Rank-ordered multifractal analysis (ROMA)
of magnetic intermittent fluctuations in the
solar wind and in the magnetospheric cusps:
evidence for global crossover behavior?
Hervé Lamy1, Marius Echim1,2, Tom Chang3
Belgian Institute for Space Aeronomy, Brussels,
Belgium
1
2
Institute for Space Sciences, Bucharest, Romania
Kavli Institute for Astrophysics and Space Research,
MIT, Cambridge, USA
3
OUTLINE OF THE TALK
• Intermittent magnetic turbulence in the cusp : previous
studies based on PDFs, flatness, … (Echim, Lamy &
Chang 2007)
• Conventional multifractal analysis and limitations
• ROMA for intermittent fluctuations in the cusp (Cluster
data)
• ROMA for intermittent fluctuations in the solar wind
(Ulysses data)
• Conclusions & Perspectives
CLUSTER data
• Outbound pass on February 26, 2001 [3:36:20 – 7:35:54 UT]
• High resolution Magnetic Field (MF) data from the FGM
magnetometer : 67 samples/sec (burst mode)  > 106
samples
Normalization and scaling of PDFs
• The traditional way of dealing with intermittency is by studying
the shapes of the PDFs of the fluctuations at varying scales :
B2() = B2(t+)-B2(t)
• Detrending of the data for the large scale variations due to the
geomagnetic dipole component via the following rescaling
procedure
2
2
b t ,  
2
B t ,   B t , 
  
•  = 2j t are the various scales or time lags (t=0.015 sec ;
j=1,2,…,15)
• If this rescaling is applied to a Gaussian variable, the PDFs at
various scales collapse onto a single master curve.
Turbulence in the cusp : PDFs
 = 2k t
Significant
departures from
Gaussians for scales
G < 61.47 sec
= hallmark of
intermittency
Echim, Lamy &
Chang (2007)
Turbulence in the cusp : PDFs
PDFs at scales > G
are approximately
Gaussians
Echim, Lamy &
Chang (2007)
Turbulence in the cusp : flatness
3
G = 61.47 sec
One-parameter rescaling of PDFs
• If fluctuations of B2 are self-similar, their PDFs at various
scales P(B2,) should collapse onto one scaled PDF Ps
according to a one-parameter scaling form (Hnat et al.
2002)
P(B2,) s = Ps(B2/s)
• The parameter Y=B2/s is a scale invariant
• The scaling exponent s may be interpreted as a
monofractal measure that characterizes the fluctuations
of all scales through the relation above
• If the one-parameter scaling is not satisfied over the full
range of the scaled variable Y, the fluctuations are
multifractal.
One-parameter rescaling of PDFs
• The scaling parameter s
may be found from a
linear fit of the variation
of the unscaled PDFs
P(0,) with scale  (Hnat
et al. 2002)
• The variation from small
to large scales is not
linear and s cannot be
determined appropriately
one-parameter
rescaling could
not be achieved
Conventional multifractal analysis
• One popular concept to quantitatively characterize intermittent
fluctuations
• The intermittent behaviour is analyzed in terms of high order
moments of the PDFs : the structure functions (SF)

  B  PB , dB
S q B , 
2
2
Bmax
2 q
2
2
 B  xi     B  xi 
2
2
q
0
• For each SF Sq, we associate a fractal exponent q for a range of
scales 
 q  d log S q B 2 ,  / d log  
• If q = 1q, the fractal properties of the fluctuating series are fully
described by the value of 1 : mono-fractal/self-similar
fluctuations. For intermittent turbulence  q is a non-linear
function of q : multifractal case
• SFs can be evaluated for any positive values of q but will
generally diverge for q < 0
Application to cusp data
q is the slope
 = 2j t with j=1,2, …, 14
Application to the cusp data
Scales between  = 1.92 sec and =245.76 sec
q is a non-linear function of q  multifractal phenomenom
Limitations of the conventional
multifractal analysis
• We visualize intermittent/multifractal fluctuations as
composed of many types, each type being characterized
by a particular fractal dimension
• What are those fractal dimensions ?
• How are the various types of fluctuations distributed
within the turbulent medium ?
• Conventional multifractal methods based on SF analyses
cannot answer those questions because they incorporate
the full set of fluctuation sizes and therefore are
dominated by the statistics of fluctuations at the smallest
sizes which are by far the more numerous.
Rank-ordered multifractal analysis
• The rank-ordered multifractal analysis (ROMA) technique
has been developed recently (Chang & Wu 2008, Chang
et al. 2008) in order to solve these problems by easily
separating the fractal characteristics of the minority
fluctuations (of larger amplitudes) from those of the
dominant population.
• We perform the same statistical analyses (based on SF)
individually for subsets of the fluctuations that
characterize the various fractal behaviors within the full
multifractal set.
Rank-order multifractal analysis
• In practice, we consider a small range Y of the scaled
variable Y = |B2|/s for which the one-parameter scaling
works, i.e. a range for which the fluctuations are monofractal.


