Self-organized branching of magnetotail current systems near the percolation threshold

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. A4, PAGES 6291-6307, APRIL 1, 2001

Self-organized branching of magnetotail current systems near the percolation threshold

Alexander V. Milovanov and Lev M. Zelenyi Department of Space Plasma Physics, Space Research Institute, Russian Academy of Sciences, Moscow, Russia

Gaetano Zimbardo and Pierluigi Veltri Dipartimento di Fisica, Universit& 'della Calabria and Istituto Nazionale di Fisica della Materia, Unit& di Cosenza, Italy

Abstract. This paper advocates an application of the fractal topology formalism to an analysis of the magnetotail current systems. Our main attention is concentrated on the magnetotail regions with the considerably stretched background magnetic field; such conditions can be the case in the distant tail, as well as in the near-Earth tail at the late substorm growth phase. Structural properties of the current system at the self-organized (quasi)stationary states are analyzed in relationship with the universal fractal geometry of the percolating fractal sets near the critical threshold. We found that the violation of the criticality character leads to the topological phase transition in the system; this transition has features of a structural catastrophe and is associated with the magnetospheric substorm onset. A comparison with various observational studies is given, and the possible directions of the future research are outlined.

1. Introduction

Structural properties of the magnetic field and plasma turbulence in the Earth's magnetotail attract a good deal of attention in many studies, both experimental and theoretical.

1.1. Turbulence Spectra: Evidence for Power Laws

As spacecraft observations have become available, it has been recognized that the turbulent coupling of plasmas and magnetic fields in the tail displays a rich va- riety of unusual features. Amongst them is the power law behavior of the Fourier energy density spectra, with the power exponents dependent on the frequency range that is analyzed.

The survey of plasmas and magnetic fields in the distant regions of the tail with the Geotail spacecraft provided the scientific community with the comprehensive data exhibiting the power law turbulence spectra [see, e.g., Hoshino et al., 1994; Nishida et al., 1994]. In fact, it was found that the energy density spectrum for the z component of the magnetic field (we are assuming below the GSM coordinate system) could be well fit- ted by the power law function P(f) ec f-• with the slope O•1 ranging between •0 0.49 and • 1.48 until the

Copyright 2001 by the American Geophysical Union.

Paper number 1999JA000446. 0148-0227/01 / 1999 JA000446509.00

frequency f is less than the characteristic turnover frequency f, •0 0.04 Hz. As soon as f exceeds the value of f,, the best fit for the spectrum becomes more steep and has the power law form P(f) ec f-•'- with the slope a2 which is unambiguously larger than C• 1 and lies between • 1.78 and • 2.43. (These results correspond to the Geotail measurements around one of its most

distant apogees, that is, approximately 200 RE.) The Fourier energy density spectrum for the z component of the field is thus a combination of the two power law functions, with the kink near f, • 0.04 Hz. As mentioned by Hoshino et al. [1994], the possible origin of this kink spectrum may be the turbulent magnetic reconnection where the characteristic scale corresponding to the turnover frequency f, is given as the reciprocal of the most unstable tearing wavenumber in the current sheet. Similar power law behavior with a kink close to f, • 0.04 Hz was also recognized in the fluctuation spectra for the x and y components of the magnetic field; the exponents of these spectra, however, differ from those for the z component of the field in the both frequency ranges f •0< f, and f ;4 f, [Hoshino et al., 1994].

The kink power law spectra are characteristic not only of the distant magnetotail measurements but can be seen in various magnetospheric satellite experiments. Consider, for instance, the results of Ohtani et al. [1995] who investigated the turbulence data obtained in the near-Earth stretched and thinned tail prior to substorm with the Active Magnetospheric Particle Tracer Ex- plorers (AMPTE). An evidence for a more fiat power law spectrum in the lower frequency range was found,

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6292 MILOVANOV ET AL.: BRANCHING OF MAGNETOTAIL CURRENT SYSTEMS

with the characteristic spectral slopes and the turnover frequency f. close to that reported by Hoshino et al. [1994]. Note, however, that the AMPTE experiments were performed at the considerably smaller geocentric distances (around • 8.8 Rs) than the Geotail magnetic field experiments discussed by Hoshino et al. [1994]. The observed similarity in the turbulence spectra from Geotail and AMPTE might be considered as a hint that the physical conditions in the considerably stretched tail prior to substorms even much closer to the Earth could be analogous to that in the distant tail.

By using magnetometer and plasma instrument data of AMPTE obtained during 3 months of plasma sheet passes, Bauer et al. [1995a] reported a kink in the energy density spectra of the magnetic field fluctuations from a computation of the average spectra of the magnetic field vectors in the relevant frequency ranges. The results of Bauer et al. [1995a] indicate that whereas for frequencies between 0.03 and 2 Hz the power law behavior with the slopes from m 2 to m 2.5 holds, in the lower frequency range the spectrum is actually more fiat, with the slopes clearly less than m 1.5 for frequencies below • 10 -3 Hz.

An evidence for the power law turbulence spectra with the characteristic slopes from 2 to 2.5 in the higher frequency range was also provided by Russell [1972] from the OGO 5 plasma sheet data. Similar power law behavior was recognized by Borovsky et al. [1997] from the ISEE 2 Fast Plasma Experiment; having performed the relevant Fourier analysis of the magnetic turbulence data over 1.5 decades of frequency, Borovsky et al. I1997] found the mean value of the characteristic spectral slopes to be • 2.2.

The existence of power laws in the magnetic turbulence data was recently supported by the results of the Interball I mission (A. Petruckovich, private communi- cation, 1998). The spectra of the magnetic field fluctuations as were seen by Interball I have the characteristic kink shape, with the typical slopes close to the value • 2.35 for frequencies higher than, approximately, f, • 10 -2 Hz, and with the relatively flat constituent of the spectrum in the frequency range below f,.

1.2. Turbulence Hierarchies: Theoretical Tools

The physical origin of the power law turbulence spectra in the magnetotail is a problem of primary importance. In many cases the observed power law behavior can be associated with the hierarchical structuring of the turbulence in a wide range of spatial scales.

"Hierarchical structuring" means that the turbulence is organized in multiscale "clumps" (i.e., "grains" of various sizes) separated by "voids" ("empty" domains between the grains). The formation of the coarse-grained turbulence hierarchies implies that the system enters into a strongly nonlinear state characterized by the coexistence of many different mutually interconnected scales affecting each other.

A thermodynamic analysis of the multiscale clumpy turbulence can be given in the framework of the generalized Lorentzian statistical mechanics based on an ex-

tension of the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchical correlation expansion of the Li- ouville equation [Treumann, 1999a,1999b]. The concep- tually equivalent approach was developed by Milovanov and Zelenyi [2000] who applied the generalized Daroczy- Tsallis thermodynamics to the coarse-grained systems with the extremely long-ranged correlations operating on many scales. .

Alternatively, one might attempt to analyze the hierarchical turbulent system geometrically. This means that the strong nonlinearity of the system is hidden in a set of topological quantities which define the inherent geometric characteristics of the turbulence. The behavior of the system is then formulated in terms of algebraic relations between the topological parameters involved; such relations (algebraic "codes") acquire the role of the principal differential equations governing the basic processes in the system.

The geometric approach was applied by Milovanov and Zelenyi [1998] to the fracton model of the inter- planetary magnetic field turbulence. Similar ideas were proposed by Zelenyi et al. [1998] for the study of turbulence hierarchies in the Earth's distant tail.

The mathematical background of the geometric description of the hierarchical turbulent systems includes the methods of the fractal topology, that is, the synthe- sis of the standard fractal geometry and the differential topology of manifolds [O'$haughnessy and Procaccia, 1985; Milovanov, 1997; Milovanov and Zimbardo, 2000]. The fractal topology deals with the two main parameters which completely describe the geometric properties of the hierarchical system, namely, the Hausdorff fractal dimension D and index of connectivity rr. (In the notations of Milovanov [1997], rr - 0.) (We restrict ourselves to only the self-similar hierarchies which possess the properties of isotropy and statistical homogeneity.)

The Hausdorff fractal dimension D determines the

minimal number iV(e) of elements (hypercubes) which cover the system with given accuracy e [Mandelbrot, 1983]' iV(e) oc e -D . This parameter is important in view of the presence of voids (empty domains) that are excluded from the covering. Roughly speaking, the Hausdorff dimension D characterizes the deviation

of the turbulence geometry from the space-filling limit (i.e., when no voids are present).

