Kraichnan 



1.6 



.8 



-.8 



-1.6 



RUN 10 Rxd) = 17.75 



tu(0)/L{0) =1.62 



2 T(k)/kd^uv2 



3 2k2T(k)/kd*u..2 



.8 



-4 



-.8 



k/k, 



1.6 



2.0 



Fig. 7 - Comparison of the dissipation spectrum 

 (1), transfer spectrum (2), and vorticity-production 

 spectrum (3) for the run shown in Fig. 6 



proportional to the instantaneous rate of vorticity production, normalized by the 

 instantaneous dissipation rate, and thereby is an important measure of the non- 

 linear transfer processes. 



Figures 9 and 10 compare the one-dimensional dissipation spectrum from 

 the nearly self-preserving run depicted in Figs. 6 and 7 with spectra obtained by 

 Stewart and Townsend [15] from measurements of decay behind grids. Again, 

 there appears to be semiquantitative agreement, which is actually within the 

 limits of experimental uncertainties. 



The apparently satisfactory performance of the direct- interaction approxi- 

 mation at the moderate Reynolds numbers discussed above deteriorates at 

 higher Reynolds numbers. At very high Reynolds numbers, the direct- 

 interaction equations yield an inertial range spectrum E(k) « k"^''^ instead of 

 the k"^/^ spectrum predicted by Kolmogorov [lO] and supported by experiment 

 [16]. The reasons for this deficiency, and the modifications of the approxima- 

 tion which correct it, are discussed in the Lagrangian Direct-Interaction Ap- 

 proximation section of this paper. 



798 



