Abstract. Transport and mixing processes in fluid flows are crucially influenced by coherent structures and the characterization of these Lagrangian objects is a topic of intense current research. While established mathematical approaches such as variational methods or transfer-operatorbased schemes require full knowledge of the flow field or at least high-resolution trajectory data, this information may not be available in applications. Recently, different computational methods have been proposed to identify coherent behavior in flows directly from Lagrangian trajectory data, that is, numerical or measured time series of particle positions in a fluid flow. In this context, spatio-temporal clustering algorithms have been proven to be very effective for the extraction of coherent sets from sparse and possibly incomplete trajectory data. Inspired by these recent approaches, we consider an unweighted, undirected network, where Lagrangian particle trajectories serve as network nodes. A link is established between two nodes if the respective trajectories come close to each other at least once in the course of time. Classical graph concepts are then employed to analyze the resulting network. In particular, local network measures such as the node degree, the average degree of neighboring nodes, and the clustering coefficient serve as indicators of highly mixing regions, whereas spectral graph partitioning schemes allow us to extract coherent sets. The proposed methodology is very fast to run and we demonstrate its applicability in two geophysical flows -the Bickley jet as well as the Antarctic stratospheric polar vortex.

We analyze large-scale patterns in three-dimensional turbulent convection in a horizontally extended square convection cell by Lagrangian particle trajectories calculated in direct numerical simulations. A simulation run at a Prandtl number Pr = 0.7, a Rayleigh number Ra = 10 5 , and an aspect ratio Γ = 16 is therefore considered. These large-scale structures, which are denoted as turbulent superstructures of convection, are detected by the spectrum of the graph Laplacian matrix. Our investigation, which follows Hadjighasem et al., Phys. Rev. E 93, 063107 (2016), builds a weighted and undirected graph from the trajectory points of Lagrangian particles. Weights at the edges of the graph are determined by a mean dynamical distance between different particle trajectories. It is demonstrated that the resulting trajectory clusters, which are obtained by a subsequent k-means clustering, coincide with the superstructures in the Eulerian frame of reference. Furthermore, the characteristic times τ L and lengths λ L U of the superstructures in the Lagrangian frame of reference agree very well with their Eulerian counterparts, τ and λU , respectively. This trajectory-based clustering is found to work for times t τ ≈ τ L . Longer time periods t τ L require a change of the analysis method to a density-based trajectory clustering by means of timeaveraged Lagrangian pseudo-trajectories, which is applied in this context for the first time. A small coherent subset of the pseudo-trajectories is obtained in this way consisting of those Lagrangian particles that are trapped for long times in the core of the superstructure circulation rolls and are thus not subject to ongoing turbulent dispersion.

Transport and mixing processes in fluid flows are crucially influenced by coherent structures and the characterization of these Lagrangian objects is a topic of intense current research. While established mathematical approaches such as variational or transfer operator based schemes require full knowledge of the flow field or at least high resolution trajectory data, this information may not be available in applications. Recently, different computational methods have been proposed to identify coherent behavior in flows directly from Lagrangian trajectory data. In this context, spatio-temporal clustering algorithms have 5 been proven to be very effective for the extraction of coherent sets from sparse and possibly incomplete trajectory data. Inspired by these recent approaches, we consider an unweighted, undirected network, where Lagrangian particle trajectories serve as network nodes. A link is established between two nodes if the respective trajectories come close to each other at least once in the course of time. Classical graph concepts are then employed to analyze the resulting network. In particular, local network measures such as the node degree serve as indicators of highly mixing regions, whereas spectral graph partitioning schemes 10 allow us to extract coherent sets. The proposed methodology is very fast to run and we demonstrate its applicability in two geophysical flows -the Bickley jet as well as the antarctic stratospheric polar vortex. IntroductionThe notion of coherence in time-dependent dynamical systems is used to describe mobile sets that do not freely mix with the surrounding regions in phase space. In particular, coherent behavior has a crucial impact on transport and mixing processes in 15 fluid flows. The mathematical definition and numerical study of coherent flow structures has received considerable scientific interest for the last two decades. The proposed methods roughly fall into two different classes, geometric and probabilistic approaches, see Allshouse and Peacock (2015) for a discussion and comparison of different methods. Geometric concepts aim at defining the boundaries between coherent sets, i.e. codimension-1 material surfaces in the flow that can be characterized by variational criteria (see Haller (2015) for a recent review). Central to these constructions is the Cauchy-Green strain tensor, 20 which is derived from the derivative of the flow map. Thus, full knowledge of the flow field or at least high resolution trajectory data is required for these methods to work successfully. This applies also to other geometric concepts such as shape coherence (Ma and Bollt (2014)). Probabilistic methods aim at defining sets that are minimally dispersive while moving with the flow. The main theoretical tools are transfer operators, i.e. linear Markov operators that describe the motion of probability densities under the action of the nonlinear, time-dependent flow. The different constructions in this family of approaches are reviewed 25 in Froyland and Padberg-Gehle (2014), also highlighti...

Barriers to the turbulent transport of heat in a convection flow can be identified by Lagrangian trajectories that stay together for a long time and thus probe spatial regions in the bulk of the fluid flow that do not mix effectively with its surroundings. They form Lagrangian coherent sets which we investigate here in direct numerical simulations of three-dimensional Rayleigh-Bénard convection at three different Prandtl numbers. The analysis is based on 524, 288 massless Lagrangian tracer particles which are advected in the time-dependent flow. Clusters of trajectories are identified by the diffusion map approach, which quantifies the connectivity of trajectory segments by a diffusion process on the data, and a subsequent sparse eigenbasis approximation (SEBA) for cluster detection. The diffusion kernel contains a cutoff that is based on the time-averaged distance between mutual Lagrangian tracers in a time window. The numerical simulations are performed in a cell at an aspect ratio Γ = 16, at fixed Rayleigh number Ra = 10 5 , and Prandtl numbers Pr = 0.1, 0.7 and 7. The combination of diffusion map and SEBA leads to a significantly improved cluster identification that is compared with the large-scale patterns in the Eulerian frame of reference. We show that the Lagrangian coherent sets contribute significantly less to the global turbulent heat transfer for all chosen Prandtl numbers as the trajectories in the spatial complement. This is realized by monitoring local Nusselt numbers, a dimensionless measure of heat transfer, along the tracer trajectories.

Coherent circulation rolls and their relevance for the turbulent heat transfer in a two-dimensional Rayleigh-Bénard convection model are analyzed. The flow is in a closed cell of aspect ratio four at a Rayleigh number Ra = 10 6 and at a Prandtl number Pr = 10. Three different Lagrangian analysis techniques based on graph Laplacians -distance spectral trajectory clustering, time-averaged diffusion maps and finite-element based dynamic Laplacian discretization -are used to monitor the turbulent fields along trajectories of massless Lagrangian particles in the evolving turbulent convection flow. The three methods are compared to each other and the obtained coherent sets are related to results from an analysis in the Eulerian frame of reference. We show that the results of these methods agree with each other and that Lagrangian and Eulerian coherent sets form basically a disjoint union of the flow domain. Additionally, a windowed time-averaging of variable interval length is performed to study the degree of coherence as a function of this additional coarse graining which removes small-scale fluctuations that cause trajectories to disperse quickly. Finally, the coherent set framework is extended to study heat transport.