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**Kolmogorov spectra of turbulence I. Wave turbulence.**
*(English)*
Zbl 0786.76002

Springer Series in Nonlinear Dynamics. Berlin: Springer-Verlag. xiii, 264 p. (1992).

The authors consider turbulence as a highly excited state of a dissipative dynamical system with a large number of degrees of freedom far beyond the limit of the Navier-Stokes equations. They divide the phenomena of turbulence into two catalogues, one of which is classified as the strong turbulence, or vortex turbulence, in which vortices of spatial extensions with the same order strongly interact in incompressible fluids, which results in energy redistribution between different modes. According to Richardson-Kolmogorov concept of turbulence cascade hypothesis, the stationary spectrum of turbulence energy \(E_ k\) in the \(k\)-space in the sub-inertial region can be expressed by dimensional analysis as \(E_ k=\lambda\) \(P^{2/3}k^{-5/3}\), where \(P\) is dissipative energy rate, \(k\) is wave number and \(\lambda\) is a constant. The vortex interaction is so strong that there exist no small parameters in fluid dynamical equations, and the system cannot be depicted by a linear approximation for an incompressible fluid against a stationary background. There are no homogeneous background waves in fluid dynamics whose amplitudes are sufficiently small. But there exist other cases in which the system has a linear approximation that describes small amplitude waves with dispersion, such as turbulent phenomena in plasma, intensive laser pulses propagating in nonlinear dielectrics, waves on fluid surface etc., and the system can be depicted efficiently by the Kolmogorov’s concept. Such turbulent phenomena are classified as the weak or wave turbulence. Starting with simple dimensional analysis, the book contains Hamiltonian formalism for waves, the derivation and stationary solutions of kinetic wave equations, stability problems and formation of Kolmogorov spectra, and physical applications of the general theory which answers main questions as follows: the existence of stationary spectra, connection between the flux and pumping characteristics, behaviour in the damping region, spectrum stability, etc. Finally, a methodological guide of this volume is given in the conclusion for readers who want to investigate wave turbulence and Kolmogorov spectra in detail. So, this book can be used as an introductory textbook on wave turbulence for graduate students and young researchers who are interested in this fields of fluid dynamics, meteorology, oceanography, astrophysics, plasma, solid-state physics, etc.

Reviewer: Wei Zhong-Lei (Beijing)

### MSC:

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

76F99 | Turbulence |