General Theory of Ocean Currents in a Homogeneous Sea 409 



between the current and the gradient direction becomes smaller and smaller as the 

 sea becomes shallower; the effect of the Earth's rotation then becomes less important 

 than that of friction. 



Other assumptions about friction. The Ekman theory assumes a constant frictional 

 coefficient. It has been used in this form in meteorology and provides an unobjec- 

 tionable explanation of the deflection of the wind direction to the right with increasing 

 height. However, it was found that the lowermost layers of the wind structure follow 

 different laws. These deviations can be attributed mainly to the assumption of a con- 

 stant frictional coefficient in the bottom layers being no longer valid. This fact 

 Ekman (1928) has taken into account by assuming in agreement with the observations 

 a current structure made up of a straight section OA, at A changing into a logarithmic 

 spiral over AB (Fig. 175). Thereby OB is thus the geostrophic wind in higher altitude. 

 The same conditions as for the surface wind must also apply to the oceanic bottom 



Fig. 175. Vertical structure in a bottom current with a boundary layer above the bottom 



(according to Ekman). 



current, and it is already known from current measurements in moving waters and 

 from laboratory experiments that the vertical structure in these, apart from the devia- 

 tion due to the Coriolis force, is somewhat different from that of the Ekman spiral. 

 The velocity curve of Fig. 175 can therefore only be given a physical m.eaning by 

 assuming the presence of a boundary layer just above the bottom in which the velocity 

 changes approximately linearly, and without change in direction from zero at the 

 bottom to the value OA = Vg at its upper limit. The water mass present above this 

 lower boundary layer flows as though gliding over the bottom ; it is retarded only by 

 the slowly moving boundary layer. Ekman assumed a constant frictional coefficient 

 in each of the two layers and investigated the thickness of the boundary layer, the 

 decrease in velocity in it and the angle of deflection which would be able to prove the 

 validity of such a concept. 



This concept can more or less accommodate the fact that the lowermost layer just 

 above the bottom has a special status, and that in practice the assumption of a constant 

 turbulent coefficient in the water masses above is quite justified. Modern hydrodynamic 

 fluid research approaches the whole problem from the point of view that the variation 

 of the frictional coefficient with distance from the solid underlaying surface changes 

 with its roughness, whereby the entire current structure takes on a different form. The 

 Prandtl theory (see especially 1942, p. 318) starts with the components of the shearing 

 stress T^ and Ty at the bottom. Taking r-positive upwards, furthermore a variable 17, 

 a pressure gradient in the direction of the positive j-axis and taking into account 

 equation (XIII. 13), then equations (XIII. 28) can be transformed into 



