Geometric description of the magnetic field and plasma coupling in the near-Earth stretched tail prior to a substorm

Journal of Atmospheric and Solar-Terrestrial Physics 63 (2001) 705–721www.elsevier.nl/locate/jastp

Geometric description of the magnetic )eld and plasmacoupling in the near-Earth stretched tail prior to a substorm

Alexander V. Milovanova ;∗, Lev M. Zelenyia, Pierluigi Veltrib, Gaetano Zimbardob,Alexander L. Taktakishvilic

aDepartment of Space Plasma Physics, Space Research Institute, Russian Academy of Sciences, 117810 Profsoyuznaya 84=32,Moscow, Russia

bDipartimento di Fisica, Universit&a della Calabria, and Istituto Nazionale di Fisica della Materia, Unit&a di Cosenza, ItalycAbastumani Astrophysical Observatory, Tbilisi, Georgia

Received 30 November 1999; accepted 16 June 2000

Abstract

Topological properties of the self-organized magnetic )eld and plasma turbulence in the near-Earth stretched tail prior tothe magnetospheric substorm onset are analyzed. We found that the magnetotail turbulence structures at the late substormgrowth phase should be associated with the hierarchical (“singular”) geometry; this geometry includes fractal objects, i.e.,self-similar “clumpy” structures with “voids” present on many scales. The fractal description of the self-organized magnetic)eld and electric current coupling in the tail has led us to the idea of the “geometric” formulation of the basic physicalprocesses in the tail; this formulation deals with certain topological “codes” (i.e., algebraic relations between the principalgeometric parameters) and is an alternative of the more traditional approach based on the direct analysis of the correspondingnonlinear di;erential equations. A “topological scenario” for the substorm onset is further proposed, and the relevant “codes”characterizing the microscopic structural properties of the turbulence are obtained. The behavior of the Fourier energy densityspectra of the magnetic )eld and cross-tail electric current >uctuations is discussed. The results of the numerical modeling ofthe turbulence at the late growth phase are given. c© 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Self-organised turbulence; Substorm onset; Fractal topology

1. Introduction

The substorm growth phases preceding the onset are char-acterized by a considerable local stretching and thinning ofthe magnetotail; the stretched regions of the tail customar-ily exhibit strong magnetic )eld and plasma turbulence, thisbeing an inherent feature of the onset location.

1.1. Turbulence spectra: evidence for power laws

Many of the recent studies indicate that the magnetotailturbulence prior to the substorm onset has an important gen-

∗ Corresponding author.E-mail address: [email protected] (A.V. Milovanov).

eral property, namely, the power-law behavior of the Fourierenergy density spectrum:

A comprehensive analysis of the turbulence spectra hasbeen performed by Ohtani et al. (1995) with use of thefractal time series technique. Ohtani et al. (1995) investi-gated the turbulence data obtained in the near-Earth stretchedtail with the Active Magnetospheric Particle Tracer Ex-plores (AMPTE). The power-law energy density spectrum,P(f)˙ f−�, was recognized in the higher frequency range(i.e., at the frequencies f&f∗ ∼ 5 × 10−2 Hz), with thespectral index � ≈ 2:4. In the lower frequency interval,f.f∗, the spectrum P(f) was unambiguously more >at,resulting in a kink around f∼f∗.

By using magnetometer and plasma instrument data ofAMPTE obtained during three months of plasma sheetpasses, Bauer et al. (1995) reported a kink in the energy

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density spectra of the magnetic )eld >uctuations from acomputation of the average spectra of the magnetic )eldvectors in the relevant frequency ranges. The results ofBauer et al. (1995) indicate that whereas for frequenciesbetween 0.03 and 2 Hz the power-law behavior with theslopes ranging from ≈2 to ≈ 2:5 holds, in the lower fre-quency range the spectrum is actually more >at, with theslopes being clearly less than ≈1:5 for frequencies below∼10−3 Hz.

An evidence for the power-law turbulence spectra withthe characteristic slopes from 2 to 2.5 in the higher frequencyrange was also provided by Russell (1972) from the OGO5 plasma sheet data. A similar power-law behavior was rec-ognized by Borovsky et al. (1997) from the ISEE 2 FastPlasma Experiment; having performed the relevant Fourieranalysis of the magnetic turbulence data over 1.5 decadesof frequency, Borovsky et al. (1997) found the mean valueof the characteristic spectral slopes to be ≈ 2:2.

The existence of the power laws in the magnetic turbu-lence data has been recently supported by the results of theInterball-1 mission (A. Petruckovich, private communica-tion, 1998). The obtained spectra of the magnetic )eld>uctuations as they were seen by Interball-1 have thecharacteristic kink shape, with the typical slopes close to thevalue ≈ 2:35 for frequencies higher than, approximately,f∗ ∼ 10−2 Hz, and with the relatively >at constituent of thespectrum in the frequency range below f∗.

1.2. Turbulence hierarchies: theoretical tools

The physical origin of the power-law spectra observedin the magnetotail could be the hierarchical structuring ofthe turbulence at the substorm onset. “Hierarchical struc-turing” means that the turbulence is organized in multi-scale “clumps” (i.e., “grains” of various sizes) separated by“voids” (“empty” domains between the “grains”). The for-mation of the “coarse-grained” turbulence hierarchies im-plies that the system enters into a strongly nonlinear statecharacterized by the coexistence of many di;erent mutuallyinterconnected scales a;ecting each other.

A thermodynamic description of the multiscale “clumpy”turbulence might be proposed in the framework of thegeneralized Lorentzian statistical mechanics based on anextension of the BBGKY hierarchical correlation expan-sion of the Liouville equation (Treumann, 1999a,b). Theconceptually equivalent approach has been developed byMilovanov and Zelenyi (2000) who applied the generalizedDaroczy-Tsallis thermodynamics to the “coarse-grained”systems with the extremely long-ranged correlations oper-ating on many scales.

Alternatively, one might attempt to analyze the hierar-chical turbulent system geometrically. This means that thestrong nonlinearity of the system is hidden in a set of geo-metric quantities which de)ne the topological characteristicsof the turbulence. The behavior of the system is then formu-lated in terms of algebraic relations between the geometric

parameters involved; such relations (“algebraic codes”) areassumed to play the role of the dynamical equations describ-ing the evolutionary processes in the system.

The geometric approach has been recently applied byMilovanov and Zelenyi (1998) to a dynamical (fracton)model of the interplanetary magnetic )eld turbulence. Sim-ilar ideas have been proposed in the papers of Milovanov etal. (1996,2000) and Zelenyi et al. (1998) who analyzed thestrong magnetic )eld and plasma turbulence in the Earth’sdistant tail.

The mathematical background of the geometric descrip-tion of the hierarchical turbulent systems includes the meth-ods of the fractal topology, i.e., the synthesis of the standardfractal geometry and the di;erential topology of manifolds(for details, see Nakayama et al., 1994; Milovanov, 1997;Milovanov and Zimbardo, 2000).

The fractal topology deals with the two main topologicalquantities which completely describe the geometric proper-ties of the turbulent system: The Hausdor; fractal dimen-sion, D, and index of connectivity, �. (In the notations ofMilovanov (1997) and Milovanov and Zimbardo (2000),� ≡ .) (We restrict ourselves to only the self-similar hier-archies which reveal the properties of isotropy and statisti-cal homogeneity. For more details, see, e.g., the monographsof Mandelbrot (1983) and Feder (1988).)

The Hausdor; fractal dimension D quanti)es the expo-nential growth rate of the minimal number of elements(hypercubes) which cover the system with given accuracy.This parameter is important in view of the presence of the“voids” (“empty” domains) that are excluded from the cov-ering. Roughly speaking, the Hausdor; dimension D char-acterizes the deviation of the turbulence geometry from thespace-)lling limit (i.e., when no “voids” are present).

The Hausdor; dimension D is customarily introducedas the e;ective (fractional) dimensionality of the set occu-pied by the turbulence. The space-)lling limit maximizesthe values of D: in this case, the quantity D is assumedto coincide with the topological (integer) dimensionality,E, of the embedding Euclidean space, i.e., D = E for thespace-)lling turbulence. The presence of the “voids” vio-lates the space-)lling approximation: The turbulence occu-pies a fraction of the embedding Euclidean space; thissituation corresponds to the values of the Hausdor; fractaldimension D that are smaller than E, i.e., D¡E in gen-eral. More precisely, the Hausdor; dimension D de)nesthe scaling of the fractal “mass” density: �( ) ˙ D−E ,where is a scale length, and D− E 6 0 as a result of the“voids” that are present.

Following Katok and Hasselblatt (1999), one can demon-strate that the fractal dimension D yields the topologicalentropy for a self-similar turbulence pattern, i.e., de)nesthe number of topological “states” occupied by the fractaldistribution.

The fractal dimension D alone is not a suNcient char-acteristic of the fractal structure since it ignores the spatial“orientation” of the “voids”. This is given by the index of

A.V. Milovanov et al. / Journal of Atmospheric and Solar-Terrestrial Physics 63 (2001) 705–721 707

connectivity �: the space-)lling limit (no “voids” present)implies � ≡ 0, whereas fractal distributions with “voids”may have nonzero (positive or negative) values of �, de-pending on the relative positions of the “empty” domains.(Descriptive illustrations can be found in, e.g., Nakayamaet al. (1994); Rodriguez-Iturbe and Rinaldo (1997) andZelenyi et al. (1998).)

The index of connectivity � can be rigorously introducedby using the concept of the geodesic line on a fractal ob-ject (i.e., the shortest line connecting two arbitrary pointsof the fractal). Indeed, the topological arguments proposedby Milovanov (1997) show that the geodesic line on aself-similar fractal object can be treated as a self-aNne frac-tal curve whose own Hausdor; fractal dimension is equal to(2+�)=2. This idea is illustrated in the paper of Milovanovand Zelenyi (1998).

