General Theory of Ocean Currents in a Homogeneous Sea 387 



(2) The depth shall be a function of the latitude only. Then dhldX = and the topo- 

 graphy of the sea bottom will be symmetrical about the poles. In that case, according 

 to (XIII. 12), there must be either dijdX = or chjcd + /? tangi^ = 0. The first condi- 

 tion leads again to zonal currents ; the second gives on integration h = H cos d where 

 H is the depth of the sea at the poles {d = 0°). 



In these cases both d^jdd and dl,jcX are free, that is, I, is also free. For a meridional 

 depth distribution of this type (decreasing gradually from a depth H at the poles to 

 a depth zero at the equator) steady currents would be possible in any direction also in 

 an ocean on the spherical Earth; conditions here are then the same as in a sea of 

 constant depth with planar co-ordinates. For this depth distribution both effects 

 balance exactly. It can therefore be deduced that in higher latitudes small changes in 

 depth will be able to compensate the effect of the curvature of the Earth, this effect 

 will therefore be small there. On the other hand, in lower latitudes larger changes in 

 depth will be required to balance this effect and therefore almost only zonal currents 

 will be possible. The critical vertical gradient in meridional direction which will be 

 able to balance the effect of the spherical shape of the Earth is given by {hIR) tan (^. 

 Table 116a gives these critical values for different latitudes and for depths of 3000 and 

 5000 m. 



The discussion of the above equation (Defant, 1929fl, p. 61) leads to an estimate of 

 the two effects on steady currents. Following Ekman (1923), these can be summarized 

 as follows : Up to 3-4° latitude — and when the changes in depth are small, even farther 

 away from the equator — the effect of the bottom relief is rather unimportant for the 

 tendency of the current to flow in zonal direction. Between 10° and 20° of latitude 

 the two effects are equal and in higher latitudes (> 40°) the effect of the bottom topo- 

 graphy gains in importance and the currents tend to follow definitely the isobaths of 

 the sea bottom. The observed fact that in reahty ocean currents do preferably follow 

 a zonal direction in lower latitudes and their direction in higher latitudes is presumably 

 more affected by the bottom topography, appears to be reasonably well explained by 

 the thf oretical results presented above, 



3. Eddy Viscosity (Turbulent Friction) in Ocean Currents 



(a) Mixing Length and Eddy Viscosity (Turbulent Frictional) Coefficient 



The movement of the water masses in ocean currents is mostly disordered and tur- 

 bulent and part of the strong variations in speed and direction of the flow which are 

 observed in quick-response recordings (see p. 347) can be attributed reasonably to this 

 internal turbulence. More or less large elements of water (water quanta) are continu- 

 ously being carried by these internal turbulent motions into the layers above, below or 



