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.

Collaborator: G. Haller

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.

Collaborators: D. Karrasch, H. Teramoto, and G. Haller

We propose here the use of variational level set methodology to capture Lagrangian vortex boundaries in 2D unsteady velocity fields. This method reformulates earlier approaches that seek material vortex boundaries as extremum solutions of variational problems. We demonstrate the performance of this technique for two different variational formulations built upon different notions of coherence. The first formulation uses an energy functional that penalizes the deviation of a closed material line from piecewise uniform stretching. The second energy function is derived for a graph-based approach to vortex boundary detection. Our level-set formulation captures an a priori unknown number of vortices simultaneously at relatively low computational cost. We illustrate the approach by identifying vortices from different coherence principles in several examples.

Collaborator: G. Haller

Coherent Lagrangian (material) structures are ubiquitous in unsteady fluid flows, often observable indirectly from tracer patterns they create, for example, in the atmosphere and the ocean. Despite these observations, a direct identification of these structures from the flow velocity field (without reliance on seeding passive tracers) has remained a challenge. Several heuristic and mathematical detection methods have been developed over the years, each promising to extract materially coherent domains from arbitrary unsteady velocity fields over a finite time interval of interest. Here we review a number of these methods and compare their performance systematically on three benchmark velocity data sets. Based on this comparison, we discuss the strengths and weaknesses of each method, and recommend minimal self-consistency requirements that Lagrangian coherence detection tools should satisfy.

Collaborators: M. Farazmand, D. Blazevski, G. Froyland and G. Haller