General Theory of Ocean Currents in a Homogeneous Sea 



411 



(current amount) which will lead to the same value for (U — u) dz. From this a 



relationship between h^ and n is obtained having the form 



h 

 log — = («+l)loge. 



(XIII.39) 



Putting the expressions for T^ equal in (XIII.35 and 37) gives a further relationship 

 between h^ and U 



A, = 0-160 4?^'^. (XIII.40) 



n{n + 1) / 



This relation shows that h-^ is directly proportional to U as was to be expected. With 

 this all the unknowns are determined. 



Numerical values can be obtained in the following way : for a given value of n, 

 which according to equation (XIII.35) fixes the angle a, and for given latitudes 4> and 

 velocities U, the equation (XIII.40) allows to compute the related h^ and (XIII.39) 

 gives the value for Cq. From Cq the roughness height k can be found quite simply and 

 finally (XIII. 37) gives the value of T^. This then fixes the current structure completely. 



Table 118 presents corresponding values for different roughness values of the sea 

 bottom as they could be expected to occur in reahty. These values are valid for ^=50° 



Table 118. Basic values for the structure of the bottom current 

 {according to the Prandtl theory); 0=50°, C/=100 cm/sec 



(/= 1-016 X 10~^ sec~^) and for [/= 100 cm/sec. The three roughness values correspond 

 to average conditions. The frictional depths are obtained in a row as 174, 94 and 76 m 

 which are plausible values. The vertical velocity distribution of u and v is shown in 

 Fig. 176 for the first case (« = 5, /?=174m) together with a vectorial representation 

 (uv); for comparison with the values given by the Ekman theory the corresponding 

 curves are shown by the dotted lines. The greatest differences, as would be expected, 

 appear in the immediate vicinity of the bottom; up to about 5 m from the bottom the 

 velocity increases linearly with distance from the bottom as was assumed by Ekman 

 to be the case inside his boundary layer. 



Similar considerations also apply for the drift current caused by the wind. Here U 

 must be zero in equation (XIII. 37) and in addition the boundary condition at the 

 surface (~=h) must be 



