7.1 Scaling exponents of structure functions
The phenomenology of turbulence developed by Kolmogorov (1941) deals with some statistical
hypotheses for fluctuations. The famous footnote remark by Landau (Landau and Lifshitz, 1971) pointed
out a defect in the Kolmogorov theory, namely the fact that the theory does not take proper account
of spatial fluctuations of local dissipation rate (Frisch, 1995). This led different authors to
investigate the features related to scaling laws of fluctuations and, in particular, to investigate the
departure from the Kolmogorov’s linear scaling of the structure functions (cf. Section 2.8).
An uptodate comprehensive review of these theoretical efforts can be found in the book by
Frisch (1995).
Here we are interested in understanding what we can learn from solar wind turbulence about the basic
features of scaling laws for fluctuations. We use velocity and magnetic fields time series, and we investigate the
scaling behavior of the highorder moments of stochastic variables defined as variations of fields separated by a
time
interval . First of all, it is worthwhile to remark that scaling laws and, in particular, the exact
relation (26) which defines the inertial range in fluid flows, is valid for longitudinal (streamwise)
fluctuations. In common fluid flows the Kolmogorov linear scaling law is compared with the moments of
longitudinal velocity differences. In the same way for the solar wind turbulence we investigate the scaling
behavior of , where represents the component of the velocity field along the
radial direction. As far as the magnetic differences are concerned , we are free for
different choices and, in some sense, this is more interesting from an experimental point of view. We can use
the reference system where represents the magnetic field projected along the radial
direction, or the system where represents the magnetic field along the local background
magnetic field, or represents the field along the minimum variance direction. As a different
case we can simply investigate the scaling behavior of the fluctuations of the magnetic field
intensity.
Let us consider the th moment of both absolute values of velocity fluctuations
and magnetic fluctuations , also called th order structure function in literature
(brackets being time average). Here we use magnetic fluctuations across structures at intervals
calculated by using the magnetic field intensity. Typical structure functions of magnetic field
fluctuations, for two different values of , for both a slow wind and a fast wind at ,
are shown in Figures 72. The magnetic field we used is that measured by Helios 2 spacecraft.
Structure functions calculated for the velocity fields have roughly the same shape. Looking at these
figures the three typical ranges of turbulence can be observed. Starting from low values at small
scales, the structure functions increase towards a region where at the largest
scales. This means that at these scales the field fluctuations are uncorrelated. A kind of “inertial
range”, that is a region of intermediate scales where a power law can be recognized for both
and , is more or less visible only for the slow wind. In this range
correlations exists, and we can obtain the scaling exponents and through a simple linear
fit.
Since the inertial range is not well
defined,
in order to compare scaling exponents of the solar wind turbulent fluctuations with other experiments, it is
perhaps better to try to recover exponents using the Extended SelfSimilarity (ESS), introduced some time
ago by Benzi et al. (1993), and used here as a tool to determine relative scaling exponents. In the fluidlike
case, the th order structure function can be regarded as a generalized scaling using the inverse of
Equation (26) or of Equation (27) (Politano et al., 1998b). Then, we can plot the th order structure
function vs. the th order to recover at least relative scaling exponents and . Quite
surprisingly (see Figure 73), we find that the range where a power law can be recovered extends well
beyond the inertial range, covering almost all the experimental range. In the fluid case the scaling exponents
which can be obtained through ESS at low or moderate Reynolds numbers, coincide with the
scaling exponents obtained for high Reynolds, where the inertial range is very well defined Benzi
et al. (1993). This is due to the fact that, since the law, by definition in the
inertial range (Frisch, 1995), whatever its extension might be. In our case scaling exponents
obtained through ESS can be used as a surrogate, since we cannot be sure that an inertial range
exists.
It is worthwhile to remark (as shown in Figure 73) that we can introduce a general scaling relation
between the th order velocity structure function and the th order structure function,
with a relative scaling exponent . It has been found that this relation becomes an exact
relation
when the velocity structure functions are normalized to the average velocity within each period used to
calculate the structure function (Carbone et al., 1996a). This is very interesting because it implies
(Carbone et al., 1996a) that the above relationship is satisfied by the following probability
distribution function, if we assume that odd moments are much smaller than the even ones:
That is, for each scale the knowledge of the relative scaling exponents completely
determines the probability distribution of velocity differences as a function of a single parameter
.
