Complexity, magnetic field topology, criticality, and metastability in magnetotail dynamics

Journal of Atmospheric and Solar-Terrestrial Physics 64 (2002) 541 – 549

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Complexity, magnetic +eld topology, criticality, and metastability in magnetotail dynamics Giuseppe Consolinia;∗ , Tom Changb a Istituto

di Fisica dello Spazio Interplanetario, Consiglio Nazionale delle Ricerche, Area di Ricercia Roma-Tor Vergata, Via del Fosso del Cavaliere 100, 00133 Rome, Italy b Center for Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract Recently, it has been shown that certain features of magnetotail dynamics in response to solar wind changes may resemble the behavior of a complex system near a dynamical critical state (Chang, Phys. Plasmas 6 (1999) 4137; Consolini and Chang, Space Sci. Rev. 95 (2001) 309), and of topological phase transitions (Chang, Phys. Scr. (2001), 80). Moreover, the impulsive part of the magnetotail response seems to be well described by cellular automata and other simulation models displaying criticality. Here, the relevance of the magnetic +eld topological disorder will be discussed in connection with observed complexity and near-criticality, showing how the impulsive character of the magnetotail response could be viewed as 5uctuation-induced c 2002 Elsevier Science Ltd. All rights topological transitions among metastable con+gurations of magnetic +eld topology.  reserved. Keywords: Earth’s magnetosphere; Magnetotail dynamics; Complexity; Criticality

1. Introduction The Earth’s magnetosphere, i.e. the near-Earth region of space where the geomagnetic +eld is con+ned by the solar wind, is a highly dynamical system, which continuously exchanges energy, mass and momentum with the solar wind and Earth’s ionosphere. In the framework of statistical mechanics, such a system can be considered as an open system in an out-of-equilibrium con+guration due to the continuous driving of the solar wind. Evidences of this out-of-equilibrium con+guration may be found in the highly structured shape of the magnetospheric cavity and in the existence of a complex system of currents which continuously dissipates energy. As a matter of fact, it is well known in complex system statistical mechanics that the condition of non-equilibrium may force a system to self-organize in a highly structured and ordered state where the appearance of dissipative structures is required to release part of the energy ∗ Corresponding author. Tel.: +39-06-49934564; fax: +39-0649934383. E-mail address: [email protected] (G. Consolini).

transferred by external constraints (Nicolis and Prigogine, 1987). The irregular behavior of the Earth’s magnetosphere is intensi+ed during magnetic storms and substorms, particularly within the dynamical region of the magnetotail. Traditional modeling eAorts for the dynamics of the magnetosphere were based on the magneto-hydrodynamic (MHD) concepts. The classical MHD-approach, dealing with a 5uid-like description of large-scale phenomena and processes, achieved relevant results in the description of global magnetospheric con+gurations and processes, of the large-scale in5uence of the solar wind to the dynamics of the magnetosphere, as well as, of a number of more localized processes, and may be considered to be equivalent to a sort of laminar continuum approximation. On the other hand, the classical MHD approach has been proven to be not always adequate to describe the highly structured and irregular behavior of the magnetospheric dynamics, especially with regard to the substorm onset and related dynamical features at smaller scales down to the kinetic scales (Lui, 2000). Furthermore, some inconsistencies have been noted with one of the most accepted and classical substorm scenario, i.e. the well-known

