Time-Dependent 2-D Vector Field Topology: An Approach Inspired by Lagrangian Coherent Structures

This paper presents an approach to a time-dependent variant of the concept of vector field topology for 2-D vector fields. Vector field topology is defined for steady vector fields and aims at discriminating the domain of a vector field into regions of qualitatively different behaviour. The presented approach represents a generalization for saddle-type critical points and their separatrices to unsteady vector fields based on generalized streak lines, with the classical vector field topology as its special case for steady vector fields. The concept is closely related to that of Lagrangian coherent structures obtained as ridges in the finite-time Lyapunov exponent field. The proposed approach is evaluated on both 2-D time-dependent synthetic and vector fields from computational fluid dynamics

Geodesic Transport Barriers in Jupiter's Atmosphere: A Video-Based Analysis

Jupiter's zonal jets and Great Red Spot are well known from still images. Yet the planet's atmosphere is highly unsteady, which suggests that the actual material transport barriers delineating its main features should be time-dependent. Rare video footages of Jupiter's clouds provide an opportunity to verify this expectation from optically reconstructed velocity fields. Available videos, however, provide short-time and temporally aperiodic velocity fields that defy classical dynamical systems analyses focused on asymptotic features. To this end, we use here the recent theory of geodesic transport barriers to uncover finite-time mixing barriers in the wind field extracted from a video captured by NASA's Cassini space mission. More broadly, the approach described here provides a systematic and frame-invariant way to extract dynamic coherent structures from time-resolved remote observations of unsteady continua

Lagrangian Based Methods for Coherent Structure Detection

There has been a proliferation in the development of Lagrangian analytical methods for detecting coherent structures in fluid flow transport, yielding a variety of qualitatively different approaches. We present a review of four approaches and demonstrate the utility of these methods via their application to the same sample analytic model, the canonical double-gyre flow, highlighting the pros and cons of each approach. Two of the methods, the geometric and probabilistic approaches, are well established and require velocity field data over the time interval of interest to identify particularly important material lines and surfaces, and influential regions, respectively. The other two approaches, implementing tools from cluster and braid theory, seek coherent structures based on limited trajectory data, attempting to partition the flow transport into distinct regions. All four of these approaches share the common trait that they are objective methods, meaning that their results do not depend on the frame of reference used. For each method, we also present a number of example applications ranging from blood flow and chemical reactions to ocean and atmospheric flows. (C) 2015 AIP Publishing LLC.ONR N000141210665Center for Nonlinear Dynamic

How coherent are the vortices of two-dimensional turbulence?

We use recent developments in the theory of finite-time dynamical systems to objectively locate the material boundaries of coherent vortices in two-dimensional Navier--Stokes turbulence. We show that these boundaries are optimal in the sense that any closed curve in their exterior will lose coherence under material advection. Through a detailed comparison, we find that other available Eulerian and Lagrangian techniques significantly underestimate the size of each coherent vortex.Comment: revised versio

Multi-scale geometric analysis of Lagrangian structures in isotropic turbulence

We report the multi-scale geometric analysis of Lagrangian structures in forced isotropic turbulence and also with a frozen turbulent field. A particle backward-tracking method, which is stable and topology preserving, was applied to obtain the Lagrangian scalar field φ governed by the pure advection equation in the Eulerian form ∂_tφ + u · ∇φ = 0. The temporal evolution of Lagrangian structures was first obtained by extracting iso-surfaces of φ with resolution 1024^3 at different times, from t = 0 to t = T_e, where T_e is the eddy turnover time. The surface area growth rate of the Lagrangian structure was quantified and the formation of stretched and rolled-up structures was observed in straining regions and stretched vortex tubes, respectively. The multi-scale geometric analysis of Bermejo-Moreno & Pullin (J. Fluid Mech., vol. 603, 2008, p. 101) has been applied to the evolution of φ to extract structures at different length scales and to characterize their non-local geometry in a space of reduced geometrical parameters. In this multi-scale sense, we observe, for the evolving turbulent velocity field, an evolutionary breakdown of initially large-scale Lagrangian structures that first distort and then either themselves are broken down or stretched laterally into sheets. Moreover, after a finite time, this progression appears to be insensible to the form of the initially smooth Lagrangian field. In comparison with the statistical geometry of instantaneous passive scalar and enstrophy fields in turbulence obtained by Bermejo-Moreno & Pullin (2008) and Bermejo-Moreno et al. (J. Fluid Mech., vol. 620, 2009, p. 121), Lagrangian structures tend to exhibit more prevalent sheet-like shapes at intermediate and small scales. For the frozen flow, the Lagrangian field appears to be attracted onto a stream-surface field and it develops less complex multi-scale geometry than found for the turbulent velocity field. In the latter case, there appears to be a tendency for the Lagrangian field to move towards a vortex-surface field of the evolving turbulent flow but this is mitigated by cumulative viscous effects

Refining Finite-Time Lyapunov Exponent Ridges and the Challenges of Classifying Them

While more rigorous and sophisticated methods for identifying Lagrangian based coherent structures exist, the finite-time Lyapunov exponent (FTLE) field remains a straightforward and popular method for gaining some insight into transport by complex, time-dependent two-dimensional flows. In light of its enduring appeal, and in support of good practice, we begin by investigating the effects of discretization and noise on two numerical approaches for calculating the FTLE field. A practical method to extract and refine FTLE ridges in two-dimensional flows, which builds on previous methods, is then presented. Seeking to better ascertain the role of a FTLE ridge in flow transport, we adapt an existing classification scheme and provide a thorough treatment of the challenges of classifying the types of deformation represented by a FTLE ridge. As a practical demonstration, the methods are applied to an ocean surface velocity field data set generated by a numerical model. (C) 2015 AIP Publishing LLC.ONR N000141210665Center for Nonlinear Dynamic