DYNAMICS OF THE SOUTHERN OCEAN 131 



equal to the difference of the dynamic depths. In a study of the dynamics of the sea the 

 dynamic depths are therefore magnitudes of the first importance and they are usually 

 tabulated with the results. The unit of dynamic depth in most common use is the 

 dynamic metre which corresponds to 10 units of work or change of potential on the 

 metre-ton-second scale. The dynamic metre is approximately 2 per cent greater than 

 the common metre near the Equator and 5 per cent greater near the poles. 



The dynamic depths are usually obtained by evaluating the integral v .dp as accur- 

 ately as possible for each vertical series of temperature and salinity observations: the 

 mean specific volume of the water in each interval of depth is multiplied by the dif- 

 ference of pressure in the interval (approximately the same as the difference of depth 

 when decibars and metres are employed as units) and the values of v . dp so obtained are 

 summed downwards from the surface to each of the isobaric surfaces whose dynamic 

 depth is required. Several methods have been devised for shortening the process of 

 integration. Sandstrom and Helland-Hansen (1903, pp. 14-35), an< ^ Sverdrup (1933) 

 worked with anomalies of specific volume referred to a standard sea of uniform tem- 

 perature o° C. and salinity 35-00 °/ 00 , so that they had to deal with numbers of not 

 more than three figures instead of with five-figure numbers, and Hesselberg and 

 Sverdrup (191 5, pp. 1-17) showed that it was also easier to work with figures for 1 — v 

 instead of v. Further time and space can also be saved if only the anomalies of dynamic 

 depth are calculated and tabulated ; these will generally serve the same purposes as 

 the dynamic depths themselves and they can easily be converted to them if necessary. 



The closed curve consisting of two verticals with their upper and lower extremities 

 joined by isobaric lines is also very convenient for the calculation of the effect of the 

 earth's rotation. In all large ocean currents the vertical movement is small compared 

 with the horizontal movement and as a rule only the horizontal movements need be 

 considered. If the closed curve is placed at right angles to such a movement in which 

 the velocities in the upper and lower isobaric surfaces are u and u x , and the distance 

 between the two verticals L, then the area projected by the curve on a horizontal surface 

 after unit time has elapsed will be (w - u x ) L, and the rate of increase of the projection 

 of the curve on the plane of the Equator will be (u Q - Uj) L sin <£, where </> is the latitude. 

 Denoting the dynamic depths of the lower isobaric surface along the verticals a and b 

 by D a and D b , equation (4) becomes 



2cu sin 4> ( u o — u i) L = D a ~ lh, 



Da- D» I (fa 



or > *"* = -L--207W ( '" 



This formula which was developed by Helland-Hansen and Nansen in 1905 (Kriimmel, 

 191 1, 11, 502) allows the difference between the velocities of the current at two isobaric 

 surfaces to be calculated, and if the water at the lower isobaric surface can be regarded 

 as motionless, it will give the absolute velocity at the upper surface. Generally when the 

 formula is used the water is assumed to be motionless at great depths, and the deepest 

 isobaric surface for which observations are available is regarded as horizontal. 



