i30 DISCOVERY REPORTS 



So far it has been found impracticable to apply this general expression to actual 



oceanographical problems, and the difficulty of evaluating the terms -~ and R has 



limited the use of the theorem to current systems in which both of these magnitudes are 

 small enough to be neglected. Li such circumstances the general theorem becomes a 

 specialized theorem which implies that the solenoid field is balanced by the influence of 

 the earth's rotation, so that 



Zw dt = A (4). 



Generally the conditions under which -~ and R can be neglected will not be found in 



the sea, but the close agreement that has been found between the actual and theoretical 

 currents in many parts of the world — in the Gulf Stream (Wiist, 1924) and in the region 

 south of Newfoundland (Smith, 1925) for example — shows that the actual conditions 

 must often approximate to such conditions. 



THE APPLICATION OF BJERKNES' THEOREM TO 

 OCEANOGRAPHICAL PROBLEMS 



The most convenient form of closed curve that can be studied dynamically is one 

 which is built up of two verticals a and b with two isobaric lines along which the pres- 

 sures are p and p x joining their upper and lower extremities. Since the pressure 

 gradients along the isobaric lines are zero the number of solenoids enclosed by the 

 curve is 



- v a .dp+ v b .dp, 



J Pa J Po 



or, if v a and v„ are the average specific volumes of the water in the two vertical columns, 



v a (po~pi)- v b (j> -pj) (5). 



If the sea surface is used as the upper isobaric surface and the pressure of the atmo- 

 sphere is disregarded, the pressure p x at a depth /?, if the water has a mean specific 



h a 

 volume v, is -=-, the weight of a water column of unit cross section and depth //. 



Substituting analogous values for the pressure difference p — p 1 in equation (5) the 

 number of solenoids enclosed by the curve, when the upper isobaric surface coincides 

 with the sea surface, is given by the equation 



A = h a g - h b g, 

 where h a and h b are the depths of the lower isobaric surface along the two verticals. 



The two terms on the right-hand side of the equation represent the work which must 

 be done against gravity in raising unit masses from the lower isobaric surface to the 

 sea surface along the two verticals a and b, and such measures of work or change of 

 potential have been called by Bjerknes the dynamic depths of the lower isobaric surface 

 along the two verticals. Using this notation the number of solenoids enclosed by the 

 curve formed by the two verticals, the lower isobaric surface, and the sea surface, is 



