where u and w are the root-mean-square turbulent components in the 



2 



horizontal and vertical directionso In such flows uw ~ 0.4 u , (Town^end, 



1956), so that the r^m.s. velocity in the minimum velocity gradient re- 

 quired to maintain turbulence is about 0.2 cm sec for A = gm cm sec 



Further insight into the turbulent diffusion can be obtained by 

 noting that particles moving in a one-dimensional random walk have a distri- 

 bution after a sufficient time which is described by the diffusion equation. 

 The diffusion coefficient is replaced by ^n-t , where n is the jump rate 

 and Z is the constant jump distance. If I is associated with a 

 characteristic eddy size, ^nt has a value which approximates the velocity. 

 For an A of 10 gm cm sec for a velocity of 0.2 cm sec , I must 

 be 50 cm , the order of size of the eddies needed in the vertical di- 

 rection. The time required for a particle to circulate around an eddy 

 is then about 250 seconds. 



An attempt was made to apply the random walk method to horizontal 

 diffusion for the case where the horizontal motion consists primarily 

 of a shear flow. Vertical fluctuations in position would move the dif- 

 fusing particles in random steps through layers of differing horizontal 

 velocities. Any particle leaving a reference plane as a result of 

 vertical motion would be replaced by another particle having a horizontal 

 velocity characteristic of a plane separated from the reference plane 

 by a distance equal to the vertical eddy scale. Then many particles in 

 the reference plane would have velocities differing from that of the 

 mean velocity of the plane by - I- -r- . As an approximation, it was 



V QZ 



assumed that all particles in this plane were engaged in a random walk 



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