  Kinetic roughening of a randomly growing surface can be modelled by the
Kardar-Parisi-Zhang equation with a time-independent (``spatially quenched'' or
``columnar'') random noise. In this paper, we use the field-theoretic
renormalization group approach to investigate how randomly moving medium
affects the kinetic roughening. The medium is described by the stochastic
differential Navier-Stokes equation for incompressible viscous fluid with an
external stirring force. We find that the action functional for the full
stochastic problem should be extended to be renormalizable: a new nonlinearity
must be introduced. Moreover, in order to correctly couple the scalar and
velocity fields, a new dimensionless parameter must be introduced as a factor
in the covariant derivative of the scalar field. The resulting action
functional involves four coupling constants and a dimensionless ratio of
kinematic coefficients. The one-loop calculation (the leading order of the
expansion in $\varepsilon=4-d$ with $d$ being the space dimension) shows that
the renormalization group equations in the five-dimensional space of those
parameters reveal a curve of fixed points that involves an infrared attractive
segment for $\varepsilon>0$.

Statistical Mechanics