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ASTROPHYSICALJOURNAL612,276(2004)[e-printastro-ph/0312046] PreprinttypesetusingLATEXstyleemulateapjv.11/12/01 SIMULATIONSOFTHESMALL-SCALETURBULENTDYNAMO ALEXANDERA.SCHEKOCHIHIN,1STEVENC.COWLEY,2SAMUELF.TAYLOR3 PlasmaPhysicsGroup,ImperialCollege,BlackettLaboratory,PrinceConsortRd.,LondonSW72BW,UK JASONL.MARON4 DepartmentofPhysicsandAstronomy,UniversityofRochester,Rochester,NY14627and CenterforMagneticReconnectionStudies,DepartmentofPhysicsandAstronomy,UniversityofIowa,IowaCity,IA52242 AND JAMESC.MCWILLIAMS 5 DepartmentofAtmosphericSciences,UCLA,LosAngeles,CA90095-1565 0 February2,2008 0 ABSTRACT 2 Wereporttheresultsofanextensivenumericalstudyofthesmall-scaleturbulentdynamo. Theprimaryfocus n isonthecaseoflargemagneticPrandtlnumbersPr ,whichisrelevantforhotlow-densityastrophysicalplasmas. a m APr parameterscanisgivenforthemodelcaseofviscosity-dominated(low-Reynolds-number)turbulence.We J m concentrate on three topics: magnetic-energy spectra and saturation levels, the structure of the magnetic-field 6 lines,andintermittencyofthefield-strengthdistribution. Themainresultsareasfollows:(1)thefoldedstructure 1 ofthefield(directionreversalsattheresistivescale, fieldlinescurvedatthescaleoftheflow)persistsfromthe 3 kinematictothenonlinearregime;(2)thefielddistributionisself-similarandappearstobelognormalduringthe v kinematicregimeandexponentialinthesaturatedstate;and(3)thebulkofthemagneticenergyisattheresistive 6 scaleinthekinematicregimeandremainsthereaftersaturation,althoughthemagnetic-energyspectrumbecomes 4 muchshallower.Weproposeananalyticalmodelofsaturationbasedontheideaofpartialtwo-dimensionalization 0 ofthevelocitygradientswithrespecttothelocaldirectionofthemagneticfolds. Themodel-predictedsaturated 2 spectra are in excellent agreement with numerical results. Comparisons with large-Re, moderate-Pr runs are m 1 carried out to confirm the relevance of these results and to test heuristic scenarios of dynamo saturation. New 3 featuresatlargeReareelongationofthefoldsinthenonlinearregimefromtheviscousscaletotheboxscaleand 0 thepresenceofanintermediatenonlinearstageofslower-than-exponentialmagnetic-energygrowthaccompanied / h by anincrease ofthe resistivescale and partialsuppressionof thekinetic-energyspectrumin the inertialrange. p Numericalresultsforthesaturatedstatedonotsupportscale-by-scaleequipartitionbetweenmagneticandkinetic - energies, with a definite excess of magnetic energy at small scales. A physical picture of the saturated state is o r proposed. t s Subjectheadings: magneticfields—MHD—plasmas—turbulence—methods:numerical a : v 1. INTRODUCTION turbulentmotionsoftheconstituentplasmasandexistatscales i X 1.1. Large-andSmall-ScaleMagneticFieldsinAstrophysics below those at which the turbulence is forced. Both types of fields are usually strong enough to be dynamically important. r a Magnetic fields are detected everywhere in the universe: Thechallengeistoconstructatheoryoftheirorigin,evolution, stars,accretiondisks,galaxies,andgalaxyclustersallcarrydy- andstructureconsistentwithobservationsandtounderstandthe namically important magnetic fields (some of the relevant re- rolethesefieldsplayinthedynamicsofastrophysicalobjects. views and observations are Kronberg 1994; Beck et al. 1996; A physically plausible scenario for the origin and mainte- Minter&Spangler1996;Zweibel&Heiles1997;Vallée1997, nanceof these fieldsis a turbulentdynamo, which wouldam- 1998; Balbus & Hawley 1998; Kulsrud 1999; Weiss & To- plify a weak seed field and culminate in a nonlinearsaturated bias 2000; Title 2000; Beck 2000, 2001; Ferrière 2001; Kro- state. Just as magnetic fields can be classified into large- and nbergetal.2001;Kronberg2002;Carilli&Taylor2002;Han small-scale varieties, there are also two kinds of dynamo re- &Wielebinski2002;Widrow2002;Tobias2002;Ossendrijver sponsible for these fields. First, three-dimensional velocity 2003; Han et al. 2004; Brandenburg & Subramanian 2004). fields sufficiently random in space and/or time will amplify There are two kinds of magnetic fields observed. First, there small-scalemagneticfluctuationsviarandomstretchingofthe are large-scale fields, i.e., fields spatially coherent at scales field lines (Batchelor 1950; Zel’dovich et al. 1984; Childress comparabletothesizeoftheastrophysicalobjectthattheyin- & Gilbert 1995). Since it is a small-scale process, we believe habit. Twoexamplesof suchfieldsarethecyclicdipolarfield thatthissmall-scaledynamocanbestudiedinahomogeneous of the Sun and the spiral fields of galaxies. Second, there are andisotropic setting. In contrast, the generationand structure small-scale fields: e.g., the fluctuating fields in the solar pho- ofthelarge-scalemagneticfieldscannotbeunderstoodwithout tosphere, galaxies, and clusters. They are associated with the goingbeyondthehomogeneouspictureandtakingintoaccount 1Present-timeaddress:DAMTP/CMS,UniversityofCambridge,WilberforceRoad,CambridgeCB30WA,UK;e-mail:[email protected]. 