124 WIND CURRENTS AND WIND WAVES 



These equations were first integrated by Ekman, on the assumption that 

 the eddy viscosity is a constant. Ekman undertook the mathematical 

 analysis on the suggestion of Fridtjof Nansen, who, during the drift of the 

 Fram across the Polar Sea in 1893-1896, had observed that the ice drift 

 deviated 20 to 40 degrees to the right of the wind, and had attributed this 

 deviation to the effect of the earth's rotation. Nansen further reasoned 

 that, in a similar manner, the direction of the motion of each water layer 

 must deviate to the right of the direction of movement of the overlying 

 water layer, because it is swept on by this layer much as the ice which 

 covers the surface is swept on by the wind ; therefore, at some depth the 

 current would run in a direction opposite to the surface flow. Nansen's 

 conclusions were fully confirmed by mathematical treatment. 



Assuming that the eddy viscosity is independent of depth, the equa- 

 tions can be directly integrated. If 



(VII, 17) 



(VII, 18) 



where Ci, C2, Ci, and c^ are constants that must be determined by means of 

 the boundary conditions. 



In this form the result is applicable to conditions in the Northern 

 Hemisphere only, because in the Southern Hemisphere sin (p is negative, 

 wherefore D is imaginary. In order to obtain a solution that is valid 

 in the Southern Hemisphere, the direction of the positive y axis must be 

 reversed. 



The solution takes the simplest form if the depth to the bottom is so 

 great that one can assume no motion near the bottom, because then Ci 

 must be zero. Assuming, furthermore, that the stress of the wind, Tq, 

 is directed along the y axis, one has 



-"• (^")„ = ^" ^""^ ^' (S)„ = °' 



from which equations C2 and Ci can be determined. Calling the velocity 

 at the surface vo, one obtains 



(VII, 19) 



VTOp D\p V2 



