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 parameter epsilon, 0 <= epsilon <= 1, representing the ratio between deterministic and random components in the coupling between N identical copies of the turbulent field. Our numerical experiments give strong evidence that in the limit N → ∞ anomalous scaling sets in proportional to epsilon⁴. This result shows consistency with the nonperturbative closure proposed by the authors in Phys. Fluids, 12 (2000) 803. In this procedure closed equations of motion for the low-order correlation and response functions are obtained, keeping terms proportional to epsilon⁰ and epsilon⁴, discarding terms of orders epsilon⁶ and higher. Moreover we give strong evidences that the birth of anomalous scaling appears at a finite critical epsilon, being epsilon_c approx 0.6.