The Hausdorff dimension D is customarily introduced as the effective (fractional) dimensionality of the set oc- cupied by the turbulence. The space-filling limit maxi- mizes the values of D: In this case, the quantity D coincides with the topological (integer) dimensionality E of the embedding Euclidean space, that is, D - E for the space-filling turbulence. The presence of the voids violates the space-filling approximation: The turbulence occupies a fraction of the embedding Euclidean space; this situation corresponds to the values of the Haus-

MILOVANOV ET AL.' BRANCHING OF MAGNETOTAIL CURRENT SYSTEMS 6293

dorff fractal dimension D that are smaller than E, so that D _< E in general. More precisely, the Hausdorff dimension D defines the scaling of the fractal "mass" (or "energy") density: P(X) cr X D-E, where X is a scale length, and D- E _< 0 as the result of the voids present.

The fractal dimension D alone is not a sufficient char-

acteristic of the fractal structure: This quantity is insensitive to the intrinsic topological connections between the constituent units of the fractal set; these connections are not in one-to-one correspondence with the mass scaling of the fractal (as determined by D) and are included in the index of connectivity 0r [O'$haughnessy and Procaccia, 1985]. The index er is generally nonzero for the fractal distributions, in contrast to Eu- clidean spaces where er -0.

The quantity er can be rigorously introduced by using the concept of the geodesic line on a fractal set (i.e., the shortest line connecting two arbitrary points of the fractal). Indeed, the geodesic line on a self-similar fractal object can be treated as a self-affine fractal curve whose own Hausdorff fractal dimension is equal to (2 + er)/2. This issue was discussed in detail by Milovanov and Ze- lenyi [1998].

The index of connectivity er often appears when describing dynamical phenomena on fractals. Together with the fractal dimension D it determines the scaling exponent for the conductivity [O'Shaughnessy and Procaccia, 1985], as well as the density of states exponent for the low-frequency Ëacton modes [Orbach, 1989]. The original important promotion of the parameter er was given in the pioneering paper of Gefen et al. [1983] where the concept of the range-dependent diffusion on percolating fractal networks was proposed.

In this study, we apply the geometric approach to an analysis of the strong magnetic field and electric current coupling in the magnetotail current sheet. In the self-consistent regime, two pairs of topological quantities describing the system should be introduced, that is, (1) the Hausdorff fractal dimension D and index of connectivity er characterizing the magnetic field structures, and (2) the Hausdorff fractal dimension D + and index of connectivity er + relevant for the electric current configurations. (The topological meaning of the quantities D + and er + is similar to that of D and

The necessity to introduce the quantities D + and in addition to the parameters D and er can be jus- tified by the following arguments: One assumes that the magnetic field fractal geometry is self-consistently maintained by the proper (fractal) distribution of the electric current links; the latter does not necessarily coincide with the distribution of the magnetic field elements (magnetic turbulence clumps) and must be associated with another fractal object whose Hausdorff fractal dimension D + and index of connectivity er + are distinguished from the magnetic field distribution parameters, D and er. The two distributions, however, cannot be arbitrary but are intrinsically interconnected between each other. (These interconnections are de-

scribed by Maxwell equations and are of the conventional electrodynamic origin.) Figuratively speaking, the two interconnected (fractal) distributions are self- consistently "pulled" one on the other: Such sets might be termed "dual," or (mutually) "conjugated." The duality property implies certain (algebraic) relations between the parameters D, D +, 0r, and 0r+; these relations provide the geometric representation of the magnetic field and electric current coupling in the turbulent system.

The analysis of the present study applies solely to the magnetotail regions with the considerably stretched magnetic field in the lobes. Such conditions can be satisfied, for example, for the distant tail, and the near- Earth stretched and thinned tail prior to a substorm. The role attributed to the self-organized turbulence in the stretched tail is discussed in section 2. Numerical

estimates of the main physical parameters are collected in section 3. Scaling behavior of the basic equations for the plasma dynamics in the tail is analyzed in section 4. Topological properties of the self-organized turbulence structures are studied in section 5 in the context of the

dual fractal distributions; the results of this section refer to the (quasi)stationary states of the stretched tail associated with the local minima of the free energy of the system. In section 6 we demonstrate that accumulation of the free energy in the tail leads to a topological phase transition in the current system; the main features of this transition resemble the structural catastrophe of the tail at the substorm onset. Spectral characteristics of the turbulence are considered in section 7. Our main

conclusions are presented in section 8.

2. Self-Organized Turbulence in the Current Sheet

The issue of the structural stability of the considerably stretched tail addresses the important role played by the turbulence-associated processes in the current sheet. In fact, assume, for a while, that the cross-tail electric currents are organized in straight parallel filaments; this corresponds to the "laminar" electromag- netic structure of the tail without turbulence. Because

parallel currents attract each other, such a current system is potentially unstable. The current instability might become energetically unfavorable in presence of a sufficiently large regular component of the magnetic field normal to the current sheet; this, however, is not the case in the considerably stretched tail where the normal field is relatively small.

An effective mechanism that might prevent the col- lapse of the magnetotail current system (initiated by, e.g., small current density inhomogeneities) might be self-organized turbulence in the current sheet. For instance, the turbulent structures might become effective "scatterers" for the current-carrying particles that otherwise would be strongly accelerated along the

6294 MILOVANOV ET AL.' BRANCHING OF MAGNETOTAIL CURRENT SYSTEMS

Figure 1. Schematic image of the magnetotail current system at the marginal (quasi)equilibrium state. The cross-tail currents are organized in a web-like network composed of the multiscale conducting elements; these elements are surrounded by the self-consistently generated magnetic field fluctuations shown as irregular clumps. Topological stability of such a system is provided by the knots retaining the current web as a connected structure.

(straight) filaments by the cross-tail (dawn-dusk) electric field.

Self-organized turbulence appears in the multiscale braiding of the filaments. In plane geometry, this leads to the formation of a web-like current network com-

posed of numerous conducting elements of irregular shape; development of such a network suppresses the particle cross-tail acceleration and saturates the col- lapse of the sheet.

The characteristic property of the web-like networks is the existence of numerous branching points giving rise to a hierarchy of the smaller-scale conducting links. The links going from one current line to another form a "knotted" configuration which is topologically stable. (Here the knots are identified with the branching points of the web.) From the geometric viewpoint, the knots retain the entire current configuration as a connected system (see Figure 1). In the self-consistent regime the electric current web gives rise to the multiscale magnetic field fluctuations assembled in clumps around the links.

2.1. Basic Electrodynamic Equations

More precisely, the network in Figure 1 carries some nonzero average current, (jy) • 0, in the cross-tail direction; the current (jy) self-consistently maintains the lobe component B• of the magnetotail field. (We imply that (j•) = (jz) - 0.) The braiding of the current filaments can be associated with the multiscale pertur- bations 5j which generally have both x and y components and support the fluctuating magnetic field 5Bz. (As usual, we apply the boldface to distinguish vector quantities, i.e., 5Bz is understood as a vector, whereas 5Bz would be a scalar.) We ignore the possible appearance of the magnetic fluctuations in x and y directions at the marginal (quasi)equilibrium state of the stretched magnetotail, restricting ourselves to the (quasi)two- dimensional topology of the turbulence [Milovanov et al., 1996] (see, however, sections 5 and 6). Thus we imply E = 2 for the turbulence (quasi)equilibrium, where E is the number of the embedding dimensions for the turbulent system (see section 1).

The set of Maxwell equations for the system reads

4• X7 x B• = -- {jy), (1)

½

V x 5Bz - 4___• 5j. (2) c

Equations (1) and (2) are supplemented by the conti- nuity condition V. [(j•) + 8j] - 0.

Equation (2) shows that the cross-tail electric currents may considerably deviate from a straight homogeneous flow along y direction when the perturbation component 8j plays a role. This is emphasized by the explicit scalar form of the vector equation (2)'

c •y {SBz ß ez}, (3)

c • {SBz-ez}. (4) •J• = 4• Here • and • denote differentiations over z and y, respectively, and e• is the unit vector in z direction.

The physical origin of the clumpy turbulence associated with the multiscale structuring of the electric currents and magnetic fields might be the modified-two- stream instability (MTSI) analyzed in detail by Lui • al. [1995]. MTSI is characterized by the generalized lower-hybrid frequencies and is one of the principal modes of the cross-field current instability. The importance of the cross-field streaming instabilities in the neutral sheet environment have been recently addressed by O•iani • al. [1998].

2.2. •rbulence Structuring Interval

Suppose the turbulence structuring interval lies between the two characteristic length scales, •min • a and •max • • • a. The quantities a and • can be interpreted as the minimal and m•imal sizes of the turbulence clumps, respectively; the inequality • •) a is verified in section 3 where a self-consistent estimate of

the parameter a is obtained. For X •( a, the turbulence configurations are as-

sumed to have no any internal substructure that might

MILOVANOV ET AL.: BRANCHING OF MAGNETOTAIL CURRENT SYSTEMS 6295

noticeably influence the cross-tail current propagation; the distribution of the turbulent field 5Bz at such fine scales is considered as (quasi)homogeneous (see, however, section 7). This motivates the term "turbulence smoothness length" for the quantity a.