2AlsoattheDepartmentofPhysicsandAstronomy,UCLA,LosAngeles,CA90095-1547. 3Present-timeaddress:DepartmentofPhysics,PrincetonUniversity,Princeton,NJ08544 4Present-timeaddress:DepartmentofAstrophysics,AmericanMuseumofNaturalHistory,West79thSt.,NewYork,NY10024-5192 1 2 SCHEKOCHIHINETAL. large-scale object-specific features: boundary conditions, ro- theoryofturbulence(e.g.,Frisch1995)givesl Re- 3/4l .The ν 0 ∼ tation, helicity, mean velocity shear, density stratification etc. hydrodynamicReynoldsnumberRe u l /ν is usuallyfairly 0 0 ∼ These effects can combine with turbulence to give rise to the largeforastrophysicalsystems,sol l (u isthetypicalve- ν 0 0 ≪ secondkindof dynamo: the large-scaledynamo. Thisis usu- locity at the outer scale, ν is the fluid viscosity). This is the ally handledin the mean-field framework(e.g., Moffatt1978; secondscaleseparationintheproblem. Itisatscalesbetween Parker1979;Ruzmaikinetal.1988;Brandenburg&Subrama- the outer and the inner scale — the inertial range — that the nian2004). Mean-fieldtheoriestendtoassume thatallsmall- universalphysicsofturbulenceiscontained. scale magnetic fluctuations result from the shredding of the If magnetic fields are present, they bring in their own dis- mean field by the turbulence. Such inducedsmall-scale fields sipation scale l associated with the magnetic diffusivity η of η doexist,buttheyvanishifthemeanfieldvanishesandaredis- the plasma (Spitzer 1962). The ratio Pr =ν/η is called the m tinctfromthedynamo-generatedsmall-scalefields. Neglecting magneticPrandtlnumber. UsingtheSpitzervaluesforthevis- thelatteris,infact,notapriorijustifiedbecausethesmall-scale cosity and magnetic diffusivityof a fully ionized plasma, one dynamoisusuallymuchfasterthanthelarge-scaleone: itam- finds Pr 10- 5T4/n, where T is temperature in K and n is m plifies magnetic energy at the rate of turbulent stretching and the particl∼e concentration in cm- 3. In the case of a partially produces dynamically significant fields before the large-scale ionizedgaswithneutral-dominatedviscosity(e.g.,warminter- fields can grow appreciably (e.g., Kulsrud & Anderson 1992; stellar medium), we can estimate ν v /σn, where v is the th th ∼ Kulsrud 1999). Procedures that have been devised for incor- thermalspeedandσtheatomiccross-section. Theformulafor poratingtheeffectofthesmall-scalefieldsintothemean-field Pr isthenPr 107T2/n.Inhotlow-densityplasmassuchas m m ∼ theory(Vainshtein&Kichatinov1983;Rädleretal.2003,and (warm) interstellar and intracluster media, as well as in some referencestherein)usuallydonotevolvethe small-scalecom- accretion disks, Pr 1 (see, e.g., Brandenburg& Subrama- m ≫ ponentself-consistently,requiringinsteadcertainstatisticalin- nian 2004). Systems with Pr 1 will be the primary focus m ≫ formationaboutthesmall-scalefieldstobesuppliedatthemod- of this paper. If a weak magnetic field is amplified by Kol- ellingstage.Thus,solvingthelarge-scaleproblemispredicated mogorovturbulence,the magnetic-energyexponentiationtime onmakingcorrectassumptionsaboutthesmallscales. isthesameastheturnovertimeoftheviscous-scaleeddies,be- Whether the magnetic fields are dynamo-generatedor stem causetheseeddiesarethefastestones. Balancingtheviscous- from primordial and/or external mechanisms (Kulsrud et al. eddyturnovertimewiththemagnetic-fielddiffusiontimegives 1997a;Kronbergetal.2001),thekeyquestionishowtheyare anestimatefortheresistivescaleinthekinematic(weak-field) maintained in the course of their interaction with the ambient regime: l Pr- 1/2l . As Pr 1, l l . Thisis the third turbulence. Unlessthelarge-scalefieldisextremelystrong,we scale sepaηra∼tionmin tνhe problemm≫. Theηs≪caleνrange in between mustalwaysaskwhyitisnotquicklychurnedupbythesmall- contains degrees of freedom that are accessible to magnetic scale turbulence. Any large-scale field configuration must be fluctuationsbutnottovelocities(Fig.1). Itishardlysurprising consistentwiththepresenceandcontinuedregenerationofcon- thatthesedegreesoffreedomarequicklyoccupied: thesmall- siderableamountsofsmall-scalemagneticenergy. Ingalaxies scalekinematicdynamospreadsmagneticenergyoverthesub- andclusters,bothlarge-andsmall-scalefieldsareinthemicro- viscousrangeandpilesitupattheresistivescale(e.g.,Kulsrud gauss range, which