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Uspekhi Mat. Nauk, 2007, Volume 62, Issue 3(375), Pages 193–206 (Mi umn6118)  

This article is cited in 8 scientific papers (total in 8 papers)

Conformal invariance in hydrodynamic turbulence

G. Falkovich

Weizmann Institute of Science

Abstract: This short survey is written by a physicist. It contains neither theorems nor precise definitions. Its main content is a description of the results of numerical solution of the equations of fluid mechanics in the regime of developed turbulence. Due to limitations of computers, the results are not very precise. Despite being neither exact nor rigorous, the findings may nevertheless be of interest for mathematicians. The main result is that the isolines of some scalar fields (vorticity, temperature) in two-dimensional turbulence belong to the class of conformally invariant curves called SLE (Scramm–Löwner evolution) curves. First, this enables one to predict and find a plethora of quantitative relations going far beyond what was known previously about turbulence. Second, it suggests relations between phenomena that seemed unrelated, like the Euler equation and critical percolation. Third, it shows that one is able to get exact analytic results in statistical hydrodynamics. In short, physicists have found something unexpected and hope that mathematicians can help to explain it.

DOI: https://doi.org/10.4213/rm6118

Full text: PDF file (811 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2007, 62:3, 497–510

Bibliographic databases:

UDC: 517.54+517.957+517.958:531.32
MSC: Primary 60K35, 76F55; Secondary 35Q35, 60J65, 76F05, 76F25, 81T40, 82B27, 82C05
Received: 05.10.2006

Citation: G. Falkovich, “Conformal invariance in hydrodynamic turbulence”, Uspekhi Mat. Nauk, 62:3(375) (2007), 193–206; Russian Math. Surveys, 62:3 (2007), 497–510

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Falkovich G., “Nodal lines in turbulence”, The European Physical Journal - Special Topics, 145:1 (2007), 211–216  crossref  mathscinet  adsnasa  isi  scopus
    2. Eyink G., Frisch U., Moreau R., Sobolevskiǐ A., “General introduction”, Proceedings of an international conference. Euler equations: 250 Years on (EE250), Phys. D, 237:14–17 (2008), xi–xv  crossref  mathscinet  isi  scopus
    3. V. N. Grebenev, M. Oberlack, A. N. Grishkov, “Infinite dimensional Lie algebra associated with conformal transformations of the two-point velocity correlation tensor from isotropic turbulence”, Z. Angew. Math. Phys, 2012  crossref  mathscinet  isi  scopus
    4. V. N. Grebenev, A. N. Grishkov, M. Oberlack, “The Extended Symmetry Lie Algebra and the Asymptotic Expansion of the Transversal Correlation Function for the Isotropic Turbulence”, Advances in Mathematical Physics, 2013 (2013), 1  crossref  mathscinet  isi  scopus
    5. Gregory Falkovich, Alexander Zamolodchikov, “Operator product expansion and multi-point correlations in turbulent energy cascades”, J. Phys. A: Math. Theor, 48:18 (2015), 18FT02  crossref  mathscinet  zmath  isi  scopus
    6. Batista-Tomas A.R., Diaz O., Batista-Leyva A.J., Altshuler E., “Classification and dynamics of tropical clouds by their fractal dimension”, Q. J. R. Meteorol. Soc., 142:695, B (2016), 983–988  crossref  isi  scopus
    7. Grebenev V.N., Waclawczyk M., Oberlack M., “Conformal Invariance of the Lungren-Monin-Novikov Equations For Vorticity Fields in 2D Turbulence”, J. Phys. A-Math. Theor., 50:43 (2017), 435502  crossref  mathscinet  zmath  isi  scopus
    8. Waclawczyk M., Grebenev V.N., Oberlack M., “Lie Symmetry Analysis of the Lundgren-Monin-Novikov Equations For Multi-Point Probability Density Functions of Turbulent Flow”, J. Phys. A-Math. Theor., 50:17 (2017), 175501  crossref  mathscinet  zmath  isi  scopus
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