In this paper, we apply the geometric approach to an anal-ysis of the strong magnetic )eld and plasma coupling inthe near-Earth stretched magnetotail prior to the substormonset. In the self-consistent regime, four topological quan-tities describing the system should be introduced, i.e. (1)the Hausdor; fractal dimension D and index of connectiv-ity � characterizing the magnetic )eld structures, and (2)the Hausdor; fractal dimension D+ and index of connectiv-ity �+ relevant for the electric current con)gurations. (Thetopological meaning of the quantities D+ and �+ is analo-gous to that of D and �.)The necessity to introduce the quantities D+ and �+ in

addition to the parameters D and � might be justi)ed by thefollowing arguments: One might assume that the magnetic)eld fractal geometry is self-consistently maintained by theproper (fractal) distribution of the electric current links; thelatter does not necessarily follow the distribution of the mag-netic )eld elements (magnetic turbulence “clumps”) andshould be generally associated with another fractal objectwhose Hausdor; fractal dimension, D+, and index of con-nectivity, �+, are distinguished from the magnetic )eld dis-tribution parameters, D and �. The two distributions, how-ever, cannot be arbitrary but are intrinsically interconnected.(The interconnections are described by the standard set ofthe Maxwell equations and are of the conventional electro-dynamic origin.) Figuratively speaking, the two intercon-nected (fractal) distributions might be considered as two(di;erent) fractal objects self-consistently “pulled” one onthe other; these objects might be termed “dual”, or (mutu-ally) “conjugated”. In the geometric language, the dualityproperty implies certain (algebraic) relations between theparameters D; D+; �, and �+; these relations provide thegeometric representation of the strong magnetic )eld andplasma coupling in the turbulent system.

The fractal geometry approximation addresses the impor-tant issue of nonlocality of the basic mechanisms maintain-ing the hierarchical structure of the turbulence. In fact, theformation of a self-similar hierarchy implies the propertyof the scaling invariance throughout the entire structuringinterval which is customarily long ranged. Such a global in-

variance property might be recognized as a sort of nonlocalsymmetry existing in the turbulence system.

The issue of nonlocality is discussed in Section 2 in rela-tionship with the connectivity properties of the fractal distri-bution. The “geometric” description of the multiscale mag-netic )eld and plasma turbulence in the Earth’s stretchedtail is further developed in Section 3 where the issue of theself-organized critical percolation is pointed out. In Section4, we demonstrate that the self-organized topology of cur-rents and magnetic )elds in the tail may become unstableunder certain conditions, and a structural catastrophe hap-pens in the system; the phenomenon has features that enableto associate it with the magnetospheric substorm onset. Thebehavior of the turbulence energy density spectra as pre-dicted by our theoretical treatment is discussed in Section 5.Relevant results of the numerical modeling of the turbulentsystem are presented in Section 6. We summarize the resultsobtained in Section 7 of the paper.

2. Turbulent saturation of the current sheet instability

Consider, )rst, the physical role attributed to the turbu-lence in the stretched tail at the late substorm growth phase.Assume, for a while, that the cross-tail electric currents areorganized in straight parallel )laments; this corresponds tothe “laminar” electromagnetic structure of the tail withoutturbulence. Since parallel currents attract each other, such acurrent system is potentially unstable. The current instabilitymight become energetically unfavorable in the presence ofa suNciently large regular component of the magnetic )eldnormal to the current sheet; this, however, is not the casein the considerably stretched tail where the normal com-ponent of the magnetic )eld could be relatively small. Ane;ective mechanism that might prevent the collapse of thecurrent system (initiated, e.g., by small current density inho-mogeneities) could be then the self-organized magnetic )eldand plasma turbulence in the current sheet. For instance, theturbulent structures might become e;ective “scatterers” forthe plasma particles that otherwise would be strongly ac-celerated by the cross-tail electric )eld along the (straight))laments. The self-organized turbulence would appear inmultiscale twisting and branching of the )laments: the for-mation of a web-like electric current network composed ofnumerous conducting links of irregular shape could sup-press the particle acceleration and saturate further instabil-ity growth. In the self-consistent regime, the electric currentweb should give rise to the multiscale magnetic )eld >uc-tuations “assembled” in “clumps” between the conductinglinks.

The physical origin of the “clumpy” turbulence associatedwith the multiscale structuring of the electric currents andmagnetic )elds might be the so-called modi)ed two-streaminstability (MTSI) which has been analyzed in detail byLui et al. (1995). MTSI is characterized by the general-ized lower-hybrid frequencies and is one of the modes of


Fig. 1. A schematic illustration of a highly branched, consider-ably “knotted” fractal network representing the cross-tail currentdensity distribution in the magnetotail current sheet. Such a dis-tribution characterizes the “basic” equilibrium of the tail. Theself-consistently generated magnetic )eld >uctuations are shownas the )eld intensity “clumps” of various sizes “attached” to themultiscale conducting elements. One might )guratively say thatthe current network is self-consistently “pulled” on the magnetic)eld >uctuations.

the cross-)eld current instability. The importance of thecross-)eld streaming instabilities in the neutral sheet envi-ronment has been recently addressed by Ohtani et al. (1998).

An electric current network generating the magnetic >uc-tuation “clumps” in the magnetotail current sheet is schemat-ically illustrated in Fig. 1. Such a network carries somenonzero average current in the cross-tail direction; let us de-note this current by 〈jy〉where the subscript “y” correspondsto the GSM coordinate system. This current self-consistentlymaintains the lobe component, Bx, of the magnetotail )eld.Considerable branching and twisting of the current )lamentscan be associated with the multiscale perturbation, �j, whichgenerally has both x- and y-component and maintains the>uctuating magnetic )eld, �Bz . (As usual, we apply the bold-face everywhere to distinguish vector quantities, i.e., �Bz isunderstood as a vector, whereas �Bz would be a scalar.)

The set of Maxwell equations for the system reads asfollows:

∇× Bx =4�c〈jy〉; (1)

∇× �Bz =4�c

�j: (2)

For the sake of simplicity, we ignore here the possible ap-pearance of the magnetic >uctuations in x- and y-direction,restricting ourselves to a quasi-two-dimensional topology ofthe turbulence in the stretched magnetotail. We also notethat the continuity condition div [〈jy〉+ �j] = 0 should sup-plement Eqs. (1) and (2).

To account for the multiscale magnetic )eld and plasmacoupling in the magnetotail current sheet, one has to extendthe Maxwell equation (2) to fractal geometry. Such an ex-tension leads to a modi)cation of the operator ∇× on the

left-hand side of Eq. (2): In fact, ∇× must be made consis-tent with the fractal topology of the set on which Eq. (2) isanalyzed.

The proper generalization of the operator ∇× to frac-tal geometry has been proposed by Zelenyi and Milovanov(1996) in the context of the fractional di;erential calculus(see, e.g., Oldham and Spanier, 1974; Le Mehaute, 1991).According to Zelenyi and Milovanov (1996), the main mod-i)cations in ∇× are as follows: (1) Each of the scalar op-erators @=@x and @=@y in the explicit expansion of ∇× onthe (xy)-plane attains the scaling factor ˙ −�=(2+�); ∼√

x2 + y2; (2) The di;erentiation order in @=@x and @=@ymust include the connectivity index �: The generalized dif-ferentiation order is equal to � = 2=(2 + �) 6 1 insteadof unity. This means that the derivatives @=@s; s = x; y, arereplaced for the fractional operators ˙ @�=@s�; the formalaction of the fractional derivatives @�=@s� on the function�Bz(s) is given by

@�

@s��Bz(s) =

1R(1− �)

@@s

∫ s

0

�Bz(w)(s− w)�

dw (3)

for 0 6 �¡ 1. (Here, R denotes the Euler gamma func-tion.) Applying the Riemann–Liouville identity (see, e.g.,Le Mehaute, 1991), it is easy to verify that the fractionalderivative (3) is reduced to the standard )rst-order deriva-tive @=@s in the limit � → 1; for � = 0, Eq. (3) yields@0�Bz(s)=@s0 ≡ �Bz(s).

The integro-di;erential form of the fractional derivative(3) is an explicit representation of the nonlocality propertyof the self-similar hierarchical system. The “degree” of non-locality is quanti)ed by the power exponent � = 2=(2 + �)in the singular kernel (s−w)−�. One notices that the indexof connectivity of a fractal object, �, explicitly character-izes the nonlocal behavior in the system, associated with theformation of the fractal structure.

The proposed extension of the operator ∇× to fractalgeometry leads one to conclude that the left-hand side ofEq. (2) behaves as �Bz · −�=(2+�).

In the self-consistent regime, the current density pertur-bations �j [which appear on the right-hand side of Eq. (2)]must be also treated as functions of �Bz; according to Milo-vanov et al. (1996), these functions are power laws �j ˙�Bz · �B�

z where � = 2(1 + �)=(2 + �). (To simplify ourconsideration, here we omit the derivation of the exact func-tional dependence between the quantities � and �; properdetails are given in the papers of Milovanov et al. (1996)and Zelenyi et al. (1998).)

The result �j˙�Bz · �B�z could be attributed the following

physical meaning. The “scattering” of the current-carryingparticles at the magnetic turbulence “clumps” leads to thedeviation, �j, of the cross-tail current density from its aver-age value, 〈jy〉; this deviation is proportional to the magnetic)eld of the “clumps”, �Bz , i.e., �j=〈jy〉 ˙ �Bz , or, equiv-alently, �j ˙ �Bz · 〈jy〉. On the other hand, the averagecross-tail current density, 〈jy〉, at which the current sheetinstability saturates, is by itself determined by the magnetic


>uctuation amplitudes, �Bz: The larger the current density〈jy〉 is, the higher the magnetic turbulence level in the sheetmust be to stabilize the instability. The equilibrium of thetail is then maintained by the increased level of the magnetic)eld turbulence in the current sheet, i.e., the magnetic >uc-tuations �Bz self-consistently grow with the average currentdensity 〈jy〉. For a fractal (scale-invariant) distribution ofthe magnetic >uctuation “clumps”, the dependence between〈jy〉 and �Bz must be a dimensionless (i.e., power-law) func-tion: 〈jy〉˙ �B�

z .Substituting �j ˙ �Bz · �B�

z ; �= 2(1 + �)=(2 + �), intoEq. (2) and considering the scaling law ∇× ˙ −�=(2+�),one )nds the energy density of the magnetic )eld >uc-tuations versus length scale , i.e., �B2

z ˙ −�=(1+�). Inthe self-consistent regime, this result must coincide withthe scaling of the fractal number density of the turbulence“clumps” on a plane, �( )˙ D−E; E =2 (see Section 1).Consequently, the self-organized topology of the magnetic)eld turbulence obeys the condition

D = E − �1 + �

; E = 2: (4)

In Euclidean geometry limit, �→ 0, so that the frac-tal dimension D tends towards the topological dimen-sionality of the plane, D→E = 2 (i.e., the space-)llingturbulence).

