112 OCEAN CURRENTS RELATED TO THE DISTRIBUTION OF MASS 



agreement has been obtained. Thus, in spite of deficiencies of the 

 method the computation of currents is a most useful tool in oceanography. 

 Bjerknes' Theorem of Circulation. The general formula for 

 computing ocean currents from the slope of the isobaric surfaces, v = 

 —gi/\, was derived by H. Mohn in 1885, but at the time when Mohn 

 presented his theory the oceanographic observations were not sufficientl}^ 

 accurate for computation of the relative field of pressure. Owing to 

 these circumstances and others depending on certain characteristics of 

 the theory, Mohn's formula received no attention. The corresponding 

 formula for computation of currents associated with the relative distri- 

 bution of pressure, 



v,-v,=^ 10 ^^-^^^ (VI, 27) 



was derived independently by Helland-Hansen from V. Bjerknes' 

 theorem of circulation. 



Bjerknes makes use of the term ''circulation along a closed curve." 

 Consider a closed curve that is formed by moving particles of fluid. The 

 velocity of each particle has the component Vt tangential to the curve c, 

 and the integral of all these components along the entire curve represents 

 the circulation along the curve \ C = ^c Vrds. 



The time change of the circulation, if friction is neglected, can easily 

 be found from the equations of motion, because 



C -= j^ Vrds = £ {vjx + Vydy + v^dz). (VI, 28) 



Consider first conditions in a coordinate system that is at rest and 

 assume that gravity is the only external force. The integral of the 

 component of gravity along a closed curve is always zero, because it 

 represents the work performed against gravity when moving a particle 

 along a certain path back to the starting point. There remains therefore 

 only 



C = - f^adp = N. (VI, 29) 



The integral on the right-hand side is zero only if the specific volume is 

 constant along the curve or is a function of pressure only. In these 

 cases no internal field of force exists, and the theorem states that the 

 circulation along a closed curve is constant if the fluid is homogeneous or 

 if isosteric surfaces coincide with isobaric surfaces. If the isosteric 

 surfaces cut the isobaric surfaces, the space can be considered as filled 

 by tubes, the walls of which are formed by isosteric and isobaric surfaces. 

 If these are entered with unit difference in specific volume and pressure, 

 respectively, the tubes are called solenoids. It can be shown that the 

 integral in equation (VI, 29) is equal to the number of solenoids, iV, 

 enclosed by the curve. Consider now a curve that runs vertically down 



