Critical dimension for hydrodynamic turbulence

Hydrodynamic turbulence exhibits nonequilibrium behavior with k^{-5/3} energy spectrum, and equilibrium behavior with k^{d-1} energy spectrum and zero viscosity, where d is the space dimension. Using recursive renormalization group in Craya-Herring basis, we show that the nonequilibrium solution is valid only for d<6, whereas equilibrium solution with zero viscosity is the only solution for d>6. Thus, d=6 is the critical dimension for hydrodynamic turbulence. In addition, we show that the energy flux changes sign from positive to negative near d=2.15. We also compute the energy flux and Kolmogorov's constants for various d's, and observe that our results are in good agreement with past numerical results.