Eq. (4) describes the turbulence properties in the mag-netotail current sheet in the form of a topological “code”,i.e., the algebraic relation between the Hausdor; fractal di-mension of the turbulence, D, and its index of connectivity,�. Such a topological “encoding” of the turbulence formsthe basis of the geometric language advocated in our study.In Section 3, we analyze further “codes” relevant for theself-consistent description of the magnetic )eld and plasmacoupling in the Earth’s stretched tail.

Numerical estimates show that the multiscale particle“scattering” at the self-consistently generated magneticturbulence “clumps” might be e;ective mostly for ions,being the heavier plasma species. Indeed, ions, havinglarger (quasi)Larmor radius rL, can more easily migratethrough the “clumpy” )eld �Bz , whereas most of electronsare trapped in magnetic “deadlocks” for relatively longtime. Assuming that the ion population in the tail can bedescribed by some typical “thermal” energy (W ∼ 1 keV,say), one arrives at the characteristic “collisional” length, a,where the bulk of the ion distribution is “scattered” (Milo-vanov et al., 1996): a∼√

LrL�Bz=Bx ∼√

vmcL=eBx. (Here,v∼√

W=m is the ion thermal velocity, and m is the protonmass.) The important role played by the ion componentof the plasma in the principal conductivity mechanismsoperating in the tail has been substantiated by Ohtani et al.(1998) (see Section 3 of their article).

The quantity a can be interpreted as the “smoothness”length of the turbulence, i.e., the minimal size of the mag-netic >uctuation “clumps”; larger-scale “clumps” might thenbe treated as “clusters” of the “clumps” of size a.

In a self-consistent model, the values of rL and a musthave the same order of magnitude, i.e., rL ∼ a. Indeed, themagnetic turbulence “clumps” are generated by the sameparticles (ions) migrating through the turbulence, so that thebasic “scattering” cross-section, ∼ a, should be compara-ble with the characteristic particle Larmor radius determinedby the energy W . Hence, the ions are unmagnetized in thestretched tail, i.e., no “frozen-in” conditions might be ap-plied to them. This theoretical conclusion agrees with theobservational results of Ohtani et al. (1998) who indicatethat “the spatial scale of the magnetic >uctuations is of theorder of the proton gyroradius or could be even shorter” (seep. 4680 of their report).

The condition rL ∼ a might be the starting point for aself-consistent estimate of the magnetic >uctuation levelin the current sheet. The evaluation made by Milovanovet al. (2000) indicates that the >uctuating )eld �Bz could beseveral 10% of the magnetic )eld in the lobes of the tail,Bx; meanwhile, the self-consistent value of the length scalea was around a ∼ 4× 107 cm.

It is relevant to remark that the fractal geometry approx-imation of the magnetic )eld and plasma turbulence in thecurrent sheet has the upper-scale limitation. This limitationis due to the global “modulation” of the turbulence systemin the tail, coming from the tearing mode scales. In fact,the self-organized turbulence structures in the vicinity ofthe current sheet could be in>uenced by the excited tearingmodes having the energy reservoir in the basic )eld-reversedcon)guration of the magnetotail (Coppi et al., 1966; Galeevand Zelenyi, 1975). One then concludes that the relevantscales should be bounded by the minimum unstable tear-ing wavelength in the current sheet, which we shall denoteby !∗. The parameter !∗ is related to the characteristic cur-rent sheet thickness, L, through the widely known estimate!∗ ∼ 2�L (Coppi et al., 1966). This relation immediatelyfollows from the inequality kL 6 1 which de)nes the un-stable tearing wavenumbers, k = 2�=!, for a current sheetof thickness L. Hence, by order of magnitude, . !∗ ∼2�L. On the basis of observations, one might invoke an es-timate L∼ 108 cm for the current sheet thickness prior to asubstorm (see, e.g., Lui, 1993), yielding . 6 × 108 cm.Hence, !∗=a∼ 15�1; this substantiates the possibility forthe hierarchical structuring of the turbulence at the scalerange from a to !∗. Below, when applying the fractal con-cepts, we assume everywhere that the length scales varybetween a and !∗. (In this regard, note that the property ofself-similarity for the turbulence structures is implied [seeSection 1].)

3. Self-organized critical states

To obtain the self-consistent estimates of the parametersD and � in the magnetotail current sheet, one must sup-plement Eq. (4) with an additional relation describing theself-organized fractal geometry of the magnetic )eld and


current density >uctuations. We deduce such a relation fromthe following consideration.

First, we note that magnetic )eld and current densitystructures in the current sheet must form percolating pat-terns. “Percolating” means that such patterns spread to arbi-trary long scales; this condition says, in particular, that thecross-tail currents do extend from one side of the tail to an-other, i.e., maintain the global magnetic con)guration of thetail.

Second, we assume that this percolating structure is criti-cal, i.e., the “minimal” possible. The critical regime impliesthat the magnetotail is principally “transparent” for the elec-tric currents, i.e., the conducting links do penetrate through-out the current sheet, but this “transparency” is relativelylow, so that the current density networks are only at thethreshold of percolation. (An introduction to the percolationtheory can be found in the monograph of Stau;er (1985).)

It must be emphasized that the critical percolating struc-tures (i.e., the structures at the threshold of percolation)could only describe the equilibrium steady state of the cur-rent systems corresponding to a free energy minimum ofthe stretched regions of the magnetotail. This might be asort of the “basic” equilibrium of the stretched tail. Such anequilibrium might be realized in the nearer-Earth stretchedtail at the initial stages of the substorm growth phase. Atthe later stages, accumulation of the free energy in the sys-tem coming from the reconnection on the Earth’s day-sidemagnetopause, results in a considerable increase of thecross-tail currents; this generally violates the critical per-colation regime, i.e., the current system tends to overcomethe marginal percolation threshold. As shown in Section 4,this might lead to the current disruption instability oftenassociated with the substorm event.

We now discuss in more detail the self-organized criticalpercolation structure of the magnetic )eld >uctuations andcross-tail electric currents in the magnetotail current sheet:

There exists an explicit algebraic relation between theHausdor; fractal dimension, D, of a percolating fractal ob-ject at criticality and its index of connectivity, �. Such arelation was originally proposed by Alexander and Orbach(1982) and recognized as an important manifestation of theuniversal behavior of the fractal geometry of percolation atthe critical threshold.

In order to describe dynamical properties of thepercolating fractal networks, Alexander and Orbach (1982)introduced a speci)c combination, d̃ ≡ 2D=(2 + �), of theparameters D and �, which is commonly referred to as spec-tral (or fracton) dimension. From a wealth of numericalstudies they noted that the value of d̃ was remarkably closeto 4=3 in all Euclidean dimensions E greater than one, eventhough the corresponding values of D and � were by nomeans constant as functions of E. This numerical evidenceled them to speculate that the spectral dimension d̃ might beexactly 4=3 for the percolating fractal sets in all embeddingdimensions E¿ 2, i.e., d̃ ≡ 2D=(2+�)=4=3; E ¿ 2. Thisassertion has come to be known as the Alexander–Orbach

(AO) conjecture. (From the AO conjecture one concludes,for instance, that larger values of the Hausdor; fractaldimension D assigned to the fractal objects at criticality,imply larger values of the index of connectivity �.)

General proof for the AO conjecture was found inrelatively high embedding dimensions E¿ 6 within themean-<eld theory (see, e.g., reviews of Havlin andBen-Avraham, 1987; Nakayama et al., 1994). In the lowerembedding dimensions 26E6 5, the mean-)eld percola-tion cannot be directly applied, and the validity of the AOconjecture for these E was an open question until recently.In an attempt to prove the AO conjecture in 26E6 5,Milovanov (1997) proposed an unconventional analyticalapproach based on the concept of the fractal manifold. Thisapproach led him to show that the AO conjecture should beimproved for 26 E 6 5 and the original AO estimate 4=3should be replaced for the smaller value, C ≈ 1:327¡ 4=3.The quantity C is the universal topological constant whichis remarkably close to 4=3 and characterizes the fractal man-ifolds at criticality in all embedding dimensions 26E6 5(Milovanov, 1997). Following the recent study of Milo-vanov and Zimbardo (2000), we refer to the parameter Cas the “percolation constant”. Although the actual di;er-ence between C ≈ 1:327 and 4=3 is slight, the conditionC ¡ 4=3 plays a fundamental role in the percolation theory(Havlin and Ben-Avraham, 1987).

We exploit below the improved form of the AO con-jecture using the percolation constant C (Milovanov, 1997;Milovanov and Zimbardo, 2000):

d̃ ≡ 2D=(2 + �) = C ≈ 1:327: (5)

[This form includes, in particular, the cases of the two-(E =2) and three- (E =3) dimensional percolation.] Whenappropriate, we also present (rough) numerical estimatesof the basic parameters involved, that would correspond toC ∼ 4=3.

Eq. (5) provides the universal geometric “code” for thecritical percolation regime in all dimensions 2 6 E 6 5.As pointed out by Zelenyi et al. (1998), relation (5) canbe also regarded as the structural stability condition forthe magnetic )eld and current density fractal patterns in themagnetotail current sheet.

