Session A29: Nonlinear Dynamics: Coherent Structures I

Correlating Lagrangian structures with forcing in two-dimensional flow

Lagrangian coherent structures (LCSs) are the dominant transport barriers in unsteady, aperiodic flows, and their role in organizing mixing and transport has been well documented. However, nearly all that is known about LCSs has been gleaned from passive observations: they are computed in a post-processing step after a flow has been observed, and used to understand why the mixing and transport proceeded as it did. Here, we instead take a first step toward controlling the presence or locations of LCSs by studying the relationship between LCSs and external forcing in an experimental quasi-two-dimensional weakly turbulent flow. We find that the likelihood of finding a repelling LCS at a given location is positively correlated with the mean strain rate injected at that point and negatively correlated with the mean speed, and that it is not correlated with the vorticity. We also find that mean time between successive LCSs appearing at a fixed location is related to the structure of the forcing field. Finally, we demonstrate a surprising difference in our results between LCSs computed forward and backwards in time, with forward-time (repelling) LCSs showing much more correlation with the forcing than backwards-time (attracting) LCSs.   

A Spectral Clustering Approach to Lagrangian Vortex Detection

One of the ubiquitous features of real-life turbulent flows is the existence and persistence of coherent vortices. Here we show that such coherent vortices can be extracted as clusters of Lagrangian trajectories. We carry out the clustering on a weighted graph, with the weights measuring pairwise distances of fluid trajectories in the extended phase space of positions and time. We then extract coherent vortices from the graph using tools from spectral graph theory. Our method locates all coherent vortices in the flow simultaneously, thereby showing high potential for automated vortex tracking. We illustrate the performance of this technique by identifying coherent Lagrangian vortices in several two- and three-dimensional flows.   

Objective Eulerian Coherent Structures and their Short-Term Prediction

We discuss a frame-invariant (objective) method for Eulerian Coherent Structure (ECS) identification in two-dimensional unsteady flows. ECSs reveal a time-varying skeleton of fluid flows that instantaneously approximates the most influential material surfaces in transport and mixing. We also describe an objective non-dimensional metric that quantifies the persistence of vortex-type ECSs. In an application to persistent eddy detection in satellite-derived ocean velocity data, we find that our ECS persistence metric significantly outperforms vortex predictions from other customary Eulerian diagnostics, such as the potential vorticity gradient and the Okubo-Weiss criterion.   

Variational approach to Lagrangian vortices in 3D unsteady flows

From a Lagrangian perspective, vortices can be viewed as material surfaces, i.e., as surfaces moving with the fluid. Lagrangian vortices are therefore directly linked to particle motion. Here, we use the objective (frame-invariant) approach of Lagrangian Coherent Structures (LCSs) to identify Lagrangian vortices as tubular, non-filamenting material surfaces (elliptic LCSs). By introducing a dual and autonomous dynamical system, we present a new method for constructing LCSs in three-dimensional unsteady flows. We illustrate the geometric significance of this approach with various examples.   

Complexity of coherent structures computed from braids of passive particles

Transport in fluids can be characterized by tracking passive particles advected by the fluid flow. When particles are distributed densely, as can be achieved in laboratory, the fluid velocity field can be reconstructed through Particle Tracking Velocimetry, enabling computation of Lyapunov exponents or other numerical analyses. When particles are sparse, as in drifter measurements of oceans, the velocity field cannot be reliably reconstructed. Nevertheless, the amount of entanglement of particle paths over time can be used to estimate the dynamical complexity of the flow by computing the Finite-Time Braiding Exponent (FTBE). The technique is based on braid dynamics and measures the rate at which particle motion stretches topological loops, i.e., the ``rubber bands'' enclosing subsets of particles. Allshouse and Thiffeault showed that minimally-stretching loops correspond to the structures coherent under material transport in flows. We extend their work and couple it to the FTBE calculations in order to characterize the spatial distribution of flow complexity. Analysis is demonstrated on the Hackborn rotor-oscillator model, which exhibits regions of chaotic and regular dynamics, and can be realized both numerically and experimentally.   

Laboratory experimental investigations of braid theory using the rotor-oscillator flow

Interpreting ocean surface dynamics is crucial to many areas of oceanography, ranging from marine ecology to pollution management. Motivated by this, we investigated the braid theory method to detect transport barriers bounding coherent structures in two-dimensional flows. Whereas most existing techniques rely on an extensive spatiotemporal knowledge of the flow field, we sought to identify these structures from sparse data sets involving trajectories of a few tracer particles in a two-dimensional flow. We present the results from our laboratory experiments, which were based on investigations using the rotor-oscillator flow, as a stepping stone towards oceanic applications.   

Lagrangian transport near perturbed periodic lines in three-dimensional unsteady flows

Periodic lines formed by continuous strings of periodic points are key organizing entities in the Lagrangian flow topology of certain three-dimensional (3D) time-periodic flows. Such lines generically consist of elliptic and/or hyperbolic points and thus give rise to 3D flow topologies made up of families of concentric closed trajectories embedded in chaotic regions. Weak perturbation destroys the periodic lines and causes said trajectories to coalesce into families of concentric tubes. However, emergence of isolated periodic points near the disintegrating periodic lines and/or partitioning of the original lines into elliptic and hyperbolic segments interrupt the tube formation. This yields incomplete tubes that interact with the (chaotic) environment through their open ends, resulting in intricate and essentially 3D flow topologies These phenomena have been observed in various realistic flows yet the underlying mechanisms are to date only partially understood. This study deepens insight into the (perturbed) Lagrangian dynamics of these flows by way of a linearized representation of the equations of motion near the periodic lines. Predictions on the basis of this investigation are in full (qualitative) agreement with observed behavior in the actual flows   

Lagrangian coherent structures as mesoscale transport barriers in atmospheric flows

Coherent structures in two-dimensional flows have long been studied in the context of transport in fluid dynamics. However, for geophysical systems a small vertical velocity can lead to nontrivial three-dimensional motion of airborne biological populations affecting agriculture or hazardous outputs from natural disasters. The pathways and barriers in the lower atmosphere, from ground level to a kilometer altitude and over a horizontal scale of several kilometers--which bridge the scale of, for example, local farmlands to the larger regional scale--are still unclear. This requires exploring relevant spatiotemporal scales related to advection in the space of 3D + time. In this talk, we will present the application of finite-time Lyapunov exponent based three-dimensional Lagrangian coherent structures (LCS) to address questions of transport using historical data sets from satellite observations, field measurements and the Weather Research and Forecasting (WRF) model.   

The domain dependence of chemotaxis in two-dimensional turbulence

Coherent structures are ubiquitous in environmental and geophysical flows and they affect reaction-diffusion processes in profound ways. In this presentation, we show an example of the domain dependence of chemotaxis process in a two-dimensional turbulent flow. The flow has coherent structures that form barriers that prohibit long-range transport of tracers. Accordingly, the uptake advantage of nutrient by motile and nonmotile species differs significantly if the process start in different locations of the flow. Interestingly, the conventional diagnostic of Finite-time Lyapunov exponents alone is not sufficient to explain the variability --- methods to extract elliptic transport barriers are essential to relate to the explanation. We also offer some explanations of the observed scalar behaviors via analyses of bulk quantities.