Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-12-01T02:02:25.977Z Has data issue: false hasContentIssue false
  • English
  • Français

Relative dispersion in generalized two-dimensional turbulence

Published online by Cambridge University Press:  24 May 2017

Alexis Foussard ,
Stefano Berti Open the ORCID record for Stefano Berti [Opens in a new window] ,
Xavier Perrot  and
Guillaume Lapeyre Open the ORCID record for Guillaume Lapeyre [Opens in a new window]
Alexis Foussard
Affiliation:
LMD/IPSL, CNRS/ENS, 24 rue Lhomond, 75005 Paris, France Ecole des Ponts Paris Tech, Cité Descartes, 6–8 Avenue Blaise Pascal, 77455 Champs-sur-Marne, France
Stefano Berti
Affiliation:
Université de Lille, CNRS, FRE 3723, Laboratoire de Mécanique de Lille, 59000 Lille, France
Xavier Perrot
Affiliation:
LMD/IPSL, CNRS/ENS, 24 rue Lhomond, 75005 Paris, France
Guillaume Lapeyre*
Affiliation:
LMD/IPSL, CNRS/ENS, 24 rue Lhomond, 75005 Paris, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The statistical properties of turbulent fluids depend on how local the energy transfers among scales are, i.e. whether the energy transfer at some given scale is due to the eddies at that particular scale, or to eddies at larger (non-local) scale. This locality in the energy transfers may have consequences for the relative dispersion of passive particles. In this paper, we consider a class of generalized two-dimensional flows (produced by the so-called $\unicode[STIX]{x1D6FC}$ -turbulence models), theoretically possessing different properties in terms of locality of energy transfers. It encompasses the standard barotropic quasi-geostrophic (QG) and the surface quasi-geostrophic (SQG) models as limiting cases. The relative dispersion statistics are examined, both as a function of time and as a function of scale, and compared to predictions based on phenomenological arguments assuming the locality of the cascade. We find that the dispersion statistics follow the predicted values from local theories, as long as the parameter $\unicode[STIX]{x1D6FC}$ is small enough (dynamics close to that of the SQG model), for sufficiently small initial pair separations. In contrast, non-local dispersion is observed for the QG model, a robust result when looking at relative displacement probability distributions. However, we point out that spectral energy transfers do have a non-local contribution for models with different values of $\unicode[STIX]{x1D6FC}$ , including the SQG case. This indicates that locality/non-locality of the turbulent cascade may not always imply locality/non-locality in the relative dispersion of particles and that the self-similar nature of the turbulent cascade is more appropriate for determining the relative dispersion locality.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Geostrophic turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 821 , 25 June 2017 , pp. 358 - 383
Check for updates
Copyright
© 2017 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

