Scaling properties of growing surfaces.

Abstract

We investigate the theory of growth of anisotropically self-similar (i.e. self-affine) rough surfaces. Such surfaces have been observed in recent experiments on fluid displacement in porous media, bacterial colony expansion, meniscus front propagation in paper and burning of paper. It has been suggested that novel types of noise might exist in some of the experiments. We studied the effects of two examples of novel noise on the scaling properties of simulated surfaces. For growth with power-law noise of a particular power, we found that the bulk of the deposit is a fractal object. The surface scaling properties in the general case are solved exactly using scaling arguments. For another novel type of noise, namely long range temporally correlated noise, dynamical renormalization group predictions of the scaling exponents are verified numerically. In both cases, much progress has been obtained in the identification, understanding and suppression of serious crossover effects which have brought about many controversies. To test the validity of the models in describing the experiments, we propose an "inverse method", which is a novel numerical technique for extracting continuum equation descriptions of experimental or simulated surfaces. The technique has already proven to have excellent performance on simulated surfaces and is expected to have a wide scope of applications in other macroscopic dynamical systems.