complexity, forcedand/or self-organized criticality, … · complexity in space plasmas 3 the...

COMPLEXITY, FORCEDAND/ORSELF-ORGANIZEDCRITICALITY, AND TOPOLOGICALPHASETRANSITIONSIN SPACE PLASMAS

TOM CHANG andSUNNY W. Y. TAMCenter for Space Research, Massachusetts Institute of Technology, Cambridge, MA, USA

CHENG-CHINWUDepartment of Physics and Astronomy, University of California, Los Angeles, CA, USA

GIUSEPPECONSOLINIIstituto di Fisica dello Spazio Interplanetario, Consiglio Nazionale delle Ricerche, Rome,Italy

Abstract. Thefirst definitive observation thatprovidedconvincing evidenceindicatingcer-tain turbulentspaceplasmaprocessesarein statesof “complexity” wasthediscovery of theapparentpower-law probabilitydistributionof solarflareintensities.Recentstatisticalstudiesof complexity in spaceplasmascamefrom the AE index, UVI auroralimagery, andin-situmeasurementsrelatedto thedynamicsof theplasmasheetin theEarth’s magnetotailandtheauroralzone.

In this review, we describea theoryof dynamical“complexity” for spaceplasmasystemsfar from equilibrium.Wedemonstratethatthesporadicandlocalizedinteractionsof magneticcoherentstructuresaretheoriginof “complexity” in spaceplasmas.Suchinteractionsgeneratetheanomalousdiffusion,transport,acceleration,andevolutionof themacroscopicstatesof theoverall dynamicalsystems.

Several illustrative examplesare considered.Theseinclude: the dynamicalmulti- andcross-scaleinteractionsof the macro-andkinetic coherentstructuresin a shearedmagneticfield geometry, thepreferentialaccelerationof theburstybulk flows in theplasmasheet,andtheonsetof “fluctuationinducednonlinearinstabilities”thatcanleadto magneticreconfigura-tions.Thetechniqueof dynamicalrenormalizationgroupis introducedandappliedto thestudyof two-dimensionalnonresonantintermittentMHD fluctuationsandan analogousmodifiedforest-firemodelexhibiting forcedand/orself-organizedcriticality [FSOC]andothertypesoftopologicalphasetransitions.

Keywords: complexity, magnetotail,plasmasheet

1. Introduction

Researchactivity in spaceplasmaphysicsis now arriving at an interestingjuncturethatbecomesapparentwhenlookingbackatwhathasbeenaccomplishedin thepastandlooking forward to what will be requiredin the next decade.In this regard,itis notedthat considerableobservationalandtheoreticalattentionhasbeendevotedtowardstheunderstandingof local, point observationsof spaceplasmaphenomenaasthey have beenidentifiedon US/European/Japaneseresearchsatellites.However,it is becomingclearthat importantquestionsthat will be receiving attentionin thecoming years(particularly with the successfullaunchesof the CLUSTER II and

© 2002Kluwer Academic Publishers. Printed in the Netherlands.

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2 Changetal.

IMAGE satellites)areaddressedtoward global andmultiscaleissues:questionsofenergy andmomentumtransportwithin themagnetosphere,ionosphereandtheSun-Earthconnectionregion, andof thenatureof the particlepopulations,their source,entry, energization,diffusion,andultimatelossfrom thesystems.Thus,observationsarebecomingmulti-spacecraftand/ormulti-point in scope,andtheoreticalmodelsarelikewisebeingforcedto confrontissuesof nonlocalityand“complexity”.

In this review, we discussa numberof suchspaceplasmaprocessesof com-plexity, with specialemphasison thedynamicsof themagnetotailandits substormbehavior. We demonstratethat the sporadicandlocalizedinteractionsof magneticcoherentstructuresarisingfrom plasmaresonancesaretheorigin of “complexity” inspaceplasmas.Suchinteractions,whichgeneratetheanomalousdiffusion,transportandevolutionof themacroscopicstatevariablesof theoveralldynamicsystem,maybe modeledby a triggeredlocalizedchaoticgrowth equationof a set of relevantorderparameters.Thedynamicsof suchintermittentprocesseswouldgenerallypavetheway for theglobalsystemto evolve into a “complex” stateof long-rangedinter-actionsof fluctuations,displayingthe phenomenonof forcedand/orself-organizedcriticality (FSOC).The coarse-graineddissipationdue to the intermittentfluctua-tions canalsoinduce“fluctuation-inducednonlinearinstabilities” that can,in turn,reconfigurethetopologiesof thebackgroundmagneticfields.

Theorganizationof thispaperis asfollows.In Section2, wediscusstheorigin ofthestochasticallydistributedcoherentmagneticstructuresin theEarth’smagnetotail.In Section3, we considerthe preferentialaccelerationof the coherentmagneticstructuresin the neutral sheetregion and their relevanceto the interpretationofthe observed fastbursty bulk flows (BBF). In Section4, we considerthe localizedmerging processesof the coherentstructuresandtheir connectionto the observedlocalized,sporadicreconnectionsignaturesin plasmas(e.g.,in theEarth’smagneto-tail). Theconceptandimplicationsof thephenomenonof FSOCarethenintroduced.Theseideascanleadnaturallyto the power-law scalingof the probabilitydistribu-tionsandfractal spectraof the intermittentturbulenceassociatedwith thecoherentstructures.In Section5, we provide someconvincing argumentsfor intermittencyandFSOCin themagnetosphericplasmasheet.In Section6, theconceptof invariantscaling,the dynamicrenormalizationgroup,and topologicalphasetransitionsarediscussed.Thesearethenrelatedto themultifractal scalingacrossvariousphysicalscalesand the fluctuation-inducednonlinearinstabilitiesthat may leadto the trig-gering of global magneticfield reconfigurationssuchas substorms.In Section7,simple phenomenologicaland analogousintermittency modelsare described.Theconclusionsectionthenfollows.

