THEORY OF UPWELLING AND COASTAL CURRENTS 563 



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dicular to the coast in addition to a coastal current parallel to the 

 direction of the wind. The vertical component of this circulation evi- 

 dently represents the upwelling. 



From the expressions (28), (29) and (30) for u, v and w, it can be 

 expected that the horizontal velocity of the water in this process is 

 approximately D^jD^ or ■\/A^IA^ times as large as the vertical velocity. 

 This result will be very useful in estimating the approximate speed of 

 upwelling. But this kind of vertical circulation can be noticed best 

 in the case of a very deep sea where there is very little current produced 

 by the slope of the surface of the sea. 



If we define a function ^ (x, z) as 



00 r 1 



^ (X, z) =^3lL-R f sin Xx/D,(l-cQs AL/D,) |^-V^' + '^'' ^/°v _ ^ k 



(36) 



we can show that this is the stream function in the plane perpendicular 

 to the coast and u and w are given by 



so that any curve 



■^ (x, z) = constant 

 represents a stream line. 



4. A Numerical Exampi^ 



So far the author has elucidated the process of upwelling in a quan- 

 titative manner and obtained the expressions representing the motion 

 of water produced by a wind blowing parallel to the coast in a belt of 

 finite width. The stream-function >t (x, z) can be computed from the 

 formula (36) for any distance x/Di^ and for any depth z/Dy below the 

 sea surface where D^, and D^ are the distance and depth specifying the 

 intensity of the horizontal and vertical mixing respectively. The result 

 of computation of the stream function is given in the Table I and illus- 

 trated by the diagram in Figure 2. The unit is given by 



^''■' -X 10-* 



pcii sin ^ 



From the table and diagiam it can be easily shown that the vertical 

 circulation is most strongly developed close to the coast and in the upper 

 layers of the sea directly below the surface swept by the wind. An in- 



