Geometric and statistical properties in the evolution of material surfaces in three-dimensional chaotic flows

In this article we analyze the invariant geometric properties of three-dimensional (3-D) chaotic flows. Attention is focused on the statistical (measure-theoretical) characterization of the asymptotic evolution of material surfaces forming the boundary between fluid elements, which can be characterized quantitatively in terms of intermaterial contact area density. The approach developed by Giona and Adrover [Phys. Rev. Lett. 81, 3864 (1998)] for diffeomorphisms (Poincare map of two-dimensional periodically forced flows) is extended to three-dimensional autonomous systems, for which a relation is obtained between intermaterial contact area density and stretching field. The Arnold-Beltrami-Childress flow is considered as a model system. The statistical and singular properties of the intermaterial contact area measure are addressed and some as yet unsolved fundamental issues related to nonautonomous three-dimensional flows are discussed. (C) 2001 American Institute of Physics.