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

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.