On Turbulence in the Ocean. 579 



of the water, a> the component of the earth's velocity of 

 rotation about the vertical at the point considered, z the 

 height above a standard eqnipotential surface, and k the 

 coefficient of eddy viscosity. Then the equations of motion 

 of the water are 



k~+2a>v=Q, ...... (1) 



ip-2o,u=0 (2) 



If we put w = u + iv , these combine into the single equation 

 k~-2ia>w = (3) 



Suppose first that k is a constant. Then for a deep ocean 

 the relevant solution of this equation is the one that remains 

 finite when z tends to — c© ; and in the northern hemisphere 

 this is 



w = Woexp-(*(l + 0\/( w /£)}> • • • (4) 



where w is the value of w at the surface, and is to be deter- 

 mined from the condition that the rate at which momentum 

 is communicated to a vertical column of water shall be equal 

 to the skin friction at its upper surface. Now a column of 

 unit cross-section gains in unit time by friction a momentum 



P 



d- 



-j-^dz, the integral being taken from the bottom of the 



ocean to the surface ; and the skin friction on it is Kp& 2 , 

 where k is a constant equal to about 0*002, p being the 

 density of the air, and S the wind velocity, supposed along 

 the axis of x. Then the surface condition is 



?<l + 0(^^) l = t 002pS 2 (5) 



As all the other factors are real, it follows that w (l + i) 

 is real. Hence in the northern hemisphere the surface 

 current is always inclined at 45° to the w T ind, flowing, towards 

 the side where the pressure is greater. As the depth 

 increases the velocity diminishes in magnitude and rotates 

 clockwise. In the southern hemisphere the surface current 

 is seen similarly to be inclined at 45° to the wind, flowing 

 towards the side of greater pressure, but the current rotates 

 counter-clockwise as the depth increases. The resultant 

 momentum for all depths together is perpendicular to the 

 wind, in accordance with the condition stated earlier. 



When o), iv , and S are known, equation (5) can be used 



