128 DISCOVERY REPORTS 



BJERKNES' THEOREM OF OCEANIC CIRCULATION 



In order to find the relation between the pressure and density distribution in the sea 

 and the water movements Bjerknes has studied the dynamics of a closed curve composed 

 of water particles in the sea. In general the velocity of each particle will have a component 

 along the curve and Bjerknes has called the sum of these components the circulation of 

 the curve. If the tangential velocity along an element ds of the curve is given as v t , then 

 the circulation C a is expressed by the integral formula 



C a = \v t . ds, 



or if the velocity is given relative to the rotating earth as u t the relative circulation C r is 

 given by the analogous formula 



C r = in, .ds. 



Between the absolute and relative circulations Bjerknes has deduced the simple relation 



C a =C r +2wS (i), 



where to is the angular velocity of the earth, and S is the area of the projection of the 

 closed curve on the plane of the Equator. 



In the deduction of his general theorem Bjerknes has calculated the change of the 



relative circulation with time —rf on the assumption that the water movements are 



affected only by gravity, the distribution of pressure and density, the earth's rotation, 

 and friction. If the component of acceleration along an element of the curve is u t , 



then —jy , usually known as the acceleration of the circulation, is given by the equation 



dC r f. j 



-dT = i Ut - ds ' 



or if ii t is expressed as the sum of a series of vectors which represent the tangential com- 

 ponents of the accelerations due to each of the factors mentioned above on the water 

 particles 



-~ = J g, . ds + | p, . ds + | d, . ds + | /, . ds (2). 



The first vector represents the tangential component of the acceleration due to 

 gravity, g, . ds is therefore the work which must be done to move a unit mass along the 



element ds in opposition to the force of gravity, and \g t .ds is that which would be done 



if a unit mass were moved round the whole of the closed curve. Since, however, such 

 a cycle would bring the unit mass back to the point from which it started the total work 



done, \gt-ds, is zero. 



The second vector in equation (2),p t , is the tangential acceleration due to the pressure 



gradient. It will be directly proportional to the gradient -±- along the curve and inversely 



