Sharp Lower Bounds for the Dimension of the Global Attractor of the Sabra Shell Model of Turbulence

• Published in: Journal of statistical physics Vol. 127; no. 6; pp. 1173 - 1192
• Main Authors: Constantin, Peter, Levant, Boris, Titi, Edriss S.
• Format: Journal Article
• Language: English
• Published: New York Springer Nature B.V 01.06.2007
• Subjects: Bifurcations; Fractal geometry; Fractals; Lower bounds; Mathematical models; Parameters; Physics; Turbulence; Upper bounds; Viscosity
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-007-9317-x

In this work we derive lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing term and for sufficiently small viscosity term ν, the Sabra shell model has a global attractor of large Hausdorff and fractal dimensions proportional to log  ν −1 for all values of the governing parameter ε, except for ε =1. The obtained lower bounds are sharp, matching the upper bounds for the dimension of the global attractor obtained in our previous work. Moreover, the complexity of the dynamics of the shell model increases as the viscosity ν tends to zero, and we describe a precise scenario of successive bifurcations for different parameters regimes. In the “three-dimensional” regime of parameters this scenario changes when the parameter ε becomes sufficiently close to 0 or to 1. We also show that in the “two-dimensional” regime of parameters, for a certain non-zero forcing term, the long-term dynamics of the model becomes trivial for every value of the viscosity.