590 Basic Principles of the General Oceanic Circulation 



and taking into account that the divergence of the total mass transport is zero and 

 C <^ d, one obtains 



/dxdp dddp\ Idxdl dddr\ ^ ^ ^ 



and from the second equation 



fM, = Igd' ^ + gpd^^ (XVIIL23) 



Equating M^ in (XVIII.22) and (XVIII.23) gives 



/^ 1 8d\ 8C/Pd 8d\ I8p nSC\ 8p\ 8d _ 



\f~ ddy) dx-^ [jl" d^rpdx^ [ddy-^ -p dyjdd " ^- (^^111.24) 



In the case of a homogeneous ocean (p = const.), equation (XV1II.24) reduces to 



B 1 8d\8^ 1 8C 8d 



This equation states that in the case of a constant depth d only zonal steady currents 

 are possible, because the first term will vanish only when 8l,j8x = 0. When the 

 depth J is variable, all current directions are possible, if d satisfies certain conditions 

 according to (XVIII.24). If the depth d is a function only of y (the latitude), then, 

 provided that 8d/8y ^ 



B 1 8d 



This equation is identical with (XVI. 19) and states that for stationary currents the decline 

 of the lower depth d of the current system towards the poles must follow a law 

 d — K?.m 4>. In a stratified ocean (p = p{x,y,zy) the interrelationship is more compli- 

 cated. Equation (XVIII.24) shows, however, that for a constant depth t/ of no horizontal 

 motion, there can be no meridional mass transport due to frictionless currents, since 

 when d = const., the equation reduces to 



(XVIII.27) 



On substitution in equation (XVIII.23) it is found that M^ = 0. 



It has been shown above (p. 497) that in the Atlantic Ocean the zonal mean of the 

 depth of no meridional motion follows the above equation. This can be interpreted 

 to mean that the planetary vorticity {^My) is compensated by a corresponding balanced 

 topography of the lower boundary of the current system. This is frequently the case 

 in the South Atlantic, and here the westward intensification, which of course is a 

 consequence of the planetary vorticity effect, is only weakly developed. 



In criticism of Neumann's arguments, Stommel has questioned the assumption 

 that the depth f/ is a depth of no motion, and has pointed out that on the contrary, the 

 greatest vertical velocities occur at this depth. Neumann's equations can also be 

 derived from the basic assumption that the potential vorticity in the large-scale 



