J. Plasma Fusion Res. SERIES, Vol. 8 (2009)

Multiscale Multifractal Intermittent Turbulencein Space Plasmas
Wies�law M. MACEK1,2,3) , Anna WAWRZASZEK3) , and Tohru HADA1)
1)

2)

Faculty of Engineering Sciences, E S S T, Kyushu University, Fukuoka 816-8580, Japan
Faculty of Mathematics and Natural Sciences, Cardinal Stefan Wyszyński University in Warsaw, Poland
3)Space Research Centre, Polish Academy of Sciences, Warsaw, Poland(Received: 1 September 2008 / Accepted: 30 December 2008)

We examine generalized dimensions and the corresponding multifractal singularity spectrum depending on one probability measure parameter and two scaling parameters, demonstrating that themultifractal scaling is often asymmetric. In particular, we analyze time series of velocities of the slow
and fast speed streams of the solar wind plasma measured in situ by Advanced Composition Explorer
spacecraft. We show that the universal shape of the multifractal spectrum results not only from thenonuniform probability of the energy transfer rate but rather from the multiscale nature of the cascade.
It is worth noting that for the model with two diﬀerent scaling parameters a better agreement withthe solar wind data is obtained. Only in the case of the multiscale cascade one can reproduce theentire multifractal spectrum, especially for the negative index of the generalized dimensions. Therefore
we argue that there is a need to use the multi-scale cascade model. Hence we propose this new moregeneral model as a useful tool for analysis of intermittent turbulence in various environments.
Keywords: multifractals, turbulence, intermittency, space plasmas

1. Introduction

In general, the spectrum of generalized dimensions Dq as a function of a continuous index q, witha degree of multifractality Δ = D−∞ − D∞ , quantiﬁes multifractality of a given system [4, 5, 16]. A
chaotic strange attractor has been identiﬁed in thesolar wind data by Macek [17] and further examinedby Macek and Redaelli [18]. We have also considered
the Dq spectrum for the solar wind attractor using a
multifractal model with a measure of the self-similar
weighted Cantor set with one parameter describing
uniform compression in phase space and another parameter for the probability measure of the attractor of
the system. The spectrum of Dq is found to be consistent with the data, at least for positive index q [19–23].
However, the full spectrum is necessary to estimate
the degree of multifractality. Notwithstanding of thewell-known statistical problems with negative q [21],we have recently succeeded in estimating the entire
spectrum for solar wind attractor using a generalized
weighted Cantor set with two diﬀerent scales describing nonuniform compression [4].
Therefore to further quantify turbulence, we haveconsidered this generalized weighted Cantor set also
in the context of turbulence cascade [5]. In this way
we have argued that there is, in fact, need to use a
multi-scale cascade model. Therefore we have alreadyinvestigated the multifractal spectrum of dimensions
depending on two scaling parameters and one probability measure parameter using Helios data and, inparticular, we have demonstrated that intermittent
pulses are stronger for asymmetric scaling and a muchbetter agreement is obtained, especially for q < 0.

Multifractality is commonly related to a probability measure that may have diﬀerent fractal dimensions
on diﬀerent parts of the support of this measure [1].
In this case the measure is multifractal. Here we propose a notion of multifractality based on an extended
self-similarity that depends on scale. We consider theconcept of the multiscale multifractality in the context of scaling properties of intermittent turbulencein astrophysical and space plasmas [2, 3]. To quantify scaling of this turbulence, we use a generalized
weighted Cantor set with two diﬀerent scales describing with various probabilities nonuniform intermittent
multiplicative process of distribution of the kinetic energy between cascading eddies of various sizes [4, 5].
The question of multifractality is of great importance for space plasmas because it allows us to lookat intermittent turbulence in the solar wind [6–12].
Starting from Richardson’s scenario of turbulence,
many authors try to recover the observed scaling exponents, using some simple and more advanced fractal and multifractal models of turbulence describing
distribution of the energy ﬂux between cascading eddies at various scales. In particular, the multifractal
spectrum has been investigated using Voyager (magnetic ﬁeld ﬂuctuations) data in the outer heliosphere[6, 7] and using Helios (plasma) data in the innerheliosphere [11]. The multifractal scaling has also
been investigated using Ulysses observations, e.g., [13]and with Advanced Composition Explorer (ACE) and
WIND data, e.g., [14, 15].
author’s e-mail: macek@cbk.waw.pl

142©2009 by The Japan Society of Plasma
Science and Nuclear Fusion Research

W.M. Macek et al., Multiscale Multifractal Intermittent Turbulence in Space Plasmas

smaller eddies (energy transfer rate) could be divided
into nonequal fractions p and 1 − p. In particular,
for non space-ﬁlling turbulence, l1 + l2 < 1 one still
could have a multifractal cascade, even for unweighted(equal) energy transfer, p = 0.5. Only for l1 = l2 = 0.5
and p = 0.5 there is no multifractality.