The maximal length scale • can be associated with the external modulation of the turbulence coming from the tearing mode scales; the quantity • is thus related to the global field-reversed geometry of the magnetotail. In fact, the fine-scale magnetic structures in the current sheet can be considerably modulated by the ex- cited tearing modes having the energy reservoir in the basic field-reversed configuration of the tail [Galeev and Zelenyi, 1977]. One might speculate that the quantity • has the same order of magnitude as the minimum unstable tearing wavelength in the sheet •,. The parameter •, is related to the characteristic current sheet thickness L through the widely known estimate [Coppi et al., 1966] •, -• 2•rL. (This relation immediately follows from the inequality kL •< 1 which defines the unstable tearing wavenumbers, k = 2•r/•, for a current sheet of thickness L.) Hence, by order of magnitude, ••, • 2•rL.

For the distant tail, a rough estimate of L • 4 x 10 s cm might be used [see, e.g., Nishida et al., 1994], yielding • • 2.5 x 109 cm. For the substorm regions in the near-Earth tail, a smaller value of L • 10 s cm would be more reasonable [see, e.g., Lui, 1993]; hence • ~ 6 x l0 s cm.

For X • •, the topology of the magnetic field in the vicinity of the current sheet is determined by the conventional chain of magnetic islands. We imply here the traditional tearing instability of the current sheet, related to the standard one-dimensional chain structure

[Coppi et al., 1966]. In the intermediate range of scales a •< X •< •, the

hierarchical structuring of the turbulence is the case. In our study this hierarchical structuring is associated with the self-organized (quasi)stationary states of the stretched tail maintained by the turbulent mechanisms. Hence an application of the fractal topology formalism (see section 1) must assume the "physical" limitation a ~• X • • on the spatial scales that are analyzed.

From the energy balance viewpoint, the turbulence- associated (quasi)stationary states obey the standard injection -• multiscale cascade -• dissipation scheme; this scheme underlies the modern theory of the developed turbulence [Frisch, 1995]. In fact, the relevant injection mechanisms at the (quasi)stationary state of the tail could be identified with the large-scale reconnections at the Earth's dayside magnetopause; the cascade of the energy to smaller scales is related to the formation of the hierarchical turbulence patterns in the range a • X • •; the role of the dissipation processes might be attributed to the energy losses at the fine scales X •< a.

The feasible mechanism responsible for the energy losses can be plasma heating and acceleration in the

inductive electric fields: Such fields can be generated by the intrinsic "breathing" ("self-pouring") of the turbulence. The self-pouring might be understood as the turbulence internal degree of freedom, that is, inherent relative motion of the turbulence clumps consistent with the hierarchical structuring processes in the system (see also section 7). The dynamics of the self-pouring can be described on the basis of the fractional wave equation

[see Milovanov and Zelenyi, 1998]. The characteristic scales of the self-pouring processes might be compared with the typical sizes • a of the turbulence clumps; hence the quantity a can be determined as the effective dissipation length at the (quasi)stationary state.

One might speculate that the particle heating in the tail due to the self-pouring leads to the spectral energy transfer toward higher (superthermal) energies and an occurrence of the superthermal tails in the corresponding particle distribution functions. The plasma particle distributions including the superthermal tails could be often approximated by the so-called "kappa" distributions. The origin of the kappa distribution functions in the systems including particle heating and acceleration was studied in detail by Milovanov and Zelenyi [2000] who substantiated their thermodynamical relevance. The existence of the kappa distributions in the Earth's magnetotail was reported by Christon et al. [1989] from a comprehensive examination of the plasma particle populations in the near-Earth environment.

3. Numerical Estimates

We now estimate basic physical parameters relevant for our study, by order of magnitude. Denote rœ to be the (effective) plasma particle Larmor radius in the magnetic field •Bz; the quantity rœ can be defined as the reciprocal curvature of the particle trajectories in the (xy) plane.

Let us assume that the particle population is characterized by some "thermal" energy W; on the basis of observations, we set W • I keV. One might antic- ipate that the Larmor radius rœ corresponding to the thermal energy W is comparable with the length scale a: Indeed, in the self-organized regime, the turbulence is generated by the very same particles that experience scattering at the turbulence clumps; hence the typical length scales associated with the turbulence (• a) and the particle trajectories (• rœ) must be of the same order of magnitude, that is, a • rœ.

The relationship a • rœ can be interpreted as the equivalence of the effective dissipation length in the system a to the basic spatial step rœ of the particle cross-tail propagation. This leads to a conclusion that the particles having rœ • a bring the dominant contribution in the self-organized cross-tail electric currents: Using the numerical values of a given below, one might verify that these are mainly ions having energies close to W • I keV; such particles are effectively scattered


by the self-consistently generated magnetic turbulence clumps.

The particles whose Larmor radii rL are much smaller than •- a (i.e., cold ions and most of electrons: rœ << a) are effectively trapped in magnetic "deadlocks" associated with the clumpy geometry of the turbulent field 5Bz; the contribution of these particles in the cross-tail electric current is negligible at the spatial scales longer than ..• a.

On the contrary, the hot particles (ions) having Lar- mor radii much larger than a (i.e., r• >> a) can easily propagate through the magnetic field turbulence in the current sheet; the relative number of such hot particles, however, is customarily small, enabling one to ignore the corresponding currents when making rough estimates.

The scattering of particles at the magnetic fluctuation clumps appears in distortions of the particle trajectories in the (xy) plane by the characteristic angle •b ..• a/rœ. During time intervals including many consequent scatterings, the resulting motion of the plasma particles would be a diffusive-type migration through the magnetic field 5Bz, with the average displacement toward the dusk flank of the tail.

It is easy to verify that the characteristic distortion angle • •- a/r• is of the order of the fluctuations in the electric current density, that is, 5j/ (jy) ..• • ..• a/r• where 5j and (jy) denote the characteristic values of the vector functions 5j and (j•), respectively. From (1) and (2) we find that (j•) ..• (c/4•r)(B•/L) and 5j ..• (c/4•r)(SB•/a). Taking ratio of these two relations and making use of the explicit expression for the effective plasma particle Larmor radius, rL ..• vmc/eSB•, where v ..• v/W/m is the typical ion velocity and m is the proton mass, we obtain the self-consistent estimate of the spatial scale a:

•/vmcL (5) a ..• ¾ eB• '

The lobe component of the magnetic field B• which appears in (5), can be evaluated from the pressure balance in the magnetotail; assuming, for simplicity, 5B• • B•, one gets B• ..• v/8•rnW where n is the average plasma number density in the current sheet. Below we are using the rough estimates n •- I cm -3 and W •- I keV of the parameters n and W in the magnetotail [see, e.g., Paterson and Frank, 1994]. One finds, approximately, v •- 3 x 107 cms -•, and B• •- 2 x 10 -4 G. Equation (5) then yields a •- 8 x 107 cm for the distant tail, and a •- 4 x 107 cm for the substorm regions in the near-Earth tail, where the corresponding estimates L •- 4 x 10 s cm and L •- 10 s cm of the current sheet

thickness L have been used (see section 2). The ratio of the parameters • •- 2•L and a becomes, numerically, •/a •. 30 in the distant tail, and •/a •. 15 in the near- Earth tail. Thus the value of • is by order of magnitude larger than the basic turbulence length scale a:

• ..• 2•rl/eLB• >> 1. (6) a ¾ vmc

Condition (6) substantiates the possibility of the hierarchical structuring of electric currents and magnetic fields in the magnetotail current sheet in the range

To estimate, self-consistently, the level of the magnetic fluctuations 5Bz/B• in the magnetotail current sheet, one must introduce effective plasma conductivity Y• associated with ion migration through the magnetic fluctuation clumps. By order of magnitude, we have Y• • ne2•'/m [Lij'shitz and Pitaevskij, 1979] where the collisional time •- is defined as the characteristic diffu-

sion time: •- •- a2/Z). (Here the parameter Z) is the particle diffusion coefficient in the (xy) plane. Without pretending to be precise, we imply that the characteristic scale of voids between the magnetic clumps is of the order of a.) We defer an accurate derivation of the diffusion coefficient Z) to section 4 and now make

use of the rough estimate Z) •- r•v/2 which will be discussed below. Hence the conductivity • becomes Y• ..• 2ne•a•/mvrœ. The average current density can be expressed in terms of Y• as <jy) •- Y•Ey where Ey is the dawn-dusk electric field. For a rough estimate of Ey, we assume a reasonable value of •- 0.1 mV m -• for the distant tail and a larger value of •- 0.2 mV m -• for the substorm regions in the near-Earth tail.

