### 7.2 Probability density functions and self-similarity of fluctuations

The presence of scaling laws for fluctuations is a signature of the presence of self-similarity in the
phenomenon. A given observable , which depends on a scaling variable , is invariant with respect
to the scaling relation , when there exists a parameter such that .
The solution of this last relation is a power law , where the scaling exponent is
.
Since, as we have just seen, turbulence is characterized by scaling laws, this must be a signature of
self-similarity for fluctuations. Let us see what this means. Let us consider fluctuations at two different
scales, namely and . Their ratio depends only on the value of ,
and this should imply that fluctuations are self-similar. This means that PDFs are related
through

Let us consider the standardized variables

It can be easily shown that when is unique or in a pure self-similar situation, PDFs are related through
, say by changing scale PDFs coincide.

The PDFs relative to the standardized magnetic fluctuations , at three different
scales , are shown in Figure 74. It appears evident that the global self-similarity in real
turbulence is broken. PDFs do not coincide at different scales, rather their shape seems to depend
on the scale . In particular, at large scales PDFs seem to be almost Gaussian, but they
become more and more stretched as decreases. At the smallest scale PDFs are stretched
exponentials. This scaling dependence of PDFs is a different way to say that scaling exponents of
fluctuations are anomalous, or can be taken as a different definition of intermittency. Note that
the wings of PDFs are higher than those of a Gaussian function. This implies that intense
fluctuations have a probability of occurrence higher than that they should have if they were
Gaussianly distributed. Said differently, intense stochastic fluctuations are less rare than we should
expect from the point of view of a Gaussian approach to the statistics. These fluctuations play a
key role in the statistics of turbulence. The same statistical behavior can be found in different
experiments related to the study of the atmosphere (see Figure 75) and the laboratory plasma (see
Figure 76).