ cm sec~i, p^ — p=10~3 

 g cm-3, a;-a:o = 4000 km, g=10^ cm sec-2, /= 10-4 sec-i and /S = 2xl0-i3 

 cm-i sec-i, into (180) yields a depth of 400 m. As the density anomaly decreases 

 as z~^, the ocean will be virtually homogeneous for depths of the order of 

 1000 m or more. 



The simple introductory model that we have considered bears little re- 

 semblance to the density structure and circulation in any real ocean. In order 



