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The effects of a randomly moving environment on a randomly growing interfaceare studied by the field theoretic renormalization group analysis. The kineticgrowth of an interface (kinetic roughening) is described by theKardar-Parisi-Zhang stochastic differential equation while the velocity fieldof the moving medium is modelled by the Navier-Stokes equation with an externalrandom force. It is found that the large-scale, long-time (infrared) asymptoticbehavior of the system is divided into four nonequilibrium universality classesrelated to the four types of the renormalization group equations fixed points.In addition to the previously established regimes of asymptotic behavior(ordinary diffusion, ordinary kinetic growth process, and passively advectedscalar field), a new nontrivial regime is found. The fixed point coordinates,their regions of stability and the critical dimensions related to the criticalexponents (e.g., roughness exponent) are calculated to the first order of theexpansion in $\epsilon=2-d$ where $d$ is a space dimension (one-loopapproximation) or exactly. The new regime possesses a feature typical to thethe Kardar-Parisi-Zhang model: the fixed point corresponding to the regimecannot be reached from a physical starting point. Thus, physical interpretationis elusive.

The Kardar-Parisi-Zhang model of a random kinetic growth: effects of a randomly moving medium