On the geometrical and statistical properties of dynamical systems : theory and applications

Citation

Mezic, Igor (1994) On the geometrical and statistical properties of dynamical systems : theory and applications. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/SM03-1C46. https://resolver.caltech.edu/CaltechETD:etd-07212005-131406

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. We develop analytical methods for studying particle paths in a class of three-dimensional incompressible fluid flows. We study three-dimensional volume preserving vector fields that are invariant under the action of a one-parameter symmetry group whose infinitesimal generator is autonomous and volume preserving. We show that there exists a coordinate system in which the vector field assumes a simple form. In particular, the evolution of two of the coordinates is governed by a time-dependent, one-degree-of-freedom Hamiltonian system with the evolution of the remaining coordinate being governed by a first-order differential equation that depends only on the other two coordinates and time. The new coordinates depend only on the symmetry group of the vector field. Therefore they are field independent. The coordinate transformation is constructive. If the vector field is time independent, then it posseses an integral of motion. Moreover, we show that the system can be further reduced to action-angle-angle coordinates. These are analogous to the familiar action-angle variables from Hamiltonian mechanics and are quite useful for perturbative studies of the class of sytems we consider. All of the above is useful in framing a perturbative setting for analyses of chaotic, volume-preserving vector fields. In particular, explicit expressions for the transformation to action-angle-angle coordinates is given. This leads to the proof of a KAM-type theorem for volume-preserving vector fields admitting a volume-preserving group of symmetries using the KAM-type result for three-dimensional maps. The proof of the persistence of finite cylinders, relevant in fluid dynamical applications is provided. Also, a Melnikov-type theory is developed, allowing for the prediction of parameter values for which the vector field possesses chaotic behavior.We discuss the integrability of the class of flows considered, and draw an analogy with Clebsch variables in fluid mechanics. Recently there has been a lot of numerical and experimental work on three dimensional, volume-preserving, chaotic fluid flows. The above theory can explain qualitative, geometric, features observed in these flows. But, the quantities of interest in those investigations are often of statistical nature. Furthermore, in most of these investigations, the flows considered are non-ergodic, with a rich structure of the phase space. The theory of statistical properties of dynamical systems developed in this thesis is based on the Birkhoff's ergodic theorem, ergodic partition, and methods of probability theory. It is shown that, in the case when the system is not ergodic, the only quantities necessary to describe the limiting behavior (when the time or the number of iterations [approaching infinity] behavior of these systems are the time averages. Using this observation, necessary and sufficient condition are derived for the ubiquitous [...] asymptotic behavior of the dispersion. A link is obtained between probability distributions of sum functions and the ergodic partition, which is used to explain the phenomenon of patchiness in fluid flows. The problem of first passage times is analyzed, and some conjectures inspired by numerical experiments proved. The theory is developed for both maps and flows, and has applications in a variety of problems related to the statistical description of chaotic motion in physical systems. Two specific applications are diffusion in two-dimensional, area-preserving maps, and shear dispersion in fluid flows. An obvious question arising from this part of the work was: how can one calculate the ergodic partition, which is an important ingredient of the statistical part of the theory. In ergodic theory, two ways of presentation of the ergodic partition exist. These two approaches can be successfully joined to provide a simple constructive algorithm for the construction of the ergodic partition. The ergodic partition of the compact metric space A, under the dynamics of a continuous automorphism T, is shown to be the product of measurable partitions of the space induced by the time averages of a dense, countable subset of the set of all continuous functions on A. As a consequence of this, closed ergodic components are shown to be uniquely ergodic. Also, a connection can be made between the ergodic partitions induced by the time averages of measurable, bounded functions, and the ergodic partition. Besides giving a method of constructing the ergodic partition, this work might give rise to numerical algorithms for computation of the ergodic partition. All of the above theory is applied to the ideal, incompressible fluid flow induced by a helical vortex filament in an axisymmetric time-dependent strain field. It is shown that the helical filament stays helical for all times. Using symmetry concepts we transform the velocity field to a particularly simple form that is convenient for the use of perturbation methods. We analyze bifurcations and the structure of particle paths in the unperturbed velocity field (no strain). The underlying geometrical structures in the unperturbed problem are cylinders and two dimensional separatrices. Away from separatrices we transform the system into coordinates that enable us to use KAM theory to show the persistence of infinite cylinders in the perturbed flow. Further, we analyze the unperturbed motion on separating manifolds, and present a three-dimensional Melnikov theory for the analysis of the motion near the separatrices under perturbation. We use this analysis to propose that chaoticity of the motion provides a physical mechanism for the Ranque effect for swirling flows in pipes. Finally, we analyse the problem of shear dispersion of passive scalars in our flow. A natural question related to the above considerations of statistics of deterministic dynamical systems is how do they affect the statistical properties of the system when noise is added. This leads to a study of the convection-diffusion equation. We establish conditions for the maximal, [...] behavior of the effective diffusivity in time periodic incompressible velocity fields for both the [...] 0 limits. Using ergodic theory, these conditions can be interpreted in terms of the Lagrangian time averages of the velocity. We reinterpret the maximal effective diffusivity conditions in terms of a Poincare map of a velocity field. The connection between the [...] asymptotic behavior of the effective diffusivity and [...] asymptotic dispersion of the nondiffusive tracer is established. Several examples are analyzed: we relate the existence of the accelerator modes in a flow with [...] effective diffusivity, and show how maximal effective diffusivity can appear as a result of a time-dependent perturbation of a steady cellular velocity field. Also, three-dimensional, symmetric, time-dependent duct velocity fields are analyzed, and the mechanism for effective diffusivity with Peclet number dependence different from [...] in time-dependent flows is established, thus generalizing the Taylor-Aris dispersion theory.

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