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14 Appendix C: Wavelets as a Tool to Study Intermittency

Following Farge et al. (1990Jump To The Next Citation Point) and Farge (1992Jump To The Next Citation Point), intermittent events can be viewed as localized zones of fluid where phase correlation exists, in some sense coherent structures. These structures, which dominate the statistic of small scales, occur as isolated events with a typical lifetime which is greater than that of stochastic fluctuations surrounding them. Structures continuously appear and disappear, apparently in a random fashion, at some random location of fluid, and they carry most of the flow energy. In this framework, intermittency can be considered as the result of the occurrence of coherent (non-Gaussian) structures at all scales, within the sea of stochastic Gaussian fluctuations.

It follows that, since these structures are well localized in spatial scale and time, it would be advisable to analyze them using wavelets filter instead of the usual Fourier transform. Unlike the Fourier basis, wavelets allow a decomposition both in time and frequency (or space and scale). The wavelet transform W {f(t)} of a function f (t) consists of the projection of f (t) on a wavelet basis to obtain wavelet coefficients w(t, t). These coefficients are obtained through a convolution between the analyzed function and a shifted and scaled version of an optional wavelet base

( ) integral ' 1 t - t' ' w(t, t) = f (t) V~ --Y ------ dt, (95) t t
where the wavelet function
( ') Y ' (t) = V~ 1-Y t---t- t ,t t t

has zero mean and compact support. Some examples of translated and scaled version of this function for a particular wavelet called “charro”, because its profile resembles the Mexican hat “El Charro”, are given in Figure 99View Image, and the analytical expression for this wavelet is

|_ ( ) ( )_ | 1 ( t- t')2 1 (t - t')2 Yt',t(t) = V~ -- |_ 1 - ------ exp - -- ------ _| . t t 2 t

Since the Parceval’s theorem exists, the square modulus 2 |w(t,t)| represents the energy content of fluctuations f (t + t) - f(t) at the scale t at position t.

View Image

Figure 99: Some examples of Mexican Hat wavelet, for different values of the parameters t and ' t.
In analyzing intermittent structures it is useful to introduce a measure of local intermittency, as for example the Local Intermittency Measure (LIM) introduced by Farge (see, e.g., Farge et al., 1990Farge, 1992)
2 LIM = -|w(t,-t)|-- (96) <|w(t,t)|2>t
(averages are made over all positions at a given scale t). The quantity from Equation (96View Equation) represents the energy content of fluctuations at a given scale with respect to the standard deviation of fluctuations at that scale. The whole set of wavelets coefficients can then be split in two sets: a set which corresponds to “Gaussian” fluctuations wg(t, t), and a set which corresponds to “structure” fluctuations ws(t,t), that is, the whole set of coefficients w(t,t) = wg(t,t) o+ ws(t,t) (the symbol o+ stands here for the union of disjoint sets). A coefficient at a given scale and position will belong to a structure or to the Gaussian background according whether LIM will be respectively greater or lesser than a threshold value. An inverse wavelets transform performed separately on both sets, namely fg(t) = W - 1{wg(t, t)} and fs(t) = W -1{ws(t, t)}, gives two separate fields: a field fg(t) where the Gaussian background is collected, and the field f (t) s where only the non-Gaussian fluctuations of the original turbulent flow are taken into account. Looking at the field fs(t) one can investigate the spatial behavior of structures generating intermittency. The Haar basis have been applied to time series of thirteen months of velocity and magnetic data from ISEE space experiment for the first time by Veltri and Mangeney (1999b).

In our analyses we adopted a recursive method (Bianchini et al., 1999Bruno et al., 1999a) similar to the one introduced by Onorato et al. (2000) to study experimental turbulent jet flows. The method consists in eliminating, for each scale, those events which cause LIM to exceed a given threshold. Subsequently, the flatness value for each scale is checked and, in case this value exceeds the value of 3 (characteristic of a Gaussian distribution), the threshold is lowered, new events are eliminated and a new flatness is computed. The process is iterated until the flatness is equal to 3, or reaches some constant value, for each scale of the wavelet decomposition. This process is usually accomplished eliminating only a few percent of the wavelet coefficients for each scale, and this percentage reduces moving from small to large scales.

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Figure 100: The black curve indicates the original time series, the red one refers to the LIMed data, and the blue one shows the difference between these two curves.
The black curve in Figure 100View Image shows the original profile of the magnetic field intensity observed by Helios 2 between day 50 and 52 within a highly velocity stream at 0.9 AU. The overlapped red profile refers to the same time series after intermittent events have been removed using the LIM method. Most of the peaks, present in the original time series, are not longer present in the LIMed curve. The intermittent component that has been removed can be observed as the blue curve centered around zero.

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