Combining Eqs. (4) and (5), one )nds the self-consistentvalues of the Hausdor; fractal dimension D and index ofconnectivity � describing the “basic” equilibrium geometryof the magnetic )eld >uctuations in the current sheet. Thesevalues are

D = (2 + C)=2 ≈ 1:66 ∼ 5=3¡ 2; (6)

and

� = (2− C)=C ≈ 0:51 ∼ 1=2¿ 0: (7)

A similar approach based on the improved AO relation,can be applied to a direct description of the current density>uctuations in the magnetotail current sheet. Indeed, let D+

and �+ denote the Hausdor; fractal dimension and index


of connectivity of the current density networks in the cur-rent sheet; these quantities should be generally distinguishedfrom the parameters D and � relevant for the magnetic )eldfractal patterns. The threshold condition in terms of D+ and�+ reads as follows:

2D+=(2 + �+) = C ≈ 1:327 (8)

and is analogous to Eq. (5).As usual, larger values of the Hausdor; fractal dimension

D+; 16D+62, correspond to a denser distribution of theconducting links across the current sheet plane; the space-)lling case is recovered by the limiting value of D+ = 2.Milovanov et al. (2000) pointed out that the index �+

could be interpreted as the Hausdor; fractal dimension of theset of the branching points of a percolating fractal networkon a plane. The limiting case �+ → 0 describes degenerated(“released”) networks without branching points; an exampleis a set of nonintersecting plane fractal curves. In view ofEq. (8), the threshold conditions imply that larger values of�+ correspond to larger values ofD+, i.e., the more branchedcurrent networks at criticality provide a denser covering ofthe plane with the conducting elements.

The branching points of the electric current networkmight be compared to “knots” of the conducting “fabric”,retaining the current )laments to each other as a whole (con-nected) system. The index of connectivity �+ then showshow “knotty” the current network is. The “knots” are intu-itively associated with the property of “durability” of thenetwork: The more “knots” are present, the more “durable”(“resistant”) the connection between the conducting)laments inside the “fabric” would be. In this context, theparameter �+ could be considered as the degree of thestructural stability of the network, i.e., larger values of �+

generally correspond to more “durable” (more “connected”)current networks on a plane.

In the self-consistent regime, the electric current net-works and magnetic )eld patterns must be topologically“conjugated”, i.e., “dual” to each other; this might serveas a geometric manifestation of the strong nonlinear cou-pling of plasma and magnetic )elds associated with theself-organized critical states of the magnetotail.

The property of duality involves the issue of the micro-scopic connectedness for both the magnetic )eld and cur-rent density structures and can be quanti)ed in the form ofa relation between the indices of connectivity � and �+. Topropose such a relation, one might speculate that the totalconnectivity, � + �+, of the “conjugated” fractal objects isan invariant quantity, i.e., the combination � + �+ is equalto a constant: � + �+ = inv; this might be supported bythe recent results of Milovanov and Zimbardo (2000) whorecognized the invariance of the index of connectivity ofa fractal under allowed topological transformations. Takinginto account that the percolating fractal sets are character-ized by the nonnegative indices of connectivity (Nakayamaet al., 1994) (i.e., the inequalities � ¿ 0 and �+ ¿ 0hold), one might assume the invariant constant on the right

of � + �+ = inv to be “of the order of unity”:

� + �+ ∼ 1: (9)

Duality relations similar to Eq. (9) are found, for instance,when analyzing the mutual dependence between the (“conju-gated”) “site” and “bond” percolation on lattices (Isichenko,1992).

From Eq. (9) one concludes that larger values of the in-dex of connectivity �+ correspond to smaller values of theindex �, and vice versa. Physically, this means that morebranched current networks (containing a larger number ofconducting elements at smaller scales) generate more “re-)ned” magnetic )eld patterns in the current sheet plane; suchpatterns are less “clumpy” (with smaller “chunks”). Hencethe indices � and �+ are in some sense “opposite” (dual) toeach other; this is quanti)ed by relation (9).

Considering Eqs. (7) and (9), we )nd the self-consistentestimate of the dual index of connectivity �+:

�+ = 2(C − 1)=C ≈ 0:49 ∼ 1=2¿ 0: (10)

The dual Hausdor; fractal dimension D+ can be nowobtained from Eqs. (8) and (10), yielding

D+ = 2C − 1 ≈ 1:65 ∼ 5=3¡ 2: (11)

One observes that the dual parameters D+ and �+ are nu-merically very close to the values of D and �, respectively,at the “basic” equilibrium of the magnetotail [cf. Eqs. (10)–(11) and (6)–(7)], i.e., D ≈ D+ and � ≈ �+: Even theexact equality might be recognized for C=4=3. Note, how-ever, that the value of the percolation constant C must de-viate from 4=3 in low embedding dimensions (Havlin andBen-Avraham, 1987); moreover, the algebraic dependenceof D and � upon the percolation constant C di;ers from thecorresponding dependence of D+ and �+.Eqs. (6) and (11) indicate that the magnetic )eld and

plasma turbulence in the magnetotail current sheet cannot bereduced to the more conventional, space-)lling, limit. On thecontrary, this turbulence implies a topological “conjugation”of two fractal objects, each having a Hausdor; dimensionsmaller than E=2. The “conjugation” condition is given byEq. (9) which might be interpreted as the geometric “code”for the multiscale magnetic )eld and plasma coupling inthe tail.

It is remarkable to note that the dual parameters D+ and�+ could be derived exactly in the form (10)–(11) froman abstract topological analysis of the two-dimensionalsign-symmetric random )elds at criticality (Milovanovand Zimbardo, 2000). This argument shows that theself-organized magnetic )eld and current density distribu-tions in the magnetotail current sheet have a deep topolog-ical origin related to the fundamental geometric propertiesof the critical percolating sets; these properties are con-tained in the universal character of the fractal geometry ofpercolation at the threshold and control self-consistently themagnetic )eld and plasma coupling in the magnetotail.


Fig. 2. A three-dimensional current )lament emerging fromthe current sheet plane. The “knots” attaching the )lament to thesheet, provide the “restoring” forces tending to “bring” the )lamentback. This is opposed to the interaction of the magnetic momen-tum of the )lament, ', with the magnetic )eld in the lobes of thetail, Bx .

4. Magnetospheric substorm onset

Eqs. (6)–(7) and (10)–(11) describe the “basic” topol-ogy of the self-organized magnetic )eld and current densitystructures in the current sheet. This topology implies that thecurrent system is at the threshold of percolation. The thresh-old is realized as a (quasi-two-dimensional) fractal currentnetwork which lies in the close (of the order of ion Larmorradius) vicinity of the current sheet. Such a network is as-sumed to provide the free energy minimum for the stretchedmagnetotail.

An important related issue is the structural evolution ofthe current system driven by a free energy accumulation inthe stretched regions of the tail. This should be relevant forthe later stages of the substorm growth phase (as well asfor the very beginning of the onset itself) when magnetic)eld reconnection at the day-side magnetopause leads to aconsiderable increase of the cross-tail current density.

As already mentioned above, the increase of the cross-tailelectric currents results in a violation of the critical perco-lation regime, i.e., the entire current system overcomes thethreshold of percolation. Generally speaking, this means thatsome conducting elements cannot “)nd enough room” in thevicinity of the current sheet and start to rise over the currentsheet plane (see Fig. 2).

Indeed, an increase of the cross-tail current density causesa formation of larger magnetic >uctuation “clumps” (i.e.,the distribution of the magnetic )eld >uctuations tends to amore “chunky” shape). Consequently, the current sheet en-vironment becomes less “transparent” for the plasma parti-cles migrating across the tail, and the particle “jumps” overthe magnetic “chunks” appear to be energetically prefer-able. This leads to an occurrence of the conducting elementsabove and below the current sheet, so that the turbulencesystem starts to swell in the third dimensionality. This e;ectis modeled numerically in Section 6.

4.1. Geometric conditions for the substorm onset

Let us )rst discuss the possible physical meaning of thebasic geometric parameters, D; D+; �, and �+, at the laterstages of the substorm growth phase, when the entire currentsystem is already above the percolation threshold. It is theo-retically important to note that the fractal geometry of perco-lation is the remarkable property of the critical percolationregime, whereas structures above and below the percolationthreshold may not be self-similar fractals. In view of this, onecannot directly apply the above fractal quantities,D; D+; �,and �+, to a description of the three-dimensional magnetic)eld and plasma coupling in the magnetotail. The simplestsolution might be to assume that only the electric currentsthat remain in the current sheet plane are at the (critical)percolation threshold; this yields a clear image of the currentsystem above the threshold as being composed of two con-stituents, the subsystem which lies in the current sheet planeand is at the threshold of percolation, and the additionalcurrent )laments emerging from the sheet into the ambientspace (see Fig. 2). The entire (three-dimensional) currentsystem obviously overcomes the threshold of percolation.One then makes use of the fractal parameters D; D+; �, and�+ when considering the “critical” (plane) constituent of theentire three-dimensional magnetic )eld and current densitycon)guration; the criticality conditions (5) and (8) can beapplied to this constituent in the usual way (see Section 3).

The numerical values of the quantities D; D+; �, and �+

would be, however, di;erent from the previously obtainedresults (6)–(7) and (10)–(11). In fact, the formation of themore “coarse” (“chunky”) magnetic >uctuations patterns inthe current sheet (resulting from the increase of the cross-tailcurrent density) implies an increase of the Hausdor; frac-tal dimension D: The magnetic )eld >uctuations �Bz tend todensely cover the current sheet plane; the fractal dimensionfor such a covering should then start to approach the Eu-clidean dimensionality of the plane, E=2. From the critical-ity condition (5) one then concludes that the index of con-nectivity � also starts to increase. On the contrary, the du-ality relation (9) indicates that the index of connectivity �+

decreases at the later stages of the substorm growth phase,i.e., the electric current network becomes less “knotty” (i.e.,less rami)ed, with less branching points) (see Section 3).Eq. (8) shows, in the meantime, that the Hausdor; fractal di-mension of the electric current network lying in the currentsheet plane, also decreases. This implies gradual “coarsen-ing” of the main current )laments composing the networkas time progresses.