corresponds to magnetic-energy densities & Anderson 1992). The key question then is what happens inapproximateequipartitionwiththekinetic-energydensityof when the small-scale fields become sufficiently strong for the theturbulentmotionsintheseobjects.Thissuggeststhatweare Lorentzforcetoreactbackontheflow. observingaself-consistentnonlinearstateofMHDturbulence. While it is surely the interplay between motions and magnetic fields at disparate scales that ultimately determines the natureoftheobservedstates,theaboveconsiderationsmotivate ustofocusonthehomogeneousisotropicsmall-scaleMHDtur- bulence. Fromthe pointof viewof theoreticalphysics, this is anattempttounderstandtheuniversalfeaturesbeforetackling the object-specificones. From the pointof view of numerical experimentation,thisapproachisunavoidablebecausesimulat- ingthefullpicturerequiresresolvingmultiplescaleseparations thatare well beyondthe capabilitiesofcurrentcomputersand arelikelytoremainsoformanyyearstocome. 1.2. TheScalesintheProblem FIG.1.—Sketchofscalerangesandenergyspectrainalarge-Prmmedium. Theissueofmultiplescalerangesisimportant,soletusex- Wehaveidentifiedthreescaleranges,L l l l . To 0 ν η amineitinmoredetail.Thehierarchyofscalesinanastrophys- give a concrete example, let us give order≫-of-m≫agnit≫udeesti- icalobjectcanbeoutlinedasfollows. Thelargestspatialscale mates for these scales in our Galaxy. The disk diameter is is the system size L. The energy that feeds the turbulence is L 104 pc; the supernova scale at which the turbulence is injectedattheouter,orforcing,scalel0. Forobjectswithnon- sti∼rredisl0 102pc;theReynoldsnumberisRe 105,sothe ttrhievifiarlsgtescoamleetsreypsauracthioansisntathrseoprrogballeamct.icItdiissuksse,dl0b≪ymLe.aTnh-fiiselids vsoistchoeursesscisatl∼ieveisslcνa∼lei1s01- 024pkc;mth,aetPinrayndditsltnaunmcebbeyriG∼saPlramct∼ic1st0a1n4-, theoriestosplitallfieldsintomeanandfluctuatingcomponents dards (note §2.6). The associated timescales are the period withaveragesbeingdoneoverscalesoforderandbelowl0. of Galactic rotation T 108 yr, the outer-eddy turnover time Theenergycascadesfroml0downtotheviscous-dissipation τ0 107 yr, and the vi∼scous-eddy turnover time τν 105 yr. scale lν. In purely hydrodynamicsystems, the latter is called Th∼emean-fielddynamotheorygivesalarge-scaleGala∼cticfield the Kolmogorovinnerscale. Kolmogorov’s1941dimensional SIMULATIONSOFTHESMALL-SCALETURBULENTDYNAMO 3 exponentiatinginarotationtimeT,whilethetimescaleforthe have Pr Re1/2 1. Astrophysicalplasmas haveno prob- m ≫ ≫ growthofthesmall-scalemagneticenergyisτ (Kulsrud&An- lem satisfying this condition, but numerically simulating this ν derson 1992). This illustrates our earlier pointthat the small- regimeisnot,asyet,possible. scaledynamotendstobefasterthanthelarge-scaleone. WebelievethatthekeyfeatureofisotropicMHDturbulence is thescale separationthatarises betweenvelocitiesand magneticfields. Inanatrophiednonasymptoticform,thisscalesep- 1.3. SimulatingtheProblem arationisdiscernibleevenatPr =1.Therefore,inmostofthis m Simultaneouslyresolvingallofthesescalesisnotachievable paper(withtheexceptionof§5),wefollowKinneyetal.(2000) innumericalsimulations.Evenifthelarge-scaleobject-specific inchoosingtoresolvethesubviscousrangewhilesacrificingthe features are forsaken and only isotropic homogeneous MHD inertial one. Namely, we set up our simulation in such a way turbulence with large Re and Prm is considered in a periodic thattheforcingandviscousscalesarecomparable,whilemost box, it is still not possible to simulate both the hydrodynamic oftheavailableresolutionisspentonthesubviscousrange.5 At inertialandthemagneticsubviscousrangesinthesamebox. resolutionsof2563,wecanstudyPrandtlnumbersupto2500, In the relevant numerical work so far, a popular course of whichjustbarelyallowsustomakesomestatementsaboutthe actionhasbeentorunsimulationswithPrm=1(Meneguzziet asymptotic Prm 1 limit. Furthermore, this approach allows al. 1981;Kida et al. 1991; Kleva & Drake 1995; Miller et al. ustostudyPr d≫ependenceofourmodel,whichistheonlypa- m 1996; Cho & Vishniac 2000; Mininni et al. 2003; Haugen et rameterscanthatcanatpresentbeafforded.Somepreliminary al.2003,ourrunsa1,a2,andAin§5). Thischoiceisusually resultsobtainedinthesamesetting(atlowerresolutions)were motivatedby the view ofMHD turbulenceas a cascade of in- publishedinSchekochihinetal.