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Defining coherent vortices objectively from the vorticity

Published online by Cambridge University Press:  13 April 2016

G. Haller ,
A. Hadjighasem ,
M. Farazmand  and
F. Huhn
G. Haller*
Affiliation:
Department of Mechanical and Process Engineering, ETH Zürich, Leonhardstrasse 21, 8092 Zürich, Switzerland
A. Hadjighasem
Affiliation:
Department of Mechanical and Process Engineering, ETH Zürich, Leonhardstrasse 21, 8092 Zürich, Switzerland
M. Farazmand
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Av., Cambridge, MA 02139-4307, USA
F. Huhn
Affiliation:
Institute of Aerodynamics and Flow Technology, German Aerospace Center, Bunsenstrasse 10, 37073 Göttingen, Germany
*
†Email address for correspondence: [email protected]
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Abstract

Rotationally coherent Lagrangian vortices are formed by tubes of deforming fluid elements that complete equal bulk material rotation relative to the mean rotation of the deforming fluid volume. We show that the initial positions of such tubes coincide with tubular level surfaces of the Lagrangian-averaged vorticity deviation (LAVD), the trajectory integral of the normed difference of the vorticity from its spatial mean. The LAVD-based vortices are objective, i.e. remain unchanged under time-dependent rotations and translations of the coordinate frame. In the limit of vanishing Rossby numbers in geostrophic flows, cyclonic LAVD vortex centres are precisely the observed attractors for light particles. A similar result holds for heavy particles in anticyclonic LAVD vortices. We also establish a relationship between rotationally coherent Lagrangian vortices and their instantaneous Eulerian counterparts. The latter are formed by tubular surfaces of equal material rotation rate, objectively measured by the instantaneous vorticity deviation (IVD). We illustrate the use of the LAVD and the IVD to detect rotationally coherent Lagrangian and Eulerian vortices objectively in several two- and three-dimensional flows.

JFM classification

Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Mathematical Foundations: Topological fluid dynamics Vortex Flows: Vortex dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 795 , 25 May 2016 , pp. 136 - 173
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Copyright
© 2016 Cambridge University Press 

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Haller et al. supplementary movie

Advection of Lagrangian vortices in the 2D turbulence example.

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Video 1.7 MB

Haller et al. supplementary movie

Advection of rotationally coherent Lagrangian vortices and their cores in the vortex interaction example

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Video 4.7 MB

Haller et al. supplementary movie

Advection of rotationally coherent Lagrangian vortices and their cores, along with inertial particles, in the 2D geophysical flow example.

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Video 2.4 MB

Haller et al. supplementary movie

Advection of a rotationally coherent Lagrangian vortex and its core in the 3D SOSE model example.

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Video 2.3 MB

Haller et al. supplementary movie

Advection of a rotationally coherent Eulerian vortex and its core in the 3D SOSE model example

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Video 4.1 MB