S. Goto

S. Kida

National Institute for Fusion Science (NIFS-Repository)

§ 17. Applicability of the Direct-interaction Approximation Goto, S. and Kida, S. The aim of this work is to clarify the validity limit of the direct-interaction approximation (DIA), which was originally proposed l) as a closure theory of incom-pressible turbulence governed by the N a vier-Stokes equation, and which is successful in the application to the Lagrangian velocity fields 2 ). Here, we consider a model equation for N variables Xi (i = 1, 2, · · ·, N) as [ :t + v l X;(t) = .t C;;• X;(t) x.(t) + F;(t), J,k=l where Cijk, v and Fi are constant coefficients, the vis-cosity and a Gaussian random force with zero-mean, respectively. This model equation has similar prop-erties to those of the Navier-Stokes equation with an appropriate choice of the coefficients Cijk which sat-isfy the following two conditions. First, the nonlinear term conserves the energy~ L:i(Xi)2 • Second, there is only one direct-interaction, at the most, between an arbitrary pair of modes. For the Navier-Stokes sys-tem, the second property holds if a periodic boundary condition is imposed. This second property of weak nonlinear couplings is essential for DIA. The DIA is based upon a decomposition called direct-interaction decomposition Xi(t) = X~1°~ . k (tito) + X~;1 ~ . k (tito) , t toJo o t toJo o where xfJ1ojoko is an artificial field without the direct-interaction between three modes { Xio, Xj0 , Xko}, and to denotes the time when this decomposition is made. By the use of this decomposition, DIA is constructed under the following two assumptions: (i) The three d {x (O) X(O) X(O) } t t• . ll . d mo es ifijk, j fijk, kfijk are s a 1st1ca y m e-pendent of each other because there is no direct-interaction between them. (ii) The magnitude of x?) is much smaller than that of Xi(o) as long as t - t0 is of the order of the autocorrelation time of xi. In order to make the applicability limit of the ap-proximation clear, we should clarify the parameter re-gion in which the above two assumptions hold. Intu-itively, these assumptions may be satisfied if the num-ber of the degrees of freedom N is large enough to ran-240 domize and cancel the contribution of the indirect in-teractions to the correlation between the three modes, and to make the influences of artificially removing a direct-interaction from the system inconspicuous. We examine the above a priori argument by the use of numerical calculations. First, in order to con-firm whether the DIA assumption (i) is actually valid for large N or not, we evaluate the triple correlation function where an overbar stands for the long-time average, by the numerical integration of the model equation. The results of the calculations are as follows: Although the triple correlation Rijk takes nontrivial value in the true field Xi irrespective of N, it vanishes in the non-direct-interaction field xfJ1ik if N is large (N ;::- 20). Next, as for the second assumption, we evaluate the ensemble average of the magnitude of the direct-interaction field Xf1) (tito). Then, it is shown to be much smaller than the magnitude of Xi(o)(tito) as long as t - t0 is of the order of the time-scale of the autocorrelation function if N is large. In conclusion, DIA for a nonlinear system with weak couplings is applicable if the number of the degrees of freedom is sufficiently large. Actually, DIA gives excellent predictions for the autocorrela-tion function of Xi in the cases of large N (Fig.1). For further detailed arguments, see ref. 3). ,..--.._ ~ ~ 0.02 ,..--.._ ~ ----~ 0.01 0.1 0.2 0.3 0.4 0.5 Fig.1 The autocorrelation function of Xi for (v, N) = (0, 40). Thick and thin solid lines are respectively due to DIA and a long-time average of direct numerical integrations of the basic equation. References 1) Kraichnan, R. H., J. Fluid Mech. Q (1959) 497. 2) Kida, S. and Goto, S., J. Fluid Mech. 345 (1997) 307. 3) Goto, S. and Kida, S., Physica D (in press). 

Applicability of the Direct-interaction Approximation

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