Instanton for the Kraichnan Passive Scalar Problem

Abstract

We consider high-order correlation functions of the passive scalar in the Kraichnan model. Using the instanton formalism we find the scaling exponents $ζ_n$ of the structure functions $S_n$ for $n\gg1$ under the additional condition $dζ_2\gg1$ (where $d$ is the dimensionality of space). At $n<n_c$ (where $n_c = dζ_2/[2(2-ζ_2)]$) the exponents are $ζ_n=(ζ_2/4)(2n-n^2/n_c)$, while at $n>n_c$ they are $n$-independent: $ζ_n=ζ_2 n_c/4$. We also estimate $n$-dependent factors in $S_n$, particularly their behavior at $n$ close to $n_c$.