Large scale spatio-temporal behaviour in surface growth

Abstract

This paper presents new findings concerning the dynamics of the slow height variations in surfaces produced by the two-dimensional isotropic Kuramoto-Sivashinsky equation with an additional nonlinear term. In addition to the disordered patterns of specific size evident at small scales, slow height variations of scale-free character become increasingly evident when the system size is increased. The surface spectrum at small wave numbers has a power-law shape with a lower cut-off due to the finite system size. The temporal properties of these long-range height variations are investigated by analysing the time series of surface roughness fluctuations. The resulting power-spectral densities can be expressed as a sum of white noise and a generalized Lorentzian whose cut-off frequency varies with system size. The dependence of this lower cut-off frequency on the smallest wave number connects spatial and temporal properties and gives new insight into the surface evolution on large scales.