(2002c). teracting Alfvén-wave packets (Kraichnan 1965;Goldreich & Sridhar1995). InanAlfvénwave,velocityandmagneticfluc- 1.4. PlanofthePaper tuationsareequal,dissipationoccursateitherviscousorresis- Our numerical set up, equations, and the limitations of the tivescale,whicheverislarger,andwhateverhappensbelowthat model are discussed in more detail in §2. Section 3 is de- scaleisusuallynotexpectedtoaffecttheinertial-rangephysics. votedtoadetailedstudyofthekinematicregime,whichwillbe In fact, this theory only appears to work for anisotropic important for understanding further developments. Section 4 MHDturbulencewithanexternallyimposedstrongmeanfield treatsthenonlinearsaturatedstate. Summariesareprovidedat (Maron&Goldreich2001;Choetal.2002;Mülleretal.2003). theendofeachofthesetwosectionstohelpthereaderidentify In(forced)simulationswithzeromeanfield,oneinvariablysees thekeypoints. In §5, ourunderstandingof thephysicsofthe thatvelocityandmagneticfieldarenotsymmetric,theirstatis- nonlineardynamowithlargeReynoldsnumbersisoutlinedand tics arenotthesame, theydonotdissipate atthesame rate or sometentativecomparisonsaremadewiththeresultsofsimu- at the same scale, and there is no scale-by-scale equipartition lationswithlargeReandPr oforderunity.Section6discusses of kinetic andmagneticenergies, as thereshouldhave beenif m thescopeofapplicabilityofthe large-Pr -dynamoresultsand theturbulencehadbeenpurelyAlfvénic(Maronetal.2004,and m discusses directions of future work. An itemized summary of see§5.1.2).Thepicturethatdoesemergeisratherthatofavery themainresultsandconclusionsofthepaperisgivenin§7. significant excess of magnetic energy at small scales and the magnetic-energyspectra resembling the resistively-dominated 2. THEMODEL spectraofthesmall-scalelarge-Pr kinematicdynamo. m AnaturalstepbeyondPr =1istolookatPrandtlnumbers 2.1. TheEquationsandtheCode m above,butoftheorderof,unity(Meneguzzietal.1981;Chou WeconsidertheequationsofincompressibleMHD: 2001; Brandenburg 2001; Maron & Blackman 2002; Maron d et al. 2004; Archontis et al. 2003a,b; Haugen et al. 2004, our u=ν∆u- p+B B+f, (1) runBin§5)moreorlessinthehopethatessentialfeaturesof dt ∇ ·∇ the Prm≫1 regimewouldbe captured. Thisway, the inertial d B=B u+η∆B, (2) rangemightberesolvable,butthesubviscousone(whosewidth dt ·∇ is Pr1/2)islargelysacrificed. Thisapproachis, tosomeex- whered/dt=∂ +u istheadvectivetimederivative,u(t,x)is ∼ m t ·∇ tent, justified (see §5). However, as the asymmetry between thevelocityfield,B(t,x)isthemagneticfield,andf(t,x)isthe magneticandvelocityfieldbecomesmorepronouncedwithin- body force. The pressure gradient p (which includes mag- ∇ creasingPr ,oneisfacedwiththerealizationthatthePr &1 netic pressure) is determined by the incompressibility condi- m m case representsa nonasymptoticmixed state aboutwhich it is tion u=0. Wehavenormalized pandBbyρand(4πρ)1/2, ∇· very hard to make any clear statements. We have pointed out respectively,whereρ=constisplasmadensity. inanearlierpaper(Schekochihinetal.2002d,seealso§5.2.3) Equations (1-2) are solved in a triply periodic box by the that, inorderfortheeffectsofthescaleseparationbetweenl pseudospectralmethod(seecodedescriptioninMaron&Gol- ν and l on the nonlinear physics to be fully present, one must dreich2001;Maronetal. 2004). Thebodyforcef is random, η 5NotethatanotherapproachinwhichlowReynoldsnumbersareoftenconsideredistostudymagnetic-fieldgrowthandsaturationin,andnonlinearmodifications to,specificdeterministictime-independentortime-periodicvelocityfieldswithchaotictrajectoriesthatareknowntobedynamosinthekinematiclimit(Childress& Gilbert1995).Thebest-documentedvelocityfieldsofthissortaretheABCflows.Thestrategythenistosetupaforcingfunctionthat,intheabsenceofthemagnetic field, wouldreproduceagivenflowasasolutiontotheNavier–Stokes equation, andtokeepRebelowthestabilitythreshold ofthissolution—ortousesome simplifiedmodelofthevelocityequationthataccomplishesthesame,—andtomodelthefast-dynamoregimebymaximizingPrm(Galantietal.1992;Cattaneoetal. 1996;Maksymczuk&Gilbert1999).Thisapproachdiffersfromoursinthatthevelocityfieldisnotrandomintime,norisithomogeneousinspace,sosuchfeatures asstagnationpointsoftheflow,transportbarriers,etc.,mayacquireadegreeofimportancethatcannotbeafeatureofrealturbulence.Theredeemingadvantagehere isinbeingabletodealwithavelocityfieldthat,unlikeinthecaseofrealturbulence,doesnotconstituteanunsolvedproblembyitself,soonehasmorecontrolover thenumericalexperimentandsomeofthenonlineareffectsmaybemorestraightforwardtodetect.Insomeofthesestudies,Reisallowedtoexceedthehydrodynamic stabilitythreshold(Zienickeetal.1998;Brummelletal.2001),inwhichcasethesmall-scaleresultsareprobablyuniversalbecauseoftheusualphysicalexpectation thatthesmall-scalepropertiesofturbulenceareinsensitivetotheparticularchoiceoflarge-scaleforcing.Ofcourse,inorderforsuchauniversalityargumenttowork, Remustbesufficientlylarge,andresolvinglargePrandtlnumbersisagainproblematic. 