Intermittency and Thermalization in Turbulence

A dissipation rate, which grows faster than any power of the wave number in Fourier space, may be scaled to lead a hydrodynamic system {\it actually} or {\it potentially} converge to its Galerkin truncation. Actual convergence we name for the asymptotic truncation at a finite wavenumber $k_G$ above which modes have no dynamics; and, we define potential convergence for the truncation at $k_G$ which, however, grows without bound. Both types of convergence can be obtained with the dissipation rate $\mu [cosh(k\mathrm{/}{k}_c)-1]$ who behaves as $k^2$ (newtonian) and $\exp \{k\mathrm{/}{k}_c\}$ for small and large $k\mathrm{/}{k}_c$ respectively. Competition physics of cascade, thermalization and dissipation are discussed with numerical Navier-Stokes turbulence, emphasizing on the intermittency growth.