SECT. 6] TURBULENCE 805 



If I is the displacement of a particle of fluid at time t from its original position 

 at time ^ = 0, it follows that 



|2 = 2^^ f r R{T)dTdt'. (10) 



Jo jo 



For small values of t, such that R{r) does not differ appreciably from 1 in the 

 interval 0<T<t, 



^ = u^t^. (11) 



If R{r) is effectively zero for r > to, then Ii{r) dr will reach a limiting value 



jo 



for t' > TO, and for large values of t, such that t^ro, 

 where 



=f 



R{t) dr. 



(12) 



If the particles had been dispersed by a diffusion process according to Fick's 

 law, with a constant diffusion coefficient K, the mean square displacement 

 would have been 



F = 2Kt. (13) 



In the special case considered above, for large values of t, the dispersion corre- 

 sponds to diffusion with a coefficient K given by 



K = u^I. (14) 



D. Richardson's " Neighbour Separation'' Theory 



Instead of considering the distance of a particle from a fixed point, Richard- 

 son (1926) approached the diffusion problem by considering the variation with 

 time of the distance separating two particles. He defined the instantaneous 

 distance I between two particles as the "neighbour separation", and introduced 

 a "neighbour concentration function" q{l) to represent the proportion of pairs 

 of particles having separations between / and l + dl. The ordinary diffusion 

 equation was then replaced by the equation 



where F{1) is the "neighbour diffusivity". From data based on observations in 

 atmospheric diffusion, Richardson deduced that 



F{1) = eZ^/3, (16) 



where e is a constant. The application of this approach to conditions in the sea 

 is considered in section 5 of this Chapter, page 818. 