On the other hand, the current density <jy) can be defined from (1) to give <j•) •- (c/4•r)(B•/L); numerical evaluation shows that <jy) •- 1.2 x 10 -3 g•/• cm -•/• s -2 in the distant magnetotail and •- 4.8 x 10-3 g•/2 cm-•/• s-2 in the near-Earth tail, where the relevant estimates of the current sheet thickness L have

been applied. The characteristic ion Larmor radius becomes r• ..• 2ne•a•E•/mv (j•); one immediately finds that r• •- 1.5 x 10 s cm in the distant tail and rL 2 x 107 cm in the substorm regions of the near-Earth tail. The ratio r•/a equals to r•/a •- 2 and r•/a • 0.5, respectively, in good agreement with the assumptions of our model.

These theoretical conclusions are in accord with the

observational results of Ohtani et al. [1998, p.4680] who indicated that "the spatial scale of the magnetic fluctuations is of the order of the proton gyroradius or could be even shorter." Moreover, Ohtani et al. [1998, p.4679] do substantiate the essential role played by the ion component of the plasma in the generation of the magnetic field fluctuations in the current sheet, that is, one of the basic points of our consideration.

Finally, the magnetic field 5B• appears to be 5Bz vmc/er•, yielding •- 2 x 10 -• G for the distant, and -• 1.6 x 10 -4 G for the near-Earth, tail. In the relative units, 5B•/B• ..• 0.1, and 5B•/B• ..• 0.8, where 5B•/B• defines the level of the magnetic field fluctuations in the current sheet. An observational support for the estimates 5B• ..• 1.6 x 10 -4 G and 5Bz/B• ..• 0.8 describing the magnetic fluctuation amplitudes in the


near-Earth stretched and thinned tail might be found in the paper of Bauer et al. [1995b].

4. Generalized Diffusion and Scaling Analysis

In this section we analyze anomalous particle transport across the magnetotail in association with the fractal distribution of the magnetic turbulence clumps in the range of scales a •< X •< •.

4.1. Generalized Diffusion Law

Following Feder [1988], we consider the generalized relation between the mean square transverse displacement of the plasma particles (SX•'• versus their diffusion time At, that is, (5X2• - 2Z)O(At/O) 2•t, where Z)is the generalized diffusion coefficient, and 0 is the basic time step of the particle motion. The quantity H is the so- called Hurst exponent which varies between 0 and 1, that is, 0 <_ H i i. The caseH- 1/2 recovers the standard Einstein relation (SX• - 2Z)At for the particle diffusion in nonfractal media; on the contrary, the cases H • 1/2 describe "strange" diffusion processes on fractal sets; such processes customarily depend on the time step •.

The diffusion process on a fractal can be persistent or antipersistent, depending on the exact value of the parameter H. A persistent diffusion process is characterized by the parameters H varying from 1/2 to 1 and implies positive correlations between past and future displacements of the test particle. An antipersistent diffusion process is described by the values of H ranging from 0 to 1/2 and leads to negative correlations between past and future particle displacements. In the standard case of H - 1/2, the correlations between past and future displacements are absent on all temporal scales; this property is crucial for the Einstein law (5X2• - 2Z)At which applies to the completely random particle walk.

The property of the diffusion process on a fractal to be persistent or antipersistent is related to the topological characteristics of the fractal substrate, contained in the index of connectivity er. This is emphasized by the explicit relation between the Hurst exponent H and the index of connectivity er: 2 + er - l/H, where 2 + er is the effective dimensionality of the diffusion process [see Zelenyi et al., 1998].

One concludes that the persistent diffusion on a fractal (1/2 • H _• 1) corresponds to the values of a varying between -1 and 0, that is, -1 _• a • 0, whereas the antipersistent diffusion (0 _• H • 1]2) implies a _• 0. The Einstein diffusion process (H - 1•2; (SX •) - 2Z)At) is recovered by the zero value of the quantity a, that is, a-0 for H- 1•2.

Substituting 2+a - 1/H into the expression (SX •) - 2Z)O(At/O) •H, one obtains [O'Shaughnessy and Procac- cia, 1985]

4.2. Generalized Diffusion Coefficient

We now estimate the generalized diffusion coefficient Z) for the plasma particles in the magnetotail current sheet from (7). Since the particle migration across the sheet is due to the consequent scatterings at the magnetic turbulence clumps (each having the typical size • a), the time step O in (7) can be evaluated as O • a/v, where v is the plasma particle thermal velocity. (This estimate holds for the particles having rœ • a, that is, for the bulk of the ion distribution.)

It is easy to verify that the mean square displacement of the particles <SX 2) becomes of the order of the Larmor radius squared r• during the time interval St/0 • •)--1 where •b • a/rœ is the characteristic distortion angle of the particle trajectories. Considering (7) one then obtains

(7_•) 2/(2+•) I v a (8)

yielding the generalized diffusion coefficient for the particles (ions) in the magnetotail current sheet. In the standard case of er - 0, (8) is reduced to Z) -• rœv/2 which is a straightforward estimate of the diffusion coefficient expressing the important role of the particle (quasi)Larmor motion (see section 3).

4.3. Scaling Analysis

Since the particle Larmor radius rL varies with the magnetic field 5Bz as rL oc 5B• -1, from (8) it follows that the generalized diffusion coefficient Z) scales with 5Bz as Z) oc 5B• -2(1+•)/(2+•) . Hence the current density 5j which appears on the right-hand side of (2) scales as 5j ,• %0 (jy) OC r• 1•)-1 (X 5Bz ß 5Bz 2(1+a)/(2+a) where we took into account that {jy) oc Z) -1.

Let us now turn to the left-hand side of (2). First of all, we draw attention to the important point that in fractal geometries with the unconventional connectiv- i? a • q, the order of differentiation in the operators Z)x and Z)y (see the explicit form (3)-(4) of the vector equation (2)) becomes fractional and is actually equal to 2/•2 + rr) rather than unity. The fractional operators Z)x,y directly account for the (anti)persistency of the cross-tail particle migration driving the cross-tail electric currents [Milovanov et al., 2001].

The issue of the fractional differentiation and the en-

suing concept of the fractional Maxwell equations (3) and (4) are mainly left beyond the scope of the present study. For our purposes we o•nly need to consider the anomalous scaling behavior, Z)•,y cr X -•/(2+•), of the operators f)•,y implied by the fractional spatial differentiation of the order (2 + o')/2 • 1 [see Zelenyi and Milovanov, •1996]. (Note that in Euclidean geometry, a - 0, and Z)•,y are independent of the length scale •.)

One thus obtains that the left-hand side of (2) be- haves as 5Bz. X: -•/(2+•). Along with the scaling law for the current density, 5j cr 5Bz . 5B• 2(1+'•)/(2+'•) this yields the amplitude of the magnetic fluctuations 5Bz

6298 MILOVANOV ET AL.' BRANCHING OF MAGNETOTAIL CURRENT SYSTEMS

as a function of X, that is, 5Bz cr X --•/2(•+•). Hence the energy density of the magnetic field 5Bz scales as 5Bz 2 cr X -•/(1+•). In the self-consistent regime this result must coincide with the scaling behavior of the fractal "energy" density, P(X)cr X D-2 (see section 1). This leads to the relation between the fractal dimension D and the index of connectivity (r at the (basic) (quasi)stationary state of the stretched tail:

D E a - - •, •- 2. (9) l+a

Equation (9) shows that the self-consistent fractal geometry of currents and magnetic fields assumes certain restrictions on the admissible values of D and (r. Set-

ting rr - 0 in (9), one recovers the conventional case of Euclidean geometry (D = E = 2) (i.e., the space-filling limit).

5. Self-Organized Critical Percolation To obtain the self-consistent estimates of the parame-

ters D and rr in the magnetotail current sheet, one must supplement (9) with an additional relation describing the self-organized fractal geometry of the magnetic field and current density fluctuations. We deduce such a relation from the following consideration.

First, we note that the cross-tail currents in the magnetotail current sheet must form percolating patterns. "Percolating" means that such patterns spread to arbitrary long scales: This condition is obvious as it just allows the cross-tail currents to flow from one side of the tail to another and thus physically maintain the global configuration of the tail.

Second, we assume that the topology of the percolating current patterns is in some sense minimal, that is, the electric current network is at the threshold of percolation. This could be considered as a sort of a topological extremum character: The self-organized turbulent current system saturates at the critical (minimal) distribution of the conducting elements for the given external ("boundary") conditions. The self-consistent balance of currents and fields in the turbulent current sheet is then achieved for the critical distribution of the

magnetic field fluctuations 5Bz generated by the percolating current web. Note that the percolating sets at criticality (i.e., at the threshold of percolation) necessarily possess the fractal properties [Nakayama et al., 1994]. An introduction to the percolation theory can be found in the monograph of Stauffer [1985].

