  We study a self-organized critical system under influence of turbulent motion
of the environment. The system is described by the anisotropic continuous
stochastic equation proposed by Hwa and Kardar [{\it Phys. Rev. Lett.} {\bf
62}: 1813 (1989)]. The motion of the environment is modelled by the isotropic
Kazantsev--Kraichnan "rapid-change" ensemble for an incompressible fluid: it is
Gaussian with vanishing correlation time and the pair correlation function of
the form $\propto\delta(t-t') / k^{d+\xi}$, where $k$ is the wave number and
$\xi$ is an arbitrary exponent with the most realistic values $\xi = 4/3$
(Kolmogorov turbulence) and $\xi \to 2$ (Batchelor's limit). Using the
field-theoretic renormalization group, we find infrared attractive fixed points
of the renormalization group equation associated with universality classes,
i.e., with regimes of critical behavior. The most realistic values of the
spatial dimension $d=2$ and the exponent $\xi=4/3$ correspond to the
universality class of pure turbulent advection where the nonlinearity of the
Hwa--Kardar (HK) equation is irrelevant. Nevertheless, the universality class
where both the (anisotropic) nonlinearity of the HK equation and the
(isotropic) advecting velocity field are relevant also exists for some values
of the parameters $\varepsilon=4-d$ and $\xi$. Depending on what terms
(anisotropic, isotropic, or both) are relevant in specific universality class,
different types of scaling behavior (ordinary one or generalized) are
established.
