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 $u(l)$ , which depends on a scaling variable $l$ , is invariant with respect to the scaling relation $l --> cl$ , when there exists a parameter $m(c)$ such that $u(l) = m(c)u(cl)$ . The solution of this last relation is a power law $u(l) = Clh$ , where the scaling exponent is $h = - logc m$ .

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 $dz± l$ and $dz± cl$ . Their ratio $dz± /dz± ~ ch cl l$ depends only on the value of $h$ , and this should imply that fluctuations are self-similar. This means that PDFs are related through

± h ± P(dzcl) = P DF (c dzl ).

Let us consider the standardized variables

± y± = ---dzl-----. l <(dz±l )2>1/2

It can be easily shown that when $h$ is unique or in a pure self-similar situation, PDFs are related through $P (y±l ) = P DF (y±cl)$ , say by changing scale PDFs coincide.

The PDFs relative to the standardized magnetic fluctuations $Db = db/ <db2>1/2 r r r$ , at three different scales $r$ , 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 $r$ . In particular, at large scales PDFs seem to be almost Gaussian, but they become more and more stretched as $r$ 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 ).

Figure 74:

Left column: normalized PDFs for the magnetic fluctuations observed in the solar wind turbulence. Right panel: distribution function of waiting times $Dt$ between structures at the smallest scale. The parameter $b$ is the scaling exponent of the scaling relation $PDF (Dt) ~ Dt-b$ for the distribution function of waiting times.

Figure 75:

Left column: normalized PDFs of velocity fluctuations in atmospheric turbulence. Right panel: distribution function of waiting times $Dt$ between structures at the smallest scale. The parameter $b$ is the scaling exponent of the scaling relation $- b PDF (Dt) ~ Dt$ for the distribution function of waiting times. The turbulent samples have been collected above a grass-covered forest clearing at $5 m$ above the ground surface and at a sampling rate of $56 Hz$ (Katul et al., 1997 ).

Figure 76:

Left column: normalized PDFs of the radial magnetic field collected in RFX magnetic turbulence (Carbone et al., 2000 ). Right panel: distribution function of waiting times $Dt$ between structures at the smallest scale. The parameter $b$ is the scaling exponent of the scaling relation $P DF (Dt) ~ Dt -b$ for the distribution function of waiting times.