VOLUME 76, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 20 MAY 1996

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Multifractal Structure of Auroral Electrojet Index Data

Giuseppe Consolini,* Maria F. Marcucci, and Maurizio CandidiIstituto di Fisica dello Spazio Interplanetario, Consiglio Nazionale delle Ricerche, via Galileo Galilei, P.O. Bo

00044 Frascati, Roma, Italy(Received 19 April 1995)

Using a multifractal approach, based on “singularity analysis,” we investigate the scaling propertiesof the auroral electrojet index (AE) time series. The existence of a multifractal structure in the AE timeseries is the signature of the occurrence of “intermittence,” which can be interpreted as an indicatioof turbulence in magnetospheric dynamics. Furthermore, a simple model, theP-model (a two-scaleCantor set), is shown in order to investigate the underlying multiplicative nature of the signal. This sedisplays many of the multifractal properties of the AE signal. [S0031-9007(96)00242-6]

PACS numbers: 94.30.Lr, 05.40.+j

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The auroral electrojet index (AE), derived from higlatitude fluctuations of the magnetic field horizontal coponent at Earth’s surface, is meant to estimate the tmaximum amplitude of the ionospheric current systeIt was introduced by Sugiura and Davis [1] to monitthe occurrence of auroral phenomena, and more genemagnetospheric substorms.

The description and modeling of the AE index timseries and the study of their scaling properties are perful tools for understanding the nature of solar winmagnetosphere-ionosphere coupling and magnetospsubstorm dynamics.

To explain the high temporal variability of magnetsubstorms, which is evident in the AE time series (sFig. 1, top panel), many authors [2–7] investigatedpossible occurrence or not of low-dimensional chain the magnetospheric response to solar wind inphowever, this is still an open question.

Nevertheless, Takaloet al. [8–10] have clearly shownthe existence of scaling properties in AE index dathat suggest that the signal is self-affine, with scalexponentH ø 0.5, up to a time of about113s69d min.Furthermore, this characteristic time is well in agreemwith the spectral break atø5.6 3 1025 Hz previouslyobserved by Tsurutaniet al. [11].

However, the irregularity of AE temporal evolutiomay suggest a more complex nature of the analyzed pnomenon than that characterized by the above mentiosimple fractal model. The AE time series “spotty” bhavior, evidenced when the signal increments are plo(see Fig. 1, bottom panel), can indeed be an indication“intermittence,” and therefore “turbulence.” Furthermointermittence involves an anomalous scaling with respto “time dilation.”

This Letter proposes a multifractal approach to thetime series, based on the so-called “singularity analyswith the aim of revealing the occurrence of intermittenin the dynamics of magnetospheric substorms.

The purpose of multifractal analysis is to reveal texistence of a hierarchy of scaling indices, which is d

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to the different local scaling properties of the data.order to do this, first of all a “positive stationary measurhas to be defined on the data set [12,13]. Since thetime series power spectral density (PSD) is characteriby power laws with spectral exponent1 , b , 3 (seeFig. 2), the AE signal is nonstationary with stationaincrements over a range of scales which is bounded ab

FIG. 1. Sample of the original time series covering a periof three days (top panel) and relative AE increments time se(bottom panel).

© 1996 The American Physical Society

VOLUME 76, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 20 MAY 1996

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and below [12]. It is important to stress that with tterm “stationary” we mean that the data setwAEstd isstatistically invariant by translation in timet. Therefore,a new scalar stationary, non-negative field´std has beendefined according to Meneveau and Sreenivasan [14the squared absolute value of the small scale differenc

´std ­ jwAEsti 1 Dtd 2 wAEstidj2, (1)

wherewAEstd is the original AE-data set, andDt is thesampling interval. There are several methods to definstationary non-negative field. However, the peculiar pcedure does not affect the results of singularity analysiLavalléeet al. [15] pointed out.

Consequently, a positive measuredm can be defined a

dmstd ­´stdT k´l

dt , (2)

where T is the total time length. According to Paladet al. [13], a multifractal measure is characterized byscaling features of its coarse-grained weight:

pistd ­ZLi

dmstd øX

ti#t0#ti 1t

Dmst0d , (3)

where t ­ 2n is the size of the segmentsLi . Thepresence of multifractality is shown by the anomaloscaling of the “partition functionGsq, td” for small t:

Gsq, td ­XLi

pistdq ø tgsqd, (4)

wheregsqd ­ sq 2 1dDq andDq is a nonconstant function. The exponentsDq, called “generalized dimension

FIG. 2. Power spectral density(PSD) or energy spectrumrelative to the period under analysis. The solid and dolines are power-law best fits. The dashed line is relato 25y3 power law predicted for the “inertial range” bKolmogorov’s theory of fully developed turbulence in absenof “intermittence.” Two spectral breaks,f1 and f2, identifya frequency region where the dependence is nearly similaKolmogorov’s one.

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[16], are independent of moment order in the case ofmogeneous fractality.

We developed our analysis applying the multifracapproach to a set of AE-index data, covering the perfrom 1.1.1975 to 19.2.1975, with 1-min time resolutiofor a total amount of 216 points. Data comes from thNational Geophysical Data Center, Boulder, CO.

To evaluate the exponentsgsqd, the partition functionGsq, td vs t has been fitted with a power law usinthe Levemberg-Marquardt nonlinear regression algorit[17].

In Figs. 3 and 4gsqd andDq are plotted as a functionof the moment orderq. It is evident thatgsqd and qare not linearly dependent; this is the consequencean underlying multifractal structure in the AE signaas is also confirmed by the existence of a hierarchydimensionDq.

The existence of a multifractal nature of the AE signin respect to time dilation is the signature of tempoinhomogeneity, or, in other words, of intermittence [13]

The occurrence and the nature of intermittence inAE signal have been further analyzed by comparingDq curve with those proposed for two typical multplicative processes, theP-model[14] and theLog-normalmodel [18]. These models were first introduced to acount for the occurrence of intermittence in fully deveoped turbulence in ordinary fluid flows.

The solid line in Fig. 4 is the nonlinear best fit othe Dq data according to theP-model, which is formallyequivalent to a “two-scale Cantour set” withl1 ­ l2 ­1y2 and represented by

Dq ­ log2fpq 1 s1 2 pdqg1ys12qd, (5)

wherep is a parameter, associated with the fragmentatprobability in the cascade process, andq is the momentorder.

The dash-dotted line representsDq behavior accordingto the log-normal model. There is agreement betweethis model andDq data only if small values ofq are

FIG. 3. Scaling exponentgsqd of the partition functionGsq, td as a function of moment orderq.

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VOLUME 76, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 20 MAY 1996

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FIG. 4. Plot ofgeneralized dimensionsDq. The solid line isthe nonlinear best fit of the data by theP-model (see text).The dash-dotted line represents the behavior of theLog-normalmodel.

considered. On the contrary, the experimentalDq curveand theP-modelbest fit are fairly well in agreement.

The agreement betweenDq data andP-modelpredic-tion must be interpreted as the evidence for partial mixduring the cascade and for an asymmetric breakdownthe fragmentation process. The parameterp, as evaluatedfrom the nonlinear best fit ofDq data using relation (5)gives

p ­ 0.746 6 0.002 . (6)

This parameter can be used to evaluate the intermittecoefficientm:

m ­ 22dDq

dqjq­0 ­ log2f4ps1 2 pdg21

­ 0.400 6 0.002 . (7)

In the case of homogeneous turbulence, Kolmogorotheory (via Taylor’s hypothesis) predicts af2b law,with b ­ 5y3, in the inertial range of energy spectrum.When intermittence is considered the exponentb must becorrected as follows:

b ) b 1m

3­ a , (8)

where m is the intermittence coefficient. From this wobtain

a ­ 1.800 6 0.001 , (9)

which is well in agreement with the PSD power-laexponent when the intermediate rangef f1, f2g,

7.3 3 1025 , f , 2.5 3 1023 Hz , (10)

is considered (see Fig. 2).Another way to characterize the multifractality is give

by the so-calledmultifractal or singularity spectrumfsad,which can be directly evaluated from thegsqd curve by a

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Legendre transformation:

a ­dgsqd

dqfsad ­ qa 2 gsqd . (11)

In Fig. 5 we report thesingularity spectrumfsadderived for the AE index data set (pointed curve). Thsolid line represents a nonlinear best fit if an analyticexpression for theP-modelis used, which can be derivedfrom that of a general two-scale Cantor set with equscalessl1 ­ l2 ­ 1y2d but unequal weights [19],

a ­ 2log2 p 1 snym 2 1d log2s1 2 pd

nym

fsad ­ 2snym 2 1d log2snym 2 1d 2 snymd log2snymd

nym(12)

eliminating nym. Once again there is good agreemebetween theory and data.

The multifractal approach in respect to time dilatiohas evidenced the existence of different local scaliproperties in the AE-time series. This is a consequencetemporal inhomogeneity that is related to the intermittecharacter of the signal.

The comparison between the multifractal structurethe AE signal and that of two typical multiplicative processes, introduced in order to explain the intermittenceturbulence, has clearly shown that the nature of the sigis analogous to intermittent turbulence. It is importantstress that intermittent turbulence involves a different eergy distribution in space and time from the predictioof Kolmogorov’s theory for ordinary turbulence. Moreover the multifractal structure of the AE index seemsbe analogous to theP-modelprediction.

FIG. 5. Multifractal or singularity spectrumas derived fromLegendre transformation. The solid line is a nonlinear regresion best fit of the data making use of an analytical expresion for a two-scale Cantor setwith equal scales but unequaweights (P-model). The dashed line is the diagonal which indicates the homogeneous fractal locus.

VOLUME 76, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 20 MAY 1996

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Furthermore, starting from the intermittence coefficiem, evaluated on the basis of theP-model, it was possibleto identify a spectral region in which the PSD is wellagreement with Kolmogorov’s spectrum corrected in tcase of intermittence.

As a consequence of this analysis, we can conclthat intermittence and turbulence must be considereda much more relevant phenomena than low-dimensiochaos in the evolution of magnetic substorms. Furthmore, a pure mathematical model, theP-model, seems tobe able to explain the underlying multifractal structurethe signal, and this is certainly useful information in thelaboration of a magnetospheric model.

Further studies are necessary in order to extethis analysis to other periods and to explain the Aintermittent and turbulent character within a genemagnetospheric-ionospheric model.

We are grateful to Professor A. Vulpiani of the Univesity of Rome “La Sapienza,” for the useful discussionG. C. is very grateful to the National Research Coun(CNR) for its financial support. We are grateful to thItalian National Project for Antarctic Research (PNRAwhich partially supported this work.

*To whom correspondence should be addressed. Electraddress: consoli@hp.ifsi.fra.cnr.it

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