Rank-ordered multifractal analysis (ROMA) of magnetic intermittent fluctuations in the solar wind and in the magnetospheric cusps: evidence for global crossover behavior? Hervé Lamy 1, Marius Echim 1,2, Tom Chang 3 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 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)  > 10 6 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 : B 2 () = B 2 (t+)-B 2 (t) Detrending of the data for the large scale variations due to the geomagnetic dipole component via the following rescaling procedure  = 2 j 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 Significant departures from Gaussians for scales  G < sec = hallmark of intermittency Echim, Lamy & Chang (2007)  = 2 k t

Turbulence in the cusp : PDFs PDFs at scales >  G are approximately Gaussians Echim, Lamy & Chang (2007)

Turbulence in the cusp : flatness 3  G = sec 3

One-parameter rescaling of PDFs If fluctuations of B 2 are self-similar, their PDFs at various scales P(B 2,) should collapse onto one scaled PDF P s according to a one-parameter scaling form (Hnat et al. 2002) P(B 2,)  s = P s (B 2 / s ) The parameter Y= B 2 / 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) For each SF S q, we associate a fractal exponent  q for a range of scales  If  q =  1 q, 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  = 2 j t with j=1,2, …, 14

Application to the cusp data Scales between  = 1.92 sec and = 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 = |B 2 |/ s for which the one-parameter scaling works, i.e. a range for which the fluctuations are mono- fractal. S q are now the range-limited SF ; a 1 = Y 1  s and a 2 = Y 2  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 A range Y  various ranges of B 2 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=B 2 / s

Rank-order multifractal analysis 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] a 1 and a 2 for a given scale  and a given value of s

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 s 1 = 0.54 s 2 = 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 |B 2 ()| For nearly each range Y, we obtain 2 solutions s 1 (Y) and s 2 (Y) which rescale the PDFs at the scales considered for the calculation of the range-limited SF. The whole spectrum of values s 1 (Y) and s 2 (Y) allows us to fully collapse the unscaled PDFs.

Spectra s(Y) s 2 (Y) s 1 (Y) Y=5 between Y=0 and Y=60 s > 0.5 Persistent s < 0.5 Anti- persistent

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 |B 2 | are used to take advantage of the symmetry of the PDFs and for the purpose of better statistical convergence

Rescaling of the PDFs with s 1 (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 s 2 (Y) Less good rescaling for Y < 20 But expected increase of P s (Y) for small values of Y No solution s 2 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) The solutions of s may be composed of parts of both branches s 1 and s 2 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 B r 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 = 1,2.10 6

Solar wind intermittency Lamy, Wawrzaszek, Macek and Chang, 2010

Range-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) Lamy, Wawrzaszek, Macek and Chang, 2010

Search of the monofractal behavior q=1 s 1 ~ 0.38 s 2 ~ 0.74 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 |B r ()| s 1 (Y) ~ 0.4  good agreement with Hnat et al (2002) : s=0.42  0.02 (for B 2 ) s 2 (Y) Real ? Lamy, Wawrzaszek, Macek and Chang, 2010

Possible origin of s 2 (Y) ? Problem with statistics for large values of s ? Y = [0.002;0.004] Further testings needed but apparently not

Possible origin of s 2 (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 2 nd 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 !