SECT. 3] DYNAMICS OF OCEAN OUKRENTS 371 



The geostrophic equations for baroclinic flow expressed in terms of x a-re 



and the density anomaly is 



1 d^x 

 PB-P = -^- (169) 



Substitution of (167) and (168) into the continuity equation (28) and integra- 

 tion with respect to z yields 



.-=/.!• (170) 



Hence, by substituting for the density and velocity components in (159), we 

 obtain the steady-state equation 



_iJXJ!]^A.3LJiL^l?x^^. V2!!x , . ^ ,171^ 



/ By dz 8x 8z^ f dx 8z 8y 8z^ P 8x 8z^ ^ " 8z^ ^^ Sz* ^ ' ' 



for the potential energy anomaly. A measure of simplicity is achieved by 

 assuming pkn and pkv to be constant. 



The boundary conditions on x cannot be applied at the surface of the ocean 

 because the flow is not geostrophic near the surface in the Ekman frictional 

 layer and is, therefore, not represented by (171). However, we can consider 

 the upper boundary to be at a level surface below the Ekman boundary layer. 

 Along this surface, we assume the vertical velocity and the diffusive flux, or 

 the density anomaly, are given. These conditions can be expressed as 



p^-P'^' = -8^ 8^ ^^^^^ 



The boundary conditions in the deep water are satisfled if we require x and its 

 derivatives with respect to z to approach zero near the bottom. For convenience 

 in handling the bottom-boundary conditions, we can assume the ocean to be 

 infinitely deep provided we consider only baroclinic flow. We can then require 

 X to approach zero asymptotically for large values of z in place of the bottom 

 conditions. As we have neglected both viscous and acceleration terms in 

 deriving (171), we cannot satisfy lateral boundary conditions except in special 

 cases for which one of the horizontal velocity components is zero. In the more 

 general case, we must assume that the potential energy function is known for 

 flow entering the region under consideration. 



In spite of the formidable non-linearity of (171), a start has been made in 

 examining certain classes of approximate solutions. Welander (1959) studied 



