Regular and anomalous scaling of a randomly advected passive scalar

Extending Kolmogorov's refined similarity hypothesis to study the inertial behavior <[T(x+r,t)-T(x,t)](2n)>~r(zeta(2n)) of a passive scalar T(x,t) advected by a rapidly changing incompressible velocity field, a random variable straight theta was introduced by Ching [Phys. Rev. Lett. 79, 3644 (1997)]. In this paper, the statistical distribution of the random variable X=straight theta/sqrt[] is investigated analytically for the scaling in two limits, n-independent scaling zeta(2n)=zeta(2) and regular scaling zeta(2n)=nzeta(2), and numerically for the scaling of the Kraichnan conjecture zeta(2n)=1 / 2[sqrt[4ndzeta(2)+(d-zeta(2))(2)]-(d-zeta(2))]. For n-independent scaling zeta(2n)=zeta(2), the statistical distribution of X tends to an exponential distribution when zeta(2)-->0 or d-->infinity and to a Gaussian distribution when zeta(2)-->2 and d=2. For regular scaling zeta(2n)=nzeta(2), the statistical distribution of X tends to a Gaussian distribution when zeta(2)-->0 or d-->infinity. In d=2, there seems to be a phase transition for the probability density function P(X) from a convex to a concave function when the value of zeta(2) is increased and the critical point is zeta(2)=4/3 where the random variable X has a uniform distribution in [-sqrt[3],sqrt[3]]. In d=3, P(X) is a convex function for all 02. For the scaling of the Kraichnan conjecture, P(X) has two peaks in d=2 for zeta(2)>1.33, but, in d=3, it has only one peak for all 02.