On the role of coherent and stochastic fluctuations in the
evolving solar wind MHD turbulence: Intermittency
R. Bruno , V. Carbone†, L. Sorriso–Valvo† and B. Bavassano
†
Istituto Fisica Spazio Interplanetario del CNR, 00133 Roma, Italy
Dipartimento di Fisica Università della Calabria, 87036 Rende (Cs), Italy
Abstract. One of the most interesting findings about solar wind turbulence is the intermittent character of its fluctuations.
So far, intermittency has been studied, within solar wind context, only for scalar quantities neglecting to include in the study
also vector fluctuations. In this paper we compare magnetic field and wind velocity intermittency computed for both scalar
and vector fluctuations. We confirm that intermittency generally increases with radial distance from the sun only for fast wind
and we suggest that at the basis of this behavior is a competing action between coherent structures imbedded in the wind and
Alfvénic stochastic fluctuations propagating within the wind. The radial evolution experienced by these two components is
rather different and consistent with the observed radial increase of intermittency in the inner heliosphere.
INTRODUCTION
Solar wind turbulence is strongly anisotropic and poorly
single scale–invariant (see [13] and references therein
for an extensive review on this and on related subjects),
two fundamental hypotheses at the base of Kolmogorov’s
theory [8]. Moreover, the study of these fluctuations by
means of conventional spectral analysis is strongly limited by the non-Gaussianity of their probability distribution function (PDF), which makes the second order
moment of the distribution no longer the limiting order. As a consequence, it becomes necessary to look at
higher order moments of the PDFs, which are built directly from the differences of the fluctuating field over
all the possible spatial scales. This is usually done in
the framework of the so-called multifractal approach [7].
The first time this new approach was employed within
the solar wind context [4], it showed that interplanetary
magnetic field and velocity fluctuations are highly intermittent. This feature implies that small turbulent eddies
are less and less space filling if turbulence is looked at
in the framework of a classical Richardson’s cascade.
Thus, the global scale invariance required in the Kolmogorov theory would become a local scale invariance
where different fractal sets are characterized by different
scaling exponents [5]. These first results obtained for the
solar wind [4] also showed an unexpected similarity to
those obtained for laboratory turbulence, unraveling consistency between observations on scales of 1 AU and laboratory observations on scales of meters. Further studies
[9] addressed the radial evolution of intermittency within
the inner heliosphere concluding that the Alfvénic tur-
bulence observed in fast streams starts from the Sun as
self–similar but then, during the expansion, decorrelates
becoming multifractal. On the contrary, it was found that
slow wind did not suffer a similar radial evolution [12].
Sorriso et al., [11] studied quantitatively the effect of
intermittency on the PDFs of the increments over different scales adopting Castaing’s model [6]. This model is
based on the idea of a log–normal energy cascade and the
non-Gaussian behavior of the PDF’s at small scales can
be represented by a convolution of Gaussians whose variances are distributed according to a log-normal distribution. As a matter of fact, large scales PDFs closely resemble Gaussian distributions but, at smaller scales the tails
of the distributions become more and more stretched.
Moreover, techniques recently adopted in the context
of solar wind turbulence and based on the wavelet decomposition of time series [14] allowed to show that
some of the events causing intermittency are either compressive phenoma like shocks or planar sheets like tangential discontinuities separating adjacent flux–tubes [3].
Intermittency can be estimated looking at the behavior of the flatness factor of the PDFs built at different
scales. If this parameter increases as the scale of interest becomes smaller our time series can be considered
intermittent [7]. Having adopted this definition of intermittency, we studied the radial behavior of the flatness
factor at different scales as a function of the solar wind
velocity regime within the inner heliosphere. The study
was performed for magnetic field and velocity fluctuations as reported in the following sections.
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
453
TABLE 1. From left to right: time interval
in dd:hh, heliocentric distance in AU, average
wind velocity in km/sec
Fast Wind:
0.9AU,
radial distance
46:00–48:00
49:12–51:12
72:00–74:00
75:12–77:12
99:12–101:12
105:12–107:12
0.90
0.88
0.69
0.65
0.34
0.29
433
643
412
630
405
729
20
flatness
time interval
<V>
25
0.7AU,
0.3AU
MAGNETIC FIELD: SCALAR FLUCTUATIONS
15
10
5
0
10
DATA ANALYSIS
2
25
10
3
10
4
10
5
MAGNETIC FIELD: VECTOR FLUCTUATIONS
F (τ ) =
ξ 4 (τ ) >
2
2
< ξ (τ ) >
<
(1)
where τ is the scale of interest and ξ p(τ ) =< jV (t +
τ ) V (t )j p > is the Structure Function of order p of the
generic function V (t ).
