THEORY OF UPWELLING AND COASTAL CURRENTS 561 



w = 



II we separate the real part P of \/X^- + 2--i from the imaginary part 

 Q, we have 



^ = .V (V'V^- + ^TT^ + y-)/2 S = , \ i\/^' + 4^^ - X"-)/2 



, (20) 



Thus the real part of V'"^' + ^tt-/ is always greater than tt. Hence, if 

 the depth of the sea is sufficiently large {h/D^ > 2), the expression 

 (19) can be given very accurately by 



(A- + 27r'-'0w sin <\>D\^ " J 



A, ^\- + 2-'Z A (21) 



Now we have, for the vertical component w, 



10 = at z = 0; z = h (22) 



since there can be no vertical motion on the surface and bottom of the 

 sea. Integrating (2) with respect to z from the surface down to the 

 bottom, we have 



' ",„ 1 iidz = 10 \ — ^ 



This means that the integral \ ""^ 



is independent of Xj, or therefore a constant. But as this integral must 

 vanish directly on the coast or at x = 0, we must have 



j; 



h 



udz = (23) 



always. Integrating (21) with respect to z from to h and equating 

 the real part of the resulting equation to zero, we have 



_ TT^r/pco 1 — cos(AL/D,) 



This determines the relation between the wind stress t and slope of the 

 sea surface induced by the former. Substitution of (24) in (21) gives 



w,, and 1^1. The substitutions of u^, c^i and y (A) thus obtained into 

 (10), (11) and (12) give u, v and the surface slope dC/d^. The vertical 

 velocity w can be derived from the equation of continuity (2) as 



