Anomalous scaling exponents of a white-advected passive scalar.

Abstract

For Kraichnan's problem of passive scalar advection by a velocity field delta correlated in time, the limit of large space dimensionality $d\ensuremath{\gg}1$ is considered. Scaling exponents of the scalar field are analytically found to be ${\ensuremath{\zeta}}_{2n}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}n{\ensuremath{\zeta}}_{2}\ensuremath{-}2(2\ensuremath{-}{\ensuremath{\zeta}}_{2})n(n\ensuremath{-}1)/d$, while those of the dissipation field are ${\ensuremath{\mu}}_{n}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}2(2\ensuremath{-}{\ensuremath{\zeta}}_{2})n(n\ensuremath{-}1)/d$ for orders $n\ensuremath{\ll}d$. The refined similarity hypothesis ${\ensuremath{\zeta}}_{2n}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}n{\ensuremath{\zeta}}_{2}+{\ensuremath{\mu}}_{n}$ is thus established by a straightforward calculation for the case considered.