Self-Similar Decay in the Kraichnan Model of a Passive Scalar

Original · EN

We study the two-point correlation function of a freely decaying scalar in Kraichnan's model of advection by a Gaussian random velocity field, stationary and white-noise in time but fractional Brownian in space with roughness exponent 0<ζ<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/γ)+1 are statistically realizable, where d is space dimension and γ=2-ζ. An infinite sequence of invariants Jℓ, ℓ=0,1,2,... is pointed out, where J₀ is Corrsin's integral invariant but the higher invariants appear to be new. We show that at least one of the first two invariants, J₀ or J₁, must be nonzero for realizable initial data. We classify initial data in long-time domains of attraction of the self-similar solutions, based upon these new invariants. Our results support a picture of ``two-scale'' decay with breakdown of self-similarity for a range of exponents (d+γ)/γ< a < (d+2)/γ, analogous to what has recently been found in decay of Burgers turbulence.