2. Stochastically Distributed Coherent Structures in the Space PlasmaEnvironment

In situobservationsindicatethatthedynamicalprocessesin theplasmaenvironment(e.g.,themagnetotailregion(Angelopouloset al., 1996;Lui, 1996;Lui, 1998;Nagaiet al., 1998)),generallyentaillocalizedintermittentprocessesandanomalousglobaltransports.It wassuggestedby Chang(1998a,b,c;1999)that insteadof considering

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Complexity in SpacePlasmas 3

the turbulenceasa mixtureof interactingwaves,suchtypeof patchyintermittencycouldbemoreeasilyunderstoodin termsof thedevelopment,interaction,merging,preferentialaccelerationandevolutionof coherentmagneticstructures.

Mostfield theoreticaldiscussionsbeginwith theconceptof propagationof waves.For example,in the MHD formulation,one can combinethe basicequationsandexpressthemin thefollowing propagationforms:

ρdV�dt � B � ∇B �� dB

�dt � B � ∇V �� (1)

wheretheellipsesrepresentthe effectsof theanisotropicpressuretensor, thecom-pressibleanddissipative effects,andall notationsarestandard.Equations(1) admitthewell-known Alfvén waves.For suchwavesto propagate,thepropagationvectork mustcontaina field-alignedcomponent,i.e., B � ∇ � ik � B � 0. However, at siteswhere the parallel componentof the propagationvector vanishes(the resonancesites),the fluctuationsare localized.That is, aroundtheseresonancesites(usuallyin the form of curves),it may be shown that the fluctuationsareheld backby thebackgroundmagneticfield, formingcoherentmagneticstructuresin theform of fluxtubes(Chang,1998a,b,c;1999). For the neutral sheetregion of the magnetotail,thesestructuresareessentiallycurrentfilamentsin the cross-taildirection,Fig. 1.Theresultsof a 2D MHD simulationshowing suchcoherentstructuresaregiveninFig. 2.

Generally, therearevarioustypesof propagationmodes(whistlermodes,lowerhybrid waves,etc.) in a spaceplasma.Thus,we envision a correspondingnumberof differenttypesof plasmaresonancesandassociatedcoherentmagneticstructuresthat typically characterizethedynamicsof a plasmamediumunderthe influenceofabackgroundmagneticfield.

Suchcoherentstructuresmay take on the forms of convective structures,prop-agatingnonlinearsolitary waves,pseudo-equilibriumconfigurations,andotherva-rieties.Someof themmay be morestablethanothers.Thesestructures,however,

Figure 1. Cross-sectionalview of flux tubesintheplasmasheetof theEarth’smagnetotail

0 π/2 π 0

π/2

π

x

z

Figure 2. Two-dimensionalMHD coherentstructuresgeneratedby initially randomlydistributed currentfilaments.Shown are themagneticfield vectors.

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4 Changetal.

generallyarenot purely laminarentitiesasthey arecomposedof bundledfluctua-tionsof all frequencies.Becauseof thevery natureof the physicsof complexity, itwill be futile to attemptto evaluateand/orstudyeachof theseinfinite varietiesofstructures;althoughsomebasicunderstandingof eachtype of thesestructureswillgenerallybe helpful in the comprehensionof the full complexity of the nonlinearplasmadynamics.

Thesecoherentstructureswill wiggle, migrate,deform and undergo differenttypesof motions(includingpreferentialacceleration),i.e.,anisotropicstochasticran-domization,underthe influenceof the local andglobalplasmaandmagnetictopol-ogy. It is this typeof stochasticevolution andinteractionof thecoherentstructuresthatcharacterizethedynamicsof turbulentplasmas,notplanewaves.

3. Preferential Acceleration of BBF

Considerthecoherentmagneticstructuresdiscussedin theprevioussection,whichintheneutralsheetregionof theplasmasheetareessentiallyfilamentsof concentratedcurrentsin the cross-taildirection.We expectthat, in a shearedmagneticfield thattypically exists in the region underconsideration,thesemultiscalefluctuationscansupplythe requiredcoarse-grainingdissipationthatcanproducenonlinearinstabil-ities leadingto “X-point-lik e” structuresof the averagedmagneticfield lines. Forexample,for a shearedmagneticfield Bx z � , upon the onsetof suchfluctuation-inducednonlinearinstabilities,the averagedmagneticfield will generallyacquirea z-component.Let uschoosex astheEarth-magnetotaildirection(positive towardtheEarth)andy in thecross-tailcurrentdirection.Thenthedeformedmagneticfieldgeometryafter the developmentof a meanfield “X-point” magneticstructurewillgenerallyhave a positive (negative) Bz componentearthward (tailward) of the X-point. Nearthe neutralsheetregion, the Lorentzforce will thereforepreferentiallyacceleratethe coherentstructures(which areessentiallycurrentfilamentswith po-laritiesprimarily pointing in thepositive y-direction)earthward if they aresituatedearthward of the X-point and tailward if they aresituatedtailward of the X-point,Fig. 3. Theseresultswould thereforematchthegeneraldirectionsof motion of theobservedBBF in themagnetotail(Nagaiet al., 1998).

