DYNAMICS OF THE SOUTHERN OCEAN 129 



proportional to the density p, so that 



1 dp 



the negative sign showing that the acceleration is directed towards the low pressure. 



If the expression is given in terms of the specific volume of the water v instead of 



density 



dp 



and \pt . ds = — \v. dp. 



The third integral in equation (2), \d t .ds, representing the influence of the earth's 



rotation, enters the expression for the change of relative circulation only, and since the 

 absolute and relative circulations are equally influenced by the other factors, gravity, 

 pressure and density differences, and friction, it follows that 



dC a dC r [ j j 



-df = -dT-} d '- ds 



and afterwards, by comparison with the differentiated form of equation (1) 



dC a dC r dS 



dt dt dt ' 



dS 

 dt' 



The last integral in equation (2), a measure of the effect of friction on the circulation, 

 has so far not been evaluated, the only practicable method being to use equation (2) 

 when all the other terms are known. To shorten the final form of the equation, however, 



the integral /< . ds has been written as — R, the negative sign showing that the friction 



has a retarding influence. 



Substituting the new values for the four integrals in equation (2) it is found that 



dC r f , dS 



that \d t . ds = — 2to 



— --j V .d P -2co^-R. 



The first term on the right-hand side of the equation is of particular importance since 

 it is a measure of the primary cause of the movements in the sea, the pressure and 

 density differences. The earth's rotation and friction represented by the last two terms 

 can only deform an existing current. 



For a closed curve in the sea v. dp is equal to the number of isobar-isostere tubes, 



tubes formed by the intersection of the isobaric and isosteric surfaces, that are enclosed 

 by the curve. Each of these tubes is known as a solenoid, and writing the number of 

 solenoids enclosed by the curve as A Bjerknes expresses his general theorem of 

 circulation 



