388 General Theory of Ocean Currents in a Homogeneous Sea 



to the side and there is thus an equalization of the momentum (current impulse) in the 

 direction of the strongest velocity gradient. There is also an associated equalization 

 of all the characteristic substances and of the water properties. This equahzation pro- 

 cess has already been discussed in detail in Pt. I, Chapter II (see p. 105). For the 

 property-pair momentum-velocity under conditions of immediate and complete 

 equalization of the flow momentum a general expression for the apparent shearing 

 stress of a turbulent flow has been derived having the form 



da 



(XIII. 13) 



where U is the mean velocity along the x-axis, :: is perpendicular to it, t] is the exchange 

 coefficient for momentum (eddy coefficient or turbulent frictional coefficient). 



In Chapter II (see p. 329) another expression was derived for the apparent shearing 

 stress occurring in turbulent flow from the analysis of the current variations in it. 

 This was given as 



T=-pi7^"'. (XIII. 14) 



The variations in velocity u' and v' are of course connected with the distribution of the 

 mean velocity which varies across the stream lines. To give a practical form to equation 

 (XIII. 14) Prandtl (see especially 1942) introduced the mixing length I, defined as the 

 length which can be regarded as the diameter of the water quanta moving with the 

 turbulent flow or as that distance that such a quantum travels before losing its identity 

 due to mixing with the surroundings. A water element with a mean velocity u(z) at a 

 point z (see Fig. 167) will have a mean velocity u(: + /) = m(z) + l{8uldz) at a distance 



777777777777777777777777777777777777777777. 



Fig. 167. 



/ across the current. If a water element is moved from one layer to another then the 

 magnitude of u is given by 



u' = u(z + /) - i7(r) = l{du\dz). 



The variations in velocity v arise from the movements of the water elements entering 

 the place under consideration from different sides, moving one behind the other and 

 approaching or receding from each other with a velocity diff'erence of ll{du\dz) and 

 thus give rise to transverse movements. Thus r' will also have the order of magnitude 

 l(dul8z). Between u' and v' there must, however, be a negative correlation. The water 



