Some Progress in Turbulence Theory 

 3T(y,s) 



(4) 



and, again, the direct- interaction approximation is obtained by dropping the 

 higher terms. .,,, - 



The most striking difference between Eqs. (1) and (4) is that the former is 

 a differential equation, local in space and time, while the latter involves inte- 

 grals over space and time and cannot be reduced to a differential equation. The 

 nonlocalness of Eq. (4) is an analytical embodiment of some simple and widely 

 held intuitive ideas about eddy transport phenomena. The effective range of 

 the space integral in Eq. (4) is the correlation length of u^j, or the character- 

 istic eddy size, and the effective range of the time integral is the correlation 

 time of U. . , or the typical eddy circulation time, whichever is shorter. Thus, 

 the nonlocalness of Eq. (4) is that which is intrinsic to the description of a mix- 

 ing process on space and time scales, i.e., the order of the effective mean free 

 path and intercoUision time. 



In the appropriate limit, Eq. (4) reduces to a local form in which an effec- 

 tive eddy-diffusivity tensor appears. Suppose that t(x, t) varies so gradually 

 with its arguments that 



BT(y,s) BT(x.t) 

 Bv- Bx. 



for y and s, where g(x, t ; y , s )u. (x, t ; y , s) is large enough to contribute ap- 

 preciably to the integral. Then Eq. (4) reduces to 



d A ~ 3 



ot / ox- 



, , BT(x,t) 

 (x,t) 



3x, 



(5) 



where 



(x,t) 



j ds J d^y G(x,t;y,s) U.. (x,t;y,s) (6) 



is the eddy-diffusivity tensor. It can be shown (as confirmed by the numerical 

 results to follow) that the elements of «. have the typical order of magnitude 

 £vq , where ? is the correlation length and vq is the root -mean -square turbu- 

 lent velocity component. If, instead of making the direct- interaction approxima- 

 tion, the full series Eqs. (3) and (4) are retained, the eddy-diffusivity limit of 

 Eq. (5) still emerges, but with an infinite series of higher-order integrals over 

 G and u added to Eq. (6). 



To provide a clean numerical test of the direct-interaction approximation, 

 the prediction for eddy diffusivity was compared with the results of computer 

 realizations of turbulent dispersion in statistically homogeneous and isotropic 

 velocity fields. Homogeneity and isotropy imply that k (x, t) has the form 

 x.jix, t) = h.^.K (t). 



789 



