General Theory of Ocean Currents in a Homogeneous Sea 423 



Using the equations previously derived to calculate 77 gives 



77 = 1 -03 vv^ for IV < 6 m/sec, 



and 77 = 4-3 u'^ for vv > 6 m/sec. 



The values calculated from these formulae are also to be regarded as only approximate 

 average values; the few directly determined values are widely scattering and indicate a 

 large dependence on the vertical stratification of the water masses, Schmidt (1917) 

 has presented some values : 



The high values for strong winds apply of course only for the especially intense tur- 

 bulence produced by the wind in the uppermost water layer; below this layer the co- 

 efficient decreases rapidly with depth. An average value for the top layer of the ocean 

 will be between 50 and 100. Its magnitude in the deep layers will be about 1-10, 



Diagrams of forces for a wind-driven, stratified ocean. With a complete knowledge 

 of the total current and pressure structure of the ocean diagrams of forces for any 

 layer can be derived in the following way (Defant, 1941 b). Denoting the sea surface 

 slopes (of the isobaric surfaces in the deeper layers) in the positive .v-direction (towards 

 east) with i^ and in the j-direction (towards north) with iy, then the equations of 

 motion for a variable 17 are of the form 



8 / 8u\ 8 / 8v\ 



fpv + gpi. + ^, [1 -^.j = 0; -fpu + gpiy+ ^ [r^ j^j = 0. (XIII.52) 



Integrating these equations from the surface to the depth D with the assumption that 

 the current falls to zero at a depth d and taking furthermore into account that for 

 z == the components of the wind stress are given by 



cu 8v 



and vanish when z = d, the following equations are obtained : 



f7v + g'pi'x+T, = and -f^u -{- gJTy + Ty = 0, (XIII.53) 



where the integrals (sums) down to the depth d are indicated by a bar. This states 

 merely that for a steady current the Coriolis force must be in equilibrium with the 

 sum of the total pressure force and the total wind stress exerted on the entire layer. 



The equations (XIII.53) can be evaluated numerically from the absolute topography 

 of the pressure surfaces and of the physical sea-level, as well as from the rather reliable 

 vertical current distribution as measured at two anchor stations in the region of the 

 South Equatorial Current in the Atlantic. Table 1 25 contains all the necessary numerical 

 values and Fig. 179 shows the vertical changes in current- and pressure-gradient 

 quantities for calculation of the integrals. It can be seen that the £'-component of 

 the velocity decreases regularly with depth, while the A^-component changes already 

 in the uppermost layers from small positive values to negative values and then falls 

 back to zero at 100 m. This distribution leads to a turn of the current vector cum sole 

 which must be the case in drift currents. Below this there is only a gradient current 



