336 Forces and their Relationship to the Structure of the Ocean 



and its change with latitude amounts to only a few per cent. In higher latitudes, how- 

 ever, taking ^ as constant is only a rough assumption since between 45° and 60° 

 the increase in ^ is about 29%. 



If the current is non-divergent then from equation (X, 62) it follows that 



|(^+/) = 0, ^« = ^+/= const. (X.66) 



In a non-divergent, barotropic current the vertical component of the absolute vorticity 

 is constant and the change in the relative vorticity must be compensated by a corre- 

 sponding displacement in latitude. 



To use the vorticity equation for a water mass of thickness h which is variable with 

 both time and space, it is necessary to take the continuity equation for the water layer 

 // into account in addition to (X. 62). For a horizontal current (w, v) it is easy to show 

 that the continuity equation must have the form 



dh dhu dhv ^ dh , ^. ^ ,^^ ^_. 



^+-^-f-p=0 or -w;-h divn v = 0. (X.67) 



dt ex dy ot 



Combined with (X. 62) this gives 



It is obvious that the relative vorticity now is variable not only with latitude, but also 

 with the thickness of the water layer under consideration. The value it, + /)/// is thus 

 invariable for a given water mass; it is termed tht potential vorticity. 



