5.20.1 A New Tool of Turbulence Analysis: Wavelet Bicoherence
Recently, a new tool for the analysis of turbulence has been introduced and investigated, it is the wavelet bicoherence (see van Milligen et al. 1995a). This tool
Fig. 5.59 (a) Logistic equation, (b) Bi-linear process
Fig. 5.59 (a) Logistic equation, (b) Bi-linear process is able to detect the phase coupling, i.e. the nonlinear interactions of the lowest "quadratic" order with time resolution. Its potential is important and it was applied in specific works (see footnote10) to numerical models of chaos and turbulence and also applied to real measures. In the first case, the van der Pol model of chaos, this tool detected the coupling interaction between two coupled van der Pol oscillators. In the second case, the drift wave turbulence model concerning the plasma physics, it detected a highly localized "coherent structure". In the case of real measures, i.e. for the analysis of reflectometry measures concerning fusion plasmas, it detected temporal intermittency and a strong increase in nonlinear phase coupling which coincide with the Low-to-High11 confinement mode transition. Three arguments plead for a new tool about the turbulent phenomenon analysis: (1) First, the tools of chaos theory (as fractal dimensions, Lyapunov exponent, etc.) are not always easily applicable to real phenomena, in particular if the noise level is high. Moreover, information recovered by these methods is not always adapted to physical understanding. Indeed, a low fractal dimension measured in a real phenomenon is helpful information, but a high dimension, in particular higher than 5 (for example often observed in fusion plasmas) does not offer interesting solution. (2) Second, the applications of the traditional analysis (i.e. which involve long-time averages of moments of data) of standard spectral analysis are limited concerning chaotic or turbulent real phenomena. Indeed, if we consider the transition from quasi-periodicity to chaos in theoretical dynamical systems, as we have explained in the first part of this book, there exist several possible routes to chaos which can be summarized as follows: period-doubling, crises and intermittency. In these three main routes (Ott 1993), the transition to chaos are abrupt. Then, an explanation of the chaotic regime by means of a superposition of a large number of harmonic modes (i.e. oscillators which correspond to the Fourier analysis12) does not seem suitable. Indeed,
70 van Milligen et al. 1995b. Asociacion EURATOM-CIEMAT, Madrid, Spain. B. Carreras. Oak Ridge National Laboratory, Oak Ridge, Tennessee, U.S.A. L. Garcia.Universidad Carlos III, Madrid, Spain.
71 Usually written by physicists: (L/H).
72 See the Navier-Stokes equations sections in the present book.
Fig. 5.60 "Coherent structures" in turbulent phenomenon (ref: Haller)
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