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near-Earth neutral line (NENL) model (McPherron, 1991; Klimas et al., 2000). In-situ satellite observations and detailed analyses of the plasma sheet dynamics have uncovered relaxation processes at spatiotemporally localized sites associated with the geomagnetic tail activity during substorms (Lui, 1998; Angelopoulos et al., 1999, and references therein). Being based on a set of diAerential equations, the MHD formulation, by itself, is not properly equipped to handle dynamical processes involving complicated topological connectivity at multi-scales (Choudhuri, 1998). This is to say, that since a set of diAerential equations deals with local derivatives (e.g. local changes of physical quantities), a description of global long-range connectivity might be diHcult to achieve using such a formulation. On the other hand, the intermittent and turbulent evolution of complex topological structures is particularly relevant for the understanding of the dynamical processes related to the energy transport and diAusion within the Earth’s magnetosphere. Recently, it has been recognized that for a global understanding of the highly dynamical features of the magnetotail, it is necessary to investigate the underlying macro-, meso-, and micro-scale processes and the space-time cross-couplings among them (Chang, 1998, 1999, 2000; Consolini and Chang, 2001; Lui, 2000). During the past decade, much attention has been devoted to the possible occurrence of low-dimensional chaos in the dynamics of the magnetosphere. Although some controversies existed in the determination of the “dimensionality” of the geomagnetic system when analyzing the spectral and fractal features of the geomagnetic indices (AE-indices, Dst index, etc.), these investigations clearly indicated that, to some extent, the magnetospheric dynamics seems sometimes to be compatible with a low-dimensional dynamical system. However, it has been noted that it is more appropriate to treat the magnetospheric dynamics in the framework of nonlinear input-output dynamical systems, instead of autonomous attractor dynamics (Tsurutani et al., 1990; Baker et al., 1990; Prichard and Price, 1992; Sharma, 1995; Klimas et al., 1996). Chang (1992a, b) suggested a somewhat diAerent approach to the magnetospheric dynamics. He pointed out that some features of the magnetotail dynamics might resemble those of a stochastic system near a dynamical critical point. In this framework, the observed low-dimensional behavior of the geomagnetic indices might be due to a reduced number of relevant eigenoperators (of the parameters characterizing the dynamical system) near a critical point. Moreover, because the correlation lengths among the 5uctuations of the random dynamical +elds are generally long ranged for a nonlinear stochastic system near criticality, the dynamical system should also exhibit scale-invariance. Such a framework when applied to the Earth’s magnetotail which is perturbed continuously by the solar wind, entails a strongly dynamically intermittent scenario, characterized by the phenomena of symmetry-breaking and +rst- and second-order-like phase

transitions among metastable con+gurations (Chang, 1992a, b). More recently, Chang (1998, 1999, 2001) proposed a physically realistic model, where “the generation, dispersing, and merging of multiscale localized coherent plasma structures” could explain the intermittent character of the multiscale dynamics of the magnetotail. Such an emergence of a complex topology of coherent plasma structures was demonstrated in a two-dimensional analog via large-scale numerical simulations (Wu and Chang, 2000). Evidences of a possible intermittent and scale-invariant dynamics for the magnetosphere have been found in analyzing the impulsive character of the magnetotail dynamics both by in situ satellite measurements (Angelopoulos et al., 1999), and by studying the statistical features of the bursty behavior of the AE indices (Consolini et al., 1996; Consolini, 1997, 1999; Consolini and De Michelis, 1998), as well as of the auroral displays (Lui et al., 2000). Other evidences come from numerical simulations of global magnetotail dynamics by means of cellular automata and coupled-map lattice calculations (Chapman et al., 1998, 1999; Takalo et al., 1999; Uritsky and Pudovkin, 1998; Klimas et al., 2000; Consolini and De Michelis, 2001) and renormalization-group studies (Tam et al., 2000). By the way, with respect to this point, very recently Freeman et al. (2000) found that the scale-free dynamics of the global Earth’s magnetosphere, as observed in the energy relaxation events, might partially re5ect a similar behavior of the solar-wind driver, at least with regard to the AE burst lifetimes. Another relevant and related issue is the role that topological disorder in the plasma and neutral sheet regions plays in magnetotail dynamics. Motivated by the Geotail observations of “kink” Fourier power-law spectra of the magnetic +eld 5uctuations in the distant tail regions (Hoshino et al., 1994), Milovanov et al. (1996) suggested that the origin of the scale invariant features of the magnetic +eld 5uctuation spectra may be due to the percolation patterns of the tail current within a turbulent magnetic +eld. The general agreement between their model predictions and the observed spectra suggests that topological disorder plays a relevant role in the dynamical features of energy and plasma transport across the geotail. Moreover, very recently Milovanov et al. (2001a, b) proposed a topological scenario for the substorm onset where the magnetic substorms may be read as a second-order phase transition in which a “gradual topological simpli+cation” of the fractal current pattern takes place due to a current reorganization phenomenon. Evidences of such a topological phase transition with a change in the symmetry features of the current network are supported by the results of Consolini and Lui (1999, 2000) and Sitnov et al. (2000). On the basis of the afore-mentioned results, the investigation of complexity, criticality, magnetic +eld topology, and cross-couplings among 5uctuations of diAerent scales must be considered to be fundamental in the understanding of the dynamics of the magnetotail. Here, our aim is to discuss the emergence and relevance of a complex magnetic +eld topology and of metastability

