Anomalous Scaling in a Model of Hydrodynamic Turbulence with a Small Parameter

The major difficulty in developing theories for anomalous scaling in hydrodynamic turbulence is the lack of a small parameter. In this Letter we introduce a shell model of turbulence that exhibits anomalous scaling with a tunable small parameter. The small parameter $ϵ$ represents the ratio between deterministic and random components in the coupling between $N$ identical copies of the turbulent field. We show that in the limit $N\to \infty$anomalousscalingsetsinproportionalto$\epsilon^4$. Moreover we give strong evidences that the birth of anomalous scaling appears at a finite critical $ϵ$, being ${ϵ}_c\approx 0.6$.