Forces and their Relationship to the Structure of the Ocean 



335 



The vertical component of the absolute vorticity is thus always equal to the sum of 

 the relative vorticity (vertical component) and the Coriolis parameter, 



(c) Vorticity and the Equations of Motion; Potential Vorticity 



Starting from the horizontal equations of motion (without frictional effects), 

 equation (X. 16) gives 



du 

 ~dt 



-fv 



1 dp 

 p dx' 



8v \ cp 



ot -^ p cy 



(X.6]) 



Taking as a first approximation that p is independent of x and y or assuming baro- 

 tropic conditions so that p — p(p) (a function of pressure only) then, by cross-wise 

 differentiation of these equations and subtraction and simple calculation considering 

 dfjct = gives 



^^^^ + a+ f) divH r = 0; ia=i-\-f (X.62) 



This is the relative vorticity theorem of Rossby (1939); it is used for the analysis of 

 stream fields in steady currents and for the analysis of moving oceanic waves. 

 The total change in the Coriolis parameter with time is 



d4> 



d<f> 



-,- = Zoj COS 6 —r and smce y = — -. . 

 dt ^ dt R dt 



The theorem of relative vorticity then takes the form 



df 

 dt 



2(jo cos (f) 2a) cos (f) 



V = pv with /3 = 



(X.63) 



(X. 64) 



R " ''^ '' ' R 



If the horizontal current (m, v) is non-divergent then equation (X. 62) reduces to 



i-^"- 



(X.65) 



The quantity /3 = cfjcy is called the '' Rossby parameter'' and represents the meridional 

 change in the Coriolis parameter (change with latitude). It is positive in both hemis- 

 pheres so that the relative vorticity always increases when small elements move 

 southward and decreases when they move northward. 



The value of /3 at different latitudes is shown in the following Table 113. 



Table 113. 



10" ^ [cm-i sec-i] = 



90° 

 00 



75° ; 60' 45° [ 30° 



I i 



0-593 1 1145 1-619 1-983 



15° 0° 

 2-212 2-290 



In theoretical practice ^ is usually taken as a constant, that is, as independent of j\ 

 This approximation is more or less justified near the equator where /S is a maximum 