Wehaveperformedpreliminarytwo-dimensionalnumericalsimulationsto verifytheseconjectures.ThesimulationsarebasedonacompressibleMHD modelthathasbeenusedby us in our previous studiesof coherentmagneticstructures(Wu andChang,2000a,b;2001;ChangandWu, 2002;Changet al., 2002).

In theexample,we have consideredthemotionsof thecoherentmagneticstruc-tures that developedin a shearedmagneticfield upon the initial introduction ofrandommagneticfluctuations.Thesestructureswere generallyaligned in the x-directionnearthe neutral line, z � 0, after someelapsedtime. A positive Bz wasthenapplied.It canbe seen(from Fig. 4; top panel:magneticfields,bottompanel:flow vectors)that,aftersomeadditionalelapsedtime,thecoherentstructures(mostlyorientedby currentsin the positive y-direction), are generallyacceleratedin thepositive x-direction; with one exceptionwherea pair of magneticstructureswithoppositelydirectedcurrentseffectively canceledout the net effect of the Lorentz

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j×B

B

j

j×B

B j

z

x

y

X

z

xX

B

B

Earth Tail

Figure 3. Schematicof a meanfield X-point magneticfield geometry

8π 7π 6π π

2π(a)

z

8π 7π 6π π

2π(b)

x

z

Figure 4. Effect of the Lorentzforce on the motionsof the coherentstructuresin a shearedmagneticfield dueto theapplicationof a uniform field Bz. Themagneticfields (a) andflowvelocities(b) areplottedin a domain6π � x � 8π andπ � z � 2π. Themaximumvelocity in(b) is 0.006.

forceon thesestructures.We have plottedthex-componentof theflow velocitiesvx

dueto thecumulative effect of theLorentz(andpressure)forcesactingon theflow(andin particularthe coherentstructures)after somedurationof time haselapsed(Fig. 5). We notethat therearea numberof peaksandvalleys in the3-dimensionaldisplay, with peakvelocitiesnearlyapproachingthatof theAlfvén speedmimickingthefastBBF thatwereobservedin themagnetotail.It is to benotedthatanindividualBBF eventmaybecomposedof one,two, or severalcoherentmagneticstructures.

In Fig.6,wedemonstratein aself-consistentpictureof whatmightoccurneartheneutralsheetregion of themagnetotail.We injectedrandomlydistributedmagneticfluctuationsin a shearedmagneticfield. The magneticfield geometrythenunder-wentafluctuation-inducednonlinearinstabilityproducinganX-point likemeanfieldmagneticstructure.Thecoherentstructures(alignedin theneutralsheetregion) are

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6 Changetal.

10π

0

π

2π

0

0.02

xz

vx

Figure 5. A 3D perspective plot of vx. Its peakvelocitiesareabout0.02,nearlyapproachingthatof theAlfvèn speed(about0.025basedon B �� 0 � 025).Themaximumvelocity is about3 timeshigherthanthatatearliertime shown in Fig. 4.

10π 5π π

2π

z

5π 0 π

2π

x

z

Figure 6. Contourplot of vx in anX-point meanfield magneticgeometry

subsequentlyacceleratedby the inducedBz away from theX-point in boththepos-itive andnegative x-directions.Contourplots of vx after someelapsedtime clearlyindicatesucheffects.

4. Localized Reconnections, FSOC and Power-Law Scaling

As thecoherentmagneticstructuresapproacheachother, randomlyor dueto externalforcing, they may merge or scatter. Near the neutralsheetregion, most of thesestructurescarry currentsof the samepolarity (in the cross-taildirection).As two

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currentfilamentsof the samepolarity migratetoward eachother, a strongcurrentsheetis generated.The instabilitiesandturbulencegeneratedby this strongcurrentsheetcantheninitiate themerging of thesestructures.The resultsof suchmergingprocessesmight betheoriginsof thesignaturesof localizedreconnectionprocessesdetectedby ISEE,AMPTE, andotherspacecraft.Theobservedlocalizedreconnec-tion signaturesto dateseemto takeplacemainly in domainsizescomparableto thatof theion gyroradius.Thus,veryprobably, mostof theseprocessesareinfluencedbymicroscopickinetic effects.During thesedynamicprocesses,the ionscanapproxi-matelybe assumedto beunmagnetizedandtheelectronsstronglymagnetized,andtheplasmanearlycollisionless.Thiscanleadto electron-inducedHall currents.Withageneralmagneticgeometry, whistlerfluctuationsmayusuallybegenerated.

Now, in analogyto theAlfvénic resonances,singularitiesof k �� k � B � 0 cande-velopat which whistlerfluctuationscannotpropagate.These“whistler resonances”canthenprovide thenuclearsitesfor theemergenceof coherentwhistler magneticstructures,whicharetheanalogof thecoherentAlfvénic magneticstructuresbutwithmuchsmallerscalesizes.Theintermittentturbulenceresultingfrom theintermixingandinteractionsof thecoherentwhistlerstructuresin theintensecurrentsheetregioncanthenprovide the coarse-grainaverageddissipationthat allows the filamentarycurrentstructuresto merge,interact,or breakup.In additionto theabove scenarios,other plasmainstabilitiesmay set in whenconditionsare favorableto initiate themergingandinteractions.

