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It was conjectured recently that Statiscally Preserved Structures underliethe statistical physics of turbulent transport processes. We analyze here indetail the time-dependent (non compact) linear operator that governs thedynamics of correlation functions in the case of shell models of passive scalaradvection. The problem is generic in the sense that the driving velocity fieldis neither Gaussian nor $\delta$-correlated in time. We show how to naturallydiscuss the dynamics in terms of an effective compact operator that displays"zero modes" which determine the anomalous scaling of the correlationfunctions. Since shell models have neither Lagrangian structure nor "shapedynamics" this example differs significantly from standard passive scalaradvection. Nevertheless with the necessary modifications the generality andefficacy of the concept of Statistically Preserved Structures are furtherexemplified. In passing we point out a bonus of the present approach, inproviding analytic predictions for the time-dependent correlation functions indecaying turbulent transport.Comment: 13 pages, 10 figures. Submitted to Phys. Rev.

Statistically preserved structures in shell models of passive scalar advection