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in connection with the observed coarse-grained intermittent and turbulent dissipation, showing how a mechanism, different from the “classical” self-organized criticality (SOC) (Bak et al., 1987) and based on 5uctuation-induced topological phase transitions among metastable states, could be able to describe some aspects of magnetotail dynamics. 2. Coherent plasma structures, topological complexity, and metastability 2.1. Emergence of a magnetic 7eld complex topology As already mentioned in the introduction, the magnetotail dynamics requires the investigation of spatial and temporal scales ranging from the microscopic plasma scales to the macroscopic ones encompassing the totality of the magnetosphere as well as the solar wind injection. This conjecture implies that the magnetotail dynamics is intrinsically a multiscale phenomenon involving the interplay of kinetic, intermediate and MHD scale 5uctuations: i.e. complexity. Furthermore, the relevant 5uctuations (in all scales) generally are in the form of relatively coherent structures rather than plane waves. To clarify these ideas based on the role of topological complexity, we +rst consider, as an example, the development of anisotropic coherent structures in a magnetized medium at MHD scales. Consider the following convective expressions in a non-relativistic MHD formulation: dV  = B · ∇B + · · · ; (1) dt @B = B · ∇V + · · · ; (2) @t where the ellipsis represent the eAects of the anisotropic pressure tensor, the compressible and dissipative eAects, and all notations are standard. It is well known that Eqs. (1), (2) allow the propagation of AlfvNen waves. For such waves to propagate the propagation vector k must contain a +eld-aligned component, i.e., B · ∇ → ik · B = 0. However, at sites where the parallel component of the propagation vector vanishes (i.e., at the resonance sites), the 5uctuations are localized. Around these AlfvNenic resonance sites (usually in the form of curves in physical space), it may be shown that the 5uctuations are held back by the background magnetic +eld, forming anisotropic coherent structures usually in the form of 5ux tubes (Chang, 1998, 1999). Although we restricted our discussion of the development of anisotropic coherent structures to AlfvNen resonances, it is apparent that other resonances and coherent plasma structures at various scales can emerge at other plasma wave resonances (whistler modes, lower hybrid waves, etc.) in a continuum plasma. As the coherent structures migrate toward each other, there will be interactions. Sometimes these interactions can lead to the merging of the structures. Moreover, as a consequence of the interactions new plasma 5uctuations and new

