Dynamical systems as models of the energy cascade
A direct numerical simulation of the Navier–Stokes equations in the turbulent regime, i.e. at high Re, is difficult since the number of degrees of freedom which is necessary to describe the flow increases as a power of the Reynolds number (see section 2.3.5). However, these degrees of freedom are probably organized in a hierarchical way, so that simplified dynamical systems could be useful for the understanding of the scaling invariance. Borrowing some ideas introduced by the renormalization group approach for second-order phase transitions, one can also argue that the statistical properties of intermittency, at least in isotropic homogeneous turbulence, might be quite independent of the detailed dynamics of the Navier–Stokes equations. If that is true, it would imply that dynamical systems sharing the same ‘symmetries’ of the Navier–Stokes equations should be characterized by the same intermittency effects. Unfortunately, it is neither clear whether we already know all the ‘symmetries’ of the Navier–Stokes equations nor whether universality arguments, á la renormalization group, can be assumed in a theory of turbulence. In order to improve our present knowledge, we need to study and hopefully solve simplified models with the same ‘phenomenological’ properties as the Navier–Stokes equations. It is thus useful to analyse chaotic dynamical systems which model the energy cascade, instead of the complete Navier–Stokes equations, using an approach to the intermittency problem proposed by many authors, such as Gledzer , Siggia , Grappin et al.  and others.