Renormalization of shell model of turbulence

Renormalization enables a systematic scale-by-scale analysis of multiscale systems. In this paper, we employ $\mathit{\text{renormalization}}\mathit{\text{group}}$ (RG) to the shell model of turbulence and show that the RG equation is satisfied by $\mathrm{\mid }{u}_n{\mathrm{\mid }}^2=K_{\mathrm{K}\mathrm{o}}{\epsilon }^{2\mathrm{/}3}{k}_n^{-2\mathrm{/}3}$ and ${\nu }_n={\nu }_{\ast }\sqrt{K_{\mathrm{K}\mathrm{o}}}{\epsilon }^{1\mathrm{/}3}{k}_n^{-4\mathrm{/}3}$, where $k_n,u_n$ are the wavenumber and velocity of shell $n$; ${\nu }_{\ast },K_{\mathrm{K}\mathrm{o}}$ are RG and Kolmogorov’s constants; and $\epsilon$ is the energy dissipation rate. We find that ${\nu }_{\ast }\approx 0.5$ and $K_{\mathrm{K}\mathrm{o}}\approx 1.7$, consistent with earlier RG works on Navier-Stokes equation. We verify the theoretical predictions using numerical simulations.