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Recently, it has been proposed that the Navier-Stokes equations and arelevant linear advection model have the same long-time statistical properties,in particular, they have the same scaling exponents of their structurefunctions. This assertion has been investigate rigorously in the context ofcertain nonlinear deterministic phenomenological shell model, the Sabra shellmodel, of turbulence and its corresponding linear advection counterpart model.This relationship has been established through a "homotopy-like" coefficient$\lambda$ which bridges continuously between the two systems. That is, for$\lambda=1$ one obtains the full nonlinear model, and the corresponding linearadvection model is achieved for $\lambda=0$. In this paper, we investigate thevalidity of this assertion for certain stochastic phenomenological shell modelsof turbulence driven by an additive noise. We prove the continuous dependenceof the solutions with respect to the parameter $\lambda$. Moreover, we show theexistence of a finite-dimensional random attractor for each value of $\lambda$and establish the upper semicontinuity property of this random attractors, withrespect to the parameter $\lambda$. This property is proved by a pathwiseargument. Our study aims toward the development of basic results and techniquesthat may contribute to the understanding of the relation between the long-timestatistical properties of the nonlinear and linear models.

Stochastic attractors for shell phenomenological models of turbulence