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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: glapeyre@lmd.ens.fr
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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

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Papers
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Journal of Fluid Mechanics , Volume 821 , 25 June 2017 , pp. 358 - 383
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© 2017 Cambridge University Press 

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