4 SCHEKOCHIHINETAL. nonhelical, applied at the largest scales in the box (k /2π = IffisGaussian,soisu,butuhasafinitecorrelationtime: 0 1,2), and white in time (i.e., statistically independent at each timestep): 1 fi(t,x)fj(t′,x′) =δ(t- t′)ǫij(x- x′). (3) hui(t,k)uj(t′,k′)i=(2π)3δ(k+k′)2νk2e- νk2|t- t′|ǫij(k), (6) h i Forawhite-noiseforcing,theaverageinjectedpowerisfixed: 1 d where ǫij(k)= d3ye- ik·yǫij(y) is the Fourier transformof the u2 =- ν u2 - BB: u +ǫ, (4) 2dth i h|∇ | i h ∇ i forcingcorrelatRiontensor[eq.(3)],andwetookt,t′ (νk 2)- 1, 0 whereǫ= u f =(1/2)ǫii(0)=const. Thecodeunitsarebased k beingtheforcingwavenumber.Thevelocitycorre≫lationtime onboxsizeh1·aindinjectedpowerǫ=1. is0thenτ (νk 2)- 1. Equations(5-6)arestrictlyvalidonlyif c 0 AllrunsarelistedinTable1.RunsS0-S6areviscositydomi- νk2 ku∼for all k. In our simulations, we chose ν so that k nated(§2.3)withS4beingthemaintime-historyrun.Runsa1, νk 2≫k u ( k k ),i.e.,justlargeenoughfortheinertial a2,A,andBhavelargerRebutsmallerPrm(see§5). ran0ge∼to0cokl0lap⇔se.νT∼heo0bviousestimateforthevelocitycorre- lationtimeinthisregimeisτ (k u )- 1. Giventhelimitation 2.2. Averaging c∼ 0 k0 ofnothavinganinertialrange,webelievethistobeasensible Mathematically speaking, all fields are randombecause the wayofmodellingturbulentmotions. forcingisrandom.Thesystemisisotropicandhomogeneousin Explicit contact can be made between the viscosity- space, soallvectorfieldshavezeromeanandallpointsin the dominatedmodelandtheanalyticalresultsfromthekinematic- boxareequallynonspecial. Intheory,theanglebracketsmean dynamo theory. One of the very few time-dependentrandom ensembleaverageswithrespectto theforcingrealizationsand flowsforwhichkinematicdynamoisexactlysolvableisaGaus- to the initial conditions. In practical measurements, one can sianwhitevelocityfieldv(t,x): (based on the usualergodicassumption)use volumeand time averaginginstead. Inwhatfollows,wherevertimehistoriesare vi(t,x)vj(t′,x′) =δ(t- t′)κij(x- x′). (7) plotted,onlyvolumeaveragingisdone,whileforquantitiesav- h i eragedoverthekinematicorsaturatedstagesofourruns,both This model was introduced by Kazantsev (1968) and, in the volumeandtimeaveragingareperformed.Errorbarsshowthe contextof passive-scalar advection, independentlyby Kraich- meansquaredeviationsofvolumeaveragesatparticulartimes nan(1968). ForthePr 1problems,manyquantitiesofin- fromthetime-averagedvalues. m ≫ terestthenturnouttodependonlyonthefirstfewcoefficients Averagesinthekinematicregimeareover20codetimeunits (51 snapshots with time separation ∆t =0.4) for runs S0–S4, intheTaylorexpansionofthevelocitycorrelationtensor: over 6 time units (31 snapshots, ∆t =0.2) for runs A and B, over 35 time units (71 snapshots, ∆t = 0.5) for run a1, and κij(y)= κ0δij- 12κ2 y2δij- 12yiyj over 15 time units (31 snapshots, ∆t =0.5) for run a2; aver- +1 κ y2 y2δ(cid:0)ij- 2yiyj +.(cid:1).. (8) agesinthesaturatedstateareover20timeunits(51snapshots, 4! 4 3 ∆t = 0.4) for runs S1–S6, over 10 time units (51 snapshots, (cid:0) (cid:1) ∆t =0.2) for runs A and B, and over 30 time units (61 snap- Increasingν in eqs. (5-6) abovethe value at which kν ∼k0 is shots, ∆t =0.5)forrunsa1 anda2. We havecheckedthatin- equivalenttodecreasingthecorrelationtimeoftheflow. Thus, the velocity field in the viscosity-dominatedmodelreducesto creasingtheaveragingintervalsdoesnotaltertheresults. One the Kazantsev velocity in the limit ν , provided that the code time unit roughly corresponds to the box crossing time →∞ force is rescaled to keep the integral of the time-correlation (roughlytheturnovertimeoftheoutereddies). function constant: ǫij(k)=ν2k4κij(k) (i.e., f=- ν∆v). Since 2.3. TheViscosity-DominatedLimit we owe much of our understanding of the kinematic dynamo to the Kazantsev model (e.g., Kazantsev 1968; Artamonova As we can only adequatelyresolve either the inertialor the & Sokoloff 1986; Kulsrud & Anderson 1992; Gruzinov et subviscousrange,butnotboth,wechoosetoconcentrateonthe al. 1996; Chertkov et al. 1999; Schekochihin et al. 2002b,d, latter. Intherunsreportedin§3and§4(butnotin§5),thevis- cosityisfixedatν=5 10- 2,whicheffectivelyleadstol l . 