As already mentioned above, the topological (structural)stability of the electric current network in the stretched tailis provided by the “knots” corresponding to the branch-ing points of the electric current web; note that the Haus-dor; measure of the “knots” coincides with the index ofconnectivity �+: The smaller the value of �+ is, the morestructurally unstable the fractal network would be. One thenconcludes that the decrease of the parameter �+ at the later


Fig. 3. An illustration for the topologically degenerated (“released”)current density network composed of disconnected )laments. Sucha network has no “knots” and is extremely unstable: Small struc-tural >uctuations (three-dimensional loops) that might occasion-ally appear in the current system, grow exponentially fast in thedirection normal to the current sheet plane (i.e., the “in>ation”).This immediately leads to the initiation of the global current dis-ruption event (i.e., the onset). The “released” network is consider-ably “coarse”; the magnetic )eld intensity “chunks” are shown asirregular gray domains between the basic current )laments.

stages of the substorm growth phase could initiate the struc-tural instability of the current system; this provides favor-able conditions for the current disruption in the magnetotailsignifying the magnetospheric substorm onset.

A simple topological condition for the substorm onsetcould be now proposed. In fact, let us identify the onsetwith the limiting (the smallest possible) value of the indexof connectivity �+, which is easily seen to be zero, i.e.,

�+ → 0: (12)

This corresponds to a degenerated (“released”) currentdensity network in the current sheet plane; the “degreeof branching” of such a network vanishes: The cross-tailcurrents fall into a set of disconnected )laments charac-terized by relatively high density amplitudes (see Fig. 3).This set (as a whole) is described by the smallest possiblevalue of the Hausdor; dimension D+ allowed for a fractalpercolating network on a plane:

D+ → C ≈ 1:327 (13)

[see Eq. (8)]; this value coincides with the percolationconstant C. An observational evidence for the )lamentarycurrent structures at the substorm onset has been recentlypresented by Ohtani et al. (1998).

The fractal distribution of the magnetic >uctuation“chunks” can be characterized by the index of connectivity

� → ∼1 (14)

as it follows from Eqs. (9) and (12). The correspondingvalue of the Hausdor; fractal dimension D becomes, ac-cording to the AO relation (5),

D → 3C=2 ∼ 2: (15)

Eqs. (14)–(15) show that the fractal dimension D ap-proaches the topological dimensionality of the plane, E=2,whereas the corresponding value of the index of connectiv-ity � does not vanish (as one might expect for an Euclideangeometry) but tends to unity. This means that the “true”number of the embedding dimensions does exceed E = 2prior to the onset (see the schematic illustration in Fig. 2),in agreement with the qualitative conclusion made in the be-ginning of this section. The e;ective dimensionality of thecurrent system at the onset can be estimated from Eq. (4)

Ee; = D +�

1 + �→ 3C + 1

2∼ 5

2; (16)

i.e., the embedding Euclidean space is three-dimensional,E = 3.

It is worth noting that the embedding dimensionality E isan integer (i.e., discrete) quantity which cannot vary con-tinuously from E = 2 (characterizing the “basic” equilib-rium of the magnetotail current system) to E=3 (associatedwith the magnetospheric substorm onset). In other words,the increase of the embedding dimensionality from E = 2to 3 must assume an abrupt change in the topology of thecurrent system; such a change would have features resem-bling a second-order phase transition in the magnetotail.We substantiate this conclusion in what follows.

4.2. Topological phase transition

Consider a structural >uctuation of the “basic” (criti-cal) current network developed in the current sheet plane.This >uctuation is a local violation of the plane geometryof the “basic” current system and can be considered as athree-dimensional current )lament emerging from the sheetinto the ambient space (see Fig. 2). (Such three-dimensional)laments have been associated with the magnetotail currentsystem above the percolation threshold.)

Let Z be the size of the )lament in the direction normalto the current sheet. The free energy of this )lament couldbe represented as the sum of the two competing terms.

One of these terms should describe the “restoring” forces(in accordance with the Le Chatelier principle) acting onthe )lament from the remaining part of the network. The“restoring” forces tend to bring the )lament back to the cur-rent sheet plane and appear as the “elasticity” of the net-work; the “elastic” properties are due to the “knots” retain-ing the network constituent elements as a whole system (seeSection 3). The free energy contribution associated with the“elastic” (“restoring”) forces might be given by the standardquadratic dependence ˙ Z2, i.e., Felast = KZ2 where K isa constant. It is intuitively clear that the quantity K shouldbe sensible to the topological properties of the network, i.e.,highly “knotted” percolating networks like that shown inFig. 1 might be associated with relatively large values ofK , whereas the degenerated (“released”) structures without“knots” (see Fig. 3) should provide only very small “elas-ticity”. [This could be made more precise by introducing an


explicit relationship between the parameter K and the in-dex of connectivity �+ which directly describes the numberdensity of the “knots” on a percolating fractal network; onemight then obtain K ≡ K(�+) where K(�+) is a power-lawfunction of �+. Derivation of such a relation, however, goesfar beyond the present study.] Hence, transformation of thecurrent density network from a highly branched, consid-erably “knotted” con)guration shown in Fig. 1, towardsthe entirely “released” structure illustrated in Fig. 3 wouldgenerally mean �+ → 0 [see Eq. (12)] and K(�+) → 0.Consequently, the “elasticity” of the current network, K ,gradually decreases during the substorm growth phase.

The other term takes into account the interaction of thecurrent )lament with the external magnetic )eld in the lobesof the tail. This interaction might be roughly estimated asthe interaction of the magnetic momentum of the )lamentemerging from the current sheet plane, with the lobe compo-nent of the magnetotail )eld (see Fig. 2). (Here, we neglectthe higher-order interaction terms, so that the dipole approx-imation is applied.) Simple algebra shows that an orienta-tion of the magnetic momentum of the )lament towards thelobe )eld leads to the energy gain of −Fdipole = SZ tanh Z ,where S is a constant and does not depend on the parameter�+, and tanh Z corresponds to the standard Harris distribu-tion of the lobe )eld.

The total free energy of the structural >uctuationbecomes

F= KZ2 − SZ tanh Z: (17)

For small Z , this yields

F ≈ (K − S)Z2 + (S=3)Z4: (18)

Expansion (18) is the widely known representation of thefree energy functional near the point of the second-orderphase transition describing spontaneous topologicalchanges in the system (see, e.g., Landau and Lifshitz, 1970).For K ¿S, the free energy F is a monotonically increas-ing function of Z and attains its minimal (zero) value forZ = 0. This corresponds to the topologically stable currentdensity network on a plane. For K ¡S, the free energyminimum implies a substantially nonzero equilibrium sizeof the )lament in the z-direction, Z �= 0, so that the cur-rent system evolves spontaneously from the basically planecon)guration to a three-dimensional geometry. The valueof K ∼ S is a sort of thermodynamical “threshold” belowwhich the “elasticity” of the current network cannot preventthe spontaneous emerging of the current )laments and theensuing “birth” of the third dimension of the turbulence.

Integrating equations of motion for the potential functionof the form (18), one immediately obtains that the expan-sion of the )laments into the third dimensionality, z, goesexponentially fast for K ¡S at the early expansion stages.Such a behavior might be compared to the evolution of thehot early Universe during the so-called “in>ationary” stage,which is also the exponential expansion (Linde, 1990). Thissuggests an application of the term “in>ation” to a de-

scription of the early evolution of the magnetotail currentsystem in three dimensions just below the transition pointK ∼ S.

The exponentially fast expansion of the current )lamentsduring the “in>ationary” stage likely enables them to reachthe boundaries of the magnetotail current layer. This makesfavorable the closure of the )laments via the ionosphere (seeFig. 2) and the initiation of the large-scale current disruptioncustomarily associated with the magnetospheric substormonset.

4.3. “Topological scenario” for substorm onset

A “topological scenario” for the substorm onset can benow proposed. Indeed, the “basic” equilibrium of the mag-netotail is realized as a self-organized critical state associ-ated with the percolating current density networks that areat the threshold of percolation. These networks lie in theclose (of the order of ion Larmor radius) vicinity of thecurrent sheet and could be treated as quasi-two-dimensionalfractal patterns. Topological stability of such patterns is pro-vided by the “knots” recognized as the branching points ofthe network. The “knots” retain the multiscale conductingelements as a whole percolating system (see Fig. 1). Thefractal geometry of the percolating current system at thethreshold is completely described by the Hausdor; fractaldimension D+ = 2C − 1 ≈ 1:65 and the index of connec-tivity �+ = 2(C − 1)=C ≈ 0:49; the self-consistently gen-erated magnetic )eld >uctuations �Bz )ll fractal sets whoseHausdor; fractal dimension and index of connectivity are,respectively, D = (2 + C)=2 ≈ 1:66 and � = (2− C)=C ≈0:51.

Accumulation of the free energy in the tail related to thereconnection processes on the day-side magnetopause leadsto an increase of the cross-tail current density. The magne-totail current system tends to overcome the self-organizedpercolation threshold, and some current )laments start to riseover the current sheet plane (see Fig. 2). (These )lamentscan be considered as structural ?uctuations in the “basic”[quasi-two-dimensional] magnetotail current system.) Phys-ically, this means that the current-carrying particles some-times )nd free “corridors” over the turbulence structureslying in the current sheet. The topology of the current sys-tem thus starts to evolve towards a three-dimensional con-)guration. One might say that the magnetic )eld >uctuationsself-consistently generated by the increased current density,tend to “push” the conducting )laments out of the currentsheet plane. The self-organized coupling of currents andmagnetic )elds in the tail generally becomes less “re)ned”,i.e., the current )laments composing the network tend tobecome more “coarse”, whereas the magnetic )eld >uctu-ations appear to have larger “chunks”. The current systemis already less branched, i.e., less “knots” connecting thecurrent elements to each other are found. Although the re-maining “knots” do provide the structural stability of thecurrent network, its “elasticity” (which is responsible for the


“restoring” forces in the system) decreases. These processescould be seen in gradual changes of the main topological pa-rameters,D,D+, �, and �+; indeed, the Hausdor; dimensionD and index of connectivity � exceed the “basic” values(6)–(7), whereas the dual quantities D+ and �+ becomesmaller than values (10)–(11).