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plasma resonances, which are the seeds of new coherent structures (which can be of various scales), will be produced. Consequently, an intricate process of merging, interaction and evolution of these coherent topological structures results. The emerging picture, +rst introduced by Chang (1998, 1999), is highly dynamical and can be pictorially summarized (Fig. 1) in a trilogy among spontaneous 5uctuations, plasma-resonances, and coherent structures. The development of such coherent structures in a two-dimensional analog has been simulated by Wu and Chang (2000), and resembles the behavior of a stirred colloidal suspension (Chang, 2001). As a result of the above picture, a complex topology, not easily envisioned in terms of the elemental (e.g., MHD and=or Vlasov) equations, emerges (Fig. 2). Such a non-trivial magnetic +eld topology can clearly act as a reservoir of con+gurational free energy that can be released during any non-ideal relaxation process that locally or globally modi+es the topology itself (MoAatt, 1978). Moreover, the evolution of such complex topology involves the intermittent turbulent mixing, diAusing, merging of the coherent structures (Fig. 3), that can cause the observed coarse-grained dissipation (Angelopoulos et al., 1999; Lui, 1998) due to energy releases during non-ideal relaxation processes (such as magnetic reconnection) in non-trivial magnetic +eld topologies. Furthermore, the topological disorder related to local coherent plasma and magnetic +eld structures involves an irreducible complexity exhibiting dynamical metastability and irreversibility. As a consequence of the continuous solar wind forcing, under favorable conditions the topological complexity may naturally evolve towards metastable con+gurations perhaps near a state of “forced and=or self-organized criticality” (FSOC) (Chang, 1999; Wu and Chang, 2000; Consolini and Chang, 2001). In this framework, cooperative 5uctuations of magnetic +eld topology may result in a change of the metastable con+guration. Moreover, the emergence of scale-invariance in the coarse-grained dissipation events might be the consequence of the scale-free spontaneous 5uctuation spectra near such a marginally stable and=or critical state. In other terms, the appearance of scale-invariance property and of long-range 5uctuations in the magnetotail dynamics may be due to magnetic +eld 5uctuations, which are not simply driven, as we should expect for a laminar system. A similar picture can be found in the case of spontaneous 5uctuations occurring up to and above an ordinary critical point, in the case of pre-onset 5uctuations in a spinodal decomposition, as well as, in the case of instabilities in non-equilibrium steady states (Chang et al., 1992; Sornette, 1994). This is to say that the origin of criticality, where the term criticality means the occurrence of scale-invariant distributions of events, may be due to many diAerent processes. We want to underline that the emerging dynamical picture does not necessarily imply the occurrence of “classical” SOC (Bak et al., 1987). As a matter of fact, the existence of a complex topology means that the dynamics of the

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Fig. 1. A schematic representation of the dynamical trilogy among spontaneous 5uctuations, plasma-resonances, and coherent structures.

magnetotail must be viewed in terms of noise-induced topological phase transitions among metastable con+gurations, where the word noise refers to both internal spontaneous 5uctuations and externally driven 5uctuations. Therefore, from a dynamical point of view the terminology “forcedand=or self-organized criticality” (FSOC) may be read in the sense of non-equilibrium 5uctuation-induced topological transitions near criticality. 2.2. Possible evidences for metastability and scale-invariance

Fig. 2. An example of a complex 5ux-tube topology (spaghetti-like 5ux tube structure).

In order to clarify the possible occurrence of metastability in the magnetotail dynamics, we have selected a period in which simultaneous data of the IMF and solar wind plasma parameters, and of AE-index are available. These data refer to a two-day period (from December 17, 1994 to December 19, 1994) of moderate activity when provisional AE-index and WIND IMF and solar wind plasma key parameters are available. Provisional AE-index comes from WDC-2, Kyoto, Japan, and is available on Web. The WIND IMF-data are available on Web on courtesy of R.P. Lepping and NASA Goddard Space Flight Center for Magnetic Field

Fig. 3. Evolution of two coherent topological structures (represented in form of 5ux tubes), that can cause the observed coarse-grained dissipation. Steps (a) – (c) refer to situation prior, during, and after merging, respectively.

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occurring in the geotail regions. Moreover, it is commonly accepted that AE-indices are able in some sense to sample the state space of the solar wind-magnetosphere-ionosphere dynamical system. In Fig. 4 (panel a) we have reported the AE-index behavior in comparison with vBs to which AE is well related. While external driving is an high 5uctuating +eld, the impulsive part of the magnetospheric response, as evidenced by AE-index, seems to be a function of the actual con+guration of the geotail. In other terms, it may depend from the metastable con+guration of the magnetotail region. In

Investigation instrument, and K.W. Ogilvie for Solar Wind Experiment instrument. The use of auroral electrojet (AE) index as an indirect measurement of magnetotail dynamics can be justi+ed by the following reasons. As well-known, AE-index is a compound index relative to two distinct dissipative processes—an unloading process, and a directly driven process—(Kamide et al., 1999), While the directly driven component is linked to the enhancement of the convective transport during the substorm expansion phase, the unloading component is generally related to the rapid and bursty energy release

