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

Abstract

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 $0<\zeta <2$, 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 $a\le (d/\gamma )+1$ are statistically realizable, where $d$ is space dimension and $\gamma =2-\zeta$. An infinite sequence of invariants $J_{\ell },\ell =0,1,2,...$ 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 first two invariants, $J_0$ or $J_1$, 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 $(d+\gamma )/\gamma <a<(d+2)/\gamma ,$ analogous to what has recently been found in decay of Burgers turbulence.

arXiv commentNow 68 pages. A proof of nonrealizability of scaling solutions with a>(d+γ)/γis added. Other minor changes and improvements. Accepted to appear in J. Stat. Phys

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Cited by in corpus (6)

• A reversible mesoscopic model of diffusion in liquids: from giant fluctuations to Fick's law
• Free Decay of Turbulence and Breakdown of Self-Similarity
• Kinetic Wave Turbulence
• Small-Scale Kinematic Dynamo and Non-Dynamo in Inertial-Range Turbulence
• Statistical Analysis of a Semilinear Hyperbolic System Advected by a White in Time Random Velocity Field
• Passive scalar mixing and decay at finite correlation times in the Batchelor regime