General Theory of Ocean Currents in a Homogeneous Sea 383 



consists only in following these changes in the elementary current or in determining 

 under stationary conditions the elementary current that satisfies the continuity equa- 

 tion, and then in evaluating the associated time-independent sea-surface slope for all 

 points of the oceanic region under consideration. Only then can the problem be 

 considered as completely solved. This second problem is the more difficult one since 

 the boundary conditions at coasthnes must also be satisfied. It does, hov/ever, help 

 to produce the total picture of the currents for a certain preassumed ocean basin. 



The starting equations for the development of the dynamics of the ocean currents 

 are the hydrodynamic equations of motion in their most general form (see equation 

 X.16). The fact that its individual terms are of quite different significance led Jeffreys 

 (1922) to put forward a terminology for air currents which could also with advantage 

 be applied to ocean currents. According to whether the horizontal pressure gradient 

 is balanced principally by the acceleration or by the Coriolis force or by friction, it is 

 possible to distinguish between (equations for the .v-axis only, those for the j-axis 

 being analogous): 



du 1 dp 



Euler current : -y — 7r\ 



at p ex 



geostrophic current : = ^ — h 2a> sm ^y ; 



... ^ \ dp d / 8u 

 antitnptic current :0= ~ — \- — [a ^r 



p ox cz \ S.v 



The Euler current will appear for rapid changes in the sea level (storm surges, etc.) ; 

 this is also the relationship on which is based the simple theory of waves, where the 

 water displacements in general have the character of a Euler current. The geostrophic 

 current corresponds to another current constituent of the "elementary" current, 

 namely to the gradient current (deep current), while, during the formation of the wind 

 drift and the bottom current, besides the Coriolis force to a considerable extent fric- 

 tion is also involved. An antitriptic current can be expected in local circulations of 

 small extent, for example, in equalization currents in sea straits where the narrow width 

 prevents an effect of the Coriolis force. 



2. Steady Currents in a Homogeneous Sea Without Friction 



(a) General Equations 



For a horizontal frictionless water movement, the equations of motion (X.16) for a 

 homogeneous sea (p = const.) (Coriolis parameter/ = 2aj sin ^) will take the form: 



du ^ \ dp dv \ dp ■ -■• ■ - 



-7: =>--/; TT. = -/«--/• (XIII.l) 



dt p dx dt p dy ^ 



In a homogeneous sea the pressure p at a depth z (counted as positive downwards 

 from the undisturbed sea level r = 0) is given by 



. p = gp(z + 0, , (XIII.2) 



