### 7.3 What is intermittent in the solar wind turbulence? The multifractal approach

Time dependence of and for three different scales is shown in Figures 77 and 78,
respectively. These plots show that, as becomes small, intense fluctuations become more
and more important, and they dominate the statistics. Fluctuations at large scales appear to
be smooth while, as the scale becomes smaller, intense fluctuations becomes visible. These
dominating fluctuations represent relatively rare events. Actually, at the smallest scales, the
time behavior of both and is dominated by regions where fluctuations are low, in
between regions where fluctuations are intense and turbulent activity is very high. Of course, this
behavior cannot be described by a global self-similar behavior. It appears more convincing a
description where scaling laws must depend on the region of turbulence we are investigating.
The behavior we have just described is at the heart of the multifractal approach to turbulence
(Frisch, 1995). In that description of turbulence, even if the small scales of fluid flow cannot be globally
self-similar, self-similarity can be reintroduced as a local property. In the multifractal description it is
conjectured that turbulent flows can be made by an infinite set of points , each set being
characterized by a scaling law , that is, the scaling exponent can depend on the
position . The usual dimension of that set is then not constant, but depends on the local value
of , and is quoted as in literature. Then, the probability of occurrence of a given
fluctuation can be calculated through the weight the fluctuation assumes within the whole flow,
i.e.,
and the -th order structure function is immediately written through the integral over all (continuous)
values of weighted by a smooth function , i.e.,

A moment of reflection allows us to realize that in the limit the integral is dominated by the
minimum value (over ) of the exponent and, as shown by Frisch (1995), the integral can be formally
solved using the usual saddle-point method. The scaling exponents of the structure function can then be
written as

In this way, the departure of from the linear Kolmogorov scaling and then intermittency, can be
characterized by the continuous changing of as varies. That is, as varies we
are probing regions of fluid where even more rare and intense events exist. These regions are
characterized by small values of , that is, by stronger singularities of the gradient of the
field.

Owing to the famous Landau footnote on the fact that fluctuations of the energy transfer rate must be
taken into account in determining the statistics of turbulence, people tried to interpret the
non-linear energy cascade typical of turbulence theory, within a geometrical framework. The old
Richardson’s picture of the turbulent behavior as the result of a hierarchy of eddies at different
scales has been modified and, as realized by Kraichnan (1974), once we leave the idea of a
constant energy cascade rate we open a “Pandora’s box” of possibilities for modeling the energy
cascade. By looking at scaling laws for and introducing the scaling exponents for the energy
transfer rate , it can be found that (being when the
Kolmogorov-like phenomenology is taken into account, or when the fluid is magnetically
dominated). In this way the intermittency correction are determined by a cascade model for
the energy transfer rate. When is a non-linear function of , the energy transfer rate
can be described within the multifractal geometry (see, e.g., Meneveau, 1991 and references
therein) characterized by the generalized dimensions (Hentschel and
Procaccia, 1983). The scaling exponents of the structure functions are then related to
by

The correction to the linear scaling is positive for , negative for , and zero for
. A fractal behavior where gives a linear correction with a slope different from
.