Finite-time Lagrangian transport analysis: Stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents

📝 Abstract

We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of “transport barriers” in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterizing transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate, in a concrete manner, the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of “flow transition” which occurs when finite-time hyperbolicity is lost, or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing, and important, new area of dynamical systems theory.

💡 Analysis

We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of “transport barriers” in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterizing transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate, in a concrete manner, the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of “flow transition” which occurs when finite-time hyperbolicity is lost, or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing, and important, new area of dynamical systems theory.

📄 Content

Finite-time Lagrangian transport analysis: Stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents. Micha l Branicki1,2 and Stephen Wiggins1 1 School of Mathematics, University of Bristol, University Walk, BS8 1TW, UK 2 College of Earth, Ocean, and Environment, University of Delaware, 111 Robinson Hall, Newark, DE 19716, USA Abstract We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of transport barriers in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterizing transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate, in a concrete manner, the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of flow transition which occurs when finite-time hyperbolicity is lost, or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing, and important, new area of dynamical systems theory. 1 Introduction Organised or ‘coherent’ structures in fluid flows have been a subject of intense study for some time, especially since the seminal paper of Brown and Roshko ([14]). The dynamical systems approach to the Lagrangian aspects of fluid transport, which became widespread in the 1980’s and 90’s, has provided a variety of techniques for determining the existence and quantifying ‘organised structures’ in fluid flows. Hyperbolic trajectories and their associated stable and unstable manifolds have provided one approach to this problem, in both the periodic and aperiodic time dependent settings, that dates back to the beginning of studies of ‘chaotic advection’ in fluid flows ([68, 3, 1, 4, 86, 47, 74]). More recently, the notion of ‘Lagrangian coherent structure’ (henceforth LCS) derived from finite-time Lyapunov exponent (FTLE) fields has provided another means of identifying coherent flow structures in fluid flows which can be used in Lagrangian transport analysis ([42, 37, 38, 78, 59]). The purpose of this paper is to compare the methods based on determination of stable and unstable manifolds of hyperbolic trajectories with LCS’s derived from FTLE’s as techniques for uncovering organised structures in fluid flows and quantifying their influence on transport. 1 arXiv:0908.1129v1 [nlin.CD] 7 Aug 2009 We begin in Section 2 by reviewing some theoretical issues associated with Lagrangian transport analysis in time-dependent vector fields defined over a finite time interval. We also and take the opportunity to clarify a number of misconceptions that have arisen in the literature concerning the applicability of hyperbolic trajectories and their stable and unstable manifolds in analysing Lagrangian transport in fluid flows, especially with respect to their comparison with LCS’s. This will naturally lead to the issue of a relationship between the stable and unstable manifolds of hyperbolic trajectories and LCS’s. The purpose of this paper is to compare the methods based on detection of stable and unstable manifolds of hyperbolic trajectories with LCS’s derived from FTLE fields as techniques for uncovering organised structures in unsteady fluid flows. We will particularly focus on the performance and applicability of these techniques in flows undergoing transitions associated with a loss or gain of finite-time hyperbolicity by some trajectories. An understanding of this relationship is essential for understanding the role that each of these structures plays in Lagrangian transport. Both methods can have drawbacks as tools for diagnosing the finite- time Lagrangian flow structure, In Section 3 we consider a series of examples which aim at providing a guide for choosing the most suitable technique for a particular application. We begin the discussion by studying a one-dimensional non-autonomous system which can be solved analytically and which provides a good illustration of issues concerning the finite-time hyperboli

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