TY - GEN
N2 - 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 k−3q power law, suggesting that large wave-number eddies are in thermodynamic equilibrium, which is characterized by q = 1, finally resulting in Kraichnan’s correct k −3 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.
DO - 10.3390/e20020109
DO - DOI
AB - 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 k−3q power law, suggesting that large wave-number eddies are in thermodynamic equilibrium, which is characterized by q = 1, finally resulting in Kraichnan’s correct k −3 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.
AD - Haute école d’ingénierie et de gestion du canton de Vaud (HEIG-VD) HES-SO // Haute Ecole Spécialisée de Suisse Occidentale
AD - Laboratory of Hydraulics, Hydrology and Glaciology, Swiss Federal Institute of Technology, ETH
T1 - Tsallis extended thermodynamics applied to 2-d turbulence :Lévy statistics and q-fractional generalized kraichnanian energy and enstrophy spectra
DA - 2018-02
DA - 2018
AU - Egolf, Peter William
AU - Hutter, Kolumban
L1 - http://hesso.tind.io/record/2379/files/Egolf_2018_Tsallis.pdf
L1 - http://hesso.tind.io/record/2379/files/Egolf_2018_Tsallis.pdf?subformat=pdfa
JF - Entropy
LA - eng
LA - English
PY - 2018-02
PY - 2018
N1 - This article belongs to the Special Issue Phenomenological Thermodynamics of Irreversible Processes
ID - 2379
L4 - http://hesso.tind.io/record/2379/files/Egolf_2018_Tsallis.pdf
L4 - http://hesso.tind.io/record/2379/files/Egolf_2018_Tsallis.pdf?subformat=pdfa
KW - Ingénierie
KW - extended thermodynamics
KW - Tsallis entropy
KW - escort probability
KW - fractional calculus
KW - 2-d turbulence
KW - spectra of Kraichnan
KW - Kolmogorov-Oboukov spectrum
KW - Lévy statistics
KW - intermittency
SN - 1099-4300
TI - Tsallis extended thermodynamics applied to 2-d turbulence :Lévy statistics and q-fractional generalized kraichnanian energy and enstrophy spectra
Y1 - 2018-02
L2 - http://hesso.tind.io/record/2379/files/Egolf_2018_Tsallis.pdf
L2 - http://hesso.tind.io/record/2379/files/Egolf_2018_Tsallis.pdf?subformat=pdfa
LK - http://hesso.tind.io/record/2379/files/Egolf_2018_Tsallis.pdf
LK - http://hesso.tind.io/record/2379/files/Egolf_2018_Tsallis.pdf?subformat=pdfa
UR - http://hesso.tind.io/record/2379/files/Egolf_2018_Tsallis.pdf
UR - http://hesso.tind.io/record/2379/files/Egolf_2018_Tsallis.pdf?subformat=pdfa
ER -