Kraichnan 



.10 



a .08 



.06 



.04 



k/k. 



.6 



.8 



1.0 



Fig. 9 - Comparison of direct-interaction re^ 

 suits from the run of Figs. 6 and 7 (curve 1) 

 with experimental dissipation spectra (Ref. 

 15). The quantity 01 (k) is the one -dimensional 

 spectrum. Curve 2 shows the result from the 

 abridged LHDI equations (see the Lagrangian 

 Direct-Interaction Approximation section of 

 this paper) for the same initial spectrum. 



ADDITIONAL APPLICATIONS OF THE DIRECT- 

 INTERACTION APPROXIMATION 



The direct- interaction equations have also been formulated for the evolu- 

 tion of the spectrum of scalar contaminants convected by turbulence [l8,19j, 

 stochastic solutions of Burgers' equation [20j, hydromagnetic turbulence [21,22], 

 turbulence in a Vlasov plasma [23], buoyant convection in a Boussinesq fluid 

 [24], and second-order chemical reactions in a turbulent fluid [25 - 27]. Further 

 applications, for example to visco-elastic turbulence, are feasible. We shall 

 give here only a brief report on the application to Boussinesq convection, which 

 is the only one in which quantitative comparisons with experiment (computer 

 experiment in this case) are available at the time of writing. 



The equations for turbulent Boussinesq convection are substantially more 

 complicated than those for isotropic turbulence, although they present the same 

 general appearance, because there is reduced spatial symmetry and an addi- 

 tional field variable, the temperature field. We refer the reader to Ref. 24 for a 

 full description. Numerical results so far have only been obtained for the case 



800 



