406 General Theory of Ocean Currents in a Homogeneous Sea 



of the previous case of "adhering" ("Haften") at the sea bottom. For a small thickness 

 an almost geostrophic current is obtained. As the thickness of the layer increases, 

 the structure of the current will of course approach that of the Ekman spiral. 



(c) Pure Gradient Currents 



Drift currents in normal form are seldom found to occur in the sea, since the water 

 transport connected with such currents will give rise to piling up of water at coast 

 lines ("Anstau") leading to inclination of the sea surface. In a homogeneous sea the 

 pressure differences produced in this way would extend their influence down to the 

 sea bottom; if there were no frictional effects a geostrophic current would be generated 

 from the sea surface down to the sea bottom. However, friction at the bottom gives 

 rise to disturbances which are of considerable importance for oceanic currents. 



The equation of motion (X.16) for a steady current will be of the form 



\ dp 7] 8^u ^ , ^ \ dp 7] 8^v 



fv- - -/ + i — = and - fu - - -^ + - 5-0 = 0. (XIII.28) 



p ox p oz^ p oy p cz^ 



Replacing the pressure gradient by the slope ^ of the sea surface (equation (XIII.2), 

 p. 383) and assuming that there is a pressure gradient only along the j^-axis 

 (dpidx) = 0, then, according to (XIII.5), the geostrophic current will flow in the 

 direction of the positive x-axis and its velocity will be 



g S^ 



Considering this in the equations (XIII.28) they can be compressed in the same way as 

 for a drift current into 



- gZ-2 («+'■") - '/(" + 'i') + //t/ = 0. (XIII.30) 



To this equation add the following boundary conditions: 



(1) no wind at the sea surface, that is 



Su cv ^ 

 forz = 0: =-=0 



cz oz 



and 



(2) at the sea bottom "adhering" occurs ("Haften") 



for z = d: u — v = 0, 



The solution given by Ekman for (XIII.30) is 



cosh(l - i){7rlD)z 



u -\- iv = U 



^ cosh{\ +i){7T I D)z 



^U(l-<f>'h i>P), 



whereby 



cosh (7rlD)(d + z) cos (-^iDXd + z) + cosh (nlD)id — z) cos (nlOXd — z), 



<f> = 



cosh 27T(dlD) + cos 27r(^/Z)) 



(XIII.31) 



