Anomalous Scaling Exponents of a White-Advected Passive Scalar

For Kraichnan’s problem of passive scalar advection by a velocity field delta-correlated in time, the limit of large space dimensionality $d\gg 1$ is considered. Scaling exponents of the scalar field are analytically found to be ${\zeta }_{2n}=n{\zeta }_2-2(2-{\zeta }_2)n(n-1)\mathrm{/}d$, while those of the dissipation field are ${\mu }_n=-2(2-{\zeta }_2)n(n-1)\mathrm{/}d$ for orders $n\ll d$. The refined similarity hypothesis ${\zeta }_{2n}=n{\zeta }_2+{\mu }_n$ is thus established by a straightforward calculation for the case considered.