a2
S q B 2 ,   B
a1
2 q


Y2
P B 2 , d B 2   sq  Y q PS Y  dY
Y1
• Sq are now the range-limited SF ; a1 = Y1s and a2 = Y2s
• Range-limited SF can be evaluated for any order including
negative values of q
• The value of s validating this scaling property has to be
found iteratively for each range of Y
Graphical explanation
Chang et al., IGPP
meeting on Astrophysics,
Kauai, March 2008
Y=B2/s
• A range Y  various ranges of B2 for various scales 
• We try to collapse the small black segments of the unscaled
probabilities within Y
Rank-order multifractal analysis
The slopes give q
Y=[5,10]
a1 and a2 for
a given scale
 and a given
value of s
We search for ranges where the range-limited SF are linear
In this example : scales between =1,92 sec and =245,76 sec
Rank-order multifractal analysis
• The range-limited SF are evaluated for 100 values of s
between s=0 and s=1
• We are looking for values of s for which q = qs
Rank-order multifractal analysis
s1 = 0.54
s2 = 0.95
Linear fit of the first-order range-limited SF gives (1) for a
given value of s
Rank-order multifractal analysis
Same as before for the order moment = -1
The same values of s are approximately found
Rank-ordered multifractal analysis
• We repeat the same operations for many ranges Y of
the scaled variable in order to cover the whole ranges of
the real fluctuations |B2()|
• For nearly each range Y, we obtain 2 solutions s1(Y) and
s2(Y) which rescale the PDFs at the scales considered for
the calculation of the range-limited SF.
• The whole spectrum of values s1(Y) and s2(Y) allows us to
fully collapse the unscaled PDFs.
Spectra s(Y)
s2(Y)
s > 0.5
Persistent
s < 0.5
Antipersistent
s1(Y)
Y=5 between Y=0 and Y=60
PDFs of the raw data
+ : =1,92 sec
 : =3,84 sec
. : =7,68 sec
 : =15,36 sec
 : =30,72 sec
• PDFs of the raw data for 5 different scales
• |B2| are used to take advantage of the symmetry of the
PDFs and for the purpose of better statistical convergence
Rescaling of the PDFs with s1(Y)
+ : =1,92 sec
 : =3,84 sec
. : =7,68 sec
 : =15,36 sec
 : =30,72 sec
 : =61,44 sec
• Excellent rescaling of the PDFs
• The fact that the rescaled PDFs are flat near Y=0 is a
bit puzzling and will be investigated in more details
Rescaling of the PDFs with s2(Y)
+ : =1,92 sec
 : =3,84 sec
. : =7,68 sec
 : =15,36 sec
 : =30,72 sec
No solution s2 for Y=[0,5]
 : =61,44 sec
• Less good rescaling for Y < 20
• But expected increase of Ps(Y) for small values of Y
Rescaling of PDFs : criteria to choose s(Y)
• The solutions of s may be composed of parts of both
branches s1 and s2
• The value of s for small scales Y may be estimated by the
value of (1) of the conventional SF analysis
• The value of s should not be small (close to 0) for small
scales Y ~ 0
Advantages of ROMA
• Full collapse of the unscaled PDFs
• Quantitative measurement of how intermittent are the
scaled fluctuations Y
• The determination of the nature of the fractal nature of the
grouped fluctuations s(Y) is not affected by the statistics of
other fluctuations that do not exhibit the same fractal
characteristics
• Natural connection between the one-parameter scaling idea
and the multifractal behavior of intermittency
ROMA of the solar wind turbulence
• 21 days sample of Br measured by Ulysses in 1994
• d = 3.8 AU, heliographic latitude = -50°
• Fast wind streams during solar minimum
t = 1 or 2 sec
Number of
points = 1,2.106
Solar wind intermittency
Lamy, Wawrzaszek, Macek
and Chang, 2010
Range-Limited Structure functions
 (q)
Y =
[0.002;0.004]
Lamy, Wawrzaszek,
Macek and Chang, 2010
Between  = 4 sec and  = 1024 sec
For each value of s  set of scaling exponents  (q)
Search of the monofractal behavior
q=1
s2 ~ 0.74
s1 ~ 0.38
We repeat the operation for several values of q
to minimize the influence of the statistics
Multifractal spectra s(Y)
• We repeat the same procedure for many ranges Y in
order to cover the whole set of fluctuations |Br()|
s2(Y)
Real ?
Lamy, Wawrzaszek,
Macek and Chang, 2010
s1(Y) ~ 0.4  good agreement with Hnat et al (2002) : s=0.42  0.02 (for B2)
Possible origin of s2(Y) ?
• Problem with statistics for large values of s ?
Y = [0.002;0.004]
Further testings needed but apparently not
Possible origin of s2(Y) ?
• Some bendings of the RLSFs for some scales could indicate
a crossover behaviour of different s(Y) from different scale
regimes.
• See presentation of Tam et al tomorrow for a detailed
example
• We will do additional tests to check this hypothesis both for
the cusp and the solar wind data
Choice of the « correct » s(Y) ?
• Try to rescale the PDFs at various scales with both
spectra.
• Another method will be discussed in detail by Wu &
Chang tomorrow
Conclusions & Perspectives
• ROMA is a new statistical technique that fully
characterizes the complex statistical characteristics of
non-Gaussian PDFs.
• We have applied the ROMA to the cusp data for the first
time and to solar wind data
• Origin of the 2nd multifractal spectrum must be further
analyzed and discussed  test to check if we have some
crossover behaviour between 2 different multifractal
spectra from different scale regimes.
• In the cusp  test of the validity of the Taylor hypothesis
utilizing the data obtained from the 4 Cluster spacecraft
• … This is just the beginning !