Four-point correlation function of a passive scalar field in rapidly fluctuating turbulence: Numerical analysis of an...

A numerical analysis is made on the four-point correlation function in a similarity range of a model of two-dimensional passive scalar field ψ advected by a turbulent velocity field with infinitely small correlation time. The model yields an exact closure equation for the four-point correlation Ψ{4} of ψ, which may be casted into the form of an eigenvalue problem in the similarity range. The analysis of the eigenvalue problem gives not only the scale dependence of Ψ{4} , but also the dependence on the configuration of the four points. The numerical analysis gives S4(R)∝R{ζ{4}} in the similarity range in which S2(R)∝R{ζ{2}} , where S_{N} is the structure function defined by S{N}(R)≡⟨[ψ(x+R)-ψ(x)]{N} and ζ{4}≠2ζ{2} . The estimate of ζ_{4} by the numerical analysis of the eigenvalue problem is in good agreement with numerical simulations so far reported. The agreement supports the idea of universality of the exponent ζ{4} in the sense that ζ_{4} is insensitive to conditions of ψ outside the similarity range. The numerical analysis also shows that the correlation C(R,r)≡[ψ(x+R)-ψ(x)]{2}[ψ(x+r)-ψ(x)]{2}> is stronger than that given by the joint-normal approximation, and scales like C(R,r)∝(r/R){χ} for r/R<<1 with R and r in the similarity range, where χ is a constant depending on the angle between R and r .