  Recently, it has been proposed that the Navier-Stokes equations and a
relevant linear advection model have the same long-time statistical properties,
in particular, they have the same scaling exponents of their structure
functions. This assertion has been investigate rigorously in the context of
certain nonlinear deterministic phenomenological shell model, the Sabra shell
model, 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 linear
advection model is achieved for $\lambda=0$. In this paper, we investigate the
validity of this assertion for certain stochastic phenomenological shell models
of turbulence driven by an additive noise. We prove the continuous dependence
of the solutions with respect to the parameter $\lambda$. Moreover, we show the
existence of a finite-dimensional random attractor for each value of $\lambda$
and establish the upper semicontinuity property of this random attractors, with
respect to the parameter $\lambda$. This property is proved by a pathwise
argument. Our study aims toward the development of basic results and techniques
that may contribute to the understanding of the relation between the long-time
statistical properties of the nonlinear and linear models.
