188 
PACIFIC SCIENCE, Vol. IX, April, 1955 
The quantity Di may be called “the frictional distance,” whereas D z is the same as 
Ekman’s “depth of frictional influence” except that it does not contain sin <j>. 
The coastal conditions which p s should satisfy are 
and fn = 0 (20) 
along the shore lines, where is the derivative of p s in the direction normal to the 
dn 
shore lines. 
If equation (19) can be solved and we can determine the functions pi (x, y), p2 (pc, y), 
Ps (x, y), ... , the sum: 
P(x, y, z ) = X ^s(x, y) cos — (21) 
will give the horizontal streamlines at any level 2 for the wind stresses t x (x, y) and 
T y (x, y). The stream function p (x, y, z) should, of course, satisfy the condition: 
* = 1 = 0 (22) 
along the shore lines and the horizontal streamlines of the currents are given by 
p (x, y, z) = constant. (23) 
Infinitely Deep Ocean 
If p\ is the solution of (19) when h = 1, the solution of (19) will be 
if s(x , y) = * p\(x, y). 
Thus we have the solution (21) of our problem in the form: 
P(x, y, z) = 2^ Ps(x, y) cos — -, 
and we may write down (25) in the form: 
i ( x,y,z) =Z*^coe{^-’(2* 
2D, 
} h h ‘ 
If we consider the depth h of the ocean increases indefinitely, we have 
(n <,D, 2 D z 
(2s ~ 1} T = v ’ ~hT = dv 
and 
P 1 (x, y) -» p l (x, y; tj), 
where pi (x, y, rj) is the solution of the equation: 
dfyi 
aVi 
2c op \dx 4 dy 4 
V ( dp i 
8 \dx 2 
d 2 Pi 
dy 2 
cos p dpi 1 / dr x 
R dx pco \dy 
D t 
dx 
= 0 
(24) 
(25) 
(26) 
(27) 
(28) 
which is derived from (19) by putting (2s — 1) — - = r\ and h = 1. The right-hand sides 
h 
