Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-25T06:19:03.031Z Has data issue: false hasContentIssue false
  • English
  • Français

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: georgehaller@ethz.ch
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, V. I. 1978 Ordinary Differential Equations. MIT Press.Google Scholar
Batchelor, B. G. & Whelan, P. F. 2012 Intelligent Vision Systems for Industry. Springer.Google Scholar
Beal, L. M., De Ruijter, W. P. M., Biastoch, A., Zahn, R.& SCOR/WCRP/IAPSO Working Group 2011 On the role of the Agulhas system in ocean circulation and climate. Nature 472, 429–436.Google Scholar
Beron-Vera, F. J. 2015 Flow coherence: distinguishing cause from effect. In Selected Topics of Computational and Experimental Fluid Mechanics Environmental Science and Engineering (ed. Klapp, J., Ruíz Chavarría, G., Ovando, A. M., López Villa, A. & Di G. Sigalotti, L.), pp. 81–89. Springer.Google Scholar
Beron-Vera, F. J., Olascoaga, M. J., Haller, G., Farazmand, M., Triñanes, J. & Wang, Y. 2015 Dissipative inertial transport patterns near coherent Lagrangian eddies in the ocean. Chaos 25, 087412.Google Scholar
Beron-Vera, F. J., Wang, Y., Olascoaga, M. J., Goni, J. G. & Haller, G. 2013 Objective detection of oceanic eddies and the Agulhas leakage. J. Phys. Oceanogr. 43, 1426–1438.Google Scholar
Bertrand, J. 1873 Théoreme relatif au mouvement d’un point attiré vers un centre fixe. C. R. Acad. Sci. Paris 77, 849–853.Google Scholar
Blazevski, D. & Haller, G. 2014 Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows. Physica D 273–274, 46–64.Google Scholar
Budišić, M. & Mezić, I. 2012 Geometry of the ergodic quotient reveals coherent structures in flows. Physica D 241, 1255–1269.CrossRefGoogle Scholar
Chakraborty, P., Balachandar, S. & Adrian, R. 2005 On the relationships between local vortex identification schemes. J. Fluid Mech. 535, 189–214.CrossRefGoogle Scholar
Chelton, D. B., Gaube, P., Schlax, M. G., Early, J. J. & Samelson, R. M. 2011 The influence of nonlinear mesoscale eddies on near-surface oceanic chlorophyll. Science 334, 328–332.CrossRefGoogle ScholarPubMed
Chong, M. S., Perry, A. E. & Cantwell, B. J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids A 2, 765–777.Google Scholar
Cucitore, R., Quadrio, M. & Baron, A. 1999 On the effectiveness and limitations of local criteria for the identification of a vortex. Eur. J. Mech. (B/Fluids) 18, 261–282.Google Scholar
Dafermos, C. M. 1971 An invariance principle for compact processes. J. Differ. Equ. 239–252.CrossRefGoogle Scholar
Dresselhaus, E. & Tabor, M. 1989 The persistence of strain in dynamical systems. J. Phys. A: Math. Gen. 22, 971–984.CrossRefGoogle Scholar
Dritschel, D. G. & Waugh, D. W. 1992 Quantification of the inelastic interaction of unequal vortices in two-dimensional vortex dynamics. Phys. Fluids A 4 (8), 1737–1744.Google Scholar
Farazmand, M. & Haller, G. 2013 Attracting and repelling Lagrangian coherent structures from a single computation. Chaos 15, 023101, 1–11.Google Scholar
Farazmand, M. & Haller, G. 2016 Polar rotation angle identifies elliptic islands in unsteady dynamical systems. Physica D 315 (2016), 1–12.CrossRefGoogle Scholar
Golub, G. H. & Van Loan, C. F. 1983 Matrix Computations. Johns Hopkins University Press.Google Scholar
Gonzalez, R. C. & Woods, R. E. 20008 Digital Image Processing. Prentice Hall.Google Scholar
Gurtin, M. E. 1982 An Introduction to Continuum Mechanics. Academic.Google Scholar
Hadjighasem, A. & Haller, G. 2016 Geodesic transport barriers in Jupiter’s atmosphere: a video-based analysis. SIAM Rev. 58 (1), 69–89.CrossRefGoogle Scholar
Haller, G. 2001 Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence. Phys. Fluids. 13, 3365–3385.Google Scholar
Haller, G. 2005 An objective definition of a vortex. J. Fluid Mech. 525, 1–26.CrossRefGoogle Scholar
Haller, G. 2015 Lagrangian coherent structures. Annu. Rev. Fluid Mech. 47, 137–162.Google Scholar
Haller, G. 2016 Dynamically consistent rotation and stretch tensors from a dynamic polar decomposition. J. Mech. Phys. Solids 86 (2016), 70–93.Google Scholar
Haller, G. & Beron-Vera, F. J. 2013 Coherent Lagrangian vortices: the black holes of turbulence. J. Fluid Mech. 731, R4.Google Scholar
Helmholtz, H. 1858 Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. J. Reine Angew. Math. 55, 25–55.Google Scholar