Relative scaling exponents, calculated by using data coming from Helios 2 at , are reported in
Table 1. As it can be seen, two main features can be noted:
 There is a significant departure from the Kolmogorov linear scaling, that is, real scaling
exponents are anomalous and seem to be nonlinear functions of , say for
, while for . The same behavior can be observed for . In
Table 1 we report also the scaling exponents obtained in usual fluid flows for velocity and
temperature, the latter being a passive scalar. Scaling exponents for velocity field are similar
to scaling exponents obtained in turbulent flows on Earth, showing a kind of universality in
the anomaly. This effect is commonly attributed to the phenomenon of intermittency in fully
developed turbulence (Frisch, 1995). Turbulence in the solar wind is intermittent, just like its
fluid counterpart on Earth.
 The degree of intermittency is measured through the distance between the curve and
the linear scaling . It can be seen that magnetic field is more intermittent than velocity
field. The same difference is observed between the velocity field and a passive scalar (in our
case the temperature) in ordinary fluid flows (RuízChavarría et al., 1995). That is the
magnetic field, as long as intermittency properties are concerned, behaves like a passive field.
In Table 1 we show scaling exponents up to the sixth order. Actually, a question concerns the validation
of highorder moments estimates, say the maximum value of the order which can be determined with a
finite number of points of our dataset. As the value of increases, we need an increasing number of points
for an optimal determination of the structure function (Tennekes and Wyngaard, 1972). Anomalous scaling
laws are generated by rare and intense events due to singularities in the gradients: the higher their
intensity the more rare these events are. Of course, when the dataset has a finite extent, the
probability to get singularities stronger than a certain value approaches zero. In that case, scaling
exponents of order higher than a certain value look to be linear function of . Actually,
the structure function depends on the probability distribution function
through
and the function is determined only when the integral converges. As increases, the function
becomes more and more disturbed, with some spikes, so that the integral
becomes more and more undefined, as can be seen for example in Figure 1 of the paper by Dudok de
Wit (2004). A simple calculation (Dudok de Wit, 2004) for the maximum value of the order
which can reliably be estimated with a given number of points in the dataset, gives the
empirical criterion . Structure functions of order cannot be determined
accurately.
As far as a comparison between different plasmas is concerned, the scaling exponents of magnetic
structure functions, obtained from laboratory plasma experiments of a ReversedField Pinch at different
distances from the external wall (Carbone et al., 2000) are shown in Table 2. In laboratory plasmas it is
difficult to measure all the components of the vector field at the same time, thus, here we show only the
scaling exponents obtained using magnetic field differences calculated from the radial
component in a toroidal device where the axis is directed along the axis of the torus. As it can be seen,
intermittency in magnetic turbulence is not so strong as appears in the solar wind, actually the degree of
intermittency increases going toward the external wall. This last feature appears to be similar to what is
currently observed in channel flows, where intermittency increases going towards the external wall
(Pope, 2000).
Scaling exponents of structure functions for Alfvén variables, velocity, and magnetic variables
have been calculated also for high resolution 2D incompressible MHD numerical simulations
Politano et al. (1998b). In this case, we are free from the Taylor hypothesis in calculating the
fluctuations at a given scale. From 2D simulations we recover the fields and at some
fixed times. We calculate the longitudinal fluctuations directly in space at a fixed time, namely
(the same are made for different fields, namely the magnetic field or the
Elsässer fields). Finally, averaging both in space and time, we calculate the scaling exponents through
the structure functions. These scaling exponents are reported in Table 3. Note that, even in
numerical simulations, intermittency for magnetic variables is stronger than for the velocity
field.
Table 3: 
Normalized scaling exponents for Alfvénic, velocity, and magnetic fluctuations
obtained from data of high resolution 2D MHD numerical simulations. Scaling exponents have been
calculated from spatial fluctuations; different times, in the statistically stationary state, have been
used to improve statistics. The scaling exponents have been calculated by ESS using Equation (27)
as characteristic scale rather than the thirdorder structure function (cf. Politano et al., 1998b for
details). 
