328 



Tides and Tidal Currents in the Proximity of Land 

 u = — r e~^ ,D ^ z cos lot— -jrA , 

 d = — r e- (nlD * )z sin lot — ^-z) , 



(XI.11) 



The quantities D x and D 2 have a meaning similar to that of Ekman's depth 

 of factional resistant D = 7i]/(2v/f) (see vol. 1/2, equation XIII. 24). However, 

 in (XI. 21) they are also dependent on the frequency a of the tide wave, and 

 for currents rotating to the right (Northern Hemisphere) D x is considerably 

 larger than D 2 for currents which rotate to the left. At 54° N. lat. (southern 

 North Sea), for instance, D x = 3-36 D 2 . Both "differential currents" (XL 20 

 and 21) s and r are represented vectorially by a logarithmic spiral; in Fig. 133 



Fig. 133. Behaviour of the differential current in the presence of tidal currents (Thorade). 



AC = s and AD = r, for a height z, are two vectors of the rotary motion 

 and AB = — s and — r respectively. In order to obtain the actual tidal 

 current, the frictionless current s and r is added geometrically to s and r 

 respectively. CE = BA = s is added geometrically to AC; in the height z 

 above the bottom, this will give a current with the vector 



AE = BC = s = s V(l + e -^' D ^ - 2e~^ D ^ cos(ti/D 1 )z (XI. 12) 

 for the current rotating to the right, 



BD = r = r }/(\+e-( 2 *l D ^ -2e-^ D ^ cos(n/D 2 )z (XL 12) 



for the current rotating to the left. Figure 134 gives for <£ = 54° and for 

 the main lunar tide M 2 , s and r as a function of z/j/v; the curves are similar 

 to those of the stationary gradient current (vol. 1/2, equation XIII.4c). 



Through adequate superposition of ^ and r , we can obtain all kinds of 

 frictionless current at the sea surface and equations (XL 9 to 12) give the current 

 diagrams with friction. For the Northern Hemisphere, the following cases are 

 possible : 



(a) Alternating frictionless current: then s = r and in the case illustrated by Fig. 134 the 

 current above the height of approximately z = 460/ i/v is very weakly rotating to the right, almost 

 alternating; below this height we have r > s, the current figures become an ellipse rotating to the 

 left and this the stronger, the closer to the bottom. 