It must be emphasized that the critical percolating structures can only support the marginal (quasi)stationary state of the turbulence system corresponding to a (local) free energy minimum of the stretched tail. The topological extremum condition describing the geometry of the turbulence patterns at criticality could be therefore associated with the fundamental thermodynamical properties of the nonlinear systems with self- organization [Nicolis and Prigogine, 1977].

Thus the percolation threshold character determines the basic (quasi)equilibrium state of the stretched magnetotail' Such an equilibrium might be permanently the case in the distant regions of the Earth's tail, as well as in the near-Earth stretched and thinned tail at the ini-

tial stages of the substorm growth phase. At the later stages, accumulation of the free energy in the system coming from the reconnections at the Earth's dayside magnetopause results in the considerable increase of the cross-tail currents; this generally violates the critical percolation regime, that is, the current system tends to overcome the marginal percolation threshold. As shown in section 6, this can lead to the current disruption instability often associated with the substorm event.

From a wealth of studies it has been clearly established that the percolating systems at criticality observe universal topological features; these features could be recognized from the invariant values of the relevant geometric parameters near the percolation threshold [see Nakayama et al., 1994]. Remarkable formulation of the universal properties of percolation near criticality was proposed by Alexander and Orbach [1982]' Based on the substantial numerical evidence, they advocated the "hyperuniversal" algebraic relation between the Haus- dorff fractal dimension of the percolating fractal object D and its index of connectivity (r' 2D/(2 + (r) - 4/3, for all embedding dimensions E _> 2, even though the quantities D and (r were by no means constant as functions of E. This assertion had come to be known as

the Alexander-Orbach (AO) conjecture. For relatively high E >_ 6, general proof of the AO conjecture was given within the mean-field theory [Havlin and Ben- Avraham, 1987]. In the lower dimensions 2 _< E _< 5, the AO conjecture was improved by Milovanov [1997] who found

2D/(2 +a)- ½ • 1.327 (10)

from the topology of fractional Riemannian sphears. The quantity ½ on the right-hand side of (10) is the universal topological constant obeying the transcenden- tal algebraic equation C•rC/2/F(C/2 + 1) - •r in all 2 _< E _< 5, where F is Euler gamma function. Following the recent paper of Milovanov and Zimbardo [2000], we refer to the quantity C as the "percolation constant." In the numerical estimates below, we apply both the exact value m 1.327 of the percolation constant ½, as well as the original AO estimate • 4/3.

Combining (9) and (10), one finds the self-consistent values of the Hausdorff dimension D and index of con-

nectivity a describing the fractal distribution of the magnetic field fluctuations near the marginal percolation threshold'

D - (2 + ½)/2 m 1.66 -0 5/3 < 2, (11)

(r - (2- C)/C ,,• 0.51 • 1/2 > 0. (12)

The criticality (threshold) condition fer the cross-tail current patterns is analogous to (10)'


2D +/(2 + a +) - C m 1.327, (13)

where D + and a + are the Hausdorff dimension and in-

dex of connectivity of the electric current web (see section 1).

The quantity a+ characterizes the degree of branching of the current networks at criticality; the larger the value of a+ is, the more developed the current structures are. Following Milovanov [1997], one can demonstrate that the index a+ coincides with the Hausdorff

fractal dimension of the set of the branching points of a percolating fractal network on a plane. The limiting case a+ -• 0 describes degenerated ("released") networks without branching points; an example is a set of nonintersecting plane fractal curves.

The branching points of the electric current network can be compared to knots of the conducting "fabric," retaining the current filaments to each other as a whole interconnected system. The index of connectivity a+ then shows how knotty the current network is. The knots might be associated with the property of "dura- bility" of the network: The more knots are present, the stronger the connections between the conducting filaments inside the fabric would be (see also section 2). In this context the quantity a+ can be considered as the measure of the structural stability of the network: Larger values of a+ correspond to more interconnected current networks on a plane.

In the self-consistent regime, magnetic field and electric current patterns must be dual to each other (see section 1). The property of duality involves the issue of the microscopic connectedness for both the magnetic field and current density distributions and can be quantified in terms of a relation between the indices of connectiv-

ity a and •r +. To propose such a relation, one might speculate that the "total connectivity," a + a+, of the dual fractal objects is an invariant quantity, that is, the combination a + a+ is equal to a constant' a + a+ =inv; this might be supported by the recent results of Milo- vanov and Zimbardo [2000] who recognized the topological invariance of the connectivity properties of fractal sets. Taking into account that the percolating fractal objects are characterized by the nonnegative indices of connectivity [Nakayama et al., 1994] (i.e., the inequal-

ities a •_ 0 and a + •_ 0 hold), one might assume the invariant constant on the right of a + a+ =inv to be "of the order of unity"'

• + •+ • 1. (14)

Equation (14) shows that larger values of the index of connectivity a + correspond to smaller values of the index a and vice versa. Duality relations similar to (14) are found, for instance, when analyzing the mutual dependence between the "site" and "bond" percolation on lattices [Isichenko, 1992].

From (12) and (14) one gets the self-consistent estimate of the dual index of connectivity a+:

a + - 2(C-1)/C • 0.49 • 1/2>0. (15)

The dual Hausdorff fractal dimension D + can be now

obtained from (13) and (15) yielding

D + - 2½-1• 1.65 ~ 5/3<2. (16)

Equations (11) and (16) indicate that the magnetic field and plasma turbulence in the magnetotail current sheet is not a space-filling one as both D and D + are smaller than E - 2.

It is remarkable to note that the dual parameters a + and D + can be derived exactly in the form (15)-(16) from an abstract topological analysis of the two-dimensional sign-symmetric random fields at criticality [Milo- vanov and Zimbardo, 2000]. This argument shows that the self-organized magnetic field and current density distribution in the magnetotail current sheet has deep topological origin related to the fundamental geometric properties of the critical percolating sets.

6. Magnetospheric Substorm Onset

As was already mentioned above, an increase of the cross-tail electric currents may result in a violation of the critical percolation regime, that is, the entire current system overcomes the threshold of percolation. Generally speaking, this means that some conducting elements cannot "find enough room" in the vicinity of

Figure 2. Structural fluctuations on a percolating fractal network. These fluctuations are associated with the magnetotail current system above the marginal percolation threshold.


the current sheet and are squeezed out of the current sheet plane (see Figure 2).

Indeed, an increase of the cross-tail currents likely causes the formation of more "chunky" magnetic fluctuation patterns; the chunks could also have stronger magnetic field 5Bz compared with the corresponding threshold values. The current sheet environment thus

appears to be less transparent for the plasma particles migrating across the tail, and the particle jumps over the magnetic chunks may become energetically prefer- able. This leads to an occurrence of the conducting elements above and below the current sheet, so that the turbulence system starts to swell in the third dimension. The particle three-dimensional jumps have been recognized numerically by Veltri et al. [1998] and Milo- vanov et al. [2001,] as a splitting of the cross-tail electric currents by the magnetic field turbulence in the current sheet.

6.1. Substorm Onset: Topological Conditions

Let us now discuss the possible meaning of the basic geometric parameters, D, D +, a, and a +, at the stage when the entire current system is above the percolation threshold. It is important to note that the fractal geometry of percolation is the remarkable property of the threshold regime, whereas structures above and below the threshold may not be self-similar fractals. In view of this, one cannot directly apply the quantities D, D +, a, and a + to the system as a whole as it overcomes the marginal percolation threshold. The simplest solu- tion might be to assume that the critical conditions can be satisfied for the subsystem that remains in the current sheet plane: This leads to the image of the current system above the threshold as composed of the two con- stituents, the plane part which lies in the current sheet and is at the threshold of percolation, and the additional current filaments beyond the sheet whose presence violates the criticality conditions in the third dimension (see Figure 2). One then applies the quantities D, D +, a, and a + to a description of only the critical (plane) constituent of the entire system; the fractal geometry of this constituent then obeys (10), (13), and (14).

Numerical values of the quantities D, D +, er, and a + would be, however, different from the previously obtained results (11), (12), (15), and (16). In fact, the threshold conditions (10) and (13) do not unambiguously determine the Hausdorff fractal dimension and index of connectivity of the percolating object at criticality and can be satisfied for a continuous set of parameters: The formation of the more dense (less transparent for the current-carrying particles) magnetic fluctuation patterns implies an increase of the Hausdorff dimension D (see section 1). From (10) one concludes that the corresponding index of connectivity a also increases. Relation (14) indicates in the meantime that the index of connectivity a + decreases, that is, the electric current network becomes less knotty (with less branch-

ing points) (see section 5); (13) shows that the Haus- dorff dimension D + decreases, too. This implies gradual "coarsening" and simplification of the conducting web in the current sheet.

As was mentioned in section 5, the topological (structural) stability of the electric current network is maintained by the knots associated with the branching points of the electric current web: The smaller the value of a +

is, the weaker the interconnections between various conducting links would be. One then concludes that the decrease of the parameter a + manifests the initiation of the structural instability of the current system; in the near-Earth stretched tail, the decrease of a + to the sufficiently small values provides favorable conditions for the decay of the turbulence (quasi)equilibrium state and the ensuing onset of the tail current disruption.

