Renormalization Approach to Blowup in MHD Shell Model of Turbulence
G. T. Goedert1, A. A. Mailybaev1
1IMPA - National Institute for Pure and Applied Mathematics
This work considers the question of existence and role of finite time singularities (blowup) in turbulence from a dynamical system standpoint. Since the rigorous analysis or simulation of fluid flow over the scales of interest is beyond our capabilities, we turn to a class of dynamical systems called shell models that have been shown to model turbulent flow qualitatively. We focus on one such model of magnetohydrodynamic turbulence. The analysis is based on a renormalization scheme which takes blowup time to infinity. This transformation associates the blowup to an attractor of the renormalized system, found from its Poincare map. This way, we describe the blowup structure and explain its universality. This approach shows that, for some parameters, the blowup has a chaotic structure. Moreover, we observe an interesting effect of coexisting blowup scenarios in a specific model, which are selected based on the initial conditions. Implications of these results for MHD are discussed.
Keywords: Turbulence, Magnetohydrodynamics, Dynamical System Approach