General Theory of Ocean Currents in a Homogeneous Sea 



407 



for ^ the functions cosh and cos are replaced m the numerator by the complementary 

 functions sinh and sin. Thus 



u = {\ -4>)U and y = 0C/, (XIII.32) 



D denotes again the frictional depth to which now, since it refers to the sea bottom, is 

 supplemented the additional word "lower". The functions (/> and </» determine the 

 vertical velocity distribution of the gradient current. For z = d one obtains ^ = 1 

 and </» = as required by the boundary conditions. Its further course is best shown by 

 evaluating the equations for different values of dID. Figure 172 presents as an example 



Fig. 172. Vertical distribution of the velocity components u and v for different values of h\D 



(for values hlD = if the course of the M-component coincides with the straight line 1-0; 



the t;-component approaches rapidly the straight line 00). 



the vertical distribution of the two components for the d\D = 1-5, 0-5 and 0-25. In 

 the curves for depths somewhat greater than the frictional depth the course of both 

 components is the same. Until a depth above the bottom is reached corresponding to 

 the frictional depth u increases rapidly and reaches here the value of the geostrophic 

 current U. The M-component increases a little further but then reverts to the tZ-value 

 and remains then almost constant. The r-component (in the direction of the pressure 

 gradient) rapidly reaches a maximum not far above the bottom, then falls almost to 

 zero and oscillates with decreasing amplitude around the zero value. For depths 

 d > 1-5 D the structure of the pure gradient current has the form shown in 

 Fig. 173; this is drawn in the same way as Fig. 168. At distances from the bottom 

 greater than D there is a practically uniform velocity at right angles cum sole to the 

 pressure gradient. This is the uniform deep current; it corresponds to the frictionless 

 geostrophic current. The bottom layer is governed by the bottom current, the velocity 

 of which decreases according to a logarithmic spiral down to the sea bottom. For 

 greater depths of the sea the only change in this structure is in the vertical thickness of 

 the deep current ; the bottom current always corresponds to the frictional depth D. 

 Since the deep current runs parallel to the topographic lines ("Niveaulinien") of 



