Renormalization of the Shell Model of Turbulence

Renormalization enables a systematic scale-by-scale analysis of multiscale systems. In this paper, we employ renormalization group (RG) to the shell model of turbulence and show that the RG equation is satisfied by |u_{n}|^{2}=K_{Ko}ε^{2/3}k_{n}^{-2/3}, where Ko is the Kolmogorov constant and ν_{n}=ν_{*}sqrt[K_{Ko}]ε^{1/3}k_{n}^{-4/3}, where k_{n}andu_{n} are the wave number and velocity of shell n; ν_{*}andK_{Ko} are RG and Kolmogorov's constants; and ε is the energy dissipation rate. We find that ν_{*}≈0.5 and K_{Ko}≈1.7, consistent with earlier RG works on the Navier-Stokes equation. We verify the theoretical predictions using numerical simulations.