Hua, B. L. & Klein, O. 1998 An exact criterion for the stirring properties of nearly two-dimensional turbulence. Physica D 113, 98–110.Google Scholar
Hua, B. K., McWilliams, J. C. & Klein, P. 1998 Lagrangian accelerations in geostrophic turbulence. J. Fluid Mech. 336, 87–108.CrossRefGoogle Scholar
Hunt, J. C. R., Wray, A. A. & Moin, P.1988 Eddies, stream, and convergence zones in turbulent flows. Center for Turbulence Research Report CTR-S88, pp. 193–208.Google Scholar
Hussain, A. K. M. F. 1986 Coherent structures and turbulence. J. Fluid Mech. 173, 303–356.CrossRefGoogle Scholar
Jeong, J. & Hussein, A. K. M. F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.CrossRefGoogle Scholar
Karrasch, D., Huhn, F. & Haller, G. 2014 Automated detection of coherent Lagrangian vortices in two-dimensional unsteady flows. Proc. R. Soc. Lond. 471, 20140639.Google Scholar
Kevlahan, N. K.-R. & Farge, M. 1997 Vorticity filaments in two-dimensional turbulence: creation, stability and effect. J. Fluid Mech. 346, 49–76.CrossRefGoogle Scholar
Lapeyre, G., Hua, B. L. & Legras, B. 2001 Comment on ‘Finding finite-time invariant manifolds in two-dimensional velocity fields’. Chaos 11, 427–430.Google Scholar
Lapeyre, G., Klein, P. & Hua, B. L. 1999 Does the tracer gradient vector align with the strain eigenvectors in 2D turbulence? Phys. Fluids 11, 3729–3737.Google Scholar
Liu, I.-S. 2004 On the transformation property of the deformation gradient under a change of frame. In The Rational Spirit in Modern Continuum Mechanics (ed. Man, C.-S. & Fosdick, R. L.), pp. 555–562. Springer.Google Scholar
Lugt, H. J. 1979 The dilemma of defining a vortex. In Recent Developments in Theoretical and Experimental Fluid Mechanics (ed. Müller, U., Riesner, K. G. & Schmidt, B.), pp. 309–321. Springer.CrossRefGoogle Scholar
Majda, A. J. & Bertozzi, A. L. 2002 Vorticity and Incompressible Flow. Cambridge University Press.Google Scholar
Mason, E., Pascual, A. & McWilliams, J. C. 2014 A new sea surface height-based code for oceanic mesoscale eddy tracking. J. Atmos. Ocean. Technol. 31, 1181–1188.Google Scholar
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26, 883–889.Google Scholar
Mazloff, M. R., Heimbach, P. & Wunsch, C. 2010 An eddy-permitting Southern Ocean state estimate. J. Phys. Oceanogr. 40, 880–899.Google Scholar
McWilliams, J. C. 1984 The emergence of isolated coherent vortices in turbulent flow. Fluid Mech. 146, 21–43.Google Scholar
Milnor, J. 1963 Morse Theory (Based on Lecture Notes by M. Spivak and R. Wells), Annals of Mathematics Studies, vol. 51. Princeton University Press.Google Scholar
Okubo, A. 1970 Horizontal dispersion of floatable particles in the vicinity of velocity singularities such as convergences. Deep-Sea Res. 17, 445–454.Google Scholar
Öttinger, D., Blazevski, D. & Haller, G.2016 Global variational approach to elliptic transport barriers in three dimensions. Chaos (in press).Google Scholar
Pérez-Muñuzuri, V. & Huhn, F. 2013 Path-integrated Lagrangian measures from the velocity gradient tensor. Nonlinear Process. Geophys. 20, 987–991.Google Scholar
Provenzale, A. 1999 Transport by coherent barotropic vortices. Annu. Rev. Fluid Mech. 31, 55–93.Google Scholar
Shapiro, A. 1961 Vorticity, US National Committee for Fluid Mechanics Film Series. MIT Press.Google Scholar
Tabor, M. & Klapper, I. 1994 Stretching and alignment in chaotic and turbulent flows. Chaos, Solitons Fractals 4, 1031–1055.CrossRefGoogle Scholar
Truesdell, C. & Noll, W. 1965 The nonlinear field theories of mechanics. In Handbuch der Physik Band III/3 (ed. Flugge, S.). Springer.Google Scholar
Truesdell, C. & Rajagopal, K. R. 2009 An Introduction to the Mechanics of Fluids. Birkhäuser.Google Scholar
Weiss, J. 1991 The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Physica D 48, 273–294.Google Scholar

Haller et al. supplementary movie

Advection of Lagrangian vortices in the 2D turbulence example.

Download Haller et al. supplementary movie(Video)
Video 1.7 MB

Haller et al. supplementary movie

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

Download Haller et al. supplementary movie(Video)
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.

Download Haller et al. supplementary movie(Video)
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.

Download Haller et al. supplementary movie(Video)
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

Download Haller et al. supplementary movie(Video)
Video 4.1 MB