General Theory of Ocean Currents in a Homogeneous Sea 399 



now no longer parallel to the isobars but is deflected at an angle proportional to k. 

 On the right-hand side of the equations of motion (XIII. 1) the components for the 

 frictional force —ku and —kv have to be added. Multiplying the first equation by u 

 and the second by v and adding, gives 



1 dW 1 dp 



2 dt- p dt 



For the movement of a water element along an isobar (dpldt = 0) this equation gives 



K= VQe-'<K 



The velocity of the current which is acted upon by Coriolis force and friction, usually 

 decreases until it vanishes. The value l//c gives the time needed by the bottom friction 

 to reduce the velocity by a factor of 2-72. For currents in shallow waters k is of the 

 order of 10"*^ to 10~'^ sec~^, so that the velocity of the water movement will fall to a 

 tenth between 2 and 25 days. 



The Guldberg-Mohn frictional principle makes no allowance for the fact that a 

 turbulent flow is aff'ected also from above by mass exchange with the layers above it, 

 in addition to the eff'ect of the bottom surface which affects the flow from below. 

 Sandstrom (1910) has taken this circumstance into account by assuming that the 

 frictional force does not exactly oppose the current, but its vector deviates by a small 

 angle to the left of the current direction (or the force acts backwards and to the right 

 of the current). 



Also this frictional principle can only be considered as a makeshift and gives ac- 

 ceptable results only for currents in very shallow waters. If all the factors involved in 

 the formation and maintenance of the ocean currents are to be taken into account it 

 is necessary to return to the hydrodynamic equations of motion in the form given in 

 (X.16). Besides friction, there must also be taken into account the effect of the Coriolis 

 force and as current producing factors, especially the tangential pressure of the 

 wind on the sea surface, the pressure gradient and gravity. For horizontal water trans- 

 ports, i.e. along the gravitational level surfaces, gravity is less important as an im- 

 pelling force. If only the wind stress, the Coriolis force and friction are acting, the 

 current will be a pure drift current; if, however, gradient force, Coriolis force and 

 friction are the decisive factors, it will be a pure gradient current. The following section 

 is concerned with these two basic forms of water movement. 



The fundamental work in this direction is due almost entirely to Ekman (1905, 1906, 

 1922) who first gave a strict mathematical form to the eff"ects of the Coriohs 

 force and friction in the theory of the ocean currents in a homogeneous sea. The great 

 significance of these two forces for the generation of drift currents had already been 

 recognized and demonstrated by means of observational data by Nansen (1902, 

 1905). These investigations opened a first way to the development of a complete 

 theory of ocean currents. 



{b) Pure Drift Currents 



A pure drift current is a result of the wind stress acting on the surface of the sea. 

 This stress is produced either by friction of the air passing over the water, or by the 

 pressure effect of the wind on waves which transfers part of the momentum of the 



