Anomalous Scaling in a Model of Hydrodynamic Turbulence with a Small Parameter
Daniela Pierotti, Victor S. L'vov, Anna Pomyalov, Itamar Procaccia
Abstract
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→∞
anomalous scaling sets in proportional to
ϵ
4
. Moreover we give strong evidences that the birth of anomalous scaling appears at a finite critical
ϵ
, being
ϵ
c
≈0.6
.