Spectral and statistical analysis of plasma turbulence: beyond linear techniques

The analysis of plasma turbulence has traditionally relied on limited repertoire of methods such as Fourier analysis, correlation analysis, etc. This text gives a short overview of what deeper insight can be obtained by using techniques that exploit nonlinear properties of the data. It covers both higher order spectra and higher order statistics. published in: Space Plasma Simulation, edited by J. Büchner, C.T. Dum and M. Scholer, Springer, Berlin (2003) 1 Linear vs nonlinear world Although linear descriptions traditionally constitute a rule of conduct in the analysis of space plasma data, there is quite some evidence that nonlinear descriptions may provide considerable added value. Linear descriptions can all be cast within the single convenient framework of Fourier analysis, which includes well known quantities such as power spectra and autocorrelation functions. There is no such single framework, however, for making nonlinear descriptions. A key problem in nonlinear world therefore is the choice of the right technique, or equivalently, the right invariant. Physical guidance and a good understanding of what the model produces are a must. The repertoire of techniques for analyzing nonlinear phenomena is huge and increases by the day, see for example [1,36,61]. Most of the texts on turbulence are essentially devoted to neutral fluids only [49,44,26,9]. As it turns out, only a subset of these techniques is really relevant for space plasma applications. By relevant we mean that a clear physical interpretation can be given to the output, and a compatibility between the underlying assumptions and properties of the model (or the observations). Several applications to problems encountered in space plasmas can nevertheless be found in [24,28,22,75,76]. Most of them deal with experimental data, but the extension to simulation data is immediate. Significant advantages of the latter are generally a smaller noise level, a more flexible size of the dataset and the possibility to access quantities that cannot always be measured. If simulation data have received comparatively less attention so far, it is not by lack of relevance, but often because we simply don’t know yet how to properly handle large ensembles of multivariate data ! 2 Thierry Dudok de Wit There exist essentially three major families of techniques for analyzing turbulence, all of which are rooted in the theory of statistics and that of dynamical systems. They are respectively: Higher order spectra (section 2), which are a generalization of Fourier analysis to include information about phase coherence. This approach is particularly relevant for wave phenomena that weakly depart from nonlinearity. Typical applications are: nonlinear wave interactions, wave-particle interactions, weak turbulence, etc. Higher order statistics (section 3), although closely related to higher order spectra, are more generally applied to stochastic processes, regardless of the degree of nonlinearity. Typical applications are fully developed turbulence, fluctuations in complex systems, etc. Phase space techniques is the name given to techniques that are generally applied to chaotic systems, whose time evolution can be approximated with a low-dimensional deterministic model. Such systems are not generic in space plasmas but the concepts that have been developed in this framework are powerful and deserve attention. Two recent textbooks are [36,9]. Phase space methods will not be considered here because would require a textbook on their own. Our selection is therefore somewhat arbitrary, even more so since the same concept can often be found in more than one family. The domain of application of each family can be characterized by comparing the degrees of stochasticity (how much is the dynamics ruled by external noise terms ? ) and nonlinearity (how nonlinear is the process ? ), see Fig. 1. Notice how the applicability of each technique is usually confined to a neighbourhood around some hypothesis. One could of course add more dimensions to the plot by introducing the extension (is the process temporal or spatio-temporal ? ), the number of degrees of freedom, etc. 2 Higher-order spectra and spectral energy transfers One of the most obvious approaches for making the transition from a linear to a nonlinear description, is to start from a linear description of the system of interest, and subsequently introduce a weak nonlinearity. How far this perturbative approach can be extended to strongly nonlinear systems depends on the type of problem one is addressing. Since one starts with a linear description, it is appropriate to describe the system in terms of its eigenmodes, namely Fourier modes. The basic assumption underlying standard Fourier analysis is that any stationary fluctuating physical quantity can be regarded as the superposition of statistically independent Fourier modes. Accordingly, all the relevant information is contained in the amplitude of these modes only, that is, the power spectral density (or alternatively, the autocorrelation function). If, however, there exists some parametrical or nonlinear physical process, then the phases of the Fourier modes are not independent Techniques for turbulence analysis 3