Ultraviolet dynamic renormalization group: Small-scale properties of a randomly stirred fluid

A dynamic renormalization-group method is developed to study the ultraviolet properties of velocity correlations generated by the Navier-Stokes equation with a random stirring force with the correlator decreasing as $k^{-y}(y\ge d)$ at $k\to \infty$. It is shown that the elimination of modes from a shell near the infrared cut-off $k_0\approx \frac{1}{L}\to 0$ results in two major effects: transition to a frame of reference moving with random velocity ${\overrightarrow{\mathrm{v}}}_L$ ($c_L^{2}a{L}^{\frac{2}{3}}$ if $y=d$) and the appearance of a negative dissipation in the Navier-Stokes equation proportional to $k^{\frac{2}{3}}$. All the divergent terms are summed up into a kinematic effect of a transfer of small eddies by large ones and the dynamics is determined by convergent series. Long-time, large-scale behavior of a fluid is identical with the one obtained by Martin and de Dominicis, Lucke, and Fournier and Frisch. It is shown that the theory is asymptotically free in the ultraviolet region for any $\epsilon =d-y\le 0$. The energy spectrum of a stirred fluid is $E\left(k\right)\mathrm{}a\left[\frac{k^{-\frac{5}{3}}}{\ln \mathrm{}\left(\mathrm{kL}\right)\mathrm{}}\right]$ for ($d=y$) and the Kolmogorov spectrum without corrections never exists in the ultraviolet limit.