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Conformal invariance in two-dimensional turbulence

Nature Physics volume 2, pages 124–128 (2006)Cite this article

Abstract

The simplicity of fundamental physical laws manifests itself in fundamental symmetries. Although systems with an infinite number of strongly interacting degrees of freedom (in particle physics and critical phenomena) are hard to describe, they often demonstrate symmetries, in particular scale invariance. In two dimensions (2D) locality often extends scale invariance to a wider class of conformal transformations that allow non-uniform rescaling. Conformal invariance enables a thorough classification of universality classes of critical phenomena in 2D. Is there conformal invariance in 2D turbulence, a paradigmatic example of a strongly interacting non-equilibrium system? Here, we show numerically that some features of a 2D inverse turbulent cascade show conformal invariance. We observe that the statistics of vorticity clusters are remarkably close to that of critical percolation, one of the simplest universality classes of critical phenomena. These results represent a key step in the unification of 2D physics within the framework of conformal symmetry.

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Figure 3: A portion of a candidate SLE trace obtained from the vorticity field.
Figure 1: Vorticity clusters.
Figure 2: Fractal dimensions and probabilities of size and boundary length for vorticity clusters.
Figure 4: The driving function is an effective diffusion process with diffusion coefficient κ=6±0.3.
Figure 5: Crossing and surrounding probability for vorticity clusters.

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Acknowledgements

This work was supported by grants from the European network and Israel Science foundation. G.F. thanks A. Zamolodchikov, A. Polyakov, E. Bogomolny and K. Gawedzki for useful discussions.

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Authors and Affiliations

  1. Service de Physique Théorique de Saclay, CEA/CNRS, Orme des Merisiers, 91191, Gif-sur-Yvette Cedex, France

    D. Bernard

  2. Dipartimento di Fisica Generale and INFN, Università di Torino, via Pietro Giuria 1, 10125, Torino, Italy

    G. Boffetta

  3. CNRS, INLN, 1361 Route des Lucioles, 06560, Valbonne Sophia Antipolis, France

    A. Celani

  4. Physics of Complex Systems, Weizmann Institute of Science, Rehovot, 76100, Israel

    G. Falkovich

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Correspondence to G. Falkovich.

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Bernard, D., Boffetta, G., Celani, A. et al. Conformal invariance in two-dimensional turbulence. Nature Phys 2, 124–128 (2006). https://doi.org/10.1038/nphys217

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