  Exploiting a Lagrangian strategy we present a numerical study for both
perturbative and nonperturbative regions of the Kraichnan advection model. The
major result is the numerical assessment of the first-order $1/d$-expansion by
M. Chertkov, G. Falkovich, I. Kolokolov and V. Lebedev ({\it Phys. Rev. E},
{\bf 52}, 4924 (1995)) for the fourth-order scalar structure function in the
limit of high dimensions $d$'s. %Two values of the velocity scaling exponent
$\xi$ have been considered: %$\xi=0.8$ and $\xi=0.6$. In the first case, the
perturbative regime %takes place at $d\sim 30$, while in the second at $d\sim
25$, %in agreement with the fact that the relevant small parameter %of the
theory is $\propto 1/(d (2-\xi))$. In addition to the perturbative results, the
behavior of the anomaly for the sixth-order structure functions {\it vs} the
velocity scaling exponent, $\xi$, is investigated and the resulting behavior
discussed.
