Rank-ordered multifractal analysis (ROMA) of magnetic intermittent fluctuations in the solar wind and in the magnetospheric cusps: evidence for global

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Rank-ordered multifractal analysis (ROMA) of Rank-ordered multifractal analysis (ROMA) of magnetic intermittent fluctuations in the magnetic intermittent fluctuations in the

solar wind and in the magnetospheric cusps: solar wind and in the magnetospheric cusps: evidence for global crossover behavior? evidence for global crossover behavior?

Hervé Lamy1, Marius Echim1,2, Tom Chang3

1 Belgian Institute for Space Aeronomy, Brussels, Belgium

2 Institute for Space Sciences, Bucharest, Romania

3 Kavli Institute for Astrophysics and Space Research, MIT, Cambridge, USA

OUTLINE OF THE TALKOUTLINE 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 dataCLUSTER 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 PDFsNormalization 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

= 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.

,,,

22

2tBtB

tb

Turbulence in the cusp : PDFsTurbulence in the cusp : PDFs

Significant departures from

Gaussians for scales G < 61.47 sec

= hallmark of intermittency

Echim, Lamy & Chang (2007)

= 2k t

Turbulence in the cusp : PDFsTurbulence in the cusp : PDFs

PDFs at scales > G are approximately

Gaussians

Echim, Lamy & Chang (2007)

Turbulence in the cusp : flatnessTurbulence in the cusp : flatness

3

G = 61.47 sec

3

One-parameter rescaling of PDFsOne-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 PDFsOne-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 analysisConventional 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)

• For each SF Sq, we associate a fractal exponent q for a range of scales

• 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

qii

Bq

q xBxBBdBPBBS 22

0

2222

2max

,,

log/,log 2 dBSd qq

Application to cusp dataApplication to cusp data

q is the slope

= 2j t with j=1,2, …, 14

Application to the cusp dataApplication 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 Limitations of the conventional multifractal analysismultifractal 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 analysisRank-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 analysisRank-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 mono-fractal.

Sq are now the range-limited SF ; a1 = Y1s and a2 = Y2s

• 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

dYYPYBdBPBBS S

Y

Y

qsqa

a

q

q 2

1

2

1

2222 ,,

Graphical explanationGraphical explanation

• A range Y various ranges of B2 for various scales

• We try to collapse the small black segments of the unscaled probabilities within Y

Chang et al., IGPP meeting on Astrophysics,

Kauai, March 2008

Y=B2/s


We search for ranges where the range-limited SF are linear

In this example : scales between =1,92 sec and =245,76 sec

The slopes give q

Y=[5,10]

a1 and a2 for a given scale and a given

value of s


• 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


s1 = 0.54

s2 = 0.95

Linear fit of the first-order range-limited SF gives (1) for a given value of s


Same as before for the order moment = -1

The same values of s are approximately found

Rank-ordered multifractal analysisRank-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)Spectra s(Y)

s2(Y)

s1(Y)

Y=5 between Y=0 and Y=60

s > 0.5

Persistent

s < 0.5

Anti-persistent

PDFs of the raw dataPDFs 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 sRescaling of the PDFs with s11(Y)(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 sRescaling of the PDFs with s22(Y)(Y)

• Less good rescaling for Y < 20

• But expected increase of Ps(Y) for small values of Y

No solution s2 for Y=[0,5]

+ : =1,92 sec

: =3,84 sec

. : =7,68 sec

: =15,36 sec

: =30,72 sec

: =61,44 sec

Rescaling of PDFs : criteria to choose s(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 ROMAAdvantages 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 turbulenceROMA 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 intermittencySolar wind intermittency

Lamy, Wawrzaszek, Macek and Chang, 2010

Range-Limited Structure functionsRange-Limited Structure functions

Between = 4 sec and = 1024 sec

(q)

Y = [0.002;0.004]

For each value of s set of scaling exponents (q)


Search of the monofractal behaviorSearch of the monofractal behavior

q=1

s1 ~ 0.38

s2 ~ 0.74

We repeat the operation for several values of q to minimize the influence of the statistics

Multifractal spectra s(Y)Multifractal spectra s(Y)

• We repeat the same procedure for many ranges Y in order to cover the whole set of fluctuations |Br()|

s1(Y) ~ 0.4 good agreement with Hnat et al (2002) : s=0.42 0.02 (for B2)

s2(Y)

Real ?


Possible origin of sPossible origin of s22(Y) ?(Y) ?

• Problem with statistics for large values of s ?

Y = [0.002;0.004]

Further testings needed but apparently not

Possible origin of sPossible origin of s22(Y) ?(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) ?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 !

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Rank-ordered multifractal analysis (ROMA) of magnetic intermittent fluctuations in the solar wind and in the magnetospheric cusps: evidence for global