FSOC. As the coherentmagneticstructuresmerge andevolve, larger coherentstructuresareformed.At the sametime, new fluctuationsof varioussizesaregen-erated.Thesenew fluctuationscanprovide thenew nuclearsitesfor theemergenceof new coherentmagneticstructures.After someelapsedtime, we expect the dis-tributionsof thesizesof thecoherentstructuresto encompassnearlyall observablescales.It hasbeenarguedby Chang(1992;1998a,b,c;1999)andbriefly reviewedinSection6 that,whenconditionsarefavorable,a stateof dynamiccriticality (FSOC)might be approached.At sucha state,the structurestake on all scalesizeswith apower-law probability distribution of the scalesizesof the fluctuations,aswell aspower-law frequency (ω) and modenumber(k) spectraof the correlationsof theassociatedfluctuations.Analysesof existing observationsin the intermittentturbu-lenceregion of the magnetotailandthoseconjecturedfrom the AE index seemtoconfirmsuchpredictions(Consolini,1997;Lui, 1998;Angelopouloset al., 1999;Luiet al., 2000;Uritsky et al., 2002).In addition,thesemultiscalecoherentstructuresmayrenderthecoarse-graineddissipationthatsometimesprovidestheseedsfor theexcitationof nonlinearinstabilitiesleadingto, for example,theonsetof substormsintheEarth’smagnetosphere.

5. Observational Evidences for FSOC and Intermittency in SpacePlasmas

Thefirst definitive observationthatprovidedconvincing evidenceindicatingcertainturbulentspaceplasmaprocessesarein statesof forcedand/orself-organizedcriti-cality (FSOC)wasthediscovery of theapparentpower-law probabilitydistribution

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8 Changetal.

2

4�68

1

2

4�68

ƒ [m

Hz]

3130Time

1500

1000

500

0

AE

[nT

]

Figure 7. AE index behavior andtheLIM measurefor thesubstormoccurredon October30,1978

of solarflareintensities(Lu, 1995).Recentstatisticalstudiesof complexity in spaceplasmascamefrom the AE index, UVI auroralimagery, andin-situ measurementsrelatedto thedynamicsof theplasmasheetin theEarth’smagnetotail.For example,the power-law spectraof AE burst occurrencesasa function of AE burst strengthhasprovided an important indication that the magnetospheresystemis generallyin a stateof FSOC(Consolini,1997,2002).More recentevaluationsof the localintermittency measure(LIM) (Fargeet al., 1990)usingtheMorlett wavelet(Figure7,from ConsoliniandChang(2001))clearlyidentifiedtheburstcontributionsto theAEindex, which appearascoherenttime-frequency structures.Theseresultsstronglysuggestthat theEarth’s plasmasheet,particularlyduringsubstormstimes,is burstyand intermittent.Someof the salientfeaturesof the complexity studiesusing theUVI imageryand in-situ flow measurementsin the plasmasheetregion verifyingthesesuggestionsarebriefly describedbelow.

5.1. AURORAL UVI IMAGES

UVI imagesprovide detailedinformationon the dynamicsof spatiallydistributedmagnetotailactivity coveringextendedobservationperiods.It hasbeenshown thatthepositionsof auroralactive regionsin thenighttimemagnetospherearecorrelatedwith thepositionof theplasmasheetinstabilities(Fairfieldet al., 1999;Lyonset al.,1999;Sergeev et al., 1999;Iedaet al., 2001;Nakamuraet al., 2001a,b),whereastim-ing of theauroraldisturbancesprovidesgoodestimatesfor bothsmall-scaleisolatedplasmoidreleases(Iedaet al., 2001)andfor the global-scalesubstormonsettimes(Germany et al., 1998;Newell et al., 2001).

Recently, Lui et al. (2000)usedglobal auroralUVI imageryfrom the POLARsatelliteto obtainthestatisticsof sizeandenergy dissipatedby themagnetosphericsystemasrepresentedby the intensityof auroralemissionon a world-wide scale.They found that the internal relaxationsof the magnetospherestatisticallyfollowpower laws that have the sameindex independentof the overall level of activity.Theanalysisrevealedtwo typesof energy dissipation:thoseinternalto themagne-tosphereoccurringat all activity levelswith no intrinsic scale,andthoseassociatedwith active times correspondingto global energy dissipationwith a characteristicscale(Chapmanet al., 1998).

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More recently, Uritsky et al. (2002)have performedanextensive analysisof theprobability distributionsof spatiotemporalmagnetosphericdisturbancesasseeninPOLAR UVI imagesof the nighttime ionosphere.This statisticalstudy indicatedstablepower-law forms for both the probability distributionsof the integratedsizeandenergy of all 12300auroralimageevents.

5.2. EVIDENCE OF INTERMITTENCY AND POWER-LAW BEHAVIOR OF

BURSTY BULK FLOW (BBF) DURATIONS IN EARTH’ S PLASMA

SHEET

Statisticalanalysesbasedon in-situ datacollectedby the GEOTAIL satellitehaveyieldedconvincingproofof intermittentturbulencewith power-law correlations(An-gelopouloset al., 1999).It wasshown thatthemagnetotailis generallyin abi-modalstate:nearlystagnant,exceptwhendriventurbulentby transportefficient fastflows.Figure 8 displaysthe probability densityfunctions(PDF) of the X-componentofthe inner plasmasheetflows for both the bursty bulk flow (BBF) and non-BBFpopulations.Bothdistributionsareclearlynon-Gaussianandbothcanbefittednicelywith theCastainget al. (1990)distributions;thus,indicatingintermittency for bothcomponentsof flow in theplasmasheet.

