GENERAL CIRCULATION OF THE PACIFIC OCEAN 521 



Coriolis parameter with latitude is taken into account. The use of cur- 

 rent velocity in place of mass transport makes the mathematical analysis 

 many times more complicated because the problem is now three-dimen- 

 sional. But the result will be of importance because it should give an 

 idea of the vertical structure of the wind-driven circulation of the oceans. 



2. Theory 

 The dynamical equations of the stationary ocean currents, taking 

 both vertical and horizontal mixing into account and neglecting the 

 non-linear terms, are 



92u , a / . 9u \ dp 



d^v 



(1) 



'd / J dv \ dp 



"I r. ^ "^ -TT' I == ■;; — I — 2<opu sm d> — - — = 0, 

 8x2 8z \ 9z / ^ ^ dy 



where u and v are components of the current velocity in x (eastward 

 positive) and y (northward positive) directions, p is the pressure, p the 

 <iensity, Ai and A^ the horizontal and vertical coefficients of eddy vis- 

 cosity of the water, while w is the angular velocity of the earth, and <^ 

 the geographical latitude. The axis of z is taken positive downward, 

 the origin being placed on the undisturbed sea level. 



The boundary conditions to be satisfied on the surface (z = 0) and 

 at the bottom (z = h) are 



9u 9v 



z = 0: — ^z^ = Tx {x,y)', —A^ — = tj {x,y) (2) 



and 



z = /i: u - i; = 0. (3) 



Here both coefficients of eddy viscosity Ai and A^ are supposed to be 

 constants. The right members of (2) are known as the wind stresses. 

 The conditions to be satisfied along the coasts are also necessary. These 

 are simply that there is no water flow across and along these coasts. 

 If the coasts consist of vertical cliffs, we have 



u = f = at the coast-lines. (4) 



In addition to the dynamical equations (1) the equation of con- 

 tinuity should be included. If we neglect the vertical component of 

 velocity, it is 



9w Zv 



9x 9y 



The 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 con- 

 fine oiurselves to the gross features of the horizontal circulation in great 



