Self-similar decay in the Kraichnan model of a passive scalar

We study the two-point correlation function of a freely decaying scalar in Kraichnan's model of advection by a Gaussian random velocity field that is stationary and white noise in time, but fractional Brownian in space with roughness exponent 0 < < 2, appropriate to the inertial-convective range of the scalar. We find all self-similar solutions by transforming the scaling equation to Kummer's equation. It is shown that only those scaling solutions with scalar energy decay exponent a (d/gamma)+ 1 are statistically realizable, where d is space dimension and gamma = 2 -zeta. An infinite sequence of invariants J(p), p = 0, 1, 2,..., is pointed out, where J(0) is Corrsin's integral invariant but the higher invariants appear to be new. We show that at least one of the invariants J(0) or J(1) must be nonzero (possibly infinite) for realizable initial data. Initial datum with a finite, nonzero invariant-the first being J(p)-converges at long times to a scaling solution Phi (p) with a = (d/gamma) + p, p, = 0, 1. The latter belongs to an exceptional series of self-similar solutions with slretched-exponentiaI decay in space. However, the domain of attraction includes many initial data with power-law decay. When the initial datum has all invariants zero or infinite and also it exhibits power-law decay, then the solution converges at long times to a nonexceptional scaling solution with the same power-law decay. These results support a picture of a two-scale decay with breakdown of self-similarity for a range of exponents (d + gamma)/gamma < a <(d + 2)/gamma, analogous to what has recently been found in the decay of Burgers turbulence.