Large scale spatio-temporal behaviour in surface growth

TL;DR

This study investigates large-scale spatio-temporal behavior in surface growth modeled by the two-dimensional Kuramoto-Sivashinsky equation, revealing scale-free height variations and their spectral-temporal characteristics as system size increases.

Contribution

It provides new insights into the large-scale dynamics and spectral-temporal properties of surface growth, highlighting the connection between spatial and temporal scales in the model.

Findings

Surface spectrum exhibits a power-law shape at small wave numbers.

Long-range height variations follow a power spectral density combining white noise and Lorentzian.

Lower cut-off frequency scales with system size, linking spatial and temporal properties.

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.