The flatness factor can be studied for both scalar and
vector fluctuations (also called compressive and directional fluctuations, respectively) of velocity and magnetic field vectors. As a matter of fact, a fluctuating vector field encompasses two distinct contributions, a compressive one that modifies the strength, or intensity, of the
fluctuations and can be expressed as δ j ~B(t ; τ )j = j~B(t +
τ )j j~B(t )j and a directional contribution that acts on the
vector orientation and can be expressed as δ ~B(t ; τ ) =
∑i=x;y;z (Bi (t + τ ) Bi(t ))2 . Obviously, δ ~B(t ; τ ) takes
into account also compressive contributions and the expression δ ~B(t ; τ ) jδ j~B(t ; τ )jj is always true. In the section that follows we will describe the radial behavior of
the flatness factor F for both scalar and vector fluctuations, for both velocity and magnetic field and for both
q
454
flatness
20
For this analysis we used 81 sec averages of magnetic
field and plasma data recorded by Helios 2 s/c during its
first solar mission in 1976. Helios orbit was such that allowed us to observe the same corotating fast stream at
different heliocentric distances during consecutive solar
rotations. Thus, in the hypothesis of stationary solar wind
source conditions, we can estimate possible radial gradients of physical parameters. Together with these fast
streams we also included in our analysis the slow wind
interval preceding each stream having care of avoiding
any stream interface. Time intervals, heliocentric distances and solar wind speed are reported in Table 1. One
of the characteristics of these fast streams is that they are
notorious for being dominated by strong Alfvénic fluctuations and offer a unique opportunity to observe the
radial evolution of MHD turbulence within the inner heliosphere [13].
We computed, scale by scale, the following flatness
estimator
15
10
5
0
10
2
10
3
10
4
10
5
scale[sec]
FIGURE 1. Flatness factor F for scalar and vector magnetic
field differences as a function of time scale expressed in seconds are shown at the top and bottom panels, respectively.The
three different symbols refer to three different heliocentric distances as illustrated at the top of the Figure. Time intervals refer
to fast wind.
fast and slow wind. From the comparison of the radial
behavior of these two quantities we can infer useful hints
that can help us to better interpret the radial evolution of
the intermittency observed in the solar wind MHD turbulence.
Radial dependence of magnetic field and
velocity intermittency
Flatness factor F for both scalar and vector differences
are shown in Figure 1 for magnetic field as a function
of time scale expressed in seconds. The three different
curves in each panel, which refer to fast wind observed
at the three heliocentric distances we chose, show that
flatness increases with distance for both kinds of fluctuations. In particular, the smallest scales are the least Gaussian while, the largest scales are rather Gaussian (flatness
close to 3) and not influenced by the radial excursion. In
addition, since F increases at small scales more slowly
for directional fluctuations than for scalar fluctuations,
these last ones can be considered more intermittent. It is
interesting to compare this behavior to that shown within
slow wind as illustrated in Figure 2. The most striking
Slow Wind:
0.9AU,
Fast Wind:
0.9AU,
0.3AU
25
MAGNETIC FIELD: SCALAR FLUCTUATIONS
20
20
15
15
flatness
flatness
25
0.7AU,
10
5
0.3AU
10
5
0
0
10
2
25
10
3
10
4
10
5
10
2
25
MAGNETIC FIELD: VECTOR FLUCTUATIONS
20
20
15
15
flatness
flatness
0.7AU,
WIND VELOCITY: SCALAR FLUCTUATIONS
10
5
10
3
10
4
10
5
10
5
WIND VELOCITY: VECTOR FLUCTUATIONS
10
5
0
0
10
2
10
3
10
4
10
5
10
scale[sec]
2
10
3
10
4
scale[sec]
FIGURE 2. Flatness factor F for scalar and vector magnetic
field differences as a function of time scale are shown in the top
and bottom panels in the same format used for Figure 1.Time
intervals refer to slow wind.
feature is that intermittency does not evolve with distance either for scalar or for vector fluctuations. Moreover, at small scales, F is generally higher than in fast
wind, suggesting a higher intermittency level. In particular, F for vector fluctuations starts to increse at much
larger scales than in the fast wind, suggesting that intermittency is already present at hourly scales. Results relative to fast wind velocity fluctuations are shown in Figure
3 in the same format as of the previous Figures. Although
the general trend shown in both panels confirms that intermittency for scalar and vector fluctuations increases
with distance also for fast solar wind velocity, we easily
recognize that scalar fluctuations (upper panel) are generally less intermittent than the corresponding magnetic
field fluctuations (Figure 1, upper panel). On the contrary, the behavior of F for velocity vector fluctuations
(lower panel) suggests that intermittency of these fluctuations is quite similar to that of magnetic field vector
fluctuations (Figure 1, lower panel). This last result, as it
will be discussed later on, clearly derives from the strong
contribution due to the Alfvénic component of these fluctuations.
Results similar to those obtained for magnetic field
observations within slow wind, are shown for velocity
fluctuations in Figure 4. The three curves, for both panels, show that F increases from the largest to the small-
455
FIGURE 3. Flatness factor F for scalar and vector velocity
fluctuations as a function of time scale expressed in seconds
are shown at the top and bottom panels, respectively. The three
different symbols refer to three different heliocentric distances
as illustrated at the top of the Figure. Time intervals refer to fast
wind.
est scales but its behavior doesn’t depend on heliocentric
distance given that these curves intermingle with one another especially at the smallest scales where the existence
of a radial trend is expected to be more clear. However,
this complicated behavior might also be due to a poor
stationarity at the source regions of the wind itself. However, also in this case, especially for scalar fluctuations,
wind velocity confirms to be less intermittent than magnetic field.
SUMMARY AND DISCUSSION
We have studied the radial evolution of intermittency associated with magnetic field and wind velocity fluctuations between 0.3 and 1 AU on the ecliptic plane for time
scales ranging between 81 sec and 24 hours . Since one of
the main effects of intermittency is that of increasing the
value of the flatness factor of the PDFs of the fluctuations
as we look at smaller and smaller scales [6, 7, 11], we
concentrated our attention on the behavior of the flatness
factor at different scales, analyzing both scalar and vector
fluctuations. This last point is important since intermittency of vector fluctuations contains also contributions
due to stochastic fluctuations like the Alfvénic ones.
Slow Wind:
0.9AU,
25
0.7AU,
tions is very low. However, as the wind expands, the
Alfvénic contribution is depleted because of turbulent
evolution [13] and, consequently, the underlying coherent structures convected by the wind, strengthen further
on by stream–stream dynamical interaction, assume a
more important role. Obviously, slow wind doesn’t show
a similar behavior because Alfvénic fluctuations have a
less dominant role than within fast wind and their turbulent evolution is much slower. An alternative view [10]
is that these coherent structures, contributing to increase
intermittency, are locally created by parametric decay instability of large amplitude Alfvén waves. Probably both
mechanisms are present at the same time and confirm
that the radial evolution of the intermittency level of interplanetary fluctuations is strongly related in any case to
the turbulent evolution of the spectrum of these fluctuations.
0.3AU
WIND VELOCITY: SCALAR FLUCTUATIONS
flatness
20
15
10
5
0
10
2
25
10
3
10
4
10
5
10
5
WIND VELOCITY: VECTOR FLUCTUATIONS
flatness
20
15
10
5
ACKNOWLEDGMENTS
0
10
2
10
3
10
4
scale[sec]
FIGURE 4. Flatness factor F for scalar and vector velocity
fluctuations as a function of time scale are shown at the top
and bottom panels in the same format used for Figure 3. Time
intervals refer to slow wind.
Our first result shows that magnetic field scalar fluctuations are always more intermittent than the corresponding velocity fluctuations as it was already noticed [2].
This is probably due to the fact that, differently from
wind velocity, magnetic field has to obey to ∇ B = 0
and it has to readjust its topology continuously to satisfy
this relation during the wind’s expansion.
We have also found that: a) generally, scalar (i.e. compressive) fluctuations are more intermittent than vector
(i.e. directional) fluctuations, b) that slow wind does not
show any radial evolution while c) fast wind intermittency, for both magnetic field and velocity, increases with
distance. The interpretation we give of our observations
derives from a view of the MHD turbulence that has already been given [1, 12] but that finds new evidence in
our analysis. In other words, we can imagine interplanetary fluctuations made of two distinct components: one
due to coherent, non propagating structures convected by
the wind and, another one made of propagating, stochastic fluctuations, namely Alfvénic modes. While the first
component tends to increase the intermittency level because of its coherent nature, the second one tends to decrease it because made of stochastic fluctuations. At 0.3
AU power associated to directional fluctuations largely
exceeds that associated to compressive fluctuations and
thus the corresponding intermittency of vector fluctua-
456
We thank F. Mariani and N. F. Ness, PI’s of the magnetic
experiment and, H. Rosenbauer and R. Schwenn, PI’s
of the plasma experiment onboard Helios 1 and 2, for
allowing us to use their data.
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