{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T10:46:54Z","timestamp":1760784414309,"version":"build-2065373602"},"reference-count":96,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2018,2,7]],"date-time":"2018-02-07T00:00:00Z","timestamp":1517961600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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 L\u00e9vy statistics with q &lt; 5\/3 (sub diffusion), q = 5\/3 (normal or Brownian diffusion) and q &gt; 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\u2212q. This corresponds well for q = 5\/3 with the Kolmogorov-Oboukov energy spectrum and for q &gt; 5\/3 to turbulence with intermittency. The enstrophy spectrum, occurring at large wave numbers k, leads to a k\u22123q power law, suggesting that large wave-number eddies are in thermodynamic equilibrium, which is characterized by q = 1, finally resulting in Kraichnan\u2019s correct k\u22123 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.<\/jats:p>","DOI":"10.3390\/e20020109","type":"journal-article","created":{"date-parts":[[2018,2,7]],"date-time":"2018-02-07T12:20:29Z","timestamp":1518006029000},"page":"109","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Tsallis Extended Thermodynamics Applied to 2-d Turbulence: L\u00e9vy Statistics and q-Fractional Generalized Kraichnanian Energy and Enstrophy Spectra"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8828-1980","authenticated-orcid":false,"given":"Peter","family":"Egolf","sequence":"first","affiliation":[{"name":"Thermal Sciences and Engineering Institute, University of Applied Sciences of Western Switzerland, CH-1401 Yverdon-les-Bains, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kolumban","family":"Hutter","sequence":"additional","affiliation":[{"name":"%Laboratory of Hydraulics, Hydrology and Glaciology, Swiss Federal Institute of Technology, ETH, H\u00f6nggerberg HIA 58D, CH 8093 Zurich, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,2,7]]},"reference":[{"key":"ref_1","first-page":"105","article-title":"The temperature of turbulent flows","volume":"6","author":"Castaign","year":"1996","journal-title":"J. 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