Tides and Tidal Currents in the Proximity of Land 327 



As shown previously, a circular tidal force S rotating cum sole commands 

 a circular frictionless current cum sole with the radius s = S/(o—f). It is to 

 be expected that frictional influences will decrease the radius of this current 

 diagram (ellipse) with increasing depth until it becomes zero at the sea bottom. 

 The water particles of thin individual layers describe circles with the radius sjo, 

 which decrease with depth, until the motion vanishes at the bottom, whereby 

 at the same time a retardation of the phase occurs increasing with depth. 

 For deriving the laws governing this decrease of the radius and increase of the 

 retardation of phase, Thorade uses the same idea which has been used by 

 Ekman for his elementary derivation of the stationary gradient current, which 

 is explained in vol. 1/2, Chapter V. Let us give the entire water layer and 

 the sea bottom an additional circular motion s , with a phase difference of 180° 

 (— u = — s cosot, —v = -i-SoSinat). If the water is sufficiently deep, both 

 movements will equalise each other at the surface, which will be at rest, 

 whereas at the bottom (z = 0) we will have the velocity components — u and 

 — v . This circular motion of the bottom is transmitted by friction to the super- 

 imposed water layers decreasing gradually in intensity until it vanishes at 

 great height (z = oo). Thorade calls this system of motion the "differential 

 current" ("Differenzstrom") ; u, o at the bottom it is— u , — v , at a large 

 height above the bottom (z = oo) it is zero. It must satisfy the equations of 

 motion (v = ft/g, the kinematic coefficient of viscosity) (see vol. 1/2). 



du . a 2 u 



a, , ^ < XL9 > 



with the boundary conditions u = — w , d = — v for z = and u = o = 

 for z = co. 



If the constant frictionless current (w , v ) is then added again to this 

 solution, we obtain the tidal current with friction, which satisfied its boundary 

 conditions 



for z = 0, u — u Q = 0, o— v Q =0 and for z = oo, u + w = Wo, v + v =0. 



The integration of (XI. 9) gives 



u = —s e-<" IDl)z cos lot— ^-z\ , 



v = +s e- {n,Dl)z sin lot— -jrz\ , 



If the original motion was contra solem: —u = r cosot, —v = —r s'mot, 

 the "differential current" has the form 



(XI.10) 