Figure9 shows theprobabilitydensityfunctionof theflow magnitudedurationsabove 400km/s in the innerplasmasheet(Angelopouloset al., 1999).Theplasmasheetwasselectedonthebasisof plasmapressure(Pi � 0 � 01nPa) but verysimilarre-sultsareobtainedby confiningthedatain thenear-neutralsheetregionusingplasmabeta(βi � 0 � 5). A powerlaw is clearlyindicated.Thepower-law behavior with spec-tral index �� 1 � 6 remainswhenthe velocity magnitudethresholdis changedto alower or highervalue,andwhendifferentspatialregionsof themagnetotailplasmasheetin the X-Y planeareconsidered,indicating that the resultsarequite robust.The resultsfor linear binning in flow duration,∆t , are identical to the onesfromlogarithmicbinningpresentedin thefigure.Thus,thereis strongindicationthat theenergy dissipationin themagnetotailadheresto thebehavior expectedfrom FSOC.

Figure 8. Probabilitydistributionsof thenon-BBFandBBF x-componentof thenormalizedflowsalongwith bestfits of GaussianandCastaingfunctions(Angelopouloset al., 1999)

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Figure 9. Probability density of BBFburstsin theplasmasheet(Angelopouloset al., 1999)

6. Invariant Scaling and Topological Phase Transitions

In theprevioussection,we mentionedthepossibilityof theexistenceof “complex”topologicalstatesthat canexhibit the characteristicphenomenonof dynamiccrit-icality similar to that of equilibrium phasetransitions.By “complex” topologicalstateswe meanmagnetictopologiesthat are not immediatelydeduciblefrom theelemental(e.g.,MHD and/orVlasov) equations(ConsoliniandChang,2001).Below,we shallbriefly addressthesalientfeaturesof theanalogybetweentopologicalandequilibriumphasetransitions.A thoroughdiscussionof theseideasmaybefoundinChang(1992;1999;2001;andreferencescontainedtherein).

For nonlinearstochasticsystemsnearcriticality, thecorrelationsamongthefluc-tuationsof the randomdynamicalfields areextremelylong-rangedandthereexistmany correlationscales.The dynamicsof suchsystemsarenotoriouslydifficult tohandleeitheranalyticallyor numerically. On the otherhand,sincethe correlationsare extremely long-ranged,it is reasonableto expect that the systemwill exhibitsomesortof invarianceunderscaletransformations.A powerful techniquethatuti-lizesthis invariancepropertyis thetechniqueof thedynamicrenormalizationgroup(Changet al., 1978;1992;andreferencescontainedtherein).As it is describedinthesereferences,basedon the pathintegral formalism,the behavior of a nonlinearstochasticsystemfar from equilibrium may be describedin termsof a "stochasticLagrangianL". Then, the renormalization-group(coarse-graining)transformationmaybeformally expressedas:

∂L�∂l � RL (2)

whereR is therenormalization-group(coarse-graining)transformationoperatorandl is thecoarse-grainingparameterfor thecontinuousgroupof transformations.It willbe convenientto considerthe stateof the stochasticLagrangianin termsof its pa-rameters{ Pn}. Equation(2), then,specifieshow theLagrangian,L, flows (changes)with l in theaffinespacespannedby { Pn}, Fig. 10.

Generally, there exist fixed points (singular points) in the flow field, wheredL�dl � 0. At a fixedpoint (L � or L �� in Fig.10),thecorrelationlengthshouldnot

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Figure 10. Renormalization-grouptrajectoriesandfixedpoints

be changing.However, the renormalization-grouptransformationrequiresthat alllengthscalesmustchangeunderthecoarse-grainingprocedure.Therefore,to satisfyboth requirements,thecorrelationlengthmustbeeitherinfinite or zero.Whenit isat infinity, thesystemis by definitionatcriticality. Thealternativetrivial caseof zerocorrelationlengthwill not beconsideredhere.

To studythestochasticbehavior of anonlineardynamicalsystemnearaparticularcriticality (e.g., the onecharacterizedby the fixed point L � ), we can linearizetherenormalization-groupoperatorR aboutL � . The mathematicalconsequenceof thisapproximationis that,closeto criticality, certainlinearcombinationsof theparam-etersthat characterizethestochasticLagrangianL will correlatewith eachotherintheform of powerlaws.This includes,in particular, the(k, ω), i.e.modenumberandfrequency, spectraof thecorrelationsof thevariousfluctuationsof thedynamicfieldvariables.In addition, it canbe demonstratedfrom sucha linearizedanalysisthatgenerallyonly a small numberof (relevant) parametersareneededto characterizethestochasticstateof thesystemnearcriticality [i.e., low-dimensionalbehavior; seeChang(1992)].