3. Solar Wind Data
We have already analyzed the Helios 2 data using plasma parameters measured in situ in the innerheliosphere [4] for testing of the solar wind attractor. The X-velocity (mainly radial) component of
the plasma ﬂow, vx , has been already investigated by
Macek [17, 19, 20] and Macek and Redaelli [18]. The
Alfvénic ﬂuctuations with longer (two-day) samples
have been studied by in Ref. [4, 21] and [22, 23]. To
study the turbulence cascade Macek and Szczepaniak
have selected four-day time intervals of vx samples in1976 (solar minimum) for both slow and fast solar
wind streams measured at various distances from theSun [5]. In this paper we analyze time series of velocities of the solar wind measured by ACE in theecliptic plane near the libration point L1, i.e., approximately at a distance of R = 1 AU from the Sun. Here
we have selected even longer (ﬁve-day) time intervalsof vx samples, each of 6750 data points, interpolatedwith sampling time of 64 s, for both slow and fastsolar wind streams during solar minimum (2006) andmaximum (2001).

Fig. 1 Generalized two-scale weighted Cantor set model
for solar wind turbulence, cf. [2, 4].

In this paper, we would like to test the degree ofmultifractality and asymmetry of the multifractal scaling for the wealth of data provided by another space
mission. Namely, we further consider the question of
scaling properties of intermittent turbulence using velocities of the slow and fast speed streams of the solar
wind measured in situ by ACE during solar minimum
and maximum at Earth’s orbit, R = 1 AU. By using
our cascade model we show that the degree of multifractality of the solar wind in the inner heliosphere isgreater for fast solar wind velocity ﬂuctuations thanthat for the slow solar wind. On the other hand, as thesolar activity decreases the slow solar wind spectrum
becomes more asymmetric. Thus we still hope that
this generalized new asymmetric multifractal model
could shed light on the nature of turbulence.

2. Two-Scale Cantor Set Model

4. Methods of Data Analysis

At each stage of construction of the weighted twoscale Cantor set we basically have two scaling parameters l1 and l2 , where l1 + l2 ≤ 1 (normalized), and two
diﬀerent weights p1 and p2 . To obtain the generalized
dimensions Dq ≡ τ (q)/(q − 1) for this interesting example of multifractals we use the following partition
function at the n-th level of construction [24, 25]
�n
�
pq2
pq1
q
+ τ (q)= 1. (1)Γn (l1 , l2 , p1 , p2 ) =
τ (q)
l1
l2

The generalized dimensions Dq as a function of index q [24–27] are important characteristics of complex
dynamical systems; they quantify multifractality of agiven system [16]. In the case of turbulence cascadethese generalized measures are related to inhomogeneity with which the energy is distributed between different eddies [3]. In this way they provide information
about dynamics of multiplicative process of cascading
eddies. Here high positive values of q emphasize regions of intense energy transfer rate, while negativevalues of q accentuate low-transfer rate regions.
Let us consider the generalized weighted Cantor
set, where the probability of providing energy for one
eddy of size l1 is p (say, p ≤ 1/2), and for the othereddy of size l2 is 1 − p as depicted in Fig. 1. For any q
one obtains Dq = τ (q)/(q − 1) by solving numericallythe following transcendental equation, e.g., [16]The resulting strange attractor of 2n closed intervals (narrow segments with various widths and probabilities) for n → ∞ is the generalized weighted twoscale Cantor set.
Here we consider a standard scenario of cascading
eddies, each breaking down into two new ones, but
not necessarily equal and twice smaller as schematically shown in Fig. 1, cf. [2, 4]. In particular, spaceﬁlling turbulence could be recovered for l1 + l2 = 1.
Naturally, in the inertial region of the system of size
L, η � l � L, we do not allow the energy to bedissipated directly, assuming p1 + p2 = 1, until the
Kolmogorov scale η is reached. However, in this range
at each n-th step of the binomial multiplicative process, the ﬂux of kinetic energy density ε transferred topq(1 − p)q

l1

l2

+
τ (q)

τ (q)

= 1.

(2)

In the inertial range the transfer rate of the energy
ﬂux ε(l) is widely estimated by the third moment ofstructure function of velocity ﬂuctuations, e.g., [11]ε(l) ∼

143|u(x + l) − u(x)|3
,
l

(3)

W.M. Macek et al., Multiscale Multifractal Intermittent Turbulence in Space Plasmas

�Fig. 2 The normalized transfer rate of the energy ﬂux p(t) = εi (t) /εi (t) obtained using data of the vx velocity
components measured by ACE at 1 AU for the slow (a) and (c) and fast (b) and (d) solar wind during solar
minimum (2006) and maximum (2001), correspondingly.

where u(x) and u(x + l) are velocity components parallel to the longitudinal direction separated from a position x by a distance l. Therefore to each ith eddy ofsize l in the turbulence cascade (i = 1, . . . , N = 2n )we associate a probability measure deﬁned byεi (l)pi (l) = �N.
i=1 εi (l)the slopes or generalized measures. Using α0 , wheref (α0 ) = 1, one can deﬁne a measure of asymmetryA ≡ (α0 − αmin )/(αmax − α0 ).

5. Results(4)

The results for the generalized dimensions Dq as
a function of q are shown in Fig. 3. The values of Dq
given in Eq. (6), for one-dimensional turbulence, areagain calculated using the radial velocity components
u = vx , cf. [22, Figure 3], and the corresponding results for the singularity spectra f (α) as a function of
α are shown in Fig. 4 for the slow (a) and (c), andfast (b) and (d) solar wind streams at solar minimum
and maximum, correspondingly. Both values of Dq
and f (α) for one-dimensional turbulence have been
computed directly from the data, by using the experimental velocity components.
In spite of statistical errors in Fig. 4 (a), b), (c)and (d), especially for q < 0, we see that the multifractal character of the measure can still clearly be discerned. Therefore one can conﬁrm that the spectrumof dimensions still exhibits the multifractal structureof the solar wind in the inner heliosphere.
For q ≥ 0 these results agree with the usual onescale p-model ﬁtted to the dimension spectra as obtained analytically using l1 = l2 = 0.5 in Eq. (2) andthe corresponding value of the parameter p � 0.21
and 0.20, 0.15 and 0.12 for the slow (a) and (c), andfast (b) and (d) solar wind streams at solar minimum
and maximum, correspondingly, as shown by dashedThis quantity can roughly be interpreted as a probability that the energy is transferred to an eddy of size
l = vsw t. In Fig. 2 we show the multifractal measure�εi (t) given by Eqs. (3) and (4) andp(t) = εi (t) /obtained using data of the velocity components u = vx
(in time domain) as measured by ACE at 1 AU for
the slow (a) and (c) and fast (b) and (d) solar windduring solar minimum (2006) and maximum (2001),
correspondingly.
Now, one can further associate a generalized average probability measure of cascading eddies�(5)μ̄(q, l) ≡ q−1 �(pi )q−1 �av ,and identify Dq as scaling of the measure with size l,μ̄(q, l) ∝ lDq .

(6)

Hence, the slopes of the logarithm of μ̄(q, l) of Eq. (6)versus log l (normalized) provides
Dq = lim

l→0

log μ̄(q, l)
.
log l

(7)

The singularity spectrum f (α) = qα − τ (q) as afunction of α = τ � (q) could also be obtained by using Legendre transformation [16, 21], or directly from

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W.M. Macek et al., Multiscale Multifractal Intermittent Turbulence in Space Plasmas

Fig. 3 The generalized dimensions Dq as a function of q. The values obtained for one-dimensional turbulence are calculated
for the usual one-scale (dashed lines) p-model and the generalized two-scale (continuous lines) model with parameters
ﬁtted to the multifractal measure μ(q, l) obtained using data measured by ACE at 1 AU (diamonds) for the slow (a)and (c) and fast (b) and (d) solar wind during solar minimum (2006) and maximum (2001), correspondingly.

