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 of a function consists of the projection of on a wavelet basis to obtain wavelet coefficients . These coefficients are obtained through a convolution between the analyzed function and a shifted and scaled version of an optional wavelet base
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 99, and the analytical expression for this wavelet is
Since the Parceval’s theorem exists, the square modulus represents the energy content of fluctuations at the scale at position .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., 1990; Farge, 1992)
In our analyses we adopted a recursive method (Bianchini et al., 1999; Bruno 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 (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 , 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.The black curve in Figure 100 shows the original profile of the magnetic field intensity observed by Helios 2 between day 50 and 52 within a highly velocity stream at . 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.
© Max Planck Society and the author(s)