2004b), it is convenient to be able to consider the nonlinear × 0∼ ν problemasawell-definedextensionofit. Thevelocityfieldintheserunsisnot,strictlyspeaking,turbulent. It is still random in time because the external forcing is random,butitissmoothinspaceasaresultofverystrongvis- 2.4. Incompressibility cousdissipation. Advectionbyarandombutspatiallysmooth flowisknownastheBatchelorregime(Batchelor1959). From We use incompressible MHD even though most astrophys- thepointofviewofrealisticturbulence,suchaflowisprobably ical flows are not incompressible at large scales. For exam- a fairly good model of the viscous-scale eddies acting on the ple, the supernovaforcingof the galactic turbulence produces magneticfluctuationsinthesubviscousrange. Inthisinterpre- sonicvelocitiesattheouterscale. However,motionsatsmaller tation,theexternalforcingfmodelstheenergysupplyfromthe scales are subsonic and mostly vortical and incompressible Kolmogorovcascadetothemotionsattheviscousscale. (e.g., Boldyrev et al. 2002; Porter et al. 2002; Balsara et al. Weemphasizethat,whilefiswhiteintime,theresultingve- 2004).Asweareinterestedinthesmall-scaledynamo,whichis locityis not. Inorderto see this, letusconsidereq.(1)in the drivenbythesmallesteddies,theincompressibilityassumption moredrasticlimitofRe 1, so thehydrodynamicnonlinear- is,therefore,reasonable. Thisviewisfurthersupportedbythe ≪ itycanbeneglected,andlettheLorentzforcesbenegligibleas satisfactory outcomeof a numberof comparisons(carriedout well. Thenthesolutionofeq.(1)inkspaceis by N. E. Haugen 2003, private communication) between our t simulations (§5) and the simulations of Haugen et al. (2003, u(t,k)=e- νk2tu(t=0,k)+ dτe- νk2τf(t- τ,k). (5) 2004),whousedacompressiblegridcode. Z 0 SIMULATIONSOFTHESMALL-SCALETURBULENTDYNAMO 5 2.5. Helicity regime.Thisisaratherdetailedstudy,soabusyreaderanxious to getto the nonlinearmattersmay skipto the shortsummary Our simulations are nonhelical in the sense that the net ki- in§3.5andonto§4. netichelicityiszero, u ( u) =0(helicityfluctuationsare h · ∇× i allowed). Thenethelicityoftheflowcanbecontrolledviathe helicalcomponentoftheexternalforcing(Maron&Blackman 2002). Helicity is usually considered important because it is presentin some of the astrophysicalobjectsof interest. How- ever,nethelicityisalarge-scalefeatureassociatedwithoverall rotation. Itseffectsshouldonlybefeltattimescalescompara- bletotherotationtimescale,whichisusuallymuchlongerthan theturnovertimeoftheturbulenteddies. Furthermore,helicity undergoesan inverse cascade from small scales to the largest scaleinthebox(Frischetal.1975;Pouquetetal.1976).While nethelicityis crucialforthe generationof thebox-scalemag- FIG.2.—Stretchingandfoldingoffieldlinesbyturbulenteddies. neticfield(essentiallythemeanfield),theaboveconsiderations suggest that it probablydoes not play a significant role in the The small-scale turbulent dynamo is caused by the random small-scaledynamo.Intheviscosity-dominatedsimulationsre- stretching of the (nearly) frozen-in field lines by the ambient portedin§3and§4,theboxscaleistheviscousscaleandthe randomflow.Stretchingleadstoexponentialgrowthofthefield externalforcingmodelsenergyinfluxfromlargerscales,soal- strength(Batchelor1950).Thegrowthrateγ uisthennat- ∼∇ lowinghelicalforcingisnotaphysicallysensiblechoice.Inthe urally of the order of the inverse eddy-turnover time. In the large-Rerunsof§5,theeffectofhelicitywouldbeinteresting case of Kolmogorov turbulence, it is the turnover time τν of to study, but it would introduce another scale separation into thesmallest(viscous-scale)eddies,becausetheyarethefastest the problem(betweenthe box size andthe forcingscale) and, (see §5.1.1). For themagneticfieldsat subviscousscales, the therefore,requiredramaticincreasesinnumericalresolution. action of these eddies is roughly equivalentto the application of a random linear shear. Without diffusion (η =0), stretch- 2.6. PlasmaDissipationProcesses ing is not opposed by any dissipation mechanism. However, stretchingisnecessarilyaccompaniedbytherefinementofthe OurchoiceofLaplaciandiffusionoperatorstodescribevis- field scale (Fig. 2), which proceedsexponentiallyfast in time cousandresistivecutoffsisonlyaverycrudemodeloftheac- and also at the eddy-turnoverrate. Thus, the field scale even- tualdissipationprocessesinastrophysicalplasmas. tually becomes comparable to the diffusion (resistive) scale, In a fully-ionizedplasma, ions become magnetized already bringing an end to the diffusion-free evolution. The charac- atrelativelylowmagnetic-fieldstrengthsandthehydrodynamic teristic time during which diffusion-free considerations apply viscosity must be replaced by a more generaltensor viscosity is t γ- 1ln(l /l ) γ- 1ln Pr1/2 , assuming the initial field (Braginskii1965,seealsoMontgomery1992),whichislocally ∼ ν η ∼ m anisotropic and very inefficient at damping velocity gradients variesatthevelocityscales.