Kraichnan 



Whereas energy conservation follows identically from the expressions for a^ 

 and bkpq , realizability is deduced from a remarkable property of the direct- 

 interaction equations which is not obvious from their final form. Although these 

 equations are only an approximation for actual turbulence, they are exact for a 

 model dynamical system that has the same energy integral as the Navier- Stokes 

 system, but different dynamical equations for the Fourier amplitudes. The 

 model system, which has been discussed in detail [3,7,13], is obtained by com- 

 bining, with random phase factors, the elementary conservative interactions of 

 triads of Fourier modes which characterize the Navier -Stokes system. 



The representation by a conservative model system also implies that the 

 direct- interaction equations yield a tendency toward absolute statistical equi- 

 librium. It can be verified directly from Eqs. (12) - (16) that, if v = 0, the 

 equipartition relation u(k; t, t') aG(k; t, t')(t > t') is consistent with the equa- 

 tions [12]. Although the direction of energy transfer in Eq. (12) is toward es- 

 tablishing the equipartition solution, the latter is never actually achieved from 

 physically admissible initial conditions. The tendency toward equipartition has 

 been verified by numerical integrations for v = [12]. 



So far, there have been no computer experiments on isotropic turbulence 

 decay which would give a clean test of the direct- interaction equations, similar 

 to the test for turbulent dispersion described in the previous section. Such ex- 

 periments now appear to be imminent [l]. Meanwhile, it is possible to compare, 

 behind grids, direct- interaction predictions with laboratory results on decay. 

 This comparison is not so clean for several reasons, the most important of 

 which are the difficulty of matching laboratory initial conditions, and the un- 

 certainties about the degree of anisotropy and instrumental error present in the 

 measurements. The most reasonable comparisons would appear to be those of 

 properties associated with the high-wave-number end of the energy spectrum, 

 where short intrinsic dynamical times should lead relatively rapidly to a state 

 that is insensitive to initial conditions. 



Numerical integrations of Eqs (12) - (16) have been carried out for a variety 

 of initial conditions [12]. Figures 3 through 8 show some typical results. Fig- 

 ures 3 through 5 show the evolution of the energy spectrum, viscous dissipation 

 (or mean- square- vorticity) spectrum, and transfer spectrum T(k, t) for the 

 initial condition 



/2~ , k^ / 2k A .... 



E(k,0) = 16 J -v^' — jexp --^ . (19) 



K ^ kg y kg J 



l(0) is the initial value of the integral scale, a length characteristic of the size 

 of the energy-containing eddies, while u(0) = vq is the initial root- mean- square 

 value of the velocity along any axis. The Reynolds number Rx(t) = k (t)u(t)A, 

 where u(t) is the root- mean- square velocity component at time t and \(t) is 

 the Taylor microscale [10], decreased from 35 to 17 over the time interval 

 shown. 



Figures 6 and 7 depict the evolution from a less peaked initial spectrum, 



794 



