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Entropy 2018, 20(2), 109; https://doi.org/10.3390/e20020109

Tsallis Extended Thermodynamics Applied to 2-d Turbulence: Lévy Statistics and q-Fractional Generalized Kraichnanian Energy and Enstrophy Spectra

1
Thermal Sciences and Engineering Institute, University of Applied Sciences of Western Switzerland, CH-1401 Yverdon-les-Bains, Switzerland
2
%Laboratory of Hydraulics, Hydrology and Glaciology, Swiss Federal Institute of Technology, ETH, Hönggerberg HIA 58D, CH 8093 Zurich, Switzerland
*
Author to whom correspondence should be addressed.
Received: 2 January 2018 / Revised: 24 January 2018 / Accepted: 1 February 2018 / Published: 7 February 2018
(This article belongs to the Special Issue Phenomenological Thermodynamics of Irreversible Processes)
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Abstract

The extended thermodynamics of Tsallis is reviewed in detail and applied to turbulence. It is based on a generalization of the exponential and logarithmic functions with a parameter q. By applying this nonequilibrium thermodynamics, the Boltzmann-Gibbs thermodynamic approach of Kraichnan to 2-d turbulence is generalized. This physical modeling implies fractional calculus methods, obeying anomalous diffusion, described by Lévy statistics with q < 5/3 (sub diffusion), q = 5/3 (normal or Brownian diffusion) and q > 5/3 (super diffusion). The generalized energy spectrum of Kraichnan, occurring at small wave numbers k, now reveals the more general and precise result k−q. This corresponds well for q = 5/3 with the Kolmogorov-Oboukov energy spectrum and for q > 5/3 to turbulence with intermittency. The enstrophy spectrum, occurring at large wave numbers k, leads to a k3q power law, suggesting that large wave-number eddies are in thermodynamic equilibrium, which is characterized by q = 1, finally resulting in Kraichnan’s correct k3 enstrophy spectrum. The theory reveals in a natural manner a generalized temperature of turbulence, which in the non-equilibrium energy transfer domain decreases with wave number and shows an energy equipartition law with a constant generalized temperature in the equilibrium enstrophy transfer domain. The article contains numerous new results; some are stated in form of eight new (proven) propositions. View Full-Text
Keywords: extended thermodynamics; Tsallis entropy; escort probability; fractional calculus; 2-d turbulence; spectra of Kraichnan; Kolmogorov-Oboukov spectrum; Lévy statistics; intermittency extended thermodynamics; Tsallis entropy; escort probability; fractional calculus; 2-d turbulence; spectra of Kraichnan; Kolmogorov-Oboukov spectrum; Lévy statistics; intermittency
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Egolf, P.W.; Hutter, K. Tsallis Extended Thermodynamics Applied to 2-d Turbulence: Lévy Statistics and q-Fractional Generalized Kraichnanian Energy and Enstrophy Spectra. Entropy 2018, 20, 109.

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