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Coherent Lagrangian vortices: the black holes of turbulence

  • G. Haller (a1) and F. J. Beron-Vera (a2)


We introduce a simple variational principle for coherent material vortices in two-dimensional turbulence. Vortex boundaries are sought as closed stationary curves of the averaged Lagrangian strain. Solutions to this problem turn out to be mathematically equivalent to photon spheres around black holes in cosmology. The fluidic photon spheres satisfy explicit differential equations whose outermost limit cycles are optimal Lagrangian vortex boundaries. As an application, we uncover super-coherent material eddies in the South Atlantic, which yield specific Lagrangian transport estimates for Agulhas rings.


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Arnold, V. I. 1973 Ordinary Differential Equations. Massachusetts Institute of Technology.
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.
Beem, J. K., Ehrlich, P. L. & Kevin, L. E. 1996 Global Lorentzian Geometry. CRC Press.
Beron-Vera, F. J., Olascoaga, M. J. & Goni, G. J. 2008 Oceanic mesoscale vortices as revealed by Lagrangian coherent structures. Geophys. Res. Lett. 35, L12603.
Beron-Vera, F. J., Wang, Y., Olascoaga, M. J., Goni, G. J. & Haller, G. 2013 Objective detection of oceanic eddies and the Agulhas leakage. J. Phys. Oceanogr. 43, 14261438.
Chelton, D. B., Schlax, M. G. & Samelson, R. M. 2011 Global observations of nonlinear mesoscale eddies. Prog. Oceanogr. 91, 167216.
Denman, K. L. & Gargett, A. E. 1983 Time and space scales of vertical mixing and advection of phytoplankton in the upper ocean. Limnol. Oceanogr. 28, 801815.
Froyland, G., Horenkamp, C., Rossi, V., Santitissadeekorn, N. & Gupta, A. S. 2012 Three-dimensional characterization and tracking of an Agulhas ring. Ocean Model. 52–53, 6975.
Goni, G. J., Garzoli, S. L., Roubicek, A. J., Olson, D. B. & Brown, O. B. 1997 Agulhas ring dynamics from TOPEX/Poseidon satellite altimeter data. J. Mar. Res. 55, 861883.
Haller, G. 2005 An objective definition of a vortex. J. Fluid Mech. 525, 126.
Haller, G. & Sapsis, T. 2008 Where do inertial particles go in fluid flows? Physica D 237, 573583.
Hawking, S. & Penrose, R. 1996 The Nature of Space and Time. Princeton University Press.
Jeong, J. & Hussain, F. 1985 On the identification of a vortex. J. Fluid Mech. 285, 6994.
Provenzale, A. 1999 Transport by coherent barotropic vortices. Annu. Rev. Fluid Mech. 31, 5593.
Truesdell, C. & Noll, W. 2004 The Nonlinear Field Theories of Mechanics. Springer.
van Aken, H. M., van Veldhovena, A. K., Vetha, C., de Ruijterb, W. P. M., van Leeuwenb, P. J., Drijfhoutc, S. S., Whittled, C. P. & Rouaultd, M. 2003 Observations of a young Agulhas ring, Astrid, during MARE in March 2000 . Deep-Sea Res. II 50, 167195.
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Coherent Lagrangian vortices: the black holes of turbulence

  • G. Haller (a1) and F. J. Beron-Vera (a2)


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