Transport, stretching, and mixing in classes of chaotic tangles

Citation

Beigie, Darin (1993) Transport, stretching, and mixing in classes of chaotic tangles. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/bk8m-a904. https://resolver.caltech.edu/CaltechTHESIS:11292012-143847218

Abstract

We use global stable and unstable manifolds of invariant hyperbolic sets as templates for studying the dynamics within classes of homoclinic and heteroclinic chaotic tangles, focusing on transport, stretching, and mixing within these tangles. These templates are exploited in the context of lobes in phase space mapping within invariant lobe structures formed out of the intersecting global stable and unstable manifolds. Our interest lies in: (a) extending the templates and their applications to fundamentally larger classes of dynamical systems, (b) expanding the description of dynamics offered by the templates, and (c) applying the templates to the study of various nonlinear physical phenomena, such as stirring and mixing under chaotically advecting fluids and molecular dissociation under external electromagnetic forcing. These and other nonlinear physical phenomena are intimately connected to the underlying chaotic dynamics, and describing these processes encourages study of finite-time, or transient, phenomena as well as asymptotics, the former being much more virgin territory from a dynamical systems perspective. Under the rubric of themes (a)-(c) we offer five studies.

(i) One of the canonical classes of dynamical systems in which these templates have been exploited is defined by 2D time-periodic vector fields, where the analysis reduces to a 2D Poincaré map. In this instance, one is well-equipped with basic elements of dynamical systems theory associated with 2D maps, such as the Smale horseshoe map paradigm, KAM-tori, hyperbolic fixed points and their global stable and unstable manifolds that define the tangle boundaries, classical Melnikov theory, and so on. Our first study performs a systematic extension of the dynamical system constructs associated with 2D time-periodic vector fields to apply to 2D vector fields with more complicated time dependences. In particular, we focus on 2D vector fields with quasiperiodic, or multiple-frequency, time dependence. Any extension past the time-periodic case requires the fundamental generalization from 2D maps to sequences of 2D nonautonomous maps. To large extent the constructs associated with 2D Poincaré maps are found to be robust under this generalization. For example, the Smale horseshoe map generalizes to a traveling horseshoe map sequence, hyperbolic fixed points generalize to points that live on invariant normally hyperbolic tori, and invariant 2D chaotic tangles generalize to sequences of 2D chaotic tangles derived from an invariant tangle embedded in a higher-dimensional phase space. It is within the setting of 2D lobes mapping within a sequence of 2D lobe structures that one has a template for systematic study of the dynamics generated by multiple-frequency vector fields. Dynamical systems tools with which to study these systems include: (i) a generalized Melnikov theory that offers an approximate analytical measure of stable and unstable manifold separation in the tangles, the basis for a variety of analytical studies, and (ii) a double phase slice sampling method that allows for numerical computation of precise 2D slices of the higher-dimensional invariant chaotic tangles, the basis for numerical work. The Melnikov function defines relative scaling functions which give an analytical measure of the relative importance of each frequency on manifold separation. With the template and tools in hand, we study multiple-frequency dynamics and compare with single-frequency dynamics. We recast lobe dynamics under a hi-infinite sequence of nonautonomous maps in closed form by exploiting underlying periodicity properties of the vector field, and present numerical simulations of sequences of chaotic tangles and lobe dynamics within these tangles. In contrast to lobes of equal area mapping within a fixed 2D lobe structure found under single-frequency forcing, we find lobes of varying areas mapping within a sequence of lobe structures that are distorting and breathing from one time sample to the next, affording greater freedom in the nature of the dynamics. Our primary focus in this new setting is on phase space transport (we consider stretching and mixing in other contexts in later studies). The non-integrable motion in chaotic tangles allows for transport between various regions of phase space, in particular, between regions corresponding to qualitatively different types of motion, such as bounded and unbounded motion. This transport is intimately connected to basic physical processes, such as the fluid mixing and molecular dissociation processes. Transport theory refers to the enterprise where one uses a combination of invariant manifold theory, Melnikov theory, numerical simulation and/or approximate models such as Markov models, to partition phase space into regions of qualitatively different behavior (such as bounded and unbounded motion), establish complete and partial barriers between the regions, identify the turnstile lobes that are the gateways for transport across partial barriers, and then study in the context of lobe dynamics such phase space transport issues as flux and escape rates from a particular region. The formal construction of a transport theory for multiple-frequency vector fields is more involved than in the single-frequency case, as a consequence of more complicated manifold geometry. This geometry is uncovered and explored, however, via theorems and numerical studies based on Melnikov theory. We then partition phase space and define turnstiles in the higher-dimensional autonomous setting, and from this obtain the sequence of partitions and turnstiles in the 2D nonautonomous setting. A main new feature of transport is its manifestation in the context of a sequence of time-dependent regions, and we argue this is consistent with a Lagrangian viewpoint. We then perform a detailed study of such transport properties as flux, lobe geometry, and lobe content. In contrast to the single-frequency case, where a single flux suffices, in the multiple-frequency case a variety of fluxes are allowed, such as different types of instantaneous, finite-time average, and infinite-time average flux. We find for certain classes of multiple-frequency forcing that infinite-time average flux is maximal in a particular single-frequency limit, but that the spatial variation of lobe areas found in multiple-frequency systems affords greater freedom to enhance or diminish finite-time transport quantities. We illustrate our study with a quasiperiodically oscillating vortical flow that gives rise to chaotic fluid trajectories and a quasiperiodically forced Duffing oscillator. We explain how the analysis generalizes to vector fields with more complicated time dependences than quasiperiodic.

(ii) Besides the destruction of phase space barriers, allowing for phase space transport, other essential features of the dynamics in chaotic tangles include greatly enhanced stretching and mixing. Our second study returns to 2D time-periodic vector fields and uses invariant manifolds as templates for a global study of stretching and mixing in chaotic tangles. The analysis here thus complements the one of transport via invariant manifolds, and can essentially be viewed as a generalization of the horseshoe map construction to apply to entire material interfaces inside the tangles. Given the dominant role of the unstable manifold in chaotic tangles, we study the stretching of a material interface originating on a segment of the unstable manifold associated with a turnstile lobe. We construct a symbolic dynamics formalism that describes the evolution of the entire material curve, which is the basis for a global understanding of the stretch processes in chaotic tangles, such as the topology of stretching, the mechanisms for good stretching, and the statistics of stretching. A central interest will be in understanding the stretch profile of the material interface, which is the graph of finite-time stretch experienced as a function of location on the interface. In a near-integrable setting (meaning we add a perturbation to the vector field of an originally integrable system) we argue how the perturbed stretch profile can be understood in terms of a corresponding unperturbed stretch profile approximately repeating itself on smaller and smaller scales, as described by the symbolic dynamics. The basic interest is in how the non-uniformity in the unperturbed stretch profile can approximately carry through to the non-uniformity in the perturbed stretch profile, and this non-uniformity can play a basic role in mixing properties and stretch statistics. After the stretch analysis we then add to the deterministic flows a small stochastic component, corresponding for example to molecular diffusion (with small diffusion coefficient D) in a fluid flow, and study the diffusion of passive scalars across material interfaces inside the chaotic tangles. For sufficiently thin diffusion zones, the diffusion of passive scalars across interfaces can be treated as a one-dimensional process, and diffusion rates across interfaces are directly related to the stretch history of the interface. Our understanding of stretching thus directly translates into an understanding of mixing. However, a notable exception to the thin diffusion zone approximation occurs when an interface folds on top of itself so that neighboring diffusion zones overlap. We present an analysis which takes into account the overlap of neighboring diffusion zones, capturing a saturation effect in the diffusion process relevant to efficiency of mixing. We illustrate the stretching and mixing study in the context of two oscillating vortex pair flows, one corresponding to an open heteroclinic tangle, the other to a closed homoclinic tangle. Though we focus here on single-frequency systems, from the previous study the extensions to multiple-frequency systems should be clear.

(iii) We then study stretching from a different perspective, focusing on rates of strain experienced by infinitesimal line elements as they evolve under near-integrable chaotic flows associated with 2D time-periodic velocity fields. We introduce the notion of irreversible rate of strain responsible for net stretch, study the role of hyperbolic fixed points as engines for good irreversible straining, and observe the role of turnstiles as mechanisms for enhancing straining efficiency via re-orientation of line elements and transport of line elements to regions of superior straining.

(iv) The remaining two studies can be viewed as applications of the material developed in the previous studies, although both applications develop new theory and/or new ideas as well. The first application studies the dynamics associated with a quasi-periodically forced Morse oscillator as a classical model for molecular dissociation under external quasiperiodic electromagnetic forcing. The forcing entails destruction of phase space barriers, allowing escape from bounded to unbounded motion, and we study this transition in the context of our quasiperiodic theory, comparing with single-frequency forcing. New and interesting features of this application beyond the subject matter of the previous quasiperiodic study includes that the relevant fixed point of the unforced system is non-hyperbolic and at infinity, and the study of additional transport issues, such as escape (implying dissociation) from a particular level set of the unforced Hamiltonian system corresponding to a quantum state. We find for example that though infinite-time average flux can be maximal in a single-frequency limit, escape from a level set, or equivalently lobe penetration, can be maximal in the multiple-frequency case.

(v) The second application studies statistical relaxation of distributions of finite-time Lyapunov exponents associated with interfaces evolving within the chaotic tangles of 2D time-periodic vector fields. Whereas recent studies claim or give evidence that distributions of finite-time Lyapunov exponents are essentially Gaussian, our previous analysis of stretching via the symbolic dynamics construction shows the wide variety of stretch processes and stretch scales involved in the tangle, motivating our further study of stretch statistics. In particular, we focus on the high-stretch tails of finite-time Lyapunov exponents, which have relevance in incompressible flows to the mixing properties and multifractal characteristics of passive scalars and vectors in the limit of small spatial scales. Previous studies of stretch distributions consider a fixed number of points, thus lacking adequate resolution to study these tails. Instead, we use a dynamic point insertion scheme to maintain adequate interfacial covering, entailing extremely good resolution at high-stretch tails. These tails show a great range in behavior, varying from essentially Gaussian to nearly exponential, and these non-Gaussian deviations can have a significant effect on interfacial stretching, one that persists asymptotically. These non-Gaussian deviations can be associated with very small probabilities, thus indicating the need for highly-resolved numerical studies of stretch statistics. We explain the nearly exponential tail in a particular limiting regime corresponding to highly non-uniform stretch profiles, and explore how the full statistics might be captured by elementary models for the stretch processes.

Thesis Files