Spatio-temporal organization of dynamics in a two-dimensional periodically driven vortex flow: A Lagrangian flow network perspective

Abstract

We study the Lagrangian dynamics of passive tracers in a simple model of a driven two-dimensional vortex resembling real-world geophysical flow patterns. Using a discrete approximation of the system's transfer operator, we construct a directed network that characterizes the exchange of mass between distinct regions of the flow domain. By studying different measures characterizing flow network connectivity at different time-scales, we are able to identify the location of dynamically invariant structures and regions of maximum dispersion. Specifically, our approach allows to delimit co-existing flow regimes with different dynamics. To validate our findings, we compare several network characteristics to the well-established finite-time Lyapunov exponents and apply a receiver operating characteristic (ROC) analysis to identify network measures that are particularly useful for unveiling the skeleton of Lagrangian chaos.

Network-based study of Lagrangian transport and mixing

“…In particular in the context of volume-preserving flows, this is only possible when fluid parcels get stretched and folded. Thus, both d and d nn are expected to be large in mixing regions and can be at least qualitatively related to finite-time Lyapunov exponents; see Donner et al (2010a), Padberg et al (2009), Froyland andPadberg-Gehle (2012), Lindner and Donner (2017), and SerGiacomi et al (2015) for related studies. However, whereas finite-time Lyapunov exponents measure the overall stretching at the final time, in our construction all intermediate times are also considered.…”

“…These and other quantities have been considered in previous work on recurrence networks by Donner et al (2010a) and Donner et al (2010b), where the authors could link network measures to properties of the underlying dynamical system. In a similar fashion, Lindner and Donner (2017) as well as Ser-Giacomi et al (2015) considered the in-and outdegrees of a weighted, directed network obtained from a discretized transfer operator and found these to highlight hyperbolic regions in the flow. We note that the node degree in our construction exactly corresponds to the trajectory encounter number very recently introduced by Rypina and Pratt (2017), a quantity that measures fluid exchange and thus mixing.…”

“…Whereas in recurrence networks, two points on a trajectory or more generally of a time series are linked when they are close, in the present work we consider a whole trajectory as a single entity. We note that the discretized transfer operator has also been viewed and treated as a network: see, e.g., Dellnitz and Preis (2003), Dellnitz et al (2005), Padberg et al (2009), Lindner andDonner (2017), and Ser- Giacomi et al (2015). The latter used the directed, weighted network to analyze model data of the Mediterranean Sea with the main focus on in-and out-degrees.…”

Lagrangian Flow Network approach to an open flow model

“…The network representation of fluid flow [19,22] uses the set-oriented approach to transport [24][25][26]28], and requires the discretization of the fluid domain D in small boxes, {B i , i = 1, 2, ..., N }, which are identified with network nodes. Then, a directed link with a weight P(t 0 , τ ) ij , the proportion of the fluid started in B i which is found in B j after a time τ , is assigned to each pair of nodes i, j:…”

“…The powerful framework of network theory has become a standard toolbox in many scientific fields ranging from social science to climate [16][17][18]. In the context of fluid dynamics, Lagrangian Flow Networks (LFNs) [19][20][21][22][23] have been introduced as a coarse-grained representation of transport in which small regions in the fluid domain are interpreted as nodes of a network, and the transfer of mass from one of these regions to another defines weighted links among them. They are based on the concept of transport operators (also called transfer or mapping operators; in fact they are the Perron-Frobenius operators of the transport dynamics) [24][25][26][27][28][29][30][31].…”