Some Progress in Turbulence Theory- 

 distortion of the small eddies associated with energy transfer. This is why the 

 direct-interaction approximation is inadequate in the inertial range. It confuses 

 the two processes and makes the convection decorrelation time the effective de- 

 correlation time for energy-transferring triple correlations built up by inter- 

 actions among the inertial- range wave numbers [28j. 



In order to correct this situation, the direct-interaction equations have been 

 modified so that the integral over the past history is taken along particle trajec- 

 tories, instead of at fixed points in space. This transforms away the spurious 

 effects of convection on energy transfer. The change requires an initial re- 

 formulation in terms of generalized Lagrangian velocities. The resulting equa- 

 tions, called the Lagrangian- History Direct-Interaction (LHDI) approximation, 

 yield Kolmogorov's form of the inertial- range spectrum, and provide a unified 

 dynamical description of both Eulerian and Lagrangian flow statistics [29]. 



A simplified formulation, called the abridged LHDI approximation, has been 

 integrated numerically in considerable detail, yielding numerical predictions for 

 Kolmogorov's constant, for the dissipation range spectrum of isotropic turbu- 

 lence at high Reynolds numbers, and for a number of Lagrangian statistics [30]. 

 At moderate Reynolds numbers, the abridged LHDI equations yield an isotropic 

 turbulence decay which is similar to that obtained from the original direct- 

 interaction approximation. Figures 9 and 10 illustrate the moderate-Reynolds- 

 number comparison in the dissipation range, where the differences are largest. 



Figures 14 and 15 compare the abridged LHDI predictions for inertial range 

 and dissipation range at high Reynolds numbers with the measurements in sea 

 water by Grant, Stewart, and Moilliet [l6j. Here e is the rate of dissipation by 

 viscosity per unit mass, k = (e/v^)^'"* is the Kolmogorov dissipation wave 

 number, and 0j (k) is the one-dimensional energy spectrum. It should be noted 

 that these comparisons are absolute in the sense that there are no adjustable 

 scaling parameters in the abridged LHDI equations. 



The LHDI and abridged LHDI approximations have also been applied to sev- 

 eral other problems: relative dispersion of two particles by turbulence [31], 

 spectrum of scalar fluctuations convected by turbulence [31,32], spectrum evolu- 

 tion of random solutions of Burgers' equation [20], turbulence in a Vlasov plasma 

 [33], and second-order chemical reactions in a turbulent fluid [34], In these 

 varied applications, the LHDI equations have had substantial success in differen- 

 tiating among qualitatively different kinds of dynamical behavior. Thus the same 

 approximation which yields the Kolmogorov k"^^-^ law in isotropic incompressi- 

 ble turbulence, yields Richardson's law in two-particle dispersion [31], a k'^^^ 

 law for the inertial- convective spectrum range of a passive scalar [31], a k"^ 

 inertial range for Burgers' equation [20], the k"^ viscous-convective- range law 

 of Batchelor for the spectrum of a passive scalar at very high wave numbers 

 [32], and a k"^^^ inertial- range law for hydromagnetic turbulence [22,35]. How- 

 ever, where quantitative accuracy has been assessable in these applications, it 

 appears mostly to be poorer than in the case of the Kolmogorov inertial and 

 dissipation range for isotropic Navier-Stokes turbulence. 



805 



