SECT. 3] DYNAMICS OF OCEAN CURRENTS 339 



The additional terms that arise from the interchange of the order of integration 

 and differentiation cancel out with the exception of the viscous terms. Here, 

 these additional terms and the variation of density are neglected in order to 

 retain a simple dissipative term. In view of the fact that eddy viscosity is a 

 relatively crude approximation to actual dissipative processes, it is not worth 

 keeping additional complicating terms. If these terms are neglected, the 

 boundary conditions at the surface and bottom of the ocean reduce to 



Tsx = iJ-v Su/dz \ TBx = /xt, dujdz } 



Tsy = jJiv CVJdz J TBy = }lv OUJdz J 



The boundary conditions on (50) depend on the assumptions made about the 

 exchange of water across the ocean surface. We may interpret Q as the rate at 

 which water is added to the ocean per unit area, i.e. the source Q is equal to 

 — piVs, where iVg is the vertical velocity at the surface. The integrals of pressure 

 gradients in (47) and (48) are expressed in terms of the potential energy by 

 transformations of the type 





^dz = ^ + PB— = ^ + E^^(^\ (53) 



dx dx dx dx g \ dx 



where {dPBlc'x)z is a component of pressure gradient along a level surface at 

 the bottom and ao{PB) is assumed to be equal to the specific volume at the 

 bottom, aB. 



We shall assume that the velocities near the bottom of the ocean are small 

 enough for tangential stresses on the bottom to be neglected. The flow near the 

 bottom will then be approximately geostrophic, i.e. 



fUB = -«^(^)^. /"B = «^(^)^- (5*) 



Consequently, we can interpret the terms containing the bottom pressure 

 gradients as transports, because 



Pbocb (^Pb\ Pb/vb 



g 



mr-"-r-^s>'-^'^- 



We have interpreted the barotropic velocity as being independent of depth and 

 equal to the deep-water velocity. By analogy, we can term the transport 

 components in (55) as the barotropic transports. Similarly, the baroclinic 

 components of velocity give rise to baroclinic transports which, if we neglect 

 non-linear terms and surface stresses, are given by : 



fV, = |. fU. = -| (56) 



The anomaly of potential energy, x, can be calculated from oceanographic 

 data. Therefore, it is of interest to interpret the relations in (56) more fully. 

 It can be seen from (56) that the baroclinic transport must be directed along 



