372 FOFONOFF [sect. 3 



(171) for purely convective flow {kH = kv = 0). In the absence of diffusion, the 

 equation admits arbitrary zonal flows in which both the vertical and meridional 

 components of velocity are zero, i.e. x ^ function of y and z only. Welander 

 then assumed x to be of the more general form P{x,y)Q[zlF{y)] and found that 

 solutions existed for arbitrary P{x,y) \iQ{zjF{y)] was of the form e^f^^lf, where 

 k is an arbitrary constant. For positive values of k, the convective circulation 

 dies away exponentially with depth. The penetration of the circulation is least 

 at low latitudes and increases polewards. The arbitrary constant k can be 

 estimated by dimensional analysis or chosen to agree with the observed decrease 

 of the density anomaly with depth at a given latitude. Welander showed that 

 if the observed density distribution at the ocean surface is used to define 

 P{x,y), the density field derived from the theoretical solution had features in 

 common with the observed density field. In particular, surfaces of constant 

 density were deepest at mid-latitudes. 



For the purely convective flows studied by Welander, the density must 

 remain constant along a stream-line of the flow. We can obtain flows of this type 

 that are not uniform zonally provided we do not insist that the vertical velocity 

 at the ocean surface be zero. Instead we apply the upper boundary condition 

 on the vertical velocity along a surface below the Ekman frictional layer and 

 assume that the flow across the surface is absorbed by the divergence of the 

 Ekman transport. Similarly, we assume that the flux of heat and salt into 

 the Ekman layer is balanced by surface sources. Hence, both the divergence of 

 Ekman transport and the surface sources are implicitly specified by the choice 

 of interior solutions of (171). The more direct approach in which we assume that 

 the wind stress and surface sources are given and then look for an interior 

 solution is considerably more difficult to carry out. As yet, no solutions using 

 the direct approach have been obtained. 



Before examining the model of the oceanic thermocline proposed by Robinson 

 and Stommel (1959), we shall consider a simpler baroclinic model to introduce 

 some of the features of convective flow in the presence of diffusion. We con- 

 cluded that in the special case, (152), in which vertical flow is independent of a; 

 and y and the zonal width of the ocean varies as 1//^, the flow will take place 

 in a meridional plane without inducing a zonal velocity component. A western- 

 boundary current in this special ocean either transports a constant volume of 

 water along the boundary or is entirely absent. We shall now see if these 

 results are consistent with (171) for purely baroclinic flow. 



Because the Coriolis parameter is present in the coefficients of the non- 

 linear terms of (171), we cannot assume that x is entirely independent of y. 

 However, we assume that x varies only slowly with y so that the zonal velocity 

 is small compared with the meridional velocity, i.e. that derivatives with 

 respect to y may be neglected in (171). We assume further that lateral diffusion 

 is negligible in comparison with vertical diffusion. Hence, we reduce (171) to 



