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 ${\ensuremath{\zeta}}_{n}$ of the structure functions ${S}_{n}$ for $n\ensuremath{\gg}1$ under the additional condition $d{\ensuremath{\zeta}}_{2}\ensuremath{\gg}1$ (where d is the dimensionality of space). At $nl{n}_{c}$ [where ${n}_{c}=d{\ensuremath{\zeta}}_{2}/2(2\ensuremath{-}{\ensuremath{\zeta}}_{2})]$ the exponents are ${\ensuremath{\zeta}}_{n}=({\ensuremath{\zeta}}_{2}/4)(2n\ensuremath{-}{n}^{2}{/n}_{c}),$ while at $ng{n}_{c}$ they are n independent: ${\ensuremath{\zeta}}_{n}={\ensuremath{\zeta}}_{2}{n}_{c}/4.$ We also estimate n-dependent factors in ${S}_{n},$ particularly their behavior at n close to ${n}_{c}.$