Anomalous scaling in random shell models for passive scalars.

Abstract

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 \ensuremath{\delta} correlated in time, and has a power-law spectrum \ensuremath{\propto}${\mathit{k}}_{\mathit{m}}^{\mathrm{\ensuremath{-}}\ensuremath{\xi}}$, where ${\mathit{k}}_{\mathit{m}}$ is the wave number. Deterministic differential equations for second- and 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 \ensuremath{\xi}=2/3 the anomalous scaling exponents ${\mathrm{\ensuremath{\zeta}}}_{\mathit{p}}$ are determined for structure functions up to p=16 by Monte Carlo simulations of the random shell model, using a stochastic differential equation scheme, validated by comparison with the results obtained for the second- and fourth-order structure functions. \textcopyright{} 1996 The American Physical Society.