General Theory of Ocean Currents in a Homogeneous Sea 443 



perform any inertia oscillations at all but will rather remain from the beginning in 

 the stationary position. Thus in the general case the amplitudes of corresponding 

 terms are no longer equal and the motion is then elliptical instead of circular. However, 

 the amplitudes of corresponding terms in the sea surface oscillations are always equal 

 and these are therefore always pure inertia movements. It has been found that currents 

 flowing into a wide area uninfluenced by coastal eff"ects usually follow a wave-form 

 course rather than a straight course. A current with an oscillatory streamline seems to 

 be a more stable type of motion than one with a linear course. Once a bulge is formed 

 in any direction, the centrifugal force draws the water further and further out and the 

 bulge produced by such disturbances will grow steadily; consequently, progressive 

 waves and vortices will be formed in which the current will oscillate about a mean 

 direction. In dealing with problems concerning these oscillating currents it is of course 

 necessary to take the Coriolis force into account (Exner, 1919). It surpasses the scope 

 of this section to penetrate more deeply into the dynamics of progressive waves of this 

 type in an infinitely extended medium ; it rather belongs to and will also be discussed 

 when dealing with the theory of progressive tidal waves (Vol. II) ; for an account of 

 progressive waves with inertia period see Fr. Defant (1940) and Ekman (1941). 



{b) Inertia Movements Associated with Drift and Gradient Currents 



In the formation of steady drift and gradient currents the state of motion changes 

 from the first motionless initial equilibrium state into a second state in which there is 

 an equilibrium between all the forces acting. It can be expected that this transfer will 

 give rise to inertia oscillations which will gradually be damped by friction until the 

 new stationary equilibrium state is reached. Ekman (1905) examined in some detail 

 the case of a suddenly starting wind over a deep, extended ocean. A comprehensive 

 treatment of all questions arising has been given by Fjeldstad (1930). The equations 

 of motion (X.16) which stand in question can be combined introducing u -f- iv = w 

 (/ = \/—\)'m order to obtain a single equation 



dw T) 8^w 



J, + '>• = I J?- (^"'-^^^ 



The boundary conditions to be satisfied are 



for t =Q: vv = 



and / , dw ^ ^ 



for all t: 



If the wind arises suddenly at a time / = with a tangential pressure Tin the direction 

 of the positive j-axis, then the velocity components u and v are given by 



IttT f^ sin Itt^ 



and 



V = 



pDf Jo V^ "^ \ ^D^ 



IttT f"^ cos 2tt^ 



pDf . 



exp -T7^,U^ (XIII.65) 



Vi ^^Pi-4^^i^^ 



