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

Figure 77:

Differences for the longitudinal velocity $dut = u(t + t )- u(t)$ at three different scales $t$ , as shown in the figure.

Time dependence of $dut$ and $dbt$ for three different scales $t$ is shown in Figures 77

and 78

, respectively. These plots show that, as $t$ 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 $dut$ and $dbt$ 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 $Sh(r)$ , each set being characterized by a scaling law $dZl± ~ lh(r)$ , that is, the scaling exponent can depend on the position $r$ . The usual dimension of that set is then not constant, but depends on the local value of $h$ , and is quoted as $D(h)$ 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.,

Figure 78:

Differences for the magnetic intensity $db = B(t + t )- B(t) t$ at three different scales $t$ , as shown in the figure.

P (dZ ±) ~ (dZ ±)h× Volume occupied by fluctuations, l l

and the $p$ -th order structure function is immediately written through the integral over all (continuous) values of $h$ weighted by a smooth function $m(h) ~ 0(1)$ , i.e.,

integral ± ph ± 3-D(h) Sp(l) = m(h)(dZ l ) (dZ l ) dh.

A moment of reflection allows us to realize that in the limit $l --> 0$ the integral is dominated by the minimum value (over $h$ ) 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

z = min [ph + 3- D(h)]. p h

In this way, the departure of $zp$ from the linear Kolmogorov scaling and then intermittency, can be characterized by the continuous changing of $D(h)$ as $h$ varies. That is, as $p$ varies we are probing regions of fluid where even more rare and intense events exist. These regions are characterized by small values of $h$ , 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 $± dzl$ and introducing the scaling exponents for the energy transfer rate $p tp <el> ~ r$ , it can be found that $zp = p/m + tp/m$ (being $m = 3$ when the Kolmogorov-like phenomenology is taken into account, or $m = 4$ when the fluid is magnetically dominated). In this way the intermittency correction are determined by a cascade model for the energy transfer rate. When $tp$ is a non-linear function of $p$ , the energy transfer rate can be described within the multifractal geometry (see, e.g., Meneveau , 1991 and references therein) characterized by the generalized dimensions $Dp = 1 - tp/(p - 1)$ (Hentschel and Procaccia, 1983 ). The scaling exponents of the structure functions are then related to $D p$ by

( p ) zp = -- - 1 Dp/m + 1. m

The correction to the linear scaling $p/m$ is positive for $p < m$ , negative for $p > m$ , and zero for $p = m$ . A fractal behavior where $Dp = const.< 1$ gives a linear correction with a slope different from $1/m$ .