8.2 Probability distribution functions
As said in Section 7.2 the statistics of turbulent flows can be characterized by the PDF of field
differences over varying scales. At large scales PDFs are Gaussian, while tails become higher than Gaussian
(actually, PDFs decay as ) at smaller scales.
Marsch and Tu (1994) started to investigate the behavior of PDFs of fluctuations against scales and
they found that PDFs are rather spiky at small scales and quite Gaussian at large scales. The same
behavior have been obtained by Sorriso-Valvo et al. (1999, 2001) who investigated Helios 2 data for both
velocity and magnetic field.
In order to make a quantitative analysis of the energy cascade leading to the scaling dependence of
PDFs just described, the distributions obtained in the solar wind have been fitted (Sorriso-Valvo
et al., 1999) by using the log-normal ansatz
The width of the log-normal distribution of is given by , while is the most
probable value of .
The Equation (42) has been fitted to the experimental PDFs of both velocity and magnetic intensity,
and the corresponding values of the parameter have been recovered. In Figures 84 and 85
the solid lines show the curves relative to the fit. It can be seen that the scaling behavior of
PDFs, in all cases, is very well described by Equation (42). At every scale , we get a single
value for the width , which can be approximated by a power law for
, as it can be seen in Figure 86. The values of parameters and obtained in the
fit, along with the values of , are reported in Table 5. The fits have been obtained in the
range of scales for the magnetic field, and for the velocity field. The
analysis of PDFs shows once more that magnetic field is more intermittent than the velocity
||Normalized PDFs of fluctuations of the longitudinal velocity field at four different scales
. Solid lines represent the fit made by using the log-normal model (adopted from Sorriso-Valvo
et al., 1999, © 1999 American Geophysical Union, reproduced by permission of American
||We show the normalized PDFs of fluctuations of the magnetic field magnitude at four
different scales as indicated in the different panels. Solid lines represent the fit made by using the
log-normal model (adopted from Sorriso-Valvo et al., 1999, © 1999 American Geophysical Union,
reproduced by permission of American Geophysical Union).
The same analysis has been repeated by Forman and Burlaga (2003). These authors used
averages of radial solar wind speed reported by the SWEPAM instrument on the ACE spacecraft,
increments have been calculated over a range of lag times from to several days. From the PDF
obtained through the Equation (45) authors calculated the structure functions and compared the free
parameters of the model with the scaling exponents of the structure functions. Then a fit on the scaling
exponents allows to calculate the values of and . Once these parameters have been
calculated, the whole PDF is evaluated. The same authors found that the PDFs do not precisely
fit the data, at least for large values of the moment order. Interesting enough, Forman and
Burlaga (2003) investigated the behavior of PDFs when different kernels , derived from
different cascade models, are taken into account in Equation (42). They discussed the physical
content of each model, concluding that a cascade model derived from lognormal or log-Lévy
modified by self-organized criticality proposed by Schertzer et al. (1997), seems to avoid all problems
present in other cascade models.
||Scaling laws of the parameter as a function of the scales , obtained by the fits
of the PDFs of both velocity and magnetic variables (see Figures 84 and 85). Solid lines represent
fits made by power laws (adopted from Sorriso-Valvo et al., 1999, © 1999 American Geophysical
Union, reproduced by permission of American Geophysical Union).