Further increase of the cross-tail current density results ina pronounced “coarse-graining” of the system and a criticalloss of its “elasticity” properties (see Fig. 3). The current)laments emerging from the current sheet plane cannot bereadily “brought” back by the “restoring” forces, and expandexponentially fast into the ambient space. This signi)es the“in>ationary” stage of the evolution of the current systemand spontaneous “birth” of the three-dimensional electriccurrent con)gurations. The three-dimensional current sys-tem thus evolves from the quasi-two-dimensional percolat-ing network as a result of the “in>ationary” expansion ofthe structural ?uctuations that appear in the system abovethe marginal percolation threshold and grow exponentiallyfast as soon as the system becomes suNciently “coarse”.The energetically favorable closure of the large-scale cur-rent )laments via the ionosphere during the “in>ationary”stage manifests the initiation of the global current disruptionevent customarily identi)ed with the magnetospheric sub-storm onset (see Fig. 2). The onset might be formally asso-ciated with the limiting (smallest) values of the parametersD+ →C ≈ 1:327 and �+ → 0 describing the topologicallydegenerated current density networks; the geometry of themagnetic )eld >uctuations is characterized, in the meantime,by the quantities D → 3C=2 ≈ E = 2 and � → 1, whosephysical realization requires more than two embedding di-mensions [see Eq. (16)] and causes the “in>ation”, the ex-ponential expansion of the system into the third embeddingdimensionality.

Our conclusion is that the magnetospheric substorm canbe treated as a structural catastrophe related to the gradualtopological “simpli)cation” of the percolating fractal net-work existing in the magnetotail current sheet. This catas-trophe has a deep topological origin and is manifested as thesecond-order phase transition in the current system abovethe marginal percolation threshold.

Dynamical mechanisms driving the magnetotail currentsheet disruption and diversion during substorms are analyzedin detail by Lui (1996) and Lui et al. (1988). The occurrenceof the second-order phase transition during which a cur-rent reorganization phenomenon takes place in the stretchedmagnetotail, could be supported by the arguments proposedby Consolini and Lui (1999).

The formation of the “chunky” structures implied by the“topological scenario” (cf. Figs. 1 and 3 which illustrate theevolution of the electric current network in the sheet beforethe onset) could be identi)ed with the energy transfer goingfrom smaller to larger scales. This would mean that theturbulence spectra become more steep during the substormgrowth phase. We calculate the spectral characteristics ofthe magnetotail turbulence in the next section.

5. Calculation of the spectral exponents

We begin with the spectra of the magnetic )eld >uctua-tions. First, the convection of the magnetic )eld structuresin the magnetotail must be taken into account. Such a con-vection originates from the plasma in- and out>ows close tothe “X -line”; the characteristic convection velocities, u, are∼102 km s−1 (see, e.g., Lui, 1993).It is easy to verify that a stationary convection of fractal

magnetic )eld patterns should be recognized by the restframe observer as a temporal variability of the )eld. Thistemporal variability is represented by the self-aNne fractalgraphs whose spectral characteristics are determined by thefractal distribution of the )eld in space.

It has been clearly established that the Fourier energydensity spectrum of a self-aNne fractal graph behaves as apower-law function of the frequency f, i.e., P(f) ˙ f−�,where � is a constant depending on the fractal properties ofthe graph (see, e.g., review of Isichenko, 1992). A simplerelation between the exponent � and the Hausdor; fractaldimension D describing the spatial distribution of the mag-netic )eld >uctuations was found by Milovanov and Zelenyi(1994a,b) to be

� = 2D − 1: (19)

Hence, larger values of the spatial fractal dimension D cor-respond to a steeper >uctuation spectrum P(f). Note thatthe slope � depends solely on the fractal dimension D andis insensitive to the more “delicate” quantity, the index ofconnectivity �.

The frequency f is related to the wavenumber k ∼ −1

characterizing the fractal distribution of the magnetic )eldin space, via f = ku. This leads to an alternative represen-tation of the Fourier energy density spectrum P(f)˙f−�

in terms of k as P(k)˙ k−�.The physical relevance of a relation between the char-

acteristic time and spatial scales of the >uctuations in thecurrent sheet environment was substantiated by Ohtani etal. (1998) from the AMPTE=CCE-SCATCHA simultane-ous observations. In addition, Ohtani et al. (1998) explicitlynote that the in situ observed signatures in the >uctuationcomponents “may be explained in terms of the earthwardplasma convection which conveys the associated perturba-tion currents and therefore the magnetic >uctuations withplasma” (p. 4680). This advocates the “mapping” k →fimplied by Eq. (19). Let us also stress that the results ofOhtani et al. (1998) could be interpreted as the direct sub-stantiation for the actual existence of the irregular spatialstructures that determine the observed temporal variabilityof the turbulence; this supports the relevance of the “geo-metric” methods proposed in our study for the descriptionof the main turbulence properties.

We found in Section 3 that the “basic” equilibrium of themagnetotail is associated with the fractal magnetic )eld pat-terns whose Hausdor; dimension D is equal to (2+C)=2 ≈1:66 [see Eq. (6)]. This dimension should be applied in the


range of scales a . . !∗ (i.e., 1=!∗ . k . 1=a). Sta-tionary convection of these patterns by the rest frame ob-server results in the power-law spectrum P(k)˙ k−� where

� = C + 1 ≈ 2:33 ∼ 7=3; (20)

according to Eq. (19). In the frequency domain, the spectrumP(f) ˙ f−� should be recognized in the range betweenthe two characteristic turnover frequencies, f∗ ∼ u=!∗ andf∗∗ ∼ u=a. Numerically, f∗ ∼ 10−2 Hz and f∗∗ ∼ 10−1

Hz, where the estimates of Section 2 have been used. Thespectrum P(f) may reveal kinks around the frequencies f∗and f∗∗ when considered in a wider frequency range; thekink around the turnover frequency f ∼ f∗ Hz was reportedby Ohtani et al. (1995) and Bauer et al. (1995) from theAMPTE data (see Section 1).

An increase of the Hausdor; fractal dimension D priorto the magnetospheric substorm onset might be recognizedas an increase of the spectral exponent � in the frequencyrange from f∗ to f∗∗ [see Eq. (19)]. (This is a manifestationof the energy transfer towards larger spatial scales at thesubstorm stages preceding the onset.) The limiting (onset)value of the fractal dimension D ≈ 3C=2 ≈ 2 [see Eq. (15)]corresponds to the maximum slope

� = 3C − 1 ∼ 3: (21)

We now turn to a brief consideration of the electric current>uctuation spectra in the frequency interval between f∗ andf∗∗. This interval corresponds to the fractal current networksself-consistently “pulled” on the magnetic )eld >uctuationpatterns. The topology of the current structures is describedby the Hausdor; fractal dimension D+ which is dual to D.

Similar to the above, the spectrum of the current density>uctuations behaves as a power law, P+(f)˙ f−�+ , withthe dual exponent �+ given by

�+ = 5− 2D+: (22)

Relation (22) is analogous to Eq. (19) yielding the slope ofthe magnetic )eld >uctuation spectrum. In the general theoryof fractals, relation (22) was )rst obtained by Berry (1979)who studied fractal properties of the self-aNne graphs.For more details, see the papers of Isichenko (1992) andMilovanov and Zimbardo (2000).

Exponent (22) might be roughly exploited when consider-ing the velocity >uctuations in the magnetotail current sheet.Indeed, our theoretical model assumes substantial nonadia-baticity of the ion dynamics in the sheet, i.e., no “frozen-in”conditions can be applied to the current-carrying particles.This means that the particle distribution in space does notfollow the magnetic )eld distribution associated with thefractal “clumps”, and is actually much more “di;usive”.This is a direct consequence of the fact that the bulk of theparticle population is associated with the characteristic Lar-mor radius comparable with the typical size of the smallest“clumps” (see Section 2). Hence, the particle number den-sity >uctuations probably do not contribute signi)cantly to

the cross-tail current inhomogeneities, so that the velocityspectra roughly approximate the current density >uctuations.

Combining expressions (19) and (22), one )nds the fol-lowing duality relation for the spectral exponents � and �+:

� − �+ = 5C − 6 ≈ 0:63 ∼ 2=3; (23)

where Eqs. (5), (8) and (9) have been used. Eq. (23) showsthat the slope of the magnetic )eld >uctuation spectrum ex-ceeds the one for the current density (velocity) >uctuationsby ≈ 0:63. The fact that � �= �+ is contrary to what onemight expect from conventional MHD models, and empha-sizes the importance of the plasma particle (quasi-)Larmormotion in the magnetotail current sheet.

According to the results of Section 3, the “basic” equi-librium of the magnetotail corresponds to the Hausdor; di-mension D+ = 2C − 1 ≈ 1:65 [see Eq. (11)]. ConsideringEq. (22), one obtains the relevant slope for the current den-sity >uctuation spectrum:

�+ = 7− 4C ≈ 1:69 ∼ 5=3: (24)

It is remarkable to note that slope (24) has been recognizedby Borovsky et al. (1997) from the ISEE 2 Fast PlasmaExperiment. Moreover, Borovsky et al. (1997) found thatthe corresponding exponent of the magnetic )eld >uctuationspectrum, �, is unambiguously larger than �+ and agreeswith value (20). These observational )ndings support theduality relation (23).

We found in Section 4 that the Hausdor; dimensionD+ decreases prior to the substorm onset. This leads to anincrease of the slope �+, i.e., the spectrum of the currentdensity >uctuations, P+(f), should become steeper as theonset approaches. This behavior of P+(f) is similar to thatof P(f) and signi)es the energy transfer from smaller tolarger scales in the magnetotail current sheet.

