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Alireza Hadjighasem

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Last Updated: 6-May-17



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Manifolds, Attractors and KAM Surfaces in Aperiodic Systems

Lagrangian Coherent Structures from Video Streams of Jupiter

We show how the theory of geodesic transport barriers for fluid flows can be used to uncover key invariant manifolds in externally forced, one-degree-of-freedom mechanical systems. Specifically, invariant sets in such systems turn out to be geodesics of the Cauchy–Green strain tensor computed from the flow map of the forced mechanical system. This approach enables the finite-time visualization of generalized stable and unstable manifolds, attractors and generalized KAM curves under arbitrary forcing, when Poincaré maps are not available.


Collaborators: M. Farazmand and G. Haller


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

A Spectral Clustering Approach to Lagrangian Vortex Detection

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

Level Set Formulation of Two-Dimensional Lagrangian Vortex Detection Methods

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

A Critical Comparison of Lagrangian Methods for Coherent Structure Detection

The research objectives of the project are to leverage recent advances in the mathematics of dynamical systems to develop tools that aids search and rescue crews in mapping the probable path of people lost at sea. The inherent challenge that this project tackles is that flows of the ocean and atmosphere are typically highly complex and difficult to predict; furthermore, modeling of such flows produces a huge amount of data to be processed and interpreted. The new Lagrangian methods that are being developed provide a means to generate a simplified and clear picture of the most important transport events occurring in a given domain over a given time interval, supporting simplified and more efficient operational decision-making.

This project is a joint effort among MIT, UC berkeley, Virginia Tech, WHOI and the U.S. Coast Guard.

Uncovering the hidden skeleton of environmental flows: Advanced Lagrangian methods for hazards prediction, mitigation and response