186 
PACIFIC SCIENCE, Vol. IX, April, 1955 
In addition to the dynamic equations (1), the equation of continuity should be included. 
If we neglect the vertical component of velocity, it is 
du dv _ 
dx + dy ~ ' 
(5) 
Equation (5) assumes that there is neither vertical current nor vertical gradient of the 
vertical velocity. Thus, our theory cannot be applied to the coastal and other regions of 
upwelling and sinking caused by local monsoons or other temporary winds. But, if we 
confine ourselves to the gross features of the horizontal circulation in great oceans, induced 
by the superincumbent, quasi-permanent wind system (westerlies or trades), the equation 
of continuity as given by (5) will not cause serious errors in the results. 
There may be some further question concerning the use of (5) for the continuity equa- 
tion. But, in treating the oceanographic data for estimating geostrophic currents, we 
always assume 
1 dp 
u — T 
v = — 
2 co sin (f>p dy 
1 
dp 
2u sin <f)p dx ’ 
provided the frictional terms are neglected. These expressions imply that geostrophic 
currents usually satisfy equation (5) which shows the absence of horizontal divergence. 
du dv 
This means that the equation 1 = 0 may be used without serious errors in treat- 
ed dy 
ing the general circulation of the oceans. 
On eliminating p between the two equations of (1) by cross-differentiation, we have 
(dju _ djA 
\dy 3 dx 3 / 
. d j d (du dz/\\ d , . . 
+ Jz)) + 2a,p (sin 
(6) 
when we take the equation of continuity (5) into account and because — (sin 6) = 0. 
dx 
This equation may also be regarded as expressing the condition that a function p (pressure) 
should exist on a level z as an exact differential with respect to x and y. And the validity 
of equation (6) suggests the possibility of determining the pressure in any level z as a 
function of x and y. 
Now suppose the coefficients of mixing A i and A z are both independent of z, aud put 
( 7 ) 
/ \ (2s 1) ttz 
= u s (x, y) cos ^ , 
where 
/ \ 2 r h , , {2s — 1) 7T 
Us(x, y)=i u ( x ’ y, D cos 2 ^ 
Jo 
and assume, in accord with Stokes, 
d u 
dZ « 
2^ (2s 
= - 2^ cos 
S=1 
1) f 
2h / 
Jo 
d\ 
dr 2 
(2s — 1) ir£ j 
COS 2h 
( 8 ) 
( 9 ) 
then we have, by substitution from (2), (3), and (8) 
