Ocean Currents in a Non-homogeneous Ocean 489 



A horizontal circulation free-curve (Cq = 0) will acquire by contraction a cyclonic 

 circulation and by expansion an anticyclonic circulation.* 



If curves extending as parallel circles all around the Earth and containing an ocean 

 covering the entire Earth are carried towards the equator by the general oceanic 

 circulation, then their projection on the equatorial plane will expand and they will 

 thus acquire a zonal anticyclonic circulation, that is, from east to west. On the other 

 hand, if they are displaced towards the poles there will be a shrinking of the areas 

 enclosed within the parallels and thus there will be a zonal cyclonic movement from 

 west to east. Considerable changes can also occur in the area enclosed by horizontal 

 curves flowing over a submarine ridge thereby causing the formation of cyclonic or 

 anticyclonic circulations. These will be superimposed on the basic current and will 

 give rise to a wave-form character of the current structure (see p. 431). 



(b) The Sandstrom Theorem 



In the ocean there exist closed circulations of greater or smaller extent, which are 

 maintained by the continuous supply of heat at certain fixed places and the continuous 

 withdrawal of heat at others. These sources of heat and cold maintain the differences 

 in specific volume. Thus circulation velocity in a frictionless medium will continuously 

 increase, since the circulation acceleration in equation (X.44) has a positive value. In 

 reahty, however, all circulations are affected by frictional forces. Another term R 

 must therefore be added to equation (X.44) containing all the frictional effects. There 

 will be a steady state only when 



- I adp-\- R^Q (XV.14) 



that is, in a steady state (disregarding the rotation of the Earth) the work done by the 

 pressure forces (i.e. — /« « dp) is used exclusively in overcoming the frictional forces. 

 This can only be the case when 



adp<0. (XV. 15) 



From this controlling equation it is easy to draw conclusions as to how the sources of 

 heat and cold should be located in space inside the circulation in order to allow for a 

 stationary state. The concept of sources of heat and cold must be given in the ocean a 



* It follows from equation (X.54) dCjdt = fcIF^dt that for an increase in the area dFJdt > 

 there will be an anticyclonic deflection and correspondingly for a decrease a cyclonic deflection. 

 This can, of course, also be derived directly from the equations of motion. For a geostrophic friction- 

 less current these are 



\ cp \ cp 



— /f = ^— and + /m = 7— . 



p (}x p cy 



By cross-wise differentiation and rearrangement 



df 



/div vh + Pv = 0, whereby j3 = -r . 



If the current is divergent (div r^ < 0) y must be positive; this indicates that the deflection will be 

 anticyclonic or to the right in the Northern Hemisphere ; for a convergence (div vh > 0) u is negative 

 with a corresponding cyclonic turn. 



