7.6 Intermittency properties recovered via a shell model

The FSGC shell model has remarkable properties which closely resemble those typical of MHD phenomena (Giuliani and Carbone, 1998

). However, the presence of a constant forcing term always induces a dynamical alignment, unless the model is forced appropriately, which invariably brings the system towards a state in which velocity and magnetic fields are strongly correlated, that is, where $Z ± /= 0 n$ and $± Z n = 0$ . When we want to compare statistical properties of turbulence described by MHD shell models with solar wind observations, this term should be avoided. It is possible to replace the constant forcing term by an exponentially time-correlated Gaussian random forcing which is able to destabilize the Alfvénic fixed point of the model (Giuliani and Carbone, 1998

), thus assuring the energy cascade. The forcing is obtained by solving the following Langevin equation:

where $m(t)$ is a Gaussian stochastic process $d$ -correlated in time $<m(t)m(t')> = 2Dd(t' - t)$ . This kind of forcing will be used to investigate statistical properties.

Figure 79:

We show the kinetic energy spectrum $|un(t)|2$ as a function of $log2 kn$ for the MHD shell model. The full line refer to the Kolmogorov spectrum $k -2/3 n$ .

Figure 80:

We show the magnetic energy spectrum $2 |bn(t)|$ as a function of $log2kn$ for the MHD shell model. The full line refer to the Kolmogorov spectrum $k -n2/3$ .

A statistically stationary state is reached by the system, as shown in Giuliani and Carbone (1998 ), with a well defined inertial range, say a region where Equation (29

) is verified. Spectra for both the velocity $|un(t)|2$ and magnetic $|bn(t)|2$ variables, as a function of $kn$ , obtained in the stationary state, are shown in Figures 79

and 80

. Fluctuations are averaged over time. The Kolmogorov spectrum is also reported as a solid line. It is worthwhile to remark that, by adding a random term like $± iknB0(t)Z n$ to a little modified version of the MHD GOY shell model ( $B0$ is a random function with some statistical characteristics), a Kraichnan spectrum, say $E(kn) ~ k-n 3/2$ , where $E(kn)$ is the total energy, can be recovered (Hattori and Ishizawa, 2001 ). The term added to the model could represent the effect of the occurrence of a large-scale magnetic field.

Intermittency in the shell model is due to the time behavior of shell variables. It has been shown (Okkels, 1997 ) that the evolution of GOY model consists of short bursts traveling through the shells and long period of oscillations before the next burst arises. In Figures 81 and 82 we report the time evolution of the real part of both velocity variables $un(t)$ and magnetic variables $b (t) n$ at three different shells. It can be seen that, while at smaller $k n$ variables seems to be Gaussian, at larger $kn$ variables present very sharp fluctuations in between very low fluctuations.

Figure 81:

Time behavior of the real part of velocity variable $u (t) n$ at three different shells $n$ , as indicated in the different panels.

Figure 82:

Time behavior of the real part of magnetic variable $b (t) n$ at three different shells $n$ , as indicated in the different panels.

The time behavior of variables at different shells changes the statistics of fluctuations. In Figure 83

we report the probability density functions $P(dun)$ and $P (dBn)$ , for different shells $n$ , of standardized variables

where $R e$ indicates that we take the real part of $un$ and $bn$ . Typically we see that PDFs look differently at different shells: At small $kn$ fluctuations are quite Gaussian distributed, while at large $k n$ they tend to become increasingly non-Gaussian, by developing fat tails. Rare fluctuations have a probability of occurrence larger than a Gaussian distribution. This is the typical behavior of intermittency as observed in usual fluid flows and described in previous sections.

Figure 83:

In the first three panels we report PDFs of both velocity (left column) and magnetic (right column) shell variables, at three different shells $ln$ . The bottom panels refer to probability distribution functions of waiting times between intermittent structures at the shell $n = 12$ for the corresponding velocity and magnetic variables.

The same phenomenon gives rise to the departure of scaling laws of structure functions from a Kolmogorov scaling. Within the framework of the shell model the analogous of structure functions are defined as

For MHD turbulence it is also useful to report mixed correlators of the flux variables, i.e.,

<[T± ]p/3> ~ k-b±p . n n

Scaling exponents have been determined from a least square fit in the inertial range $3 < n < 12$ . The values of these exponents are reported in Table 4. It is interesting to notice that, while scaling exponents for velocity are the same as those found in the solar wind, scaling exponents for the magnetic field found in the solar wind reveal a more intermittent character. Moreover, we notice that velocity, magnetic and Elsässer variables are more intermittent than the mixed correlators and we think that this could be due to the cancellation effects among the different terms defining the mixed correlators.

Table 4:

Scaling exponents for velocity and magnetic variables, Elsässer variables, and fluxes. Errors on $b ± p$ are about one order of magnitude smaller than the errors shown.







$p$	$z p$	$j p$	$q+ p$	$q- p$	$b+ p$	$b - p$









$1$	$0.36 ± 0.01$	$0.35 ± 0.01$	$0.35 ± 0.01$	$0.36 ± 0.01$	$0.326$	$0.318$
$2$	$0.71 ± 0.02$	$0.69 ± 0.03$	$0.70 ± 0.02$	$0.70 ± 0.03$	$0.671$	$0.666$
$3$	$1.03 ± 0.03$	$1.01 ± 0.04$	$1.02 ± 0.04$	$1.02 ± 0.04$	$1.000$	$1.000$
$4$	$1.31 ± 0.05$	$1.31 ± 0.06$	$1.30 ± 0.05$	$1.32 ± 0.06$	$1.317$	$1.323$
$5$	$1.57 ± 0.07$	$1.58 ± 0.08$	$1.54 ± 0.07$	$1.60 ± 0.08$	$1.621$	$1.635$
$6$	$1.80 ± 0.08$	$1.8 ± 0.10$	$1.79 ± 0.09$	$1.87 ± 0.10$	$1.91$	$1.94$

Time intermittency in the shell model generates rare and intense events. These events are the result of the chaotic dynamics in the phase-space typical of the shell model (Okkels, 1997 ). That dynamics is characterized by a certain amount of memory, as can be seen through the statistics of waiting times between these events. The distributions $P (dt)$ of waiting times is reported in the bottom panels of Figures 83 , at a given shell $n = 12$ . The same statistical law is observed for the bursts of total dissipation (Boffetta et al., 1999 ).