Locating coherent material vortices in three-dimensional unsteady flows

Recent work has shown that coherent material vortices in two-dimensional unsteady flows are bounded by closed stationary curves of the averaged material strain [1]. These material vortex boundaries are objective (frame-invariant) Lagrangian coherent structures (LCSs) of the elliptic type, which turn out to stretch uniformly under the flow. We extend this approach to three-dimensional unsteady flows to locate toroidal and cylindrical material vortex boundaries as two-dimensional, elliptic LCS surfaces. We provide a detailed numerical procedure building on the approach in [2] and discuss several examples.\\[4pt] [1] G. Haller and F.J. Beron-Vera, Coherent Lagrangian vortices: The black holes of turbulence. J. Fluid Mech. 731 (2013) R4 \newline [2] D. Blazevski and G.Haller, Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows, Physica D, 273--274 (2014), 46-62.