A simple topological condition for the onset could be now proposed. In fact, one might naturally identify the onset with the limiting (the smallest possible) value of the index of connectivity a +, which is easily seen to be zero:

a + 0.

This corresponds to the degenerated (released) current network whose degree of branching vanishes: The conducting web evolves into a set of disconnected filaments characterized by relatively high density amplitudes (see Figure 3). This set (as a whole) has the lowest Hausdorff dimension D + allowed for a percolating fractal network (see (13))'

D + -• ½ • 1.327 • 4/3. (18)

The onset distribution of the magnetic fluctuation clumps can be described by the index of connectivity

• -• 1 (19)

as it follows from (14) and (17). The corresponding value of the Hausdorff fractal dimension D becomes, according to the AO relation (10),

D • 3½/2 ..• 2. (20)

Equations (19) and (20) show that the fractal dimension D approaches the topological dimensionality of the plane, E - 2, whereas the corresponding value of the index of connectivity rr does not vanish (as one might expect for an Euclidean geometry) but tends to unity. This means that the "true" number of the embedding dimensions exceeds E - 2 prior to the onset (see Figure 3). The effective dimensionality of the current system at the onset can be estimated from (9)'

rr 3C+1 5

Eef f -- Dq l+rr 2 2 The embedding space is thus three-dimensional: E - 3.

It is worth noting that the embedding dimensionality E is an integer (discrete) quantity which cannot

MILOVANOV ET AL' BRANCHING OF MAGNETOTAIL CURRENT SYSTEMS 6301

Figure 3. The released topology of currents and magnetic fields in the tail. Such a configuration might be characteristic of the magnetotail current system just prior to (or during) the inflation.

grow continuously from the value E - 2 (characterizing the marginal (quasi)equilibrium state of the magnetotail current system) to E -- 3 (associated with the substorm onset). In other words, the increase of the embedding dimensionality from E = 2 to E = 3 must assume an abrupt change in the topology of the current system; such a change would have features resembling a second other phase transition in the magnetotail. We substantiate this conclusion in what follows.

6.2. Topological Phase Transition Consider a structural fluctuation of the basic (crit-

ical) conducting network in the current sheet plane. This fluctuation is a local violation of the plane geometry of the current system near the marginal percolation threshold, physically viewed as a three-dimensional current filament squeezed to the outer regions of the plasma sheet (see Figure 4).

Let Z be the size of the filament in the direction normal to the current sheet. The free energy of this

filament can be represented as the sum of the two com- peting terms'

One of these terms describes the "restoring" forces (in accordance with the Le Chatelier principle) acting on the filament from the remaining part of the network. The restoring forces tend to bring the filament back to the current sheet plane and appear as the "elasticity" of the network; the elastic properties are due to the knots retaining the network constituent elements as a whole system. The free energy contribution associated with the elastic (restoring) forces is given by the standard quadratic dependence o• Z 2, that is, •(K) - KZ2 where K is a constant.

It is clear that the quantity K must be sensible to the topological properties of fractal networks: The highly branched, considerably knotted networks like that shown in Figure I should be associated with relatively large values of K (say, K • 1), whereas the degenerated structures without knots (i.e., those revealing only very small elasticity) (see Figure 3) could be char-

Figure 4. A three-dimensional current filament emerging from the current sheet plane. The knots attaching the filament to the sheet provide the restoring forces tending to bring the filament back. These forces counteract the interaction of the magnetic momentum of the filament y with the magnetic field Bx in the outer plasma sheet.

6302 • MILOVANOV ET AL.: BRANCHING OF MAGNETOTAIL CURRENT SYSTEMS

acterized by K • 0. (This issue can be made more precise by introducing an explicit relation between the parameter K and the index of connectivity a + which defines the number density of the knots on a fractal network. In fact, one finds K = K(a +) where Tay- lor expansion of the function K(a +) near zero starts with the term proportional to a +. Derivation of the functional relationship K(a +) goes, however, beyond the scope of the present study.) Hence evolution of the electric current web from a highly branched, considerably knotted confi•.uration shown in Figure 1, toward the degenerated structure illustrated in Figure 3 would generally mean a + --> 0 (see (17)) and K(a +) --+ O. Consequently, the elasticity of the fractal network K gradually decreases during the late substorm growth phase.

The other term takes into account the interaction

of the current filament with the "external" magnetic field in the outer plasma sheet. This interaction can be roughly estimated as the interaction of the magnetic momentum of the filament emerging from the current sheet plane, with the Bx component of the magnetotail background field (see Figure 4): It is easy to verify that the orientation of the magnetic momentum of the filament in the Bx direction is energetically favorable, yielding the energy gain of -•r(s ) = $Z tanh Z, where $ is a constant and does not depend on the parameter a +, and tanh Z corresponds to the standard Harris distribution of the background field: B•(Z) cr tanh Z.

Hence the total free energy of the structural fluctuation in the current system becomes

.T' - .T'( K ) + .T'( $ ) -- K Z 2 - $ZtanhZ. (22)

For small Z this yields

.T' • (K- S)Z 2 q-(S/3)Z 4. (23)

Expansion (23) is the widely known representation of the free energy functional near the point of the second order phase transition describing spontaneous topological changes in the system [see, e.g., Landau and Lif- shitz, 1970]. For K > S the free energy •r is a mono- tonically increasing function of Z and attains its minimal (zero) value for Z - 0. This corresponds to the topologically stable electric current network on a plane. For K < $ the free energy minimum implies the sub- stantially nonzero equilibrium size of the filament in z direction, Z • 0, so that the current system evolves spontaneously from the basically plane configuration to a three-dimensional geometry. The value of K -• $ is a sort of the thermodynamical threshold below which the elasticity of the current network cannot already prevent spontaneous emerging of the current filaments and the ensuing "birth" of the third dimension of the turbulence.

Integrating equations of motion for the potential function of the form (23), one immediately obtains that the expansion of the filaments into the third dimensionality

z goes exponentially fast for K < $ at the early expansion stages. Such a behavior might be compared to the evolution of the hot early Universe during the so-called inflationary stage, which is also the exponential expansion [Linde, 1990]. This suggests an application of the term "inflation" to a description of the early evolution of the magnetotail current system in three dimensions just below the transition point K -• $.

The exponentially fast expansion of the current filaments during the inflationary stage likely enables them to get out of the plasma sheet and reach the lobe regions of the magnetotail. This makes favorable the closure of the filaments via the ionosphere (see Figure 4) and the initiation of the large-scale current disruption customarily associated with the magnetospheric substorm onset.

6.3. "Topological Scenario" for Substorm Onset

A "topological scenario" for the substorm onset can be now proposed. Indeed, the basic (quasi)equilibrium of the magnetotail is realized as a self-organized critical state associated with the web-like current networks at

the threshold of percolation. These networks lie in close (of the order of ion Larmor radius) vicinity of the current sheet and can be treated as (quasi)two-dimensional fractal patterns. Topological stability of such patterns is provided by the knots coinciding with the branching points of the web. The knots retain the multiscale conducting elements as a whole percolating system (see Figure 1). The fractal geometry of the percolating current system at the threshold is completely described by the Hausdorff fractal dimension (16) and index of connectivity (15); the self-consistently generated magnetic field fluctuations 5Bz fill fractal sets whose Hausdorff fractal dimension and index of connectivity are given by (11) and (12).

Accumulation of the free energy in the tail related to the reconnection processes at the dayside magnetopause leads to an increase of the cross-tail electric currents.

The magnetotail current system tends to overcome the self-organized percolation threshold: Some current filaments are squeezed out of the current sheet (see Fig- ure 2). These filaments can be considered as structural fluctuations of the basic (quasi)two-dimensional geometry of the current system. Physically, this means that the current-carrying particles sometimes find free "cor- ridors" over the turbulence structures lying in the current sheet plane. The topology of the current system thus starts to evolve toward a three-dimensional configuration. One might say that the magnetic field fluctuations self-consistently generated by the increased electric currents tend to "push" the conducting filaments out of the current sheet. The self-organized coupling of the currents and fields generally becomes more coarse: The magnetic field patterns appear to be more clumpy, whilst the current system is less branched (with less knots connecting the current elements into the whole conducting web). Although the remaining knots do yet


support the structural stability of the current network, its elasticity (maintaining the restoring forces in the system) decreases. These processes can be seen in gradual changes of the main topological parameters, D, D +, a, and a+; indeed, the Hausdorff dimension D and index of connectivity cr exceed the basic values (11) and (12), whereas the dual quantities D + and a + become smaller than the values (16) and (15).

