Kraichnan 



Fig. 11 - Comparison of direct-interaction and 

 computer -experiment results for evolution of the 

 energy in three interacting shear waves (Ref. 17). 

 Curves 1, 3, and 5 show the energy in the three 

 waves according to the computer experiment, while 

 curves 2, 4, and 6 show the respective direct- 

 interaction results. 



and quasinormal approximations [24]. Herring estimates that statistical uncer- 

 tainties in the numerical experiment curve, due to the finite density of modes 

 in the horizontal and finite ensemble size, amount to about 3% where they are 

 maximum, while the numerical error in the integration of the direct-interaction 

 equations is smaller. The graph therefore suggests the excellent performance 

 of the direct- interaction approximation. 



The truncation in wave-number space to three vertical modes and a single 

 octave in the horizontal is physically artificial (although not seriously so at the 

 Rayleigh number taken), but this does not weaken the test of the direct- 

 interaction equations, since the same truncation is used in both the numerical 

 experiment and the statistical approximation. 



Figure 13 shows the evolution of the spectrum of temperature fluctuations 

 in the gravest vertical mode as a function of horizontal wave number. Again 

 the direct-interaction results seem to agree excellently with the numerical 

 experiment. Here, however, the comparison is less sharp, because the statisti- 

 cal fluctuations in the numerical experiment show up more prominently in the 

 spectrum results than in the Nusselt number curve. 



Numerical results from the work at lower Prandtl numbers and higher 

 Rayleigh numbers and also for non-slip boundary conditions are expected in 



802 



