334 Forces and their Relationship to the Structure of the Ocean 



In the three-dimensional case analogously 



dw 8v £u dw .. 8v du 



i = 



8y 



C 



Ox 



cy 



(X.59) 



If the velocity has a potential (see p. 325) the vorticity will vanish and the movement is 

 irrotational (vorticity-free potential current). 



y+by 



y - 



dF 



Xi-dX 



Fig. \36d. Rectangular surface element for the derivation of vorticity. 



The vorticity for polar co-ordinates can be derived in a similar way and it can be 

 assumed that the Earth and the co-ordinate system which is rigidly connected with it 

 rotate with constant angular velocity a>. The vorticity is then made up of the vorticity 

 of the rotating Earth and the relative vorticity of the water moving relative to the Earth. 

 To derive the vertical component Ca of the absolute vorticity it is necessary to consider 

 further a surface element 8F formed by the intersection of two latitude circles and two 

 meridians. If the latitudinal difference is d(f> and the longitudinal ^A, then the total 



area SF is 



8F = R^ cos cf> 8<f> SA. 



The zonal velocity along a latitude circle 4> is u = RQ cos ^, where i3 = tu + dXjdt. 

 However, along a meridian A the meridional velocity is v = R(8<f>l8t) and some 

 simple calculations give for the vertical component of the absolute vorticity 



S/^ 1 S2JL 1 ^ 



L = 



8C 



1 



8^ 



1 



8F cos </) 8X8t cos ^ 8(f) 



- [Q cos2 cf>]. 



(X.57a) 



For a small water column at rest relative to the Earth 8Xj8t = 8<f)l8t = 0, Q = oj 

 and the vertical component the vorticity t,E of the rotating Earth can be derived from 



Ca as 



^E - 2cu sin </.=/, (X.58a) 



thus equal to the Coriolis parameter. 



The relative vorticity c, of the water movement relative to the Earth (u zonal, positive 

 towards the east; v meridional, positive towards the north) is then 



L'-f = 



1 



8v 



1 



(m cos (f)). 



(X.59a) 



R cos ^ ^A R cos (f> 



For small oceanic areas in which the latitude can be regarded as approximately 

 constant, equation (X. 59a) reduces to 



'8v 



ta-f + 



8u\ 



8x cy' 



(X.60) 



