  We study effects of turbulent mixing on the random growth of an interface in
the problem of the deposition of a substance on a substrate. The growth is
modelled by the well-known Kardar--Parisi--Zhang model. The turbulent advecting
velocity field is modelled by the Kraichnan's rapid-change ensemble: Gaussian
statistics with the correlation function $\langle vv\rangle \propto
\delta(t-t') \, k^{-d-\xi}$, where $k$ is the wave number and $0<\xi<2$ is a
free parameter. Effects of compressibility of the fluid are studied. 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 asymptotic behaviour, associated with
four possible fixed points of the renormalization group equations. In addition
to known regimes (ordinary diffusion, ordinary growth process, and passively
advected scalar field), existence of a new nonequilibrium universality class is
established. Practical calculations of the fixed point coordinates, their
regions of stability and critical dimensions are calculated to the first order
of the double expansion in $\xi$ and $\varepsilon=2-d$ (one-loop
approximation). It turns out that for incompressible fluid, the most realistic
values $\xi=4/3$ or 2 and $d=1$ or 2 correspond to the case of passive scalar
field, when the nonlinearity of the KPZ model is irrelevant and the interface
growth is completely determined by the turbulent transfer. If the
compressibility becomes strong enough, the crossover in the critical behaviour
occurs, and these values of $d$ and $\xi$ fall into the region of stability of
the new regime, where the advection and the nonlinearity are both important.