Further increase of the cross-tail currents results in

the pronounced coarse-graining of the system and the critical loss of its elasticity properties (see Figure 3). The current filaments emerging from the current sheet plane cannot be already brought back by the restoring forces and expand exponentially fast into the ambient plasma due to the interaction with the background field of the tail. This signifies the inflationary stage of the evolution of the current system and spontaneous birth of the three-dimensional electric current configurations. The three-dimensional current system thus evolves from the (quasi)two-dimensional percolating network as the result of the inflationary expansion of the structural fluctuations in the system; these fluctuations appear above the marginal percolation threshold and grow exponentially fast as soon as the system becomes sufficiently coarse. The energetically favorable closure of the large-scale current filaments via the ionosphere during the inflationary stage manifests the initiation of the global current disruption event customarily identified with the magnetospheric substorm onset (see Figure 4). The onset might be formally associated with the limiting (smallest) values of the parameters D + and a + (see (17)-(18)) describing the topologically degenerated current networks; the self-consistent geometry of the magnetic field fluctuations is characterized, in the meantime, by the quantities D and cr whose physical realization requires more than two embedding dimensions (see (19)-(21)) and causes the inflation, the exponential expansion of the current system in the third dimension.

Our conclusion is that the magnetospheric substorm can be treated as a structural catastrophe related to the gradual topological simplification of the percolating fractal network in the magnetotail current sheet. This catastrophe has deep topological origin and is mani- fested as the second-order phase transition in the current system above the marginal percolation threshold.

Dynamical mechanisms driving the magnetotail current sheet disruption and diversion during substorms are analyzed in detail by Lui [1996] and Lui et al. [1988]. The occurrence of the second-order phase transition during which a current reorganization phenomenon takes place in the stretched magnetotail can be supported by the arguments proposed by Consolini and Lui [1999] and Sitnov et al. [2000].

Gradual coarse-graining of the magnetic field and electric current structures during the late substorm growth phase can be identified with the energy transfer going from smaller to larger spatial scales. This cor-

responds to the inverse energy cascade, in contrast to the (quasi)stationary turbulence states when the energy goes from larger to smaller scales (i.e., the direct cascade) (see section 2). The existence of the inverse energy cascades at the late phase would mean that the turbulence spectra become steeper as the onset approaches. We calculate the spectral characteristics of the magnetotail turbulence in the following section.

7. Spectral Exponents

We begin with the spectra of the magnetic field fluctuations. First, the convection of the magnetic field structures in the magnetotail must be taken into account. In the distant tail, such a convection origi- nates from the plasma outflows with the decelerated solar wind velocity u. According to Paterson and Frank [1994], the typical values of u are of the order of •0 200 km s -•. In the near-Earth tail prior to a substorm, the characteristic velocities of the magnetic field and plasma convection are significantly smaller' u •0 50 km s -• [Lui,

It is easy to verify that the stationary convection of fractal magnetic field patterns can be recognized by a rest frame (satellite) observer as a temporal variability of the field. This temporal variability is represented by the self-affine fractal graphs whose spectral characteristics are determined by the fractal distribution of the field in space.

It has been clearly established that the Fourier energy density spectrum of a self-affine fractal graph be- haves as a power law function of the frequency f, that is, P(f) c• .f-•, where a is a constant depending on the fractal properties of the graph [see, e.g., Isichenko, 1992]. A simple relation between the exponent a and the Hausdorff fractal dimension D describing the spatial distribution of the magnetic field fluctuations was found by Milo,ano, and Zelenyi [1994].

a- 2D- 1. (24)

Hence larger values of the spatial fractal dimension D correspond to steeper fluctuation spectra P(f). Note that the slope a depends solely on the fractal dimension D and is insensitive to the more delicate quantity, the index of connectivity

The frequency f is related to the wavenumber k X -1 characterizing the fractal distribution of the magnetic field in space, via f - ku. This leads to an al- ternative representation of the Fourier energy density spectrum P(f) oc f-•' in terms of k as P(k) oc k-".

The physical relevance of a relation between the characteristic timescales and spatial scales of the fluctuations in the current sheet environment was substanti-

ated by Ohtani et al. [1998] from the AMPTE/CCE - SCATHA simultaneous observations. In addition, Ohtani et al. [1998, p.4680] explicitly note that the actually observed signatures in the fluctuation components "may be explained in terms of the earthward


plasma convection which conveys the associated perturbation currents and therefore the magnetic fluctuations with plasma." This supports the mapping k -• f implied by (24).

It should be emphasized that the relationship between the spectra P(f) and P(k) can be actually more deep, in spite of the formal simplicity of the linear mapping f = ku. In fact, the condition f = ku is a straightforward implication of the assumed stationary convection of the turbulence patterns in the magnetotail, which helps transform the Fourier energy density spectrum in the frequency domain P(f) into the spectrum in the wavenumber domain P(k). Making such a transformation, one, however, formally ignores the self- pouring of the turbulence (see section 2).

The importance of the self-pouring in the Earth's stretched tail might be recognized from the recent findings of Borovsky et al. [1997]. Although a detailed treatment of the self-pouring effects in the tail should be a subject of a separate study, we might neverthe- less try to address the impact of the self-pouring on the spectral characteristics of the magnetotail turbulence. Our basic idea is that the inclusion of the self-

pouring probably does not influence the spectral exponent a when the transformation from the frequency to the wavenumber domain is performed; hence the above "naive" linear mapping via f = ku might indeed yield satisfactory results. (This would be the case only when the average fractal properties of the turbulence remain unchanged at the timescales involved, that is, the Haus- dorff fractal dimension D can be considered, approximately, as a constant of time.) The possible explanation might be that calculation of the Fourier power spectra is a relatively rough instrument (contrary to, e.g., the wavelet analysis), and reveals only some average turbulence characteristics; for instance, a good deal of information about the phases of the intrinsic turbulence motion can be lost. Thus the Fourier spectrum alone cannot completely distinguish between stationary turbulence convection and the self-pouring process; this "fortunately" enables one to exploit the same spectral exponent a in both the frequency and wavenumber domains by using the formal analogy with the stationary convection of the turbulence fractal structures.

We found in section 5 that the basic (quasi)equilibrium of the magnetotail could be associated with the fractal magnetic field patterns whose Hausdorff fractal dimension D is given by (11). This dimension applies in the range of scales a •0< X •0< • (i.e., 1/• ,0< k •0< I/a). Stationary convection of these patterns by a rest frame observer results in the power law spectrum P(k) cr k -'• where

c• = ½ + 1 m 2.33 •0 7/3, (25)

according to (24). In the frequency domain the spectrum P(f) cr f-'• should be recognized in the range between the two characteristic turnover frequencies, f, ,,0 u/• and f** •0 u/a. Simple numerical estimates

yield f, • 10 -2 Hz and f** • 0.25 Hz in the distant tail; and f, • 10 -2 Hz and f** • 0.1 Hz in the substorm regions of the near-Earth tail, where the results of sections 2 and 3 have been used.

For the frequencies smaller than f, (i.e., X • •), the relevant value of the fractal dimension D should

be unity, D • 1: This value accounts for the modulation of the current sheet structures by the tearing modes and is associated with the chains of islands elon-

gated in x direction (see section 2). From (24) one then obtains a • 1, leading to a more fiat spectrum P(f) below f • f,. This theoretical conclusion is substantiated by the direct spacecraft observations of the magnetotail turbulence, showing a kink around the turnover frequency f, (see section 1).

Our theoretical treatment leads to a possibility of the second kink in the spectrum P(f), around the turnover frequency f**. In fact, at the frequencies greater than f** (i.e., X .•< a), the observer enters the interior of the magnetic fluctuation clumps. The distribution of the magnetic field 5B• at such fine scales might be probably influenced by the dynamics of the electron constituent of the plasma, in view of relatively small electron Lar- mot radius as compared with that for ions. For the sake of simplicity, we neglect the possible existence of very fine scale structures inside the clumps and assume the magnetic field distribution to be (quasi)homogeneous at the scales less than -• a. This gives D • 2 for X .•( a. Hence from (24) one gets a • 3 where f • f**, that is, the spectrum P(f) reveals relatively steep component in the higher frequency range. Experimental veri- fication of this conclusion might be addressed to future studies of the magnetotail turbulence; such a verifica- tion, in particular, might help deeper understand the role played by electrons in the self-organized dynamical structuring of the current sheet. We also note that the turnover frequency f** -• u/a is practically beyond the typical Nyquist frequency in the spacecraft data presently available; in the Geotail observations, for instance, the Nyquist frequency was found to be • 0.17 Hz [Hoshino et al., 1994].

