ATLANT. DEEP-SEA EXPED. 1910. VOL. i] PHYSICAL OCEANOGRAPHY AND METEOROLOGY 



97 



10 (p^ — pn) — m.a — q.l.a. The force per unit mass 

 (which is equal to tlie acceleration), is then : 



= 10 Pa^P2_ 



10 



Q1—Q2 



.D 



(c) 



q.l q.l 



It can be shown that the force of friction per unit 



mass is: 



/^=«L,^ + ^^\ (d) 



V ^z^ dz dz) 



« being the specific volume and /< the virtual coefficent 

 of viscosity. The friction usually acts against the current 

 but in a very different degree according to the variations 

 in /( and the vertical variations of velocity. The con- 

 ditions depend greatly upon the turbulence. At present 

 it is difficult to take the frictional force into account in 

 numerical calculations. In an e.xact treatment it should 

 be combined with the other physical forces into a resultant 

 force. We have reason to believe, however, that the 

 friction is of small importance in comparison with the 

 other forces; so in most cases it may be left out of 

 account, and we shall neglect the friction in our dis- 

 cussion here. 



Along a level surface the only force to be regarded 

 is, then, the force due to differences in pressure. Under 

 stationary conditions this force must have the same value 

 as the Coriolean force. From equations (a) and (c) we 

 obtain: 



Vd 



P1—P2 



-Pi 



2 w. q. L. sin <p 



q.L 



— cm. /sec. 



(e) 



where v'd is the average component of velocity relatively 

 to the surface water (see below), the component being 

 reckoned at a right angle to the line between the stations 

 1 and 2 at the level surface at the dynamic depth D. 

 Pi andpo denote the pressure at the level surface at the two 

 stations, L the distance between the stations in kilometres 

 and If the mean latitude, q is the mean density (com- 

 pression included) along the level surface between the 

 two stations, as mentioned above. 



In a similar manner we find, by means of equations 

 (a) and {b), the following expression for the component 

 of the relative velocity normal to the line between the 

 stations 1 and 2 in an isobaric surface where the pressure 

 \s =. p: 



Dx-D2 ^ ^ Dx-O, 

 2 w. L. sin tp L 



Vu = 



cm./sec. (f) 



We have assumed that the pressure at the level D 

 is greater at Stat. 1 than at Stat. 2. The force due to 

 pressure is, therefore, directed from 1 to 2. The distance 

 in dynamic metres from the surface of the sea to the 



isobaric surface representing a pressure --^ p^ is greater 

 at Stat. 2 than at Stat. 1, /. e. D^ -c D^- The force of 

 gravity is directed from the smaller to the greater dyna- 

 mic depth, which here means from Stat. 1 to Stat. 2 at 

 the depths in question. With (D, — D^) in the numer- 

 ator, therefore, we must put a minus before the whole 

 expression in equation (/). 



v'd and v'p are components of the velocities — re- 

 lative to the surface water — in a direction at a right 

 angle cum sole to the line from Stat. 1 to Stat. 2. They 

 tell us nothing regarding the real components of the cur- 

 rents but only that they differ, by the amounts v'd and 

 v'p in the direction mentioned, from the velocity com- 

 ponent at the sea surface. If the component of the ac- 

 tual current is at the level surface for D dynamic metres 

 the actual current at the surface of the sea has a com- 

 ponent = — v' j^, i.e. W cum sole from the direction 

 from Stat. 2 to Stat. 1. In general we have: 



— v'd =Co — CD, 



and — v'p = c^ — Cp, 



where the real components of the actual currents are c^ at 

 the surface of the sea, cd at the level surface for D 

 dynamic metres and Cp at the isobaric surface for p de- 

 cibars, the components being taken in a direction 90° 

 cum sole from the direction from Stat. 2 to Stat. 1. The 

 pure wind-current in the uppermost water-layers is then 

 disregarded. 



The results arrived at here in an elementary fashion 

 agree perfectly with those which may be found by start- 

 ing from the well-known circulation theorem formulated 

 by Professor V. Bjerknes [1898, 1901]. The equation 

 developed by Bjerknes makes it possible to calculate the 

 acceleration of circulation when the distribution of mass 

 and velocity in the ocean is known. Sandstrbm and the 

 author [1902] transformed Bjerknes's equation into a shape 

 which was more convenient for such a calculation. The 

 next step was taken by the author on the supposition 

 that the accelerations of circulation in the sea may be 

 neglected. Practically no error is introduced by taking 

 the acceleration of circulation as equal to nil. Thus a 

 formula was arrived at which is identical with our formula 

 (f) above [Helland-Hansen, 1905; cf. KrOmmel, 1911, 

 p. 502], 



The formulae developed above do not a priori assert 

 anything about the causality. They only show the con- 

 nection between a solenoidal field and the velocities of 

 the current. Such a field may not only be the cause 

 of a current but also the effect of it. As to the latter, 

 we have an example in the effect of the winds mentioned 

 above. We may also, as an example, consider the con- 

 ditions in the Baltic current. The excess of water from 