(cid:0) (cid:1) perpendiculartothemagneticfield: theperpendicularcompo- nentsoftheviscositytensorvanishinthelimitofzeroLarmor radius (the magnetic diffusivity also becomes anisotropic, but the difference between its perpendicular and parallel compo- nents is only a factor of 2). This can alter the physics at ∼ subviscousscales(Malyshkin&Kulsrud2002;Balbus2004). Inpartiallyionizedplasmas(e.g.,intheinterstellarmedium), viscousdissipationiscontrolledbyneutralspeciesandcanbe assumed isotropic. However, in the presence of neutrals, am- bipolardampingmightenhancemagnetic-fielddiffusion(e.g., Zweibel1988;Kulsrud& Anderson1992;Proctor& Zweibel 1992;Brandenburg&Zweibel1994;Brandenburg&Subrama- nian2000;Zweibel2002;Kim&Diamond2002). Finally,sincethemeanfreepathinastrophysicalplasmasis oftenmuchlargerthanthescaleassociatedwiththeSpitzerre- sistivity, the magnetic-field diffusion could be superseded by kineticdampingeffectsnotcontainedin theMHDdescription (Foote&Kulsrud1979;Kulsrudetal.1997b). These and other effects outside the MHD paradigm can be important. However,inthispaper,wehaveoptedto studythe probleminthesimplestavailableformulation:thatprovidedby eqs.(1-2).Webelievethatthisminimalmodelalreadycontains theessentialsofthesmall-scaledynamophysics. FIG. 3.—Exponential growthandsaturation ofthemagneticenergy. Full 3. THEKINEMATICDYNAMO timehistoryisonlyshownforrunS4.SeeTable1fortheindexofruns. The kinematic dynamo is a natural starting point for a nu- Oncediffusionentersthepicture,itwill(partially)offsetthe mericalexperiment.Whatwelearnaboutmagneticfieldsinthe effectofrandomstretching(see§3.3).Ifmagneticenergycon- kinematiclimitwillmotivateourinvestigationofthenonlinear 6 SCHEKOCHIHINETAL. FIG.4.—(a)Evolutionofthemagnetic-energyspectrumforrunS4.Thespectraforthediffusion-freeregime(0≤t<2)aregivenattimeintervalsof∆t=0.4. Thesubsequentevolution(2≤t≤40)isrepresentedbyspectraattimeintervals∆t=2.(b)Magnetic-energyspectra(normalizedbythetotalmagneticenergyand averaged)duringthekinematicstage.NotethatforrunS0dynamoisresistivelysuppressed. tinuesto growexponentiallyinthis regimewith a growthrate etal.1996;Schekochihinetal.2002a) thatdoesnotvanishasη +0,thedynamoiscalledfast(Vain- → shtein & Zel’dovich 1972). Whether any given flow is a fast γ¯ ∂ ∂ dynamo is a problem that very rarely has an analytical solu- ∂M= k2 M- 4kM +2γ¯M- 2ηk2M, (9) t tion (Childress & Gilbert 1995). In practice, since the advent 5 ∂k(cid:18) ∂k (cid:19) ofnumericalsimulations,turbulentflowshavemoreoftenthan not been found to support fast dynamo action (see references whereγ¯=(5/4)κ uiscomparabletotheinverseturnover in§1.3anddiscussionin§6.1). Inwhatfollows,weshallsee 2 ∼∇ timeoftheviscouseddies. Integratingeq.(9),gives thatmagneticfluctuationsthatresultfromthisdynamohavea distinctivestructure(see§3.2and§3.3). Inallourruns,theinitialmagneticfieldhadaspectrumcon- ∂ B2 =2γ¯ B2 - 2ηk2 B2 , (10) centratedatvelocityscalessothatbothdiffusion-freeanddiffu- th i h i rmsh i siveregimesofthekinematicdynamoarerealized. Asaresult ofresolutionconstraints,thediffusion-freestageisneverlonger so γ¯ is the growth rate of the rms magnetic field in the ab- than2timeunits. Thediffusivestageisquitelong,soitispos- sence of diffusion (the stretching rate). Without diffusion, sibletoextractstatisticalinformationbyaveragingover20time eq. (9) describes a spreading lognormal profile with a peak units.Themagneticenergygrowsexponentiallyfromtheinitial moving exponentially fast toward ever larger wavenumbers, valueof 10- 10tosaturatedvaluesoforderunity(Fig.3). k exp[(3/5)γ¯t],theamplitudeofeachmodegrowingex- Inthis∼Section,westudythepropertiesofthegrowingfield. ppoenaken∝tiallyattherate(3/4)γ¯,andthedispersionoflnkincreas- We concentrateon three main questions. First, whatis its en- inglinearlyintimeas(4/5)γ¯t. ergyspectrum? Thisisconsideredin§3.1. Second,whatdoes Once the resistive scale is reached and diffusion becomes the field “look” like? The field structure is studied in §3.2. important,furtherincreaseofk stops. Thespectrumisasolu- Third, howdoesitfill thevolume? Thisquestionistreatedin tionofaneigenvalueproblemforeq.(9)withM(k) exp(λγ¯t). §3.4intermsofthefield-strengthstatistics. §3.3isaqualita- Here λ<2 because diffusiondissipatessome of th∝e dynamo- tiveinterludediscussingthephysicalreasonsfortheexistence generatedenergy: eq.