Transition to chaos in a shell model of turbulence

Abstract

We study a shell model for the energy cascade in three-dimensional turbulence by varying the coefficients of the non-linear terms in such a way that the fundamental symmetries of Navier-Stokes are conserved. When the control parameter ϵ related to the strength of backward energy transfer is small enough, the dynamical system has a stable fixed point corresponding to the Kolmogorov scaling. By using the bi-orthogonal decomposition, the transition to chaos is shown to follow the Ruelle-Takens scenario. For ϵ > 0.3953… there exists a strange attractor which remains close to the Kolmogorov fixed point. The intermittency of the chaotic evolution and of the scaling can be described by an intermittent one-dimensional map. We introduce a modified shell model which has a good scaling behaviour also in the infrared region. We study the multifractal properties of this model for large number of shells and for values of ϵ 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.