Symmetry Breaking and Topological Phase Transitions. As the dynamicalsystemevolves in time (autonomouslyor underexternal forcing), the stateof thesystem(i.e., the valuesof the set of the parameterscharacterizingthe stochasticLagrangian,L) changesaccordingly. A numberof dynamicalscenariosarepossible.For example,the systemmay evolve from a critical stateA (characterizedby L �� )to anothercritical stateB (characterizedby L � ) asshown in Fig. 10. In this case,thesystemmayevolve continuouslyfrom onecritical stateto another. On theotherhand,the evolution from the critical stateA to critical stateC asshown in Fig. 10would probably involve a dynamicalinstability characterizedby a first-order-liketopologicalphasetransitionbecausethedynamicalpathof evolutionof thestochasticsystemwould have to crossover a coupleof topological(renormalization-group)separatrices.Alternatively, adynamicalsystemmayevolve from a critical stateA toa stateD (asshown in Fig. 10) which may not be situatedin a regime dominatedby any of the fixed points; in such a case,the final stateof the systemwill nolongerexhibit any of the characteristicpropertiesthat areassociatedwith dynamiccriticality. Alternatively, the dynamicalsystemmay deviate only moderatelyfromthedomainof a critical statecharacterizedby a particularfixedpoint suchthat thesystemmaystill displaylow-dimensionalscalinglaws,but thescalinglawsmaynow

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12 Changetal.

bededucedfrom straightforwarddimensionalarguments.Thesystemis thenin aso-calledmean-fieldstate.(For generalreferencesof symmetrybreakingandnonlinearcrossover, seeChangandStanley (1973);Changet al., (1973a;1973b);Nicoll et al.(1974;1976).)

7. Modeling of Dynamic Intermittency

7.1. TRIGGERED LOCAL ORDER-DISORDER TRANSITIONS

As notedin the previous section,the “complex” behavior associatedwith the in-termittent turbulencein magnetizedplasmasmay generallybe traceddirectly tothe sporadicand localizedinteractionsof magneticcoherentstructures.Coherentstructuresmay merge and form moreenergetically favorableconfigurations.Theymayalsobecomeunstable(eitherlinearly or nonlinearly)andbifurcateinto smallerstructures.Theoriginsof theseinteractionsmayarisefrom theeffectsof thevariousMHD and kinetic (linear and nonlinear)instabilitiesand the resultantfiner-scaleturbulences.

Suchlocalizedsporadicinteractionsmaybemodeled(ChangandWu, 2002)bythe triggered(fast) localizedchaoticgrowth of a setof relevant orderparameters,Oi i � 1 � 2 �� N � :

∂Oi�∂t � ψi O � P;c1 � c2 �� cn;τ1 � τ2 �� τn �� (3)

where “ψi” are functionalsof (O, P), P x � t �� Pj j � 1 � 2 �� M � are the statevariables(or control parameters), x � t � are the spatialand temporalvariables,thecn’s area setof triggeringparametersandthe τ’s arethe correspondingrelaxationtime scalescharacterizingthelocalizedintermittencies,which, in general,aremuchsmallerthanthosecharacterizingtheevolutionarytimescalesfor thestatevariables.

For thespecialcaseof asingleone-dimensionalorderparameterO, let usassumethatthedriving termsmaybeexpressedin termsof a real-valuedstatefunction(thelocal configurationalfreeenergy), F O � P;c1 � c2 �� cn;τ1 � τ2 �� τn � , anda noisetermγ, suchthat

∂O�∂t �� ∂F O � � ∂O � γ � (4)

Generally, the topologyof thevalueof thestatefunctionF will containvalleysandhills in therealspacespannedby theorderparametersO at eachgiven x � t � . IfF is real,continuouswith continuousderivatives,thenwemayvisualizeF locally intermsof a real polynomial functionof O. The topologyof the statefunction F O �may take on forms that are (1) locally stable,Figure 11a, (2) locally nonlinearlyunstable(bifurcation), Figure 11b, and (3) locally linearly unstable(bifurcation),Figure11c.States(2) and(3) maybetriggeredby a critical parameterc � 0 (with acorrespondingrelaxationtimescaleτ with or withoutnonlinearfluctuations.For thetriggeringof a locally nonlinearinstability, thetriggeringparameterwould typicallybe somesort of a measureof the amplitudeof the local nonlinearfluctuations.Forthe triggeringof a local linear instability, the triggeringparameterwould generallybesomesortof ameasureof theamplitudeof thelocalgradientof thestatevariable,P. (We note that a similar but symmetricalmodel basedon the Landau-Ginsburg

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Figure 11. Topologiesof statefunctionF � O �

expansionhasbeenconsideredby Gil andSornette(1996).)Alternatively, a locallybifurcatedstatesuchasstate(3) maybeinfluencedby anothertriggeringparameterc � 0 suchthat a more preferredlocal statesimilar to that of state(1) becomesenergeticallymorefavorablethanthatof thebifurcatedstate.SeeFigure11d.

Onemayeasilyconnectsuchlocal triggeringbehavior of order-disordertransi-tionsto thoseof the localized,sporadicinteractionsof themagneticcoherentstruc-turesdiscussedin theprevioussection.For example,the situationdepictedby Fig-ure11dcanbeinterpretedasthemergingof two coherentstructures.Also, states(2)and(3) canbeinterpretedasthebifurcationof onecoherentstructureinto two smallercoherentstructuresdue to certainnonlinearor linear plasmainstability. Becausemost of such processesare probably due to the result of certain kinetic effects,theinteractiontime-scaleswould generallybeof theorderof kinetic reaction-timesand thereforemuch shorterthan the systemevolution-time to be describedin thefollowing section.

