400 General Theory of Ocean Currents in a Homogeneous Sea 



wind to the water. Both effects usually act in the same direction and can be com- 

 bined as a single tangential force. If there is no pressure gradient within the water mass 

 the surface of the sea must be level {(dpidx) = (dpidy) = 0}. With this the condition 

 of an infinite extent of the ocean is basically connected, since otherwise the currents 

 produced will give rise to a piling up of water at the coast lines which will tend to form 

 gradient currents. Such currents will, however, for the moment be disregarded here. 

 In the case of a steady acceleration-free horizontal current {{dujdt = {dvjdt) = 

 and vt' = 0} and for constant frictional coefficients the equations of motion (X.16) 

 will take the form (/= 2w sin ^, z positive downwards): 



d'^u 8^v 



pfv + V-f:2-^ and -pfu + v^2 = ^- (XIII.23) 



Multiplying the second equation by / = \/—\ and adding to the first gives 



1^2 (" + iv) = — (m + iv). (XIII.23fl) 



For practically unlimited ocean depths the general solution can be taken in the form 



u + iv =^ A e-Ci+'X-^/'D), (XIII.24) 



where 



\l\fpj ~ ^ \J [poj sin cj^J- 



The boundary condition that the velocity of the drift current vanishes for large depths 

 (z = co) is already satisfied by (XIII.24). At the surface of the sea (z = 0), a wind in 

 the direction of the positive j'-axis will give rise to a shearing stress T, which can be 

 represented by the relation 



^(" + iv) 



for r = 0. The solution then takes the form 



M + /y = (1 + /) J^ ^-(i+»>/z)^ (XIII.25) 



Ittt] 



From this the two velocity components of the drift current are then obtained 

 M = Fo e--"D cos 1^45° - ^-) and v = V^e--''^ sin (45° - ^-) (XIII.26) 



D = TT 



with 



" \/(2)Dpoj sm(f> \J \pco sm 



At the sea surface the water in a pure drift current moves with a velocity V^ in a 

 direction 45° cum sole from the wind direction. At increasing depth the angle of de- 

 flection increases while at the same time the velocity of the current rapidly decreases. 

 At a depth D the deflection will amount to a full 180° and the velocity will have fallen 