Artale, V., Boffetta, G., Celani, A., Cencini, M. & Vulpiani, A. 1997 Dispersion of passive tracers in closed basins: beyond the diffusion coefficient. Phys. Fluids A 9, 3162–3171.Google Scholar
Aurell, E., Boffetta, G., Crisanti, A., Paladin, G. & Vulpiani, A. 1997 Predictability in the large: an extension of the concept of Lyapunov exponent. J. Phys. A 30, 1–26.Google Scholar
Babiano, A., Basdevant, C., Roy, P. L. & Sadourny, R. 1990 Relative dispersion in two-dimensional turbulence. J. Fluid Mech. 214, 535–557.CrossRefGoogle Scholar
Batchelor, G. K. 1950 The application of the similarity theory of turbulence to atmospheric diffusion. Q. J. R. Meteorol. Soc. 551, 133–146.Google Scholar
Batchelor, G. K. 1952a Diffusion in a field of homogeneous turbulence II. The relative motion of particles. Proc. Camb. Phil. Soc. 48, 345–362.CrossRefGoogle Scholar
Batchelor, G. K. 1952b The effect of homogeneous turbulence on material lines and surfaces. Proc. R. Soc. Lond. A 213, 349–366.Google Scholar
Bennett, A. 2006 Lagrangian Fluid Dynamics. Cambridge University Press.Google Scholar
Bennett, A. F. 1984 Relative dispersion: local and nonlocal dynamics. J. Atmos. Sci. 41, 1881–1886.Google Scholar
Berti, S. & Lapeyre, G. 2014 Lagrangian reconstructions of temperature and velocity in a model of surface ocean turbulence. Ocean Model. 76, 59–71.CrossRefGoogle Scholar
Berti, S., dos Santos, F., Lacorata, G. & Vulpiani, A. 2011 Lagrangian drifter dispersion in the Southwestern Atlantic Ocean. J. Phys. Oceanogr. 41, 1659–1672.Google Scholar
Boffetta, G. & Celani, A. 2000 Pair dispersion in turbulence. Physica A 280, 1–9.CrossRefGoogle Scholar
Boffetta, G. & Ecke, R. E. 2012 Two-dimensional turbulence. Annu. Rev. Fluid Mech. 44, 427–451.Google Scholar
Boffetta, G. & Sokolov, I. M. 2002 Statistics of two-particle dispersion in two-dimensional turbulence. Phys. Fluids 14, 3224–3232.Google Scholar
Bourgoin, M., Ouellette, N. T., Xu, H., Berg, J. & Bodenschatz, E. 2006 The role of pair dispersion in turbulent flow. Science 311, 835–838.Google Scholar
Burgess, B. H., Scott, R. K. & Shepherd, T. G. 2015 Kraichnan–Leith–Batchelor similarity theory and two-dimensional inverse cascades. J. Fluid Mech. 767, 467–496.Google Scholar
Er-El, J. & Peskin, R. L. 1981 Relative diffusion of constant-level balloons in the Southern Hemisphere. J. Atmos. Sci. 38, 2264–2274.Google Scholar
Falkovich, G., Gaw edzki, K. & Vergassola, M. 2001 Particles and fluids in turbulence. Rev. Mod. Phys. 73, 913–975.Google Scholar
Fung, J. C. H. & Vassilicos, J. C. 1998 Two-particle dispersion in turbulentlike flows. Phys. Rev. E 57, 1677–1690.Google Scholar
Garrett, C. 1983 On the initial streakiness of a dispersing tracer in two- and three-dimensional turbulence. Dyn. Atmos. Oceans 7, 265–277.Google Scholar
Graff, L. S., Guttu, S. & LaCasce, J. H. 2015 Relative dispersion in the atmosphere from reanalysis winds. J. Atmos. Sci 72, 2769–2785.Google Scholar
Held, I. M., Pierrehumbert, R. T., Garner, S. T. & Swanson, K. L. 1995 Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20.Google Scholar
Hua, B. L. 1994 The conservation of potential vorticity along Lagrangian trajectories in simulations of eddy-driven flows. J. Phys. Oceanogr. 24, 498–508.Google Scholar
Hua, B. L., McWilliams, J. C. & Klein, P. 1998 Lagrangian accelerations in geostrophic turbulence. J. Fluid Mech. 366, 87–108.Google Scholar
Juckes, M. 1995 Instability of surface and upper-tropospheric shear lines. J. Atmos. Sci. 52, 3247–3262.Google Scholar
Jullien, M.-C. 2003 Dispersion of passive tracers in the direct enstrophy cascade: experimental observations. Phys. Fluids 15, 2228–2237.Google Scholar
Jullien, M.-C., Paret, J. & Tabeling, P. 1999 Richardson pair dispersion in two-dimensional turbulence. Phys. Rev. Lett. 82, 2872–2875.CrossRefGoogle Scholar
Klein, P., Hua, B. L. & Lapeyre, G. 2000 Alignment of tracer gradient vectors in 2D turbulence. Physica D 146, 246–260.Google Scholar
Klein, P., Lapeyre, G., Roullet, G., Le Gentil, S. & Sasaki, H. 2011 Ocean turbulence at meso and submesoscales: connection between surface and interior dynamics. Geophys. Astrophys. Fluid Dyn. 105, 421–437.Google Scholar
Koszalka, I., LaCasce, J. H. & Orvik, K. A. 2009 Relative dispersion in the Nordic Seas. J. Mar. Res. 67, 411–433.Google Scholar
Kowalski, A. D. & Peskins, R. L. 1981 Numerical simulation of relative dispersion in two-dimensional, homogeneous, decaying turbulence. J. Fluid Mech. 109, 45–61.CrossRefGoogle Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.CrossRefGoogle Scholar
LaCasce, J. H.1996 Baroclinic vortices over a sloping bottom. PhD thesis, MIT/WHOI Joint Program in Physical Oceanography, p. 216.Google Scholar
LaCasce, J. H. 2008 Statistics from Lagrangian observations. Prog. Oceanogr. 77, 1–29.Google Scholar
LaCasce, J. H. 2010 Relative displacement probability distribution functions from balloons and drifters. J. Mar. Res. 68, 433–457.Google Scholar
