Turbulence and shell models.

*(English)*Zbl 1229.76001
Cambridge: Cambridge University Press (ISBN 978-0-521-19036-7/hbk; 978-0-511-91120-0/ebook). x, 152 p. (2011).

The book, structured in eight chapters, gives first an introduction to turbulence in the spirit of Kolmogorov’s phenomenology, before it discusses shell models. Then the reader can find a compact text intended for researchers and professionals who want a fast introduction to isotropic homogeneous turbulence using dynamical systems theory. The author hopes that the book should be accessible to advanced undergraduate and graduate students.

The first chapter gives an introduction to turbulence, and reviews the main characteristics and unknowns in turbulence theory. The second chapter focuses on 2D turbulence and considers the flows in Earth atmosphere as an example of flows constrained to two dimensions. Shell models form the subject of the third chapter. Here the author analyses the transfer of energy in turbulent flows from large to small scales, describing the transfer as an energy flux from small wave numbers to large wave numbers in the spectral representation of the Navier-Stokes equations.

The fourth chapter is devoted to scaling and symmetries. Regardless of the realism of shell models, they have an advantage over Navier-Stokes equations that their simplicity makes the analysis much more transparent. Taking into consideration that the case of fluid turbulence, where the number of degrees of freedom is proportional to \(Re^{9/4}\), can be described by a dynamical system when the dimension becomes large, the fifth chapter concerns the chaotic dynamics and investigates chaotic properties for shell models, such as the attractor dimension. The sixth chapter is devoted to an inviscid invariant of 3D flows, namely helicity, which is a measure of density of helices in turbulent flows. The seventh chapter investigates the intermittency in turbulence, and the final eighth chapter formulates some basic results from equilibrium statistical mechanics.

The first chapter gives an introduction to turbulence, and reviews the main characteristics and unknowns in turbulence theory. The second chapter focuses on 2D turbulence and considers the flows in Earth atmosphere as an example of flows constrained to two dimensions. Shell models form the subject of the third chapter. Here the author analyses the transfer of energy in turbulent flows from large to small scales, describing the transfer as an energy flux from small wave numbers to large wave numbers in the spectral representation of the Navier-Stokes equations.

The fourth chapter is devoted to scaling and symmetries. Regardless of the realism of shell models, they have an advantage over Navier-Stokes equations that their simplicity makes the analysis much more transparent. Taking into consideration that the case of fluid turbulence, where the number of degrees of freedom is proportional to \(Re^{9/4}\), can be described by a dynamical system when the dimension becomes large, the fifth chapter concerns the chaotic dynamics and investigates chaotic properties for shell models, such as the attractor dimension. The sixth chapter is devoted to an inviscid invariant of 3D flows, namely helicity, which is a measure of density of helices in turbulent flows. The seventh chapter investigates the intermittency in turbulence, and the final eighth chapter formulates some basic results from equilibrium statistical mechanics.

Reviewer: R. Militaru (Craiova)

##### 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 |