(10)isanexpressionofthiscompetition ofthesmall-scaledynamo. between stretching and diffusion. If λ tends to a positive η- independentconstantin the limitη +0, stretchingwinsand → the dynamois fast. Inorderto solvethis eigenvalueproblem, we must specify the boundaryconditionat small k. In princi- 3.1. TheMagnetic-EnergySpectrumandGrowthRate ple,thisboundaryconditioncanbedeterminedonlybysolving Theevolutionoftheangle-integratedmagnetic-energyspec- the originalintegrodifferentialequation of which eq. (9) is an trum,M(k)=(1/2) dΩ k2 B(k)2 ,inthelarge-Pr regimeis asymptoticformfor k k . This integrodifferentialequation k m ν h| | i ≫ thebestunderstoodRpartofthekinematicdynamophysics. For containsthevelocitystatisticsatallscalesandcannotbesolved the Kazantsevvelocitymodel, M(k) can be shownto satisfy a inageneralandmodel-independentway. However,theeigen- closedintegrodifferentialequation,which,fork k ,reduces valuesarenotverysensitivetothe boundarycondition: anat- ν ≫ to a simple Fokker–Planck equation (Kazantsev 1968; Vain- uralchoiceistoimposeazero-fluxconstraintatsomeinfrared shtein1972,1980,1982;Kulsrud&Anderson1992;Gruzinov cutoff k (Schekochihin et al. 2002d), but a number of other ∗ SIMULATIONSOFTHESMALL-SCALETURBULENTDYNAMO 7 reasonableoptionsgivethesamesolution6 short-scale correlations across. The simplest approach is, in M(k) (const)eλγ¯tk3/2K (k/k ), (11) fact,tomeasureaveragecharacteristicscalesatwhichthefield ≃ 0 η variesin the directionsperpendicularand parallelto itself. In λ 3- π2 , (12) twodimensions,thiswasdonebyKinneyetal.(2000).Inthree ≃ 4 5[ln(k /k )]2 dimensions, the characteristic scales are studied in §3.2.1. A η ∗ moredetaileddescriptionofthefield’s“statisticalgeometry”is wherek =(γ¯/10η)1/2andK istheMacdonaldfunction.Inthe η 0 providedbythedistributionofthefield-linecurvature(§3.2.4) limitη +0,wegetλ 3/4,afastdynamo.Notethat(3/4)γ¯ → → andbyitsrelationtothefieldstrength(§3.2.3). Theadvantage is the rate atwhich individualk-space modesgrow as a result ofthisapproachisthatcurvatureisalocalquantity,soweonly ofrandomstretching,sotheeffectofdiffusionissimplytostop havetolookatone-pointstatistics. Itisalsodirectlyinvolvedin themagneticenergyfromspreadingtoeverlargerk(Kulsrud& theLorentztensionforce[eq.(27)]thusprovidinginformation Anderson1992;Schekochihinetal.2002a). abouttheonsetofthenonlinearback-reaction(§4). Ananalyt- The numerical results broadly confirm this theoretical pic- icaltheorybasedontheKazantsevmodelwasdevelopedinan ture. In Figure 3, we show the evolution of the magnetic en- earlierpaper(Schekochihinetal.2002b).Thenumericalstudy ergy for several values of Pr . The growth is exponential in m belowismotivatedbythequantitativepredictionsmadethere. time and, for Pr =500, the growth rate after resistive scales m are reached (t &2) is reduced compared to the diffusion-free regimebya factorthatisactuallyquiteclose to3/8predicted bytheKazantsevmodel.7 Thegrowthrateatsmallervaluesof Pr issmaller, asa resultofnonnegligiblereductionbydiffu- m sionassuggestedbythelogarithmiccorrectionineq.(12). ThePrandtlnumbermustexceedacertaincriticalvaluePr m,c inorderforthedynamotobepossible. Indeed,ineq.(12),for k k ,wehavek /k Pr1/2andsettingλ=0givesarough ∗∼ ν η ∗∼ m estimateofthethresholdbelowwhichthedynamoisresistively suppressed: Pr 26. In our runs, we have found growth m,c ∼ at Pr =50 and decay at Pr =25, so Pr (25,50). Note m m m,c ∈ that both the Kazantsev velocity with smooth correlation ten- sor(8)andthevelocityfieldinoursimulationsaresingle-scale flows with Re 1, so Pr simply corresponds to the criti- m,c ∼ cal magnetic ReynoldsnumberRm =RePr . Since for our c m,c viscosity-dominatedrunsRe 2,wehaveRm (50,100). c ≃ ∈ Thediffusion-freespreadingofthe magneticenergytoward resistive scales and the subsequent self-similar growth of the spectrumwith a peak atthe resistive scale [eq.(11)]areman- ifest in Figure 4a. The k3/2 scaling appears to be correct (Fig. 4b), although resolving a broader scaling range is nec- essaryfordefinitecorroboration.Theoccasionalstrongdisrup- tions of the self-similar growth (Fig. 4a) are discussed at the FIG. 5.—Crosssectionofthemagneticfieldstrength(grayscale)andin- endof§3.4.3. planefielddirection(redarrows)att=20duringthekinematicstageofrunS4. Lighter regions correspond to stronger fields. Note that what appears to be 3.2. TheFoldedStructure strong-field“clumps”are,infact,crosssectionsoffoldstransversetothepage. Thesmall-scalemagneticfieldsproducedbythedynamoare 3.2.1. CharacteristicScales not, in fact, completely devoid of large-scale coherence. The smallness of their characteristic scale is due primarily to the Wedefinethecharacteristicparallelwavenumberofthefield rapid (in space) field reversals transverse to the local field di- B B2 1/2 rection.Alongthemselves,thefieldsvaryatscalescomparable k = h| ·∇ | i (13) k (cid:20) B4 (cid:21) to the size of the eddies. This folded structure is schemati- h i callyillustratedinFigure2andisevidentinfieldvisualizations anditscharacteristicperpendicularwavenumber (Fig.5). B J2 1/2 There are many ways to diagