332 Forces and their Relationship to the Structure of the Ocean 



the left relative to the isobars and an anticyclonic circulation will be present when 

 i8 < 90° and so the isosteres decline to the right. 



The circulation theorem gives the change in absolute circulation C, i.e. the circula- 

 tion referred to a co-ordinate system at rest. For oceanographic problems, however, 

 it is the change in the circulation relative to the Earth which is of interest. The abso- 

 lute velocity Va referred to a fictitious Earth at rest can always be represented as the 

 sum of the relative velocity Vr relative to the rotating Earth and the velocity V^ of 

 Earth rotation. Thus in the direction of the tangent / to the curve s 



Va,t = Vr,t + n., 



and thus 



C, = Cr-\- Ce. (X.51) 



The circulation Ce can be calculated. If the curve s lies in the equatorial plane then the 

 velocity Vg for each point on the curve will be cor where r is its distance from the Earth 

 centre. The component of it coinciding with the direction of the tangent / to the curve 

 s will be given by 



Ve, t= rw cos P, 



where )S is the angle between the tangents to the circle r and to the curve s. Thus 



Ce, t = 60 r cos ^ ds =^ 2co \ r cos P ds = 2io F, (X.52) 



where Fis the area enclosed by the curve s. If the curve s does not lie in the equatorial 

 plane it can be resolved into its projections on the equatorial plane and on the meri- 

 dional plane. Since the velocity V^ is perpendicular to the meridional plane it will have 

 no component in the direction of the tangent to the projection of the curve on the 

 meridional plane and its contribution to Cg.t will therefore be zero. The contribution 

 of the projection of the curve on the equatorial plane is identical with equation 

 (X. 52); F is now the area within the projection of the curve s on the equatorial 

 plane. Thus for the relative circulation acceleration is obtained 



dCr ^ dF .,, ^^^ 



-^ = N - 2aj -J-. (X.53) 



dt dt ^ ' 



As a first approximation, if the area is not too large, the latitude ^ is assumed constant 

 and Fcan be put equal to F^ sin <j>,'\ where F^ is the area within the projection of the 

 curve s on the sea surface. Thus 



^ = TV - 2co sin ^. (X.54) 



The acceleration is made up of two terms; the first is the number A^ of solenoids en- 

 closed by the curve and acts always in the direction from the ascendent dajdn to the 



t More exactly 



dF dF„ . , ^ dtp dFm . , ^ V 



-^- = -^- sm (f + Fm cos f ^1= ^i sin (p + Fm cos (p ^. 



Here v is the south-north velocity; in middle and higher latitudes the second term is insignificant but 

 towards the equator the first vanishes and the second becomes important. 



