Kraichnan 



DIRECT- INTERACTION APPROXIMATION 

 FOR TURBULENT DIFFUSION 



Turbulent diffusion by a random velocity field provides an example, simpler 

 than Navier-Stokes turbulence, for introducing the direct-interaction statistical 

 equations. We shall suppose that the mean velocity v vanishes, and that u. (x, t) 

 is incompressible and has a multivariate-normal statistical distribution over the 

 ensemble of realizations of the flow. Suppose that a scalar quantity, e.g., tem- 

 perature T (x, t), is passively convected according to 



3 o\ 3T(x,t) /.x 



--- r;V2 T(x,t) = -u. (x,t) ^ (1) 



where 77 is the molecular diffusivity. K the initial temperature field t(x, 0) is 

 the same in all realizations [t(x, 0) = T(x, 0)], then the mean temperature field 

 T (x, t ) = < T (x, t ) > at later times is 



T(x,t) 



jG(x,t;y,0) T(y,0)d3y , (2) 



where g(x, t; x', t') is the mean Green's function for the temperature field; i.e., 

 g(x, t; x', t') = t(x, t) for the special initial condition t(x, t') =B^ (x - x'). 



From Eq. (1) and the assumption of normality of u, an infinite -series in- 

 tegrodifferential equation for G (x, t; x', t') can be developed in the following 

 form: 



f-^- 7jV2j G(x,t;x',t') =^— J ds J d^y G(x,t;y,s) 



BG ^',s;x' , t' ) .„. 



• U.3(x,t;y,s)-^— ^+ ... , (3) 



G(x,t';x',t') = 3^ (x-x') . 



The higher terms in Eq. (3), indicated by the dots, are an infinite series of in- 

 creasingly complicated multiple integrals with g and u in the integrands. For 

 the derivation and detailed analysis of Eq. (3), the reader is referred to the 

 original papers [3,6,7]. In brief, the starting point for Eq. (3) is an iteration 

 expansion in which the right-hand side of Eq. (1) is treated as a perturbation on 

 the left-hand side. The iteration expansion is then reworked by partial summa- 

 tion to all orders, to yield Eq. (3). 



Equation (3) is a formally exact equation for G. The direct-interaction ap- 

 proximation consists of dropping all the terms indicated by the dots. Equations 

 (2) and (3) yield 



(-^ - ^^') T(x,t) = ^ r ds fdV G(x,t;y,s) U.. (x,t;y,s) 

 ^ 



788 



