Order and complexity in the Kuramoto-Sivashinsky model of weakly turbulent interfaces

Abstract

We present a large number of new geometric, ergodic and statistical properties of the Kuramoto-Sivashinsky equation modeling interfacial turbulence in various physical contexts. In addition, this equation has the remarkable property of inertial manifolds where some finite-dimensional dynamical system is rigorously equivalent to this infinite-dimensional partial differential equation. In moderate size domains (up to ten periods in length) a low-dimensional vector field skeleton underpins even strongly chaotic regimes and controls the bifurcations of the inertial manifold. The extreme numerical sensitivity of chaos in this dissipative PDE requires very high precision methods. Despite the geometrical complexities of the bifurcation structure, some statistical properties remain remarkably simple. There is overwhelming evidence that for some parameter values a permanent unsteady state exists. An unexpectedly simple diffusive relaxation of the large-scale fluctuations is extracted from extensive numerical simulations. In these calculations we observe long time tails for the correlation functions of relevant quantities. We propose an explanation in terms of an effective viscosity and compare the transport in the weakly turbulent interface with related theories for random interfaces and developed turbulence.