Dynamical systems approach to turbulence.

*(English)*Zbl 0933.76002
Cambridge Nonlinear Science Series. 8. Cambridge: Cambridge Univ. Press. xx, 350 p. (1998).

Contrary to the more traditional phenomenological or statistical approach to turbulence in fluids, the present monograph treats turbulence as a manifestation of deterministic chaos. Correspondingly, the word ‘turbulence’ is used in a much broader sense for chaotic systems which are spatially incoherent, including chemical reactions and flame fronts. The authors show that the concepts and techniques developed for chaotic dynamical systems play a key role in the understanding of turbulent states in spatially extended systems. Mainly paradigmatic dynamical models are considered which are much simpler than the Navier-Stokes equations, but nevertheless have turbulent states. This allows bridging the gap between the traditional statistical description and the dynamical behavior in phase space.

The book begins with a definition of the standard tools of analysis of deterministic chaos, such as Lyapunov spectrum, fractal dimensions, dynamical intermittency, multifractality, etc. After reviewing the phenomenological properties of fully developed hydrodynamical turbulence along the ideas of Kolmogorov, simplified shell models are introduced which capture the basic properties of the Navier-Stokes equations with a reasonable number of equations. The even simpler coupled map lattices, large systems of coupled nonlinear maps which are discrete in space anbd time, prove to be helpful in understanding the features of chaotic systems as the size of the systems becomes large, such as coherence versus disorder or the spreading of turbulent spots.

A separate section is devoted to turbulence in amplitude equations, such as the complex Ginzburg-Landau and the Kuramoto-Sivashinsky equation. The interplay between periodic states, spirals, turbulent states and so-called vortex-glass states (disordered states with many spirals) is demonstrated. It is shown that the simple-minded approach to predictability in chaotic systems via the Lyapunov exponent fails in realistic situations, where many characteristic times are involved. The similarities between turbulence and the static and dynamical properties of rough surfaces are discussed and applied to the motion of noise-driven fronts or interfaces. In the final part, a review of Lagrangian chaos and its relation to mixing and transport diffusion phenomena in fluids is given, and the problem of chaotic diffusion is addressed.

This book should be of considerable interest to physicists and more mathematically inclined readers, which are assumed to have a certain background in dynamical system theory.

The book begins with a definition of the standard tools of analysis of deterministic chaos, such as Lyapunov spectrum, fractal dimensions, dynamical intermittency, multifractality, etc. After reviewing the phenomenological properties of fully developed hydrodynamical turbulence along the ideas of Kolmogorov, simplified shell models are introduced which capture the basic properties of the Navier-Stokes equations with a reasonable number of equations. The even simpler coupled map lattices, large systems of coupled nonlinear maps which are discrete in space anbd time, prove to be helpful in understanding the features of chaotic systems as the size of the systems becomes large, such as coherence versus disorder or the spreading of turbulent spots.

A separate section is devoted to turbulence in amplitude equations, such as the complex Ginzburg-Landau and the Kuramoto-Sivashinsky equation. The interplay between periodic states, spirals, turbulent states and so-called vortex-glass states (disordered states with many spirals) is demonstrated. It is shown that the simple-minded approach to predictability in chaotic systems via the Lyapunov exponent fails in realistic situations, where many characteristic times are involved. The similarities between turbulence and the static and dynamical properties of rough surfaces are discussed and applied to the motion of noise-driven fronts or interfaces. In the final part, a review of Lagrangian chaos and its relation to mixing and transport diffusion phenomena in fluids is given, and the problem of chaotic diffusion is addressed.

This book should be of considerable interest to physicists and more mathematically inclined readers, which are assumed to have a certain background in dynamical system theory.

Reviewer: Werner Koch (GĂ¶ttingen)

##### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76F20 | Dynamical systems approach to turbulence |

37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |

82D30 | Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) |