We define objective Eulerian Coherent Structures (OECSs) in two-dimensional, non-autonomous dynamical systems as instantaneously most influential material curves. Specifically, OECSs are stationary curves of the averaged instantaneous material stretching-rate or material shearing-rate functionals. From these objective (frame-invariant) variational principles, we obtain explicit differential equations for hyperbolic, elliptic and parabolic OECSs. As illustration, we compute OECSs in an unsteady ocean velocity data set. In comparison to structures suggested by other common Eulerian diagnostic tools, we find OECSs to be the correct short-term cores of observed trajectory deformation patterns.

Collaborator: G. Haller

We propose an objective non-dimensional metric to assess the short-term persistence of Eulerian vortices identified by elliptic objective Eulerian Coherent Structures (OECSs). Elliptic OECSs, equipped with this metric, give a continuously updated picture of the coherent vortices in the flow, distinguishing them in a quantitative fashion. Our method offers a kinematical (model-independent) tool for detecting the instantaneous signature of long lived Lagrangian vortices and could forecast flow topological changes like vortex disruption and formation.

Collaborator: G. Haller

The polar vortices play a crucial role in the formation of the ozone hole and can cause severe weather anomalies. Their boundaries, known as the vortex ‘edges’, are typically identi- fied via methods that are either frame-dependent or return non-material structures, and hence are unsuitable for assessing material transport barriers. Using two-dimensional velocity data on isentropic surfaces in the northern hemisphere, we show that elliptic Lagrangian Coherent Structures (LCSs) identify the correct outermost material surface dividing the coherent vortex core from the surrounding incoherent surf zone. Despite the purely kinematic construction of LCSs, we find a remarkable contrast in temperature and ozone concentration across the identified vortex boundary. We also show that potential vorticity-based methods, despite their simplicity, misidentify the correct extent of the vortex edge.

Collaborators: P. Sathe, F. Beron-Vera and G. Haller

We propose a control logic, called Dependent Modal Space Control (DMSC), for vibration reduction in flexible structures. The well-known independent modal space control (IMSC) allows to change the frequencies and damping ratios of the controlled system, leaving the mode shapes unaltered. The DMSC, instead, allows to change also the closed loop mode shapes. We illustrate the advantages of the proposed logic on a cantilevered beam, showing the response of the system controlled with IMSC and DMSC.

Collaborators: F. Ripamonti and F. Resta

The goal of this project is to employ an integrated theoretical, computational and observational approach to develop, implement and utilize cutting-edge methods with data-driven modeling for the purpose of uncovering, quantifying and predicting key transport processes and structures during regional flow-based hazards in the ocean and atmosphere.

Joint with: MIT, UC Berkeley, Virginia Tech, WHOI.

Collaborators: P. Sathe and G. Haller

Recent results suggest that boundaries of coherent fluid vortices (elliptic coherent structures) can be identified as closed null geodesics of appropriate Lorentzian metrics defined on the flow domain. Here we derive a fully automated method for computing such null geodesics based on the geometry of geodesic flows and basic topological properties of closed planar curves.

Collaborator: G. Haller

We develop a frame-invariant theory of material spike formation during flow separation over a no-slip boundary in two-dimensional flows with arbitrary time dependence. Based on topological properties of material lines, our theory identifies both fixed and moving flow separation, is effective also over short-time intervals, and admits a rigorous instantaneous limit. The material backbone we identify acts as the first precursor, and the latter centerpiece, of unsteady Lagrangian flow separation. We also discover a previously undetected spiking point where the backbone of separation connects to the boundary, and derive wall-based analytical formulae for its location. Finally, our theory explains the perception of off-wall separation in unsteady flows and provides conditions under which such a perception is justified.

Collaborator: J. Vétel and G. Haller