  We study effects of random fluid motion on a system in a self-organized
critical state. The latter is described by the continuous stochastic model,
proposed by Hwa and Kardar [{\it Phys. Rev. Lett.} {\bf 62}: 1813 (1989)]. The
advecting velocity field is Gaussian, not correlated in time, with the pair
correlation function of the form $\propto \delta(t-t') / k_{\bot}^{d-1+\xi}$,
where $k_{\bot}=|{\bf k}_{\bot}|$ and ${\bf k}_{\bot}$ is the component of the
wave vector, perpendicular to a certain preferred direction -- the
$d$-dimensional generalization of the ensemble introduced by Avellaneda and
Majda [{\it Commun. Math. Phys.} {\bf 131}: 381 (1990)]. Using the field
theoretic renormalization group we show that, depending on the relation between
the exponent $\xi$ and the spatial dimension $d$, the system reveals different
types of large-scale, long-time scaling behaviour, associated with the three
possible fixed points of the renormalization group equations. They correspond
to ordinary diffusion, to passively advected scalar field (the nonlinearity of
the Hwa--Kardar model is irrelevant) and to the "pure" Hwa--Kardar model (the
advection is irrelevant). For the special choice $\xi=2(4-d)/3$ both the
nonlinearity and the advection are important. The corresponding critical
exponents are found exactly for all these cases.
