TRANSITION TO CHAOS IN A SHELL-MODEL OF TURBULENCE

We study a shell model for the energy cascade in three-dimensional tur bulence by varying the coefficients of the non-linear terms in such a way that the fundamental symmetries of Navier-Stokes are conserved. Wh en the control parameter epsilon related to the strength of backward e nergy transfer is small enough, the dynamical system has a stable fixe d point corresponding to the Kolmogorov scaling. By using the bi-ortho gonal decomposition, the transition to chaos is shown to follow the Ru elle-Takens scenario. For epsilon > 0.3953... there exists a strange a ttractor which remains close to the Kolmogorov fixed point. The interm ittency of the chaotic evolution and of the scaling can be described b y an intermittent one-dimensional map. We introduce a modified shell m odel which has a good scaling behaviour also in the infrared region. W e study the multifractal properties of this model for large number of shells and for values of epsilon slightly above the chaotic transition . In this case by making a local analysis of the scaling properties in the inertial range we found that the multifractal corrections seem to become weaker and weaker approaching the viscous range.