Anomalous scaling in a model of hydrodynamic turbulence with a small parameter

The major difficulty in developing theories for anomalous scaling in hydrod ynamic turbulence is the lack of a small parameter. In this letter we intro duce a shell model of turbulence that exhibits anomalous scaling with a tun able parameter epsilon, 0 less than or equal to epsilon less than or equal to 1, representing the ratio between deterministic and random components in the coupling between N identical copies of the turbulent field. Our numeri cal experiments give strong evidence that in the limit N --> infinity anoma lous scaling sets in proportional to epsilon(4) This result shows consisten cy with the nonperturbative closure proposed by the authors in Phys. Fluids , 12 (2000) 803. In this procedure closed equations of motion for the low-o rder correlation and response functions are obtained, keeping terms proport ional to epsilon(0) and epsilon(4), discarding terms of orders epsilon(6) a nd higher. Moreover we give strong evidences that the birth of anomalous sc aling appears at a finite critical epsilon, being epsilon(c) approximate to 0.6.