The limiting (onset) value of the slope �+ can be obtainedfrom Eqs. (13) and (22):

�+ = 5− 2C ≈ 2:35 ∼ 7=3; (25)

which is numerically close to result (20). SubstitutingEq. (25) into the duality relation (23), one immediatelyrecovers relation (21).

It is worth emphasizing that the noticeable increase in thepower exponents � and �+ can occur just prior to or evenduring the “in>ationary” stage of the evolution of the cur-rent system. This means that the corresponding changes inthe slopes of the spectra happen within a relatively shorttime period (of the order of, say, few minutes) and shouldbe associated with fast dynamical processes preceding thelarge-scale current disruption event. To pinpoint these fastprocesses from the actually detected signals coming fromthe magnetotail system at the “in>ationary” stage mightbe a problem of considerable complexity: The diNcultiesmay arise when analyzing the short-living, nonstationarysignals whose frequency content is varying with time; forinstance, the standard signal-processing techniques basedon the Fourier transforms might become invalidated. In


this context, the study of Lui and Najmi (1997) should berecognized, who proposed the continuous wavelet trans-forms (CWT) as the relevant instrument in dealing with theshort-duration, time-varying signals. Having performed theCWT analysis of the signals observed in the current disrup-tion events, Lui and Najmi (1997) reported the existence ofthe energy cascades from high to low frequencies (i.e., fromsmaller to larger spatial scales) as time progresses (see Plate2 of their paper). This supports the indication of our modelregarding the formation of “chunky” structures (cf. Figs. 1and 3) and the ensuing increase of the spectral exponents �and �+ before the onset.

6. Numerical simulation

In this section, we model the basic properties of the mag-netic )eld turbulence in the current sheet environment atthe late substorm growth phase. Our main goal is to verifythe principal implication of the “topological scenario” pro-posed in Section 4: The occurrence and progressive growthof the structural >uctuations in the magnetotail current sys-tem when the regular magnetic )eld normal to the currentsheet plane ceases to play a stabilizing role in the dynamicsof the plasma and )elds (i.e., the situation typical for theconsiderably stretched tail).

6.1. Magnetic <eld model

Geometric characteristics of the magnetic )eld >uctua-tions �B in the vicinity of the current sheet could be modelednumerically by means of the fractional Brownian functions(fBf). The standard numerical realization of the fBf in threedimensions reads as follows:

�B(r) =∑k;

�B(k)e(k) exp[i(k · r + -k)]: (26)

Here, �B(k) is the amplitude of the mode with wave vectork and polarization (=1; 2), e(k) is the polarization unitvector, -

k are random phases simulating the irregular natureof the magnetic )eld �Bz , and r is the radius vector withthe components (x; y; z). The condition ∇ · �B= 0 impliesk · e(k) = 0.

The amplitude �B(k) is given by the spectrum

�B(k) =A(1=R)(2�=L)3=2

(k2x l2x + k2yl2y + k2z l2z + 1)�=4+1=2: (27)

whereA is the normalizing constant, � is the spectral expo-nent, and lx, ly, and lz are the numerical correlation lengthsin the x-, y-, and z-direction, respectively. To model theproperties of the magnetic )eld turbulence in the stretchedtail in a more realistic way, we assume lz�lx; ly. The quan-tity L in Eq. (27) is the unit length in the z-direction, andRL is the unit length in the x- and y-direction. In otherwords, the periodicity of the simulation box is RL, ratherthan L, in the (xy)-plane; this choice better corresponds tothe actual conditions in the magnetotail.

The components of the wave vectors are, kx =2�nx=RL,ky =2�ny=RL, kz =2�nz=L, where nx, ny, and nz are inte-ger numbers identifying the wave modes. Spectrum (27) istruncated when k2x l

2x+k2yl

2y+k2z l

2z 6 (2�)2 and k2x l

2x+k2yl

2y+

k2z l2z ¿ (2�N )2, where N�1 emulates the ratio !∗=a corre-

sponding to the turbulence structuring interval. The spectralexponent � in expansion (27) reveals the power-law behav-ior of the turbulence energy density spectrum, P(k)˙ k−�,implied by the multiscale geometry of the magnetic )eld�B; below, we assume �∼7=3, in accordance with Eq. (20).

Topological properties of fBfs are discussed in detailby, e.g., Isichenko (1992) and Milovanov and Zimbardo(2000). The fBfs have been applied to numerical simulationof anomalous plasma transport across stochastic magnetic)elds by Zimbardo and Veltri (1995) and Zimbardo et al.(1995). E;ects of magnetic turbulence on plasma dynam-ics in the Earth’s distant tail have been studied by Veltriet al. (1998) who used the fBf approximation in a simpli-)ed model of the current sheet (without z-component of themagnetic )eld).

Perturbation (26) is imposed on the background )eld

B(r) = B0ex tanh[z=V] + Bnez (28)

which is the combination of the Harris )eld-reversed dis-tribution ∼ tanh[z=V] (Harris, 1962) and the regular )eldBn normal to the current sheet plane. [The quantity V inEq. (28) is the current sheet half thickness.] We emphasizethat Eq. (28) is an adopted reasonable approximation of thebackground magnetotail pro)le describing all the essentialgeometric features of the magnetic )eld reversal. We donot imply that the Harris equilibrium actually holds in oursystem.

6.2. Ion dynamics and the moments of the distributionfunction

Our further interest lies in the simulation of the ion nona-diabatic dynamics in )eld (26)–(28), according to the ba-sic aspects of our theoretical treatment (see Section 2). Asa )rst approximation, we ignore below the )ne-scale elec-tron e;ects and consider the electron constituent of plasmaas only the charge neutralizing background.

For the sake of simplicity, we also assume that the elec-tric )eld of the tearing perturbations (26) is negligible inmost cases; this is equivalent to the assumption that the ionthermal velocity, v ∼ √

W=m, is large compared with theAlfven velocity, vA, in the )eld �B. (In fact, the ratio v=vA isof the order of

√�; the value of the parameter � is known to

be large in the central parts of the current sheet, i.e., ��1.)Note that in this case the magnetic >uctuations can be con-sidered as static, since the ions move much faster than thewaves propagate. Note, also, that the electric )eld >uctua-tions could have an e;ect only on the particles whose ve-locities are close to the typical phase velocity of the waves,i.e., v vA. Since no resonance condition is required fora signi)cant acceleration in the static electric )eld E0, just


this )eld (and not the >uctuating )eld �E) plays, on the av-erage, a dominant role in the acceleration of the bulk of theparticles, even when E0 is much smaller than �E.In our simulation, we )x the cross-tail electric )eld

E0 ∼ 0:2 mV=m; and B0∼√8�nW ∼ 20 nT; yielding the

characteristic drift velocity vE∼cE0=B0∼10 km=s.The ion injection scheme corresponds to the in>ow in the

Alfven sheet, i.e., the particles principally entering from themagnetotail lobes (Alfven, 1968).

In the close vicinity of the neutral sheet, the particles startto bounce between the magnetic walls while being acceler-ated in the y-direction by the )eld E0. (The typical excursionin z is equal to W=

√�L0V, where �L0 is the ion “thermal”

Larmor radius in the )eld B0.)In the numerical simulation, we trace the particle tra-

jectories and record their positions ri(t) and velocitiesvi(t) at the time moments t when they cross the planesy=V = −10;−9:95;−9:90; : : : ; 9:95; 10 (i.e., when the par-ticles pass from one cell of size Wy = 0:05V to another inthe y-direction). Following Klimontovich (1967), we intro-duce the formal de)nition of the )ne-grained distributionfunction of an ensemble of M particles:

f(r; v; t) =M∑i=1

�[r− ri(t)]�[v − vi(t)]: (29)

Expansion (29) is the starting point for a calculation ofthe moments of the particle distribution; for our purposes,we construct the plasma number density, cross-tail velocity,current density, and temperature pro)les, assuming di;erentvalues for both the magnetic >uctuation level �B=B0 in thevicinity of the current sheet and the regular component Bn

of the magnetic )eld normal to the current sheet plane.

6.3. Basic results of numerical simulations

The principal e;ects revealed by the simulations areshown in Figs. 4 and 5. In fact, Fig. 4 characterizesthe properties of the current system when magnetic )eldturbulence is absent, i.e., �B=B0 = 0; for each momentof the plasma distribution function (29), three pro)lesare given, corresponding to Bn=B0 = 0:02; 0:05; and 0.1.Fig. 5, on the other hand, implies a nonzero turbulence level�B=B0 = 0:3¿ 0 but the same set of values for the normal)eld Bn=B0 = 0:02; 0:05; 0:1.Considering Figs. 4 and 5, one immediately concludes

that the regular magnetic )eld Bn normal to the current sheetsuppresses the cross-tail current intensity. On the contrary,a noticeable decrease of the normal )eld Bn (i.e., the situa-tion typical before the onset) provides favorable conditionsfor a considerable growth of the cross-tail electric currents.This e;ect is recognized in both the cases �B=B0 = 0 and�B=B0 ¿ 0.

Meanwhile, an increase of the perturbation level �B=B0

leads to the progressive broadening of the current densitypro)les, i.e., the conducting elements tend to be less con-

Fig. 4. The plasma number density, cross-tail velocity, current den-sity, and temperature pro)les for the current system without tur-bulence: �B=B0 = 0. For each moment of the plasma distributionfunction, three di;erent graphs are shown, corresponding to the fol-lowing values of the normal component Bn=B0: Bn=B0 =0:02 (solidline); Bn=B0 = 0:05 (dotted line); and Bn=B0 = 0:1 (dashed line).Note that the current density pro)les are always single-peaked,i.e., the cross-tail currents tend to concentrate in the vicinity ofthe neutral plane. The current system is suppressed as the normalcomponent Bn=B0 increases.

