Invariant manifold templates for chaotic advection

Abstract

Invariant manifolds can serve as a geometric template for the study of chaotic advection. In particular, global stable and unstable manifolds of invariant hyperbolic or normally hyperbolic sets form the boundaries of chaotic tangles in physical space, and these manifolds criss-cross one another to form an intricate network of lobes in the tangles. Manifold invariance implies that these lobes of fluid evolve from one to another in a well-defined manner, and it is in the context of lobe dynamics in the tangles that one exploits the invariant manifold template as a skeletal backbone for the study of transport, stretching, and mixing of fluid under chaotic advection, all of which are intimately connected. More concretely, in the study of fluid transport, one can use segments of the invariant manifolds to partition the irregular flow into regions of qualitatively different fluid motion, such as open flow versus closed flow, or different rolls, and then perform a lobe-dynamic study of fluid transport into, and out of, these regions (e.g. entrainment and detrainment) in a geometrically exact context, specified in terms of images and pre-images of a set of turnstile lobes. Mixing can be thought of as a consequence of barrier destruction, and transport across partially destroyed barriers can be studied in a lobe-dynamic context, providing a basic measure of mixing. A practical example of a transport calculation is the use of Melnikov theory to obtain analytical expressions for lobe areas, and hence a measure of flux in the mixing region, which allows for efficient searches through parameter space in order to enhance or diminish a mixing process. The stretching of fluid elements can be studied in the context of the lobes repeatedly stretching, folding, and wrapping around one another in an approximately self-similar manner. Such a picture can be viewed as a generalization of the horseshoe map paradigm for stretching in 2D chaotic tangles, which restricts its interest to a Cantor set near hyperbolic fixed points. A symbolic description of the evolution of lobe boundaries provides a framework for studying the global topology of, and mechanisms for, enhanced stretching in chaotic tangles. In particular, it is found that, though stretching under chaotic flows can be viewed in terms of products of weakly correlated events, there is a range of stretch, spatial, and temporal scales associated with these products, leading to a range of stretch processes. Invariant manifold geometry plays a central role in these stretch processes. For example, the invariant hyperbolic or normally hyperbolic sets have special relevance as engines of good stretching, the turnstile lobes play a fundamental role of re-orienting line elements between each successive pass by the hyperbolic sets, and intersections of stable and unstable manifolds act as partitions between segments of lobe boundaries that experience qualitatively different stretch histories. A notable consequence of the range of scales in the stretch processes is the range of statistics, and hence multifractal properties, associated with the high-stretch tails of finite-time Lyapunov exponent distributions, which has significant impact on interfacial evolution and striation width in the small-scale limit. Regions in physical space of vanishingly small initial measure associated with these tails are thus shown to be able to play a significant role under the flow. The mixing of passive scalars or vectors as a result of such stochastic effects as molecular diffusion or chemical reactions tends to wash out the structure associated with invariant manifolds. However, one can study this interplay between advection and mixing in the context of lobe evolution. In particular, mixing efficiency is seen to depend not only on the stretch experienced by lobe boundaries, but also on the thickness of the lobes, and their separation from other lobes. Though essentially all chaotic advection studies have been in the context of 2D time-periodic velocity fields, we show how an invariant manifold template can apply to quite general circumstances, such as quasiperiodic and indeed aperiodic time dependences, and 3D velocity fields. In all cases, the invariant manifold templates are seen to have physical reality, and the unstable manifolds act as a dominant structure in the regions of irregular flow.