7.2. TRANSPORT EQUATIONS OF STATE VARIABLES

The (slow time) evolution equationsof the statevariableswill generallyconsistofconvective, forcing, randomstirring, andtransportterms.Typically, they may takeon thefollowing genericform (ChangandWu, 2002):

∂Pj�∂t � ζ j P � O;ν1 � ν2 �� νm � (5)

whereζ j j � 1 � 2 �� M � arefunctionalsof (P, O) andtheν’s areasetof timescalescharacterizingthelongtimesystem-wiseevolutionundertheinfluenceof thechaoticandanomalousgrowthsof thetriggeredorderparameters.

For a one-dimensionalsinglereal-valuedstatevariable,onemight envisagethefollowing typical transportequationin differentialform:

∂P�∂t � f � h P� O � ∂P

�∂x � D∂2P

�∂x2 � g (6)

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14 Changetal.

where f is the forcing term,h P� O � is a convective function,g is a randomnoise,D � D0H O � is theanomaloustransportcoefficientwith H O � , apositive-definitivegrowing functionof themagnitudeof theabsolutevalueof O, andD0, a constant.

From this example (as well as the generalexpression,(5)), we note that theanomaloustransportand convectioncan take on varying magnitudes(becauseoftheir dependenceon theorderparameter)andit is generallysporadicandlocalizedthroughoutthesystem.Suchbehavior naturallyleadsto theevolutionof fluctuationsinto all spatialandtemporalscales,andwould generallyleadthe global systemtoevolve into a “complex” stateof long-rangeinteractionsexhibiting thephenomenonof forcedand/orself-organizedcriticality (FSOC)ashasbeendiscussedextensivelyin several of our previous publications(Chang,1998a,b,c;1999; 2001; Wu andChang,2000a,b;2001).

7.3. AN ILLUSTRATIVE EXAMPLE

We considerbelow a simple 2-D phenomenologicalmodel to mimic nonresonantMHD turbulence.We introducethe flux function ψ for the nonresonanttransversefluctuationssuchthat � ∂ψ

�∂x � ∂ψ

�∂y �!� δBy � δBx � and∇ � δB � 0. Thecoherent

structuresfor sucha systemaregenerallyflux tubesnormalto the 2-D planesuchasthosesimulatedin Fig. 2. Insteadof invoking thestandard2-D MHD formalism,herewe simply considerψ asa dynamicorderparameter. As the flux tubesmergeandinteract,they maycorrelateover longdistances,which, in turn,will inducelongrelaxationtimesnearFSOC.Assuminghomogeneity, wemodelthedynamicsof theflux tubes,in the crudestapproximation,in termsof a classicalTime-DependentLandau-Ginsburg modelasfollows:

∂ψk�∂t �� Γk ∂F

�∂ψ " k � fk (7)

whereψk aretheFouriercomponentsof theflux function,Γk ananalyticfunctionofk2, and fk arandomnoisewhichincludesall theothereffectsthathadbeenneglectedin thiscrudemodel.Weshallassumethestatefunctionto dependontheflux functionψ andthe local “pseudo-energy” measureξ. For nonresonantfluctuations,we shallassumethatdiffusiondominatesover convection.Thus,in additionto the dynamicequation(7), wenow includea diffusionequationfor ξ. In Fourierspace,we have

∂ξk�∂t �� Dk2 ∂F

�∂ξ " k � hk (8)

where ξk are the Fourier componentsof ξ, D k � is the diffusion coefficient,F ψk � ξk � k � is thestatefunction,andhk is a randomnoise.By doingso,weseparatetheslow transportdueto diffusionof thelocal “pseudo-energy” measureξ from thenoiseterm fk.

Underthedynamicrenormalizationgroup(DRG) transformation,thecorrelationfunctionC for ψk shouldscaleas:

eaclC k � ω �#� C kel � ωeaωl � (9)

whereω is the Fourier transformof the time t, l the renormalizationparameterasdefinedin the previous section,and ac � aω � the correlationanddynamicexpo-nents.Thus,C

�ωac $ aω is anabsoluteinvariantunderthe DRG, or, C � ω " λ, where

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λ �� ac�aω. DRG analysesperformedfor Gaussiannoisesfor severalapproxima-

tions(Chang,2002)yield thevaluefor λ to beapproximatelyequalto 1.88– 1.66,andavalueof 1.0for theω-exponentfor thetraceof thetransversemagneticcorrela-tion tensor. Interestingly, MatthaeusandGoldstein(1986)hadsuggestedthatsuchamagneticcorrelationexponentmightrepresentthedynamicsof discretestructuresinpseudo-2DMHD nonresonantturbulence;thus,giving somecredenceto themodelandtheDRGanalysis.

7.4. AN ANALOGOUS EXAMPLE — THE GLOBAL MODIFIED

FOREST-FIRE MODEL

Wedemonstratein thissectionananalogousmodel,whichseemsto encompasssomeof the basic characteristicsof what are expectedfor the dynamicalmagnetotail.The model is the global generalization(Tam et al., 2000) of a “modified forest-fire model” originally introducedby Bak et al. (1990)andmodifiedby DrosselandSchwabl (1992).Let usconsiderarectangulargrid of land,onwhich treesmaygrowat any givensite(i, j) andany time stepn with a probability p. At siteswherethereis a tree, thereis a finite probability f that it might be hit by lightning andcatchfire. If a treecatchesfire at certaintime stepn, thenits neighborwill catchfire at asubsequenttime stepn � 1. It is obviousthatat any giventime stepn, therewill bepatchesof greentrees,patchesof burningtreesandpatchesof emptyspaces,Fig. 12.Onemay associatethe growth of treesas the developmentof coherentstructures,theburningtreesaslocalizedmerging sitesandemptyspacesasquiescentstatesinthemagnetotail,a picturesimulatingthesporadicallygrowing of coherentmagneticstructureswith localizedmerging.

