Kraichnan 



In the limits where v is very large or the correlation time of u^j is very 

 small compared to the typical eddy circulation time, Eq. (6) can be shown to be 

 asymptotically exact [8j. Therefore, the most critical test is provided by taking 

 V = and making u(x, t) a random time- independent field in each realization. 

 The choice v = also makes the computer realization easier, because instead 

 of solving the field Eq. (1), we can equivalently [9] trace the motion of fluid 

 particles in a Monte Carlo calculation. The eddy diffusivity then is expressed 

 by 



!< (t) 



1 



<x. (t) V. (t)), 



(7) 



where, in each realization, x(t) and v(t) are the position and velocity of the 

 fluid element which starts at x = at time t = 0. 



To perform the numerical experiment economically, the velocity field in 

 each realization was constructed only along the particle trajectory, by synthesis 

 from stored Fourier amplitudes. A set of N -complex, vector Fourier coeffi- 

 cients was chosen, by use of pseudorandom numbers, from a normal probability 

 distribution such that the synthesized velocity field was incompressible and, 

 for N - CO, the spectrum function [lO] 



E(k) 



00 



— U. . (x,t;x',t) kr sin (kr) dr 



(8) 



where r 



took an assigned functional form. For the two forms of E (k) which were investi- 

 gated, N = 100 was found to give good statistics: 



E(k) = 16-y/— Vq^— -J exp 



(9a) 



E(k) = - Vo^S (k-ko) 



(9b) 



Here, ko denotes the peak k and 3vo^/2 is the kinetic energy per unit mass. In 

 each realization, the fluid element was started off at x =0, its initial velocity 

 synthesized, then the successive positions of the particle were found, and the ve- 

 locity synthesized all along the trajectory. A simple predictor-corrector 

 scheme was used for the integration. The average of Eq. (7) was taken over en- 

 sembles of approximately 20,000 realizations. 



The direct-interaction equations were also Fourier-transformed before in- 

 tegration. The transforms of Eqs. (3) and (6) when the velocity field is time- 

 independent, homogeneous, and isotropic are 



790 



