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

A numerical analysis is made on the four-point correlation function in a similarity range of a model of two-dimensional passive scalar field psi advected by a turbulent velocity field with infinitely small correlation time. The model yields an exact closure equation for the four-point correlation Psi(4) of psi, 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 Psi(4), but also the dependence on the configuration of the four points. The numerical analysis gives S-4(R) alpha R-zeta 4 in the similarity range in which S-2(R) alpha R-zeta 2, where S-N is the structure function defined by S-N(R) equivalent to <[psi(x + R) - psi(x)](N) and zeta(4) not equal 2 zeta(2). The estimate of zeta(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 zeta(4) in the sense that zeta(4) is insensitive to conditions of psi outside the similarity range. The numerical analysis also shows that the correlation C(R, r) equivalent to <[psi(x + R) - psi(x)](2) X[psi(x+r) - psi(x)](2) is stronger than that given by the joint-normal approximation, and scales like C(R, r) alpha (r/R)(chi) for r/R << 1 with R and r in the similarity range, where chi is a constant depending on the angle between R and r.