2.6 Dynamical system approach to turbulence

In the limit of fully developed turbulence, when dissipation goes to zero, an infinite range of scales are excited, that is, energy lies over all available wave vectors. Dissipation takes place at a typical dissipation length scale which depends on the Reynolds number $Re$ through $-3/4 lD ~ LRe$ (for a Kolmogorov spectrum $-5/3 E(k) ~ k$ ). In 3D numerical simulations the minimum number of grid points necessary to obtain information on the fields at these scales is given by $N ~ (L/lD)3 ~ Re9/4$ . This rough estimate shows that a considerable amount of memory is required when we want to perform numerical simulations with high $Re$ . At the present, typical values of Reynolds numbers reached in 2D and 3D numerical simulations are of the order of $4 10$ and $3 10$ , respectively. At these values the inertial range spans approximately one decade or little more.

Given the situation described above, the question of the best description of dynamics which results from original equations, using only a small amount of degree of freedom, becomes a very important issue. This can be achieved by introducing turbulence models which are investigated using tools of dynamical system theory (Bohr et al., 1998 ). Dynamical models, then, represent minimal set of ordinary differential equations that can mimic the gross features of energy cascade in turbulence. These studies are motivated by the famous Lorenz’s model (Lorenz, 1963 ) which, containing only three degrees of freedom, simulates the complex chaotic behavior of turbulent atmospheric flow, becoming a paradigm for the study of chaotic systems.

Up to the Lorenz’s chaotic model, studies on the birth of turbulence dealt with linear and, very rarely, with weak non-linear evolution of external disturbances. The first physical model of laminar-turbulent transition is due to Landau and it is reported in the fourth volume of the course on Theoretical Physics (Landau and Lifshitz, 1971 ). According to this model, as the Reynolds number is increased, the transition is due to a serie of infinite Hopf bifurcations at fixed values of the Reynolds number. Each subsequent bifurcation adds a new incommensurate frequency to the flow which, in some sense, is obliged to become turbulent because the infinite number of degrees of freedom involved.

The Landau transition scenario is, however, untenable because few incommensurate frequencies cannot exist without coupling between them. Ruelle and Takens (1971 ) proposed a new mathematical model, according to which after few, usually three, Hopf bifurcations the flow becomes suddenly chaotic. In the phase space this state is characterized by a very intricate attracting subset, a strange attractor. The flow corresponding to this state is highly irregular and strongly dependent on initial conditions. This characteristic feature is now known as the butterfly effect and represents the true definition of deterministic chaos. These authors indicated as an example for the occurrence of a strange attractor the old strange time behavior of the Lorenz’s model. The model is a paradigm for the occurrence of turbulence in a deterministic system, it reads

where $x(t)$ , $y(t)$ , and $z(t)$ represent the first three modes of a Fourier expansion of fluid convective equations in the Boussinesq approximation, $Pr$ is the Prandtl number, $b$ is a geometrical parameter, and $R$ is the ratio between the Raylaigh number and the critical Raylaigh number for convective motion. The time evolution of the variables $x(t)$ , $y(t)$ , and $z(t)$ is reported in Figure 13

. A reproduction of the Lorenz butterfly attractor, namely the projection of the variables on the plane $(x,z)$ is shown in Figure 14

. A few years later, Gollub and Swinney (1975 ) performed very sophisticated experiments⁴ , concluding that the transition to turbulence in a flow between co-rotating cylinders is described by the Ruelle and Takens (1971 ) model rather than by the Landau scenario.

Figure 13:

Time evolution of the variables $x(t)$ , $y(t)$ , and $z(t)$ in the Lorenz’s model (see Equation (18

)). This figure has been obtained by using the parameters $Pr = 10$ , $b = 8/3$ , and $R = 28$ .

Figure 14:

The Lorenz butterfly attractor, namely the time behavior of the variables $z(t)$ vs. $x(t)$ as obtained from the Lorenz’s model (see Equation (18

)). This figure has been obtained by using the parameters $Pr = 10$ , $b = 8/3$ , and $R = 28$ .

After this discovery, the strange attractor model gained a lot of popularity, thus stimulating a great number of further studies on the time evolution of non-linear dynamical systems. An enormous number of papers on chaos rapidly appeared in literature, quite in all fields of physics, and transition to chaos became a new topic. Of course, further studies on chaos rapidly lost touch with turbulence studies and turbulence, as reported by Feynman et al. (1977 ), still remains …the last great unsolved problem of the classical physics. Actually, since the solar wind is in a state of fully developed turbulence, the topic of the transition to turbulence is not so close to the main goal of this review. On the other hand, we like to cite recent theoretical efforts made by Chian and coworkers (Chian et al., 1998 , 2003 ) related to the onset of Alfvénic turbulence. These authors, numerically solved the derivative non-linear Schrödinger equation (Mjolhus, 1976 ; Ghosh and Papadopoulos, 1987 ) which governs the spatio-temporal dynamics of non-linear Alfvén waves, and found that Alfvénic intermittent turbulence is characterized by strange attractors. Turbulence can evolve via two distinct routes: Pomeau-Manneville intermittency and crisis-induced intermittency. Both types of chaotic transitions follow episodic switching between different temporal behaviors. In one case (Pomeau-Manneville) the behavior of the magnetic fluctuations evolve from nearly periodic to chaotic while, in the other case the behavior intermittently assumes weakly chaotic or strongly chaotic features.