Skip to main content Accessibility help
×
×
Home

Defining coherent vortices objectively from the vorticity

  • G. Haller (a1), A. Hadjighasem (a1), M. Farazmand (a2) and F. Huhn (a3)

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.

Copyright

Corresponding author

Email address for correspondence: georgehaller@ethz.ch

References

Hide All
Arnold, V. I. 1978 Ordinary Differential Equations. MIT Press.
Batchelor, B. G. & Whelan, P. F. 2012 Intelligent Vision Systems for Industry. Springer.
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, 429436.
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. 8189. Springer.
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.
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, 14261438.
Bertrand, J. 1873 Théoreme relatif au mouvement d’un point attiré vers un centre fixe. C. R. Acad. Sci. Paris 77, 849853.
Blazevski, D. & Haller, G. 2014 Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows. Physica D 273–274, 4664.
Budišić, M. & Mezić, I. 2012 Geometry of the ergodic quotient reveals coherent structures in flows. Physica D 241, 12551269.
Chakraborty, P., Balachandar, S. & Adrian, R. 2005 On the relationships between local vortex identification schemes. J. Fluid Mech. 535, 189214.
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, 328332.
Chong, M. S., Perry, A. E. & Cantwell, B. J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids A 2, 765777.
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, 261282.
Dafermos, C. M. 1971 An invariance principle for compact processes. J. Differ. Equ. 239252.
Dresselhaus, E. & Tabor, M. 1989 The persistence of strain in dynamical systems. J. Phys. A: Math. Gen. 22, 971984.
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), 17371744.
Farazmand, M. & Haller, G. 2013 Attracting and repelling Lagrangian coherent structures from a single computation. Chaos 15, 023101, 111.
Farazmand, M. & Haller, G. 2016 Polar rotation angle identifies elliptic islands in unsteady dynamical systems. Physica D 315 (2016), 112.
Golub, G. H. & Van Loan, C. F. 1983 Matrix Computations. Johns Hopkins University Press.
Gonzalez, R. C. & Woods, R. E. 20008 Digital Image Processing. Prentice Hall.
Gurtin, M. E. 1982 An Introduction to Continuum Mechanics. Academic.
Hadjighasem, A. & Haller, G. 2016 Geodesic transport barriers in Jupiter’s atmosphere: a video-based analysis. SIAM Rev. 58 (1), 6989.
Haller, G. 2001 Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence. Phys. Fluids. 13, 33653385.
Haller, G. 2005 An objective definition of a vortex. J. Fluid Mech. 525, 126.
Haller, G. 2015 Lagrangian coherent structures. Annu. Rev. Fluid Mech. 47, 137162.
Haller, G. 2016 Dynamically consistent rotation and stretch tensors from a dynamic polar decomposition. J. Mech. Phys. Solids 86 (2016), 7093.
Haller, G. & Beron-Vera, F. J. 2013 Coherent Lagrangian vortices: the black holes of turbulence. J. Fluid Mech. 731, R4.
Helmholtz, H. 1858 Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. J. Reine Angew. Math. 55, 2555.
Hua, B. L. & Klein, O. 1998 An exact criterion for the stirring properties of nearly two-dimensional turbulence. Physica D 113, 98110.
Hua, B. K., McWilliams, J. C. & Klein, P. 1998 Lagrangian accelerations in geostrophic turbulence. J. Fluid Mech. 336, 87108.
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.
Hussain, A. K. M. F. 1986 Coherent structures and turbulence. J. Fluid Mech. 173, 303356.
Jeong, J. & Hussein, A. K. M. F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 6994.
Karrasch, D., Huhn, F. & Haller, G. 2014 Automated detection of coherent Lagrangian vortices in two-dimensional unsteady flows. Proc. R. Soc. Lond. 471, 20140639.
Kevlahan, N. K.-R. & Farge, M. 1997 Vorticity filaments in two-dimensional turbulence: creation, stability and effect. J. Fluid Mech. 346, 4976.
Lapeyre, G., Hua, B. L. & Legras, B. 2001 Comment on ‘Finding finite-time invariant manifolds in two-dimensional velocity fields’. Chaos 11, 427430.
Lapeyre, G., Klein, P. & Hua, B. L. 1999 Does the tracer gradient vector align with the strain eigenvectors in 2D turbulence? Phys. Fluids 11, 37293737.
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. 555562. Springer.
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. 309321. Springer.
Majda, A. J. & Bertozzi, A. L. 2002 Vorticity and Incompressible Flow. Cambridge University Press.
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, 11811188.
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26, 883889.
Mazloff, M. R., Heimbach, P. & Wunsch, C. 2010 An eddy-permitting Southern Ocean state estimate. J. Phys. Oceanogr. 40, 880899.
McWilliams, J. C. 1984 The emergence of isolated coherent vortices in turbulent flow. Fluid Mech. 146, 2143.
Milnor, J. 1963 Morse Theory (Based on Lecture Notes by M. Spivak and R. Wells), Annals of Mathematics Studies, vol. 51. Princeton University Press.
Okubo, A. 1970 Horizontal dispersion of floatable particles in the vicinity of velocity singularities such as convergences. Deep-Sea Res. 17, 445454.
Öttinger, D., Blazevski, D. & Haller, G.2016 Global variational approach to elliptic transport barriers in three dimensions. Chaos (in press).
Pérez-Muñuzuri, V. & Huhn, F. 2013 Path-integrated Lagrangian measures from the velocity gradient tensor. Nonlinear Process. Geophys. 20, 987991.
Provenzale, A. 1999 Transport by coherent barotropic vortices. Annu. Rev. Fluid Mech. 31, 5593.
Shapiro, A. 1961 Vorticity, US National Committee for Fluid Mechanics Film Series. MIT Press.
Tabor, M. & Klapper, I. 1994 Stretching and alignment in chaotic and turbulent flows. Chaos, Solitons Fractals 4, 10311055.
Truesdell, C. & Noll, W. 1965 The nonlinear field theories of mechanics. In Handbuch der Physik Band III/3 (ed. Flugge, S.). Springer.
Truesdell, C. & Rajagopal, K. R. 2009 An Introduction to the Mechanics of Fluids. Birkhäuser.
Weiss, J. 1991 The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Physica D 48, 273294.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed