Forces and their Relationship to the Structure of the Ocean 333 



pressure gradient dpjdn (Fig. 136c); the second represents the product of the CorioUs 

 parameter with the change in time of the projection on the sea surface of the area en- 

 closed by the curve. This term gives rise to a cyclonic circulation for a decrease in the 

 area. 



If the vertical stratification of the sea is autobarotropic (see p. 308) then N = and 

 a change of the circulation with time can only be caused by the effect of the Earth's 

 rotation. If a small horizontal layer of water (area F) moves polewards, its projection 

 on the equatorial plane F^ will increase. If N — there will be an acceleration in anti- 

 cyclonic circulation according to equation (X. 54). If, on the other hand, it moves to- 

 wards the equator it will be subject to a cyclonic circulation acceleration. The Bjerknes 

 circulation theorem shows clearly the importance of the baroclinic stratification of the 

 sea for the dynamics of ocean currents. For application see Chap. XV, 5. 



(b) Vorticity for an Earth at Rest and for a Rotating Earth 



A further important quantity in the dynamics of ocean currents is the vortichy. 

 The horizontal area F enclosed by the curve s can be divided by two arbitrary sets of 

 curves into a large number of very small surface elements 8F. It can readily be seen 

 that the sum of all the circulations SC, in the same direction along the boundaries 

 of these surface elements 8F, is equal to the circulation along the outer boundary s 

 around the entire area F. 



Thus 



c = £ac. 



The limiting value of the ratio SC/SF is termed the vorticity and is denoted by ^. 

 It is thus given by 



C = Hm 1^. (X.55) 



The vorticity is thus the circulation around a horizontal surface unit and therefore 



C = (h {u dx -}- V dy) = 



idxdy = \ t 8F. (X.56) 



F 



The circulation around a closed curve s is equal to the integral of the vorticity over 

 the surface F enclosed by the curve s (Stokes's law). This is the two-dimensional case 

 and C is thus only the vertical component of the total three-dimensional vorticity 

 vector (curl V). 



For a horizontal surface element 8j.8y (see Fig. \36d), along the boundary (in a 

 positive sense of rotation) of a horizontal surface element 



8C — u dx -i- \v -\- 

 and from (X. 55) 



8v 

 dx 



\ / Su \ (8v du\ 



8,j - |„ + _ Syj - „ S^ = (- - -j 8.V Sy (X.57) 



' 8v du\ 