lines. On the contrary, for q < 0 the p-model cannot
describe the observational results [11]. Here we showthat the experimental values are consistent also withthe generalized dimensions obtained numerically from
Eqs. (5-7) for the weighted two-scale Cantor set using an asymmetric scaling, i.e., using unequal scalesl1 �= l2 , as is shown in Figs. 3 and 4 (a), (b), (c),and (d) by continuous lines. We also conﬁrm the universal shape of the multifractal spectrum, Fig. 4. In
our view, this obtained shape of the multifractal spectrum results not only from the nonuniform probabilityof the energy transfer rate but mainly from the multiscale nature of the cascade.
It is well known that the fast wind is associatedwith coronal holes, while the slow wind mainly originates from the equatorial regions of the Sun. Consequently, the structure of the ﬂow diﬀers signiﬁcantly
for the slow and fast streams. Hence the fast wind isconsidered to be relatively uniform and stable, while

the slow wind is more turbulent and quite variable invelocities, possibly owing to a strong velocity shear[28]. We see from Table 1 that the degree of multifractality Δ and asymmetry A of the solar wind in the
inner heliosphere are diﬀerent for slow (Δ = 1.2 − 1.6)
and fast (Δ = 2.3 − 2.6) streams; the velocity ﬂuctuations in the fast streams seem to be more multifractal
than those for the slow solar wind (the generalized dimensions vary more with the index q). On the other

Table 1Degree of multifractality Δ and asymmetry A forsolar wind data in the heliosphere

Solar Min.
Solar Max.

145

Slow Solar WindΔ = 1.22, A = 2.21
Δ = 1.60, A = 1.33

Fast Solar WindΔ = 2.56, A = 0.95
Δ = 2.31, A = 1.25

W.M. Macek et al., Multiscale Multifractal Intermittent Turbulence in Space Plasmas

Fig. 4 The corresponding singularity spectrum f (α) as a function of α.

model will be a useful tool for analysis of intermittent
turbulence in space plasmas.

hand, it seems that in the slow streams the scaling ismore asymmetric than that for the fast wind. In ourview this could possibly reﬂect the large-scale scale
velocity structure. Further, the degree of asymmetryof the dimension spectra for the slow wind is rather
anticorrelated with the phase of the solar magnetic
activity and only weakly correlated for the fast wind:(A decreases from 2.2 to 1.3) only the fast wind during solar minimum exhibits roughly symmetric scaling, A ∼ 1, and one-scale Cantor set model applies.
We see that the multifractal spectrum of the solar wind is only roughly consistent with that for themultifractal measure of the self-similar weighted symmetric one-scale weighted Cantor set only for q ≥ 0.
On the other hand, this spectrum is in a very good
agreement with two-scale asymmetric weighted Cantor set schematically shown in Fig. 1 for both positive and negative q. Obviously, taking two diﬀerent
scales for eddies in the cascade, one obtains a moregeneral situation than in the usual p-model for fully
developed turbulence [2], especially for an asymmetricscaling, l1 �= l2 . Hence we hope that this generalized

6. Conclusions
We have studied the inhomogeneous rate of thetransfer of the energy ﬂux indicating multifractal andintermittent behavior of solar wind turbulence in the
inner heliosphere. In particular, we have demonstrated that for the model with two diﬀerent scaling parameters a much better agreement with the real
data is obtained, especially for q < 0. By investigating
the ACE data we have shown that the degree of multifractality of the solar wind in the inner heliosphereis greater for fast solar wind velocity ﬂuctuations thanthat for the slow solar wind; the generalized dimensions varies more with the index q. As the solar activity increases the slow solar wind becomes somewhat
more multifractal, and the fast wind is slightly lessmultifractal. On the other hand, it seems that the degree of asymmetry of the dimension spectrum for theslow wind is rather anticorrelated with the phase of
the solar activity.

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W.M. Macek et al., Multiscale Multifractal Intermittent Turbulence in Space Plasmas

Basically, the generalized dimensions for solar
wind are consistent with the generalized p-model forboth positive and negative q, but rather with diﬀerent
scaling parameters for sizes of eddies, while the usualp-model can only reproduce the spectrum for q ≥ 0. In
general, the proposed generalized multi-scale weighted
Cantor set model should also be valid for non spaceﬁlling turbulence. Therefore we propose this cascade
model describing intermittent energy transfer for analysis of turbulence in various environments.
We would like to thank the plasma instruments
team of Advanced Composition Explorer for providing
velocity data. This work has been supported by theJapanese Society for the Promotion of Science and
the Polish Ministry of Science and Higher Education(MNiSW) through Grant NN 202 4127 33.

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