Random interface growth in a random environment: Renormalization group analysis of a simple model

We study the 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 modeled by the well-known Kardar–Parisi–Zhang model. The turbulent advecting velocity field is modeled by the Kraichnan 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 $\xi$ is a free parameter, $0<\xi<2$. We study the effects of the fluid compressibility. Using the field theory renormalization group, we show that depending on the relation between the exponent $\xi$ and the spatial dimension $d$, the system manifests different types of large-scale, long-time asymptotic behavior associated with four possible fixed points of the renormalization group equations. In addition to the known regimes (ordinary diffusion, the ordinary growth process, and a passively advected scalar field), we establish the existence of a new nonequilibrium universality class. We calculate the fixed-point coordinates and their stability regions and critical dimensions to the first order of the double expansion in $\xi$ and $\varepsilon=2-d$ (one-loop approximation). It turns out that for an incompressible fluid, the most realistic values $\xi=4/3$ or $\xi=2$ and $d=1$ or $d=2$ correspond to the case of a passive scalar field, where the nonlinearity of the Kardar–Parisi–Zhang model is irrelevant and the interface growth is completely determined by the turbulent transfer. If the compressibility becomes sufficiently strong, then a crossover occurs in the critical behavior, and these values of $d$ and $\xi$ are in the stability region of the new regime, where the advection and nonlinearity are both important. But the coordinates of the fixed point for this regime are in the unphysical region, and its physical interpretation hence remains an open problem.