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with a threshold of 70 nT, but selecting only the event whose duration is less than 100 min. Data used in this analysis come from WDC-2, Kyoto (Japan) and refer to the AE-index for a period of about 12 years from 1978 to 1988 plus the year 1975. A simple power-law behavior with an exponential cut-oA

order to separate these contributions, we have applied the local intermittency measure (LIM) analysis, introduced by Farge et al. (1990) to study intermittency eAects in turbulent 5uid 5ows. This technique consists in the visualization of that part of the signal satisfying the condition LIM 2 ¿ 1;

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is recovered in this case. Scaling exponent  is approx. 1 (=[1:09±0:06]). No-extra bump is observed at larger burst size s suggesting the absence of any characteristic scale for these short-time scale dissipative events. Moreover, the inset shows the distribution function relative to the burst lifetime T . A power-law distribution function with scaling exponents  = [1:52 ± 0:03] is found in this case. Regarding the scaling exponents, we may notice that the scaling exponent  is well in agreement with the exponent relative to the quiet-time power dissipation distribution function found by Lui et al. (2000), and that the burst lifetime exponent  is consistent with the one found by Angelopoulos et al. (1999) analysing the BBF lifetimes. This suggests that AE-index bursts belong to the same universality class of phenomena like BBFs and auroral blobs. Moreover, this result seems to con+rm the above picture about a possible near-criticality con+guration and dynamics in the tail regions. Very recently (BoAetta et al., 1999) it has been realized that a crucial quantity to understand if the observed scale-invariance of event size distributions could be ascribed to a “classical” SOC phenomenon or not, is the waiting time statistics. As a matter of fact, BoAetta et al. (1999) showed that in the case of solar 5ares the existence of a power-law waiting time distributions should contrast the hypothesis of an SOC dynamics, conversely supporting the idea that 5ares

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where  (t) is a wavelet function. Here, we used the Morlett wavelet. This technique is able to extract coherent time-frequency structures that are responsible for intermittency in the 5uctuations. Fig. 4 (panel b) shows the obtained results of LIM analysis for the selected period. We may observe the occurrence of periods where such coherent dissipative structures seem directly triggered by external driving and period when no correspondence is observed. Moreover, the structures of the response to solar wind driver seems to be not one-to-one, i.e. similar external input gives rise to diAerent responses. This is clearly the evidence of metastability in the system, if we linked this coherent time-frequency intermittent enhancement of dissipation with unloading processes occurring in the tail regions. Another relevant feature of such coherent dissipative events is that they seem to be characterized by a maximum time-scale length of the order of ≈100 min. Fig. 5 shows the results of a statistical analysis of the size of these coherent dissipative events performed using the same technique described in Consolini (1997, 1999) -1

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magnetic +eld topology of the tail regions. Furthermore, we have looked at some possible indirect evidence of metastability and scale-invariant coarse-grained dissipation events, by investigating the short-time scale features of the AE-index. By considering the complex magnetic +eld topology in the magnetotail regions we introduced new perspectives to the understanding of the magnetotail dynamics, where the topological disorder could play a relevant role in the evolution of the neutral and plasma sheet regions. In the following, we will discuss the relevance of the afore-mentioned concepts in connection with the magnetotail substorm dynamics. As already stated, many features of magnetic substorm dynamics may be understood on the grounds of classical MHD-based phenomenological models, which provide a simpli+ed picture of substorm onset by means of a sort of con+gurational instability associated with reconnection. Such an approach may be viewed as a laminar continuum approach to magnetotail dynamics. Conversely, the understanding of the physical micro-, meso-, and macro-scale processes related to substorm onset would be beyond such an approach, generally requiring the investigation of dissipative events involving 5uctuations above, below and near the ion gyroradius; i.e. multiscale 5uctuations. Speci+cally, in the case of magnetotail dynamics at high values of topological disorder and complexity, multi-scale 5uctuations and topological changes can occur, in5uencing each other to produce unexpected stochastic behavior. The dynamics of such long-range correlated systems is notoriously diHcult to handle either analytically or numerically. Returning to the topological magnetic +eld complexity in the central tail regions, the nonlinear interactions among the coherent structures can be the origin of a chain of events, yielding an intrinsic instability, which is responsible for the substorm onset. As a matter of fact, the local topological

are better described in terms of turbulent dissipation phenomena. To address such a point, using the same thresholding technique described in Consolini (1997, 1999), we have plotted in Fig. 6 the waiting time statistics (i.e. the statistics of the quiescent times) in the case of AE-index for the 12 years. This +gure clearly shows a more or less power-law distribution for the waiting times  at least over ≈1:5 order of magnitude, characterized by a scaling exponent  =[1:36±0:01] (although this exponent is not stable as shown by the inset in Fig. 6). The existence of a waiting times power-law statistics is evidence of an underlying complex dynamics, perhaps time correlated, that could be in contradiction with the “classical” SOC point of view where as underlined by BoAetta et al. (1999) one should expect a Poisson-like statistics of the waiting times. However, in the light of previous considerations on a complex magnetic +eld topology, a reasonable framework to explain the existence of such a temporal correlation among bursts in connection with the previous discussion on complex magnetic +eld topology could be the existence of a fractal set of accessible metastable states where the dynamics of the system occurs via jumps among the diAerent con+gurations in a hierarchical patchy manner (Klafter et al., 1997). Again such an interpretation supports the idea of metastability. Clearly, an alternative explanation may be that the observed time correlation is due to the external solar-wind driver. We shall postpone a detailed discussion of this fractal time statistics in a future work. 3. Discussion and conclusions In previous sections, we have presented a framework for the emergence of complexity and metastability in the

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disorder, induced by the internal and external 5uctuations, may evolve toward a critical point (perhaps like a percolative critical point) where cooperative 5uctuations of the overall topology may take place due to the long-range connectivity. A similar framework was proposed by Milovanov et al. (2001a, b) that describes the magnetic substorm onset in terms of a 2-d → 3-d percolative transition of the magnetotail current topology in a turbulent magnetic +eld, hierarchically structured. The above picture is quite similar to what is observed in a diAerent framework: the origin of biological function in biochemistry (Careri, 1982, 1998). Enzymatic activity is indeed due to strong correlation among the various spontaneous 5uctuations. This set of correlation has been called “functional order”, and involves a chain of events for the emergence of the biological function. In other terms, the temporal and spatial coherence, observed at the largest scale in the magnetotail dynamics, may be due to the occurrence of “functional order”, resulting from the multiscale spontaneous 5uctuations that mediate the interactions among the processes of diAerent scales. In conclusion, we have presented a new phenomenological framework for the magnetotail dynamics, where topological magnetic +eld disorder plays a relevant role. Moreover, a description of the magnetotail dynamics in terms of 5uctuation-induced out-of-equilibrium topological transitions among metastable con+gurations near criticality has been proposed, along with some possible indirect experimental evidences for this metastability. We remark, that the proposed scenario, involving criticality and scale-free dissipation, is not inconsistent with the occurrence of intermittent turbulence, and is more general than the “classical” SOC point of view because a certain amount of tuning would be required for the emergence of complexity and criticality. Clearly, the understanding of the interplay of the diAerent processes and 5uctuations at all scales proposed in this new phenomenological model calls for further investigations by means of numerical simulations, analyses, analytical studies, as well as in situ multipoint satellite observations.

Acknowledgements We are indebted to S. Chapman, C.F. Kennel, A. Klimas, A.T.Y. Lui, M.I. Sitnov, S.W.Y. Tam, J. Takalo, D.J. Tetreault, N. Watkins, C.C. Wu, and L. Zelenyi for useful discussions. G. Consolini is also strongly indebted to Prof. G. Careri of the University of Rome “La Sapienza”. We thank the following people and organizations for providing (on Web) the data used in this work: R.P. Lepping and NASA Goddard Space Flight Center for WIND IMF-data, K.W. Ogilvie for solar wind plasma data from WIND—Solar Wind Experiment instrument, and the WDC-2, Kyoto, Japan, for AE-index.

This work was partially supported by the Italian PNRA, by the Italian CNR by the Italian ASI, and grants from the AFOSR, NSF and NASA of the US government.

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