)ned in the close vicinity of the neutral sheet in the pres-ence of the magnetic )eld turbulence. The intriguing e;ectis the clear splitting of the current system for �B=B0 exceed-ing, approximately, �B=B0 ∼ 0:3. In fact, the current den-sity pro)les are single-peaked for �B=B0 . 0:3, and doublepeaked for �B=B0 & 0:3; the double peaked structure of thepro)les becomes more pronounced as the perturbation levelincreases. The splitting occurs for only relatively small val-ues of the normal )eld Bn and is strongly suppressed if Bn

starts to grow: Bn=B0 & 0:1.In the context of our study, the perturbation level �B=B0 ∼

0:3 might be associated with the critical percolation regimein the system. The occurrence of the double peaked pro)lesabove the threshold value∼0:3 implies an appearance of theconducting elements beyond the current sheet plane; theseelements should be identi)ed with the structural >uctuationsin the current system above the marginal percolation thresh-old. The structural >uctuations disappear in the presence ofa suNciently large )eld Bn normal to the current sheet butstrongly grow if Bn vanishes (i.e., the typical situation priorto the onset).


Fig. 5. The plasma number density, cross-tail velocity, currentdensity, and temperature pro)les for the turbulent current sys-tem: �B=B0 = 0:3¿ 0. Like in Fig. 4, three di;erent graphs foreach moment of the plasma distribution function are constructed,corresponding to three di;erent values of the normal )eld Bn:Bn=B0=0:02 (solid line); Bn=B0=0:05 (dotted line); and Bn=B0=0:1(dashed line). The e;ect of the turbulence appears in the consid-erable broadening of the current system (cf. Fig. 4). Moreover,for relatively small values of Bn, a clear splitting of the currentdensity and velocity pro)les occurs. This might be interpreted asan appearance of the structural >uctuations in the system (i.e., theconducting elements above and below the current sheet plane);the existence of such >uctuations is predicted by the “topologicalscenario” for substorm onset. The phenomenon is suppressed asthe normal component Bn increases, revealing the saturation of thecurrent sheet instability in the presence of the normal )eld.

The development of considerable structural >uctuationsin the stretched tail naturally follows from the “topologicalscenario” for substorm onset: The necessary conditions are,the over-critical cross-tail current percolation, and consid-erable decrease of the normal )eld Bn (see Section 4). Thee;ect is thus clearly seen in the numerical realization of themagnetotail current properties (see Figs. 4 and 5).

7. Summary and conclusions

This paper advocates an unconventional, “geometric”approach to studying the self-organization of plasma andmagnetic )elds in the near-Earth stretched tail at the latesubstorm growth phase. Such an approach makes it possi-ble to describe the principal mechanisms operating in thetail in terms of topological “codes”, i.e., algebraic rela-

tions between the main topological parameters explicitlycharacterizing the geometry of the system.

This actually points out an important philosophical issue:We know that the laws of nature could be expressed in termsof a mathematical “code”, although we can hardly explainto ourselves why. We also believe that the relevant “code”is generally a di;erential equation, but we are not used toasking, Are there others? In the meanwhile, the global ten-dency of “geometrization” of the modern physics (see, e.g.,Devis, 1985) gives us a hint that the right answer is yes. Suchan alternative “encoding” of nature might be topology in allits numerous aspects. The fundamental idea is that physicalprocesses generally have an impact on the geometric prop-erties of the medium; the self-organized topology of the as-sociated structures would then contain all the informationabout the process itself. This information is “hidden” in thecorresponding parameters that identify the proper topolog-ical object. “Decoding” the self-organized topology mightbe principally an alternative way of solving the relevant dif-ferential (“physical”) equations. This approach is now in-timately connected to the foundations of general relativity,quantum chromodynamics, string theory, etc. For more de-tails, see the monograph of Devis (1985).

The comprehension of the fractal geometry of na-ture (Mandelbrot, 1983) opens new perspectives on ourabilities to describe a multitude of phenomena. One of themis the developed turbulence. An application of the fractalgeometry to a description of the self-organized turbulenceprocesses is a new step towards the creation of the globalgeometric language of physics.

The fractal geometry “codes” have led us to the “topo-logical scenario” of the substorm event. First, we found thatthe “basic” equilibrium of the stretched magnetotail is char-acterized by the web-like, highly branched structure of thecross-tail electric currents; this structure could be describedas a percolating fractal network which is critical, i.e., atthe threshold of percolation. The fractal geometry of thecross-tail electric currents implies the hierarchical structur-ing of the magnetic )eld >uctuations in the current sheet; inthe self-organized regime, the )eld >uctuations are assem-bled in “clumpy” fractal patterns maintained by the multi-scale percolating current networks.

The self-organized fractal topologies of the magnetic )eldpatterns and electric current networks are “conjugated”, i.e.,their fractal characteristics are related to each other through aset of duality equations. These characteristics are, the Haus-dor; fractal dimension D and the index of connectivity �,applied to the description of the magnetic )eld fractal pat-terns; and the two “conjugated” (i.e., dual) parameters, theHausdor; fractal dimension D+ and the index of connectiv-ity �+, relevant for the current density structures.

The property of criticality (i.e., the threshold character)is quanti)ed by the improved Alexander–Orbach (AO) re-lation established both for the magnetic )eld and currentdensity topologies. This relation could be interpreted asthe structural stability condition for the magnetic )eld and


plasma coupling in the magnetotail current sheet. Alongwith the appropriate duality equation, the AO conditions de-scribe the topologically “admissible” turbulence patterns inthe current sheet, whose physical realizations range from the“basic” equilibrium of the stretched tail, to some “limiting”con)guration characterizing the magnetospheric substormonset.

We recognized that the turbulence patterns in the mag-netotail current sheet should become more “coarse” duringthe substorm growth phase. In particular, the magnetic )eld“clumps” tend to )ll a larger volume of the current sheetplane, whereas the current structures become less “rami-)ed” (less branched). The branching points of the electriccurrent network could be treated as “knots” connecting themultiscale conducting elements into the whole current sys-tem; hence, the less “knotty” the current networks are, themore potentially unstable they would be. Our results indi-cate that the critical loss of the “knots” caused by a con-siderable increase of the cross-tail electric current leads tothe structural catastrophe of the current network; this catas-trophe is a second order phase transition from the “basic”,quasi-two-dimensional topology of the current system, to asubstantially three-dimensional con)guration.

As the catastrophe approaches, the magnetotail cur-rent system overcomes the percolation threshold, and thecurrent )laments spontaneously emerge from the currentsheet plane. Although the remaining “knots” at the feet ofthe emerged )laments could still provide some equilibrium,their further loss is critical and leads to an exponentiallyfast expansion of the )laments into the ambient space (i.e.,the “in>ation”). Considerable expansion of the )lamentstowards the boundaries of the magnetotail current layermakes favorable their closure via the Earth’s ionosphere,signifying the magnetospheric substorm onset.

Our basic theoretical conclusions are supported by theresults of the numerical modeling of the magnetotail cur-rent system. We could clearly demonstrate that the struc-tural >uctuations do appear in the system above the marginalpercolation threshold: The cross-tail current density and ve-locity pro)les become double peaked as soon as the follow-ing two conditions are satis)ed: (i) the magnetic >uctuationlevel exceeds some critical value (�B=B0 ∼ 0:3 in our simu-lation), and (ii) the component of the magnetic )eld normalto the current sheet plane, Bn, becomes very small (de)nitelyless than ∼ 0:1B0). The double peaked shape of the currentdensity and velocity pro)les indicates that the conducting)laments cannot be con)ned in only a small vicinity of theneutral sheet and tend to rise over the current sheet plane.This e;ect is suppressed in the presence of a suNcientlylarge Bn (exceeding ∼ 0:1B0) and is more pronounced thesmaller Bn is. The over-critical regime (�B=B0 ¿ 0:3; thedouble peaked shape of the current density pro)les) precedesthe large-scale current disruption event associated with the“in>ationary” expansion of the current )laments and the en-suing closure of the currents via the Earth’s ionosphere (i.e.,the onset) (see, also, Section 4).

We could self-consistently calculate both the equilibriumand the onset values of the principal geometric parametersof our treatment, D, D+, �, and �+, as algebraic functionsof the so-called “percolation constant” C ≈ 1:327, a uni-versal topological parameter describing the fractal geometryof percolation at the threshold. This has made it possible todevelope a quantitative description of the magnetotail dy-namics in terms of general topological concepts.

The fractal nature of the magnetic )eld and currentdensity distributions in the magnetotail current sheet leadsto the power-law behavior of the corresponding Fourierenergy density spectra; numerical estimates for the spectralindices deduced from our model are con)rmed throughobservations.

Our results show that the slopes of the spectra in the mag-netic )eld and current density >uctuations satisfy some du-ality relation and must be distinguished from each other; thisconclusion is in contrast with the predictions of the conven-tional MHD models and emphasizes the importance of theplasma particle (quasi-)Larmor motion in the magnetotailcurrent sheet. The slopes of both the spectra increase as thesubstorm onset approaches; this signi)es the energy trans-fer towards larger scales during the substorm growth phase.This theoretical conclusion is in accord with the intriguingstudy of Lui and Najmi (1997) who could pinpoint theenergy cascades towards lower frequencies (i.e., smallerwave numbers) from the continuous wavelet transform(CWT) technique.

Acknowledgements

The authors thank two anonymous referees for carefulevaluation of the paper and encouraging suggestions. Enor-mous computer work of Antonella Greco is gratefully ac-knowledged, to whom the authors are deeply indebted. Thisstudy was made possible by the INTAS project No. 97-1612.Partial support was received from the Russian Foundationof Fundamental Research, grants No. 00-02-17127 and00-15-96631. In Italy, the work was sponsored by the Ital-ian MURST, Italian CNR, contracts No. 98.00129.CT02and 98.00148.CT02, and Agenzia Spaziale Italiana (ASI),contract No. ARS 98-82.

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Documents

Geometric description of the magnetic field and plasma coupling in the near-Earth stretched tail prior to a substorm