Anomalous scaling in random shell models for passive scalars.

A shell-model version of Kraichnan’s @Phys. Rev. Lett. 72, 1016 ~1994!# passive scalar problem is introduced which is inspired by the model of Jensen, Paladin, and Vulpiani @Phys. Rev. A 45, 7214 ~1992!#. As in the original problem, the prescribed random velocity field is Gaussian and d correlated in time, and has a power-law spectrum }k m , where km is the wave number. Deterministic differential equations for secondand fourth-order moments are obtained and then solved numerically. The second-order structure function of the passive scalar has normal scaling, while the fourth-order structure function has anomalous scaling. For j5 2 3 the anomalous scaling exponents zp are determined for structure functions up to p516 by Monte Carlo simulations of the random shell model, using a stochastic differential equation scheme, validated by comparison with the results obtained for the secondand fourth-order structure functions. @S1063-651X~96!13510-8#