General Theory of Ocean Currents in a Homogeneous Sea 389 



elements entering from below will have too small a velocity, those entering from above 

 will have correspondingly too large a velocity as compared with the velocity at the 

 point under consideration ; positive v' will thus occur together with negative u' and vice 

 versa. The product ii'v' is then always negative. The apparent shearing stress is thus 

 always positive and of the order of magnitude p{I(8ilj8zy}. The proportionality factor 

 is here arbitrarily taken as 1 ; this means only a slight change in the meaning of /. To 

 express in this relation that positive cii/cz will accompany a positive shearing stress 

 and negative ciijdz corresponds to a negative shearing stress, the eddy stress must 

 be re-written in the form 



cii 



cz 



= pP 



cu 



— . cxin.15) 



These turbulent shearing stresses change proportional to the square of the velocity and 

 this has been shown experimentally in investigations in hydraulics. The mixing length / 

 is not a constant here, but depends on the conditions in the current and will vary from 

 place to place. At a solid boundary it is zero and increases with distance from the 

 boundary. 



Comparison of the two equations (Xin.13 and 15) leads to 



cil 



= pP 



dz 



(XIII. 16) 



The eddy viscosity coefficient depends not only on the mixing length / but also on the 

 velocity and density and is thus less susceptible to clarity than the concept of mixing 

 length. However, oceanic turbulence problems can only be handled numerically using 

 the quantity t], the eddy viscosity coefficient, especially for a freely developed turbu- 

 lence remote from solid boundaries (coasts and sea bottom). In the layers near the 

 bottom, however, there are considerable advantages in the introduction of the mean 

 mixing length as a characteristic number giving the degree of the turbulence as a 

 function of the distance from the bottom and of its roughness. 

 From relation (XIII. 15) it can be seen that the quantity 



V P 



cu 

 l—^ (XIII. 17) 



has the dimension of a velocity. It is termed the friction velocity (shearing stress velo- 

 city) w,, so that T = puj which as mentioned above gives the flow resistance as a 

 quadratic function of the velocity. 



The behaviour of a turbulent flow above a rough surface can be judged upon using 

 equation (XIII. 17), making an assumption about the mixing length / (Prandtl, 

 1942, p. 108). Since /increases with the distance from the underlying surface (z = 0), 

 it can be put equal to kz and if w, is constant, (XIII. 1 7) gives the solution 



u = u, (-Inr + c). (XIII.18) 



As has been shown in numerous investigations, the observed profiles are rather well 

 approximated by such logarithmic velocity profiles; for the number k the universal 

 value 0-40 was obtained. If ordinary decadic logarithms are used instead of natural 

 ones, equation (XIII.18) becomes 



M = 5-75M, log -. (XIII. 19) 



