Distinguished hyperbolic trajectories in time-dependent fluid flows: analytical and computational approach for velocity fields defined as data sets

Abstract

Abstract. In this paper we develop analytical and numerical methods for finding special hyperbolic trajectories that govern geometry of Lagrangian structures in time-dependent vector fields. The vector fields (or velocity fields) may have arbitrary time dependence and be realized only as data sets over finite time intervals, where space and time are discretized. While the notion of a hyperbolic trajectory is central to dynamical systems theory, much of the theoretical developments for Lagrangian transport proceed under the assumption that such a special hyperbolic trajectory exists. This brings in new mathematical issues that must be addressed in order for Lagrangian transport theory to be applicable in practice, i.e. how to determine whether or not such a trajectory exists and, if it does exist, how to identify it in a sequence of instantaneous velocity fields. We address these issues by developing the notion of a distinguished hyperbolic trajectory (DHT). We develop an existence criteria for certain classes of DHTs in general time-dependent velocity fields, based on the time evolution of Eulerian structures that are observed in individual instantaneous fields over the entire time interval of the data set. We demonstrate the concept of DHTs in inhomogeneous (or "forced") time-dependent linear systems and develop a theory and analytical formula for computing DHTs. Throughout this work the notion of linearization is very important. This is not surprising since hyperbolicity is a "linearized" notion. To extend the analytical formula to more general nonlinear time-dependent velocity fields, we develop a series of coordinate transforms including a type of linearization that is not typically used in dynamical systems theory. We refer to it as Eulerian linearization, which is related to the frame independence of DHTs, as opposed to the Lagrangian linearization, which is typical in dynamical systems theory, which is used in the computation of Lyapunov exponents. We present the numerical implementation of our method which can be applied to the velocity field Correspondence to: S. Wiggins (s.wiggins@bristol.ac.uk) given as a data set. The main innovation of our method is that it provides an approximation to the DHT for the entire time-interval of the data set. This offers a great advantage over the conventional methods that require certain regions to converge to the DHT in the appropriate direction of time and hence much of the data at the beginning and end of the time interval is lost.

Barriers in the transition to global chaos in collisionless magnetic reconnection. I. Ridges of the finite time Lyapunov exponent field

“…2 in terms of stable and unstable manifolds associated with uniformly hyperbolic trajectories, 3 which are also called distinguished hyperbolic trajectories. 4 These hyperbolic lines are the generalisations in phase space (x, y, z) of X-lines in a two-dimensional geometry. The associated manifolds intersect at the hyperbolic trajectory and form invariant surfaces.…”

A Lagrangian analysis of a developing and non-developing disturbance observed during the PREDICT experiment

“…. The stable and unstable manifolds of the DHT are the particle trajectories that approach the DHT forward/backward in time (Ide et al, 2002). The manifolds of the time-dependent flow are not crossed by trajectories, and form flow boundaries in the same manner that velocity streamlines do for steady flows (Malhotra and Wiggins, 1999;Coulliette and Wiggins, 2000).…”

“…These finite-length manifolds are the material lines which show maximal linear stability/instability for finite times. Hyperbolic trajectories and manifolds were shown by Duan and Wiggins (1996) and Ide et al (2002) to control particle transport and persist in time-dependent flows. The stable and unstable manifolds of a pair of hyperbolic fixed points were found by Rogerson (1999) to control transport in and out of a cat's eye region for a barotropic jet.…”

A simple kinematic model for the Lagrangian description of relevant nonlinear processes in the stratospheric polar vortex

“…Among others, techniques developed are finite-size Lyapunov exponents (FSLEs) (Aurell et al, 1997) and finitetime Lyapunov exponents (FTLEs) (Haller, 2000;Haller and Yuan, 2000;Haller, 2001;Shadden et al, 2005). Other techniques include DHTs (Ide et al, 2002;Ju et al, 2003) and the direct calculation of manifolds as material surfaces (Mancho et al, 2003(Mancho et al, , 2004(Mancho et al, , 2006b, the geodesic theory of Lagrangian coherent structures (LCS) (Haller and Beron-Vera, 2012) and the variational theory of LCS (Farazmand and Haller, 2012), etc. Our choice in this work will be the use of the Lagrangian descriptor (LD) function M introduced by Madrid and Mancho (2009) and Mendoza and Mancho (2010).…”

“…Nevertheless, manifolds can still be defined constructively with the following procedure. At the beginning time, these curves are approximated by segments with short length, aligned with the stable and unstable subspaces of the DHT identified with algorithms described in Ide et al (2002) and Madrid and Mancho (2009). This starting step aims to build a finite-time version of the asymptotic property of manifolds.…”