An increase of the Hausdorff fractal dimension D

prior to the onset could be recognized as an increase of the spectral exponent a in the frequency range between f, and f** (see (24)). The limiting (onset) value of the fractal dimension D • 3C/2 -• 2 as given by (20) corresponds to the maximum slope

c• = 3C- 1 •0 3. (26)

This estimate coincides with the value of a already obtained for the frequency range f • f**. Hence the onset might be generally characterized by a rectification (in the log-log coordinates) of the spectrum P(f) around the second kink frequency f** and appearance of the unique (maximum) slope a m 3 in the entire frequency range f • f,. Consequently, the limiting (onset) behavior of P(.f) would be a one-kink spectrum, with the turnover frequency around f,.


We now turn to a brief consideration of the current

density fluctuation spectra in the magnetotail. We restrict ourselves to the self-organized regime corresponding to the frequencies f that are between f, and f**. This regime is characterized by the fractal geometry of the electric current networks whose Hausdorff dimen-

sion D + is dual to D. Similar to the above, the spectra of the current density fluctuations behave as a power law, P+(f) ocf -s+ with the dual exponent c• + given by

a + - 5- 2D +. (27)

Relation (27) is analogous to (24)•yielding the slope of the magnetic field fluctuation spectra. The functional difference between expressions (24) and (27) is due to the diversing topological properties of the magnetic field and electric current patterns' The former are composed of the two-dimensional clumps, whilst the latter are assembled of the one-dimensional links. In the general theory of fractals, relation (27) was first obtained by Berry [1979]; relevant topological issues were discussed by Milovanov and Zimbardo [2000].

Exponent (27) might be roughly exploited when considering velocity fluctuations in the magnetotail current sheet. Indeed, our theoretical model assumes substantial nonadiabaticity of the ion dynamics in the sheet, that is, no "frozen-in" conditions are applied to the current-carrying particles. This means that the particle distribution in space does not follow the magnetic field distribution associated with the fractal clumps and is actually much more diffusive. (This follows from the estimates of section 3: The typical values of the Larmor radii for the current-carrying particles are comparable with the sizes of the magnetic fluctuation clumps.) The diffusive character of the density distribution in the current sheet was recognized by Borovsky et al. [1998] who found the "degree of lumpiness" of the sheet to be relatively •mall.

Hence the contribution from the particle number density inhomogeneities in the ,cr,3ss-tail current fluctuations is not a significant one, that is, the velocity spectra might roughly approximate the current density fluctuations.

Combining expressions (24) and (27), one finds the duality relation for the spectral exponents c• and c• +'

c• - c• + - 5C - 6 • 0.63 • 2/3, (28)

where (10), (13), and (14) have been used. Equation (28) shows that the slope of the magnetic field fluctuation spectrum exceeds the one for the current density (velocity) fluctuations by • 0.63. The fact that c• • c• + is contrary to what one might expect from the conventional MHD models and addresses the importance of the nonadiabatic plasma particle motion in the magnetotail current sheet (see section 3).

According to the results of section 5, the basic (quasi) equilibrium of the magnetotail corresponds to the Haus-

dorff dimension D + - 2½- 1 • 1.65 (see (16)). Consid- ering (27), one obtains the relevant slope for the current density fluctuation spectrum:

a + - 7-4C • 1.69-• 5/3. (29)

The value a + = 7- 4C appears in the topological description of the sign-symmetric random fields on a plane [Milovanov and Zimbardo, 2000]. It is remarkable to note that the slope (29) was recognized by Borovsky et al. [1997] from the ISEE 2 Fast Plasma Experiment. Moreover, Borovsky et al. [1997] found that the corresponding exponent of the magnetic field fluctuation spectrum a is unambiguously larger than a + and agrees with the value (25). These observational findings support the duality relation (28).

We found in section 6 that the Hausdorff dimension D + decreases prior to the substorm onset. This leads to an increase of the slope a +, that is, the spectrum of the current density fluctuations, P+(f), should become steeper as the onset approaches. This behavior of P+ (f) is similar to that of P(f) and signifies the dominant role of the inverse energy cascades in the current system beyond the marginal (quasi)equilibrium state (i.e., the coarsening of the magnetotail conducting web at the late phase).

The limiting (onset) value of the slope a + can be obtained from (18) and (27):

c• + = 5 - 2C • 2.35 -• 7/3, (30)

which is numerically close to the result (25). It is worth emphasizing that the noticeable increase

in the power exponents c• and c• + can occur just prior to or even during the inflationary stage of the evolution of the current system. This means that the corresponding changes in the slopes of the spectra happen within a relatively short time period (of the order of, say, few minutes) and should be associated with the nonstationary processes preceeding the large-scale current disruption event. To pinpoint these processes from the actually detected signals coming from the magnetotail system at the inflationary stage might be a problem of considerable complexity: The main difficulties could be due to the short-living signals whose frequency con- tent varies with time; for instance, the standard signal- processing techniques based on the Fourier transforms might become invalidated. In this context, the study of Lui and Najmi [1997] should be recognized, who proposed the continuous wavelet transforms (CWT) as the relevant instrument in dealing with the short-duration, time-varying signals. Having performed the CWT analysis of the signals observed in the current disruption events, Lui and Najmi [1997] reported the existence of the energy cascades from high to low frequencies (i.e., from smaller to larger spatial scales) as time progresses (see Plate 2 of their paper). This supports the indica- tion of our model regarding the formation of the coarse


structures (compare Figures 1 and 3) and the ensuing increase of the spectral exponents c• and c• + before the onset.

An important information about the nonstationary physical processes in the tail just prior to the inflation and the consequent current disruption event was obtained by Bauer et al. [1995a] who observed a local short-duration decrease in the magnetic fluctuation amplitudes a few minutes before the onset. Similar behavior was recognized by Ohtani et al. [1995] who found that the magnetic fluctuations are suppressed in each of the components just before the onset of the tail current disruption. This effect could be naturally incorporated in our geometric model: The evolution of the current system from the considerably knotted (see Figure 1) to the released (see Figure 3) configuration implies that the currents tend to concentrate in filaments elongated in y direction, whereas the fluctuating x component of the currents tends to decrease. From Maxwell equations (1) and (2) one then concludes that the magnetic fluctuations 5Bz can loose in the amplitude prior to the onset. This phenomenon might be figuratively termed "calm before substorm." The filamentary structure of the cross-tail currents before the onset was confirmed

by Ohtani et al. [1998]. The feasible existence of the filamentary (released)

configurations (like that shown in Figure 3) might be also substantiated by the observational results of Bauer et al. [1995b]. In fact, the band-like magnetic structures lying between the basic cross-tail current filaments, when conveyed earthward by a rest frame observer, can be seen as a series of peaks in the frequency domain. The characteristic period of the peaks can be estimated as r, • •/u, where • • 6 x 10 s cm is the upper turbulence structuring scale (see section 2), and u • 50 km s -• is the characteristic convection velocity. Hence r, • fj • • 102 s. It is remarkable to note that the intense oscillations with periods of 1-2 min (• 102 s) do occur in the lower frequency range prior to the onset [Bauer et al., 1995b], in accord with the implications of our theoretical study.

8. Conclusions

Our principal conclusion is the efficiency of the geometric language in analyzing the basic properties (both structural and dynamical) of the Earth's magnetotail. The substance of this language is the formulation of the relevant phenomena in terms of algebraic relations between certain geometric quantities which completely describe the topology of the system. The implication of such an approach is the fundamental interconnection between the observable physical processes and the intrinsic topological characteristics of the system; this interconnection establishes important links between physics and topology for the strongly nonlinear dynamical systems such as the Earth's tail, enabling one to analyze the corresponding physical mechanisms

by means of the relevant topological instruments. We found that the proper theoretical basis for the introduction of the geometric language in the magnetotail studies might be the fractal topology, the unconventional formalism combining the ideas of the standard fractal geometry and the topology of manifolds. We believe the possible applications of the fractal topology can open new promising perspectives in the space physics research.

Acknowledgments. We are grateful to Prof. Yu. I. Galperin for the genuine interest in our study. One of the authors (A.V.M.) gratefully acknowledges the very warm hospitality in the University of Calabria where the final ver- sion of this paper was written. This work was made possible by the INTAS project 97-1612. Partial support was received from the RFBR grants 00-02-17127 and 00-15-96631. In Italy the work was sponsored by the Italian MURST, Ital- ian CNR, contracts 98.00129.CT02 and 98.00148.CT02, and Agenzia Spaziale Iraliana (ASI), contract ARS 98-82.

Janet G. Luhmann thanks Masha M. Kuznetsova, Alex J. Klimas, and S. Peter Gary for their assistance in evaluating this paper.

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P. Veltri and G. Zimbardo, Dipartimento di Fisica, Uni- versit• della Calabria, Arcavacata-di-Rende (CS) 87036, Italy. (e-mail: [email protected])

(Received December 6, 1999; revised September 13, 2000; accepted September 13, 2000.)

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Self-organized branching of magnetotail current systems near the percolation threshold