Theincrementalchangesof theprobabilities p � f � andthedensitiesof thegreentrees,burning treesandemptyspaces ρ1 � ρ2 � ρ3 � , characterizethedynamicsof theforestfire. In particular, p � f � arethedynamicparameters,andthedensitiesarethedynamicstatevariables.Following the discussionsgiven in the previous section,we perform a renormalization(coarse-graining)transformationof the parameters

Figure 12. A snapshotof forest-firemodelsimulation.Black, grayandwhite pixels refer toemptysites,trees,andburning trees,respectively. Thedimensionof thesquarelattice is 100% 100siteswith fixedboundaries.

Chang.tex; 17/12/2002; 18:18; p.15

16 Changetal.

p � f � suchthat the phenomenonretainsits essentialbasiccharacteristics.Let ussymbolicallydenotethis transformationas:

p � F p � f ;ρ1 � ρ2 � ρ3 � ; f � G p � f ;ρ1 � ρ2 � ρ3 � (10)

Whenthesystemreachesa steadystate,thedensities, ρ1 � ρ2 � ρ3 � , maybeobtainedfrom the meanfield theory at any level of coarse-grainingand are expressibleintermsof thedynamicparameters p � f � .

In Figure13,weshow theaffinespaceof transformationsof p � f � for aparticularchoiceof renormalization-groupprocedure(Tam et al., 2000) (a global general-ization of the coarse-grainingproceduresuggestedby Loreto et al. (1995)) in thephysicallymeaningfulregion of p � f � between(0, 1). We notethat thereare4 dis-tinct fixedpoints.Eachfixedpoint hasits own distinctdynamiccritical behavior. Inthefigure,therenormalizationtrajectoriesaredisplayedassolid curveswith arrowsindicatingthe directionof coarse-graining.If the dynamicstateof the forest-fireisat a stateA nearthe fixedpoint ii � , thenthe systemwill exhibit the characteristiccritical behavior prescribedby theinvariancepropertiesof thisfixedpointuntil suffi-cientcoarse-graininghastakenplacesuchthat therenormalizedsystemapproachesa point suchas B in a region dominatedby the fixed point i � . At this point, inthecoarse-grainedview, thedynamicalsystemwill now exhibit thecritical behaviorcharacterizedby the fixed point i � . We have calculatedthis crossover (symmetrybreaking)behavior for the power-law index of the probability distribution P s � ofthe scalesizesof the burnt trees,Figure 14. In termsof the magnetotailanalogy,suchtype of crossover behavior might be associatedwith the changeof physicalbehavior from thekinetic stateto theMHD state.

We note that thereis a distinct separatrixconnectingthe fixed points iii � and iv � . Thus,transitionfrom a dynamicstatesuchasB to a dynamicstatesuchasCgenerallycannotbe smoothand is probablycatastrophic.This would be the ana-log of the triggeringof the onsetof substormsdueto coarse-graineddissipationin

Figure 13. Renormalization-grouptrajecto-ries for the modified forest-firemodel.Thefour fixed pointsareindicatedby the labels(i) to (iv).

Figure 14. Crossover between the fixedpoints(ii) and(i), asindicatedby thebreakin the probability distribution of the clustersizeof burnt trees

Chang.tex; 17/12/2002; 18:18; p.16


themagnetotail.Althoughnot detailedhere,therearemany otherpropertiesof thisanalogousmodelthatresemblethemagnetotaildynamics.

8. Summary

We have presenteda dynamicaltheoryof “complexity” for spaceplasmasfar fromequilibrium.Thetheoryis basedon thephysicalconceptsof mutualinteractionsofcoherentmagneticstructuresthatemergenaturallyfrom plasmaresonances.

Models are constructedto representthe local intermittenciesand global evo-lutional processes.Dynamicalrenormalization-grouptechniquesare introducedtohandlethestochasticnatureof themodelequationsthatexhibit “complexity”. Exam-plesareprovided to illustratethe phenomenaof anomaloustransports,preferentialaccelerations,forced and/orself-organizedcriticality [FSOC], symmetrybreakingand generaltopological phasetransitionsincluding the reconfigurationsof meanmagneticfield geometriesdueto coarse-graineddissipation.

Both thephysicalconceptsandmathematicaltechniquesdiscussedin this reviewarenontraditional.Thereadersareencouragedto consulttheoriginal referencesforfurtherin-depthstudiesof thesenew emerging ideas.

Acknowledgements

Theauthorsareindebtedto V. Angelopoulos,S. C. Chapman,P. De Michelis, C. F.Kennel,A. Klimas, A. T. Y. Lui, D. Tetreault,N. Watkins, D. Vassiliadis,A. S.Sharma,M. I. Sitnov, andG. Ganguli for usefuldiscussions.The work of TC andSWYT waspartially supportedby AFOSR,NSFandNASA. Theresearchof CCWwassupportedby NASA. GC thanksthe Italian NationalResearchCouncil (CNR)andthe Italian NationalProgramfor AntarcticaResearch(PNRA) for thefinancialsupport.

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