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

We investigate the fixed points of a shell model for the turbulent advection of passive scalars introduced in Jensen, Paladin, and Vulpiani [Phys. Rev. A 45, 7214 (1992)]. The passive scalar field is driven by the velocity field of the popular Gledzer-Ohkitani-Yamada (GOY) shell model. The scaling behavior of the static solutions is found to differ significantly from Obukhov-Corrsin scaling straight theta(n) approximately k(-1/3)(n), 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. We estimate Lyapunov exponents and the intermittency parameters using a definition proposed by Benzi, Paladin, Parisi, and Vulpiani [J. Phys. A 18, 2157 (1985)]. The full model is found to be as chaotic as the GOY model, measured by the maximal Lyapunov exponent, but is more intermittent.