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 $d\gg1$ is considered. Scaling exponents of the scalar field are analytically found to be $\zeta_{2n}=n\zeta_2-2(2-\zeta_2)n(n-1)/d$, while those of the dissipation field are $\mu_{n}=-2(2-\zeta_2)n(n-1)/d$ 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.