Fixed points, stability and intermittency in a shell model for advection of passive scalars

Abstract

We investigate the fixed points of a shell model for the turbulent advection of passive scalars introduced Jensen, Paladin and Vulpiani. The passive scalar field is driven by the velocity field of the popular GOY shell model. The scaling behavior of the static solutions is found to differ significantly from Obukhov-Corrsin scaling theta_n ~ k_n^(-1/3) which is only recovered in the limit where the diffusivity vanishes, D -> 0. From the eigenvalue spectrum we show that any perturbation in the scalar will always damp out, i.e. the eigenvalues of the scalar are negative and are decoupled from the eigenvalues of the velocity. Furthermore we estimate Lyapunov exponents and the intermittency parameters using a definition proposed by Benzi et al. The full model is as chaotic as the GOY model, measured by the maximal Lyapunov exponent, but is more intermittent.