LaCasce, J. H. & Bower, A. 2000 Relative dispersion in the subsurface North Atlantic. J. Mar. Res. 58, 863–894.Google Scholar
LaCasce, J. H. & Ohlmann, C. 2003 Relative dispersion at the surface of the Gulf of Mexico. J. Mar. Res. 61, 285–312.Google Scholar
Lacorata, G., Aurell, E., Legras, B. & Vulpiani, A. 2004 Evidence for a k -5/3 spectrum from the EOLE Lagrangian balloons in the low stratosphere. J. Atmos. Sci. 61, 2936–2942.Google Scholar
Lapeyre, G. 2002 Characterization of finite-time Lyapunov exponents and vectors in two-dimensional turbulence. Chaos 12, 688–698.Google Scholar
Lapeyre, G. 2017 Surface quasi-geostrophy. Fluids 2, 7.Google Scholar
Lin, J.-T. 1972 Relative dispersion in the enstrophy cascading inertial range of homogeneous two-dimensional turbulence. J. Atmos. Sci. 29, 394–396.2.0.CO;2>CrossRefGoogle Scholar
Lumpkin, R. & Elipot, S. 2010 Surface drifter pair spreading in the North Atlantic. J. Geophys. Res. 115, C12017.Google Scholar
Lundgren, T. S. 1981 Turbulent pair dispersion and scalar diffusion. J. Fluid Mech. 111, 27–57.CrossRefGoogle Scholar
Morel, P. & Larcheveque, M. 1974 Relative dispersion of constant-level balloons in the 200-mb general circulation. J. Atmos. Sci. 31, 2189–2196.2.0.CO;2>CrossRefGoogle Scholar
Nicolleau, F. & Yu, G. 2004 Two-particule diffusion and locality assumption. Phys. Fluids 16, 2309–2321.CrossRefGoogle Scholar
Oetzel, K. G. & Vallis, G. F. 1997 Strain, vortices and the enstrophy range in two-dimensional turbulence. Phys. Fluids A 9, 2991–3004.Google Scholar
Ohkitani, K. 1990 Nonlocality in a forced two-dimensional turbulence. Phys. Fluids A 2, 1529–1531.CrossRefGoogle Scholar
Okubo, A. 1970 Horizontal dispersion of floatable particles in the vicinity of velocity singularity such as convergences. Deep-Sea Res. 17, 445–454.Google Scholar
Ollitrault, M., Gabillet, C. & Colin de Verdière, A. 2005 Open ocean regimes of relative dispersion. J. Fluid Mech. 533, 381–407.Google Scholar
Ouellette, N. T., Xu, H., Bourgoin, M. & Bodenschat, E. 2006 An experimental study of turbulent relative dispersion models. New J. Phys. 8, 109.Google Scholar
Özgökmen, T. M., Poje, A. C., Fischer, P. F., Childs, H., Krishnan, H., Garth, C., Haza, A. C. & Ryan, E. 2012 On multi-scale dispersion under the influence of surface mixed layer instabilities and deep flows. Ocean Model. 56, 16–30.Google Scholar
Pierrehumbert, R. T., Held, I. M. & Swanson, K. L. 1994 Spectra of local and nonlocal two-dimensional turbulence. Chaos, Solitons Fractals 4, 1111–1116.Google Scholar
Poje, A. C., Hazab, A. C., Özgökmen, T. M., Magaldi, M. G. & Garraffo, Z. D. 2010 Resolution dependent relative dispersion statistics in a hierarchy of ocean models. Ocean Model. 31, 36–50.Google Scholar
Richardson, L. F. 1926 Atmospheric diffusion on a distance-neighbour graph. Proc. R. Soc. Lond. A 110, 709–737.Google Scholar
Rivera, M. K. & Ecke, R. E. 2005 Pair dispersion and doubling time statistics in two-dimensional turbulence. Phys. Rev. Lett. 95, 194503.Google Scholar
Salazar, J. P. L. C. & Collins, L. R. 2009 Two-particle dispersion in isotropic turbulent flows. Annu. Rev. Fluid Mech. 41, 405–432.Google Scholar
Sawford, B. 2001 Turbulent relative dispersion. Annu. Rev. Fluid Mech. 33, 289–317.Google Scholar
Schroeder, K., Chiggiato, J., Haza, A. C., Griffa, A., Özgökmen, T. M., Zanasca, P., Molcard, A., Borghini, M., Poulain, P. M., Gerin, R. et al. 2012 Targeted Lagrangian sampling of submesoscale dispersion at a coastal frontal zone. Geophys. Res. Lett. 39, 1168.Google Scholar
Smith, K. S., Boccaletti, G., Henning, C. C., Marinov, I. N., Tam, C. Y., Held, I. M. & Vallis, G. K. 2002 Turbulent diffusion in the geostrophic inverse cascade. J. Fluid Mech. 469, 14–47.Google Scholar
Thomson, D. J. & Devenish, B. J. 2005 Particle pair separation in kinematic simulations. J. Fluid Mech. 526, 277–302.Google Scholar
Von Kameke, A., Huhn, F., Fernández-García, G., Muñuzuri, A. P. & Pérez-Muñuzuri, V. 2011 Double cascade turbulence and Richardson dispersion in a horizontal fluid flow induced by Faraday waves. Phys. Rev. Lett. 107, 074502.Google Scholar
Watanabe, T. & Iwayama, T. 2004 Unified scaling theory for local and non-local transfers in generalized two-dimensional turbulence. J. Phys. Soc. Japan 73, 3319–3330.Google Scholar
Watanabe, T. & Iwayama, T. 2007 Interacting scales and triad enstrophy transfers in generalized two-dimensional turbulence. Phys. Rev. E 76, 046303.Google Scholar
Weiss, J. 1991 The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Physica D 48, 273–294.Google Scholar
Zouari, N. & Babiano, A. 1990 Expériences numériques lagrangiennes à partir de modèles eulériens. Atmos.-Ocean 28, 345–364.Google Scholar
Supplementary material: File

Foussard supplementary material

Foussard supplementary material 1

Download Foussard supplementary material(File)
File 5.1 MB