146 DISCOVERY REPORTS 



Assuming, furthermore, that the wind blows in the direction of the channel, the 

 boundary condition at the surface takes the form 



where T represents the tangential stress of the wind. 



Conditions can be stationary only when no transport takes place across the channel, 

 or when ?;^ = o at all levels. From equations (i) and (3) it follows that this condition is 

 fulfilled when ^^ 



V -r- = — T = constant. 

 dz 



Since the coefficient v is always positive, it follows that the velocity must decrease at all 

 depths, and since it is improbable that v increases with depth it also follows that the 

 velocity decreases more and more rapidly with increasing depth. Furthermore, the 

 pressure gradient and the velocity must be zero at the bottom. If the velocity at the 

 bottom differs from zero the influence of the friction at the bottom must give rise to a 

 transversal current, in which case the original condition that v^ shall be zero at all levels 

 cannot be fulfilled. 



The solution which gives stationary conditions can be written in the following form 

 if h is the depth to the bottom : 



~"E=-^^^^' ^ = ''''/(-) = °' -'^1 = °' 



2a>sm<^-' 



dVy _ _ rp 



dz 



The condition that the pressure gradient shall be zero at the bottom leads, as already 

 stated, to the conclusion that a slope current cannot exist, because within the slope 

 current the gradient remains unchanged from the surface to the bottom. 



It can easily be shown that such currents, as the above solution demonstrates, are not 

 met with in the sea. In order to do this it is necessary to introduce some function which 

 shows the relationship between the tangential stress of the wind and the wind velocity. 

 According to Ekman's and Taylor's investigations we have approximately 



T= 3-2 X 10-6 (IF- Vf, 



where W is the velocity of the wind and V the surface velocity of the water. The latter 

 is usually so small compared with the former that it can be disregarded, but in the 

 present case it has to be considered. Furthermore, it should be noted that with given 

 values of T, and .'o the coeflicient of eddy viscosity at the surface, the smallest surface 

 velocity is reached if the velocity is a linear function of depth, which means that the 

 eddy viscosity is constant. This follows from the assumption that the eddy viscosity 



