S. Goto

S. Kida

National Institute for Fusion Science (NIFS-Repository)

§25. Nonlinear Coupling Density Goto, S., Kida, S. There are two independent parameters which charac-terize nonlinearity of a dynamical system. One is the strength of nonlinearity, which expresses magnitude of the nonlinear term comparing that of the linear one, e.g., the Reynolds number in the Navier-Stokes system. Another parameter is nonlinear coupling density, which will be introduced in the next paragraph. The purpose of this study is to demonstrate that we can construct a closure theory, called direct-interaction approximation (D IA), even for a very strong nonlinear system, on con-dition that its nonlinear coupling is sparse. In order to introduce the nonlinear coupling density, we consider a dynamical system {Xiii = 1,2"" ,N} governed by d N N dtXi(t) = n L L Cijk Xj (t)Xk (t) - Xi(t) + fi(t) , j=lk=1 (1) where constant coefficients Cijk (rv 0(1)) satisfy three conditions: Cijk + Cjki + Ckij = 0 (conservation of en-ergy), Cijk = CHm,j+m,k+m (homogeneity) and Cijj = 0 (no self interaction). The coupling density p is defined by the number of direct interactions (i.e., the nonlinear interactions which appear explicitly on the right-hand side of (1)), between a pair of modes. In other words, p is the number of non-zero components of Cijk for fixed i and j. Thus, this system has three parameters: n (the strength of nonlinearity), N (the number of degrees of freedom) and p (:s N - 2). The purpose of closure theories is to predict statisti-cal quantities, e.g., the autocorrelation function V(r) = Xi(t + r)Xi(t) (an overbar denotes the ensemble aver-age), based on the governing equation (1) of the sys-tem. The DIAl} was proposed for this purpose, and its main assumption is summarized as "if there were no di-rect interaction between a triplet of modes, they would be statistically independent of each othe~'. It is intuitively expected and numerically confirmed2} (see below) that this assumption (then, DIA) is valid when the nonlinear coupling is sparse, even if n is large. Under the DIA assumption stated above, we can eas-ily derive a closed equation for V(r) in a statistically stationary state as [ d] 2Cn (r dr + 1 V(r) = - V(O) 10 dr' V(r')2 V(r - r') . (2) Prediction VDIA by this closure equation coincides excel-lently with VDNS by direct numerical simulation of (1), when p rv 1 and N = 103 (Fig.1(a), a sparse coupling 334 case). However, the overlap between two curves is much worse, when p = 8 and N = 10 (Fig.l(b), the densest coupling case). These results are consistent with the con-jecture. For a quantitative estimation of accuracy of the DIA equation, we define (see the figure caption) a pa-rameter ~, and plot it in Fig.2 for various combinations of p and N in the case of n -+ 00. This figure suggests that DIA is valid when p «: Vii, The DIA assumption requires smallness, in a statistical sense, of contribution from indirect interactions between the triplet of modes to the correlation between them. The validity condition can be regarded as the condition that such a randomization of indirect interactions does occur; it can be shown that if p ~ Vii, probability of existence of non-randomized indirect interactions gets larger. (a) 0 .001 ~ ___________ --, (p ,... 1, N = 103, 'R. -+ 00) o 0.02 (b) r 0.1,...,..---------------, (p = 8, N = 10, 'R. -+ 00) o r Fig.1 A utocorrelation function, VDIA (thick line) and VDNS (thin line). (Without any adjustable parameter.) 1.0,--------,--------,.-------, 00 0.8 o o 0.6 Ll 0.4 0.2 0 0 2 3 p/.JN Fig.2 Integral difference parameter, Reference 1) Kraichnan, R.H., Phys. Rev. 109 (1958) 1407. 2) Goto, S. & Kida, S., Physica D 117 (1998) 191; Non-linearity (submitted). 

Nonlinear Coupling Density

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