VOLUME 59, NUMBER 17 PHYSICAL REVIEW LETTERS 26 OCTOBER 1987 Application of Kraichnan s Decimated-Amplitude Scheme to the Betchov Model of Turhulence Timothy Williams, E. R. Tracy, and George Vahala Department of Physics, College of William and Mary, Williamsburg, Virginia 23185 (Received 27 July 1987) The decimated-amplitude scheme (DAS) devised by Kraichnan is applied to a random-coupling model of turbulence originally introduced by Betchov to test the direct-interaction approximation (DIA). By use of a system of 32 variables, forced stochastically under appropriate statistical constraints, it is shown that the DAS can accurately represent the autocorrelation function of a full Betchov system of 96 vari- ables. A comparison is also made between the DAS and the DIA. PACS numbers: 47. 25.Cg In strong homogeneous turbulence (Reynolds number R» 1), a large range of scales is excited; the ratio of dissipation-scale eddy size to energy-containing eddy size is O(R 't ). Since the number of numerical operations required to simulate the IIow is O(R ), one is confronted with the problem of unresolvable small scales. The many attempted solutions of this problem have all relied on symmetries (isotropy and homogeneity) among groups of modes. Each of these previous attempts has its own diffi- culties. For example, the direct interaction approxima- tion (DIA) ' results in complicated coupled nonlinear integro-differential equations v hich violate statistical Galilean invariance and have not yielded easily to nu- merical solution. (Attacking these difficulties has led to an even more complicated set of equations. ) Renor malization-group theory has difficulties with closure which are not resolved either in the e-expansion meth- od, which requires t. =4 to recover the Kolmogorov energy spectrum, or in the recursion method. Recently, Kraichnan has introduced a new approach to turbulence which is applicable to any stochastic prob- lem that exhibits dynamical or statistical symmetries among group of variables. The decimated-amplitude scheme (DAS) exploits symmetries among groups of variables to reduce a large system to a much smaller sys- tem which has the same statistical properties. This reduction is called decimation. The decimated system is driven by stochastic forces; statistical constraints on these forces insure agreement between the two systems on chosen statistical properties. Determining the ap- propriate statistical constraints for the stochastic forces is a major difficulty of the DAS. The theory behind the DAS is difficult and subtle, but Kraichnan has demonstrated that both the DIA and renormalization-group theories can be recovered from the DAS as special cases. The only explicit numerical example of the DAS discussed by Kraichnan was a very small system of equations (five variables decimated to three variables) which had a significant number of unwanted constants of the motion. It is desirable to ap- ply the DAS to a nonlinear system with many degrees of Cj ik + Cj kl + Ckij (2) This cyclic condition forces the system to have a quadra- tic constant of the motion (the "energy") '': 1V E— : —, ' g x; (t) =const. In addition, the coupling coefficients are chosen to satisfy the following three conditions: (a) Of the possi- ble N coefficients, only O(N ) coefficients are nonzero. (b) C~k =0 for any repeated indices. (c) The coefficients are statistically similar and are chosen from a Gaussian ensemble with zero mean and unit variance. Betchov proposed these conditions to mimic partially the spectral representation of the inviscid Navier-Stokes equations, whose modes obey a cyclic equation like Eq. (2) and also obey (a) and (b). Condition (c) makes the system "iso- tropic. " It is useful to consider a model with random coupling coe%cients because Kraichnan has shown that the DIA is exact for many-dimensional random sys- tems. ' However, the couplings of the Navier-Stokes equations are not random; on this point the Betchov equations diAer significantly. Equation (1) is solved for a large number of realiza- tions of the initial values [x;(0) ~i =1, 2, . .. , Nj. These realizations are chosen from a Gaussian distribution with freedom. This work examines a simple many-variable system as a step toward decimation of the Navier-Stokes equations. In this work, the DAS is applied to the Betchov' model of turbulence. Since Betchov originally intro- duced this model to test the DIA, it is possible not only to judge the success of the DAS, but also to compare it to the DIA. The Betchov model is a set of N ordinary diAerential equations (with N » 1): dx; Cjkxjxt„ i =1, 2, ... , N. j, k =1 The coupling coefficients are generated at random and chosen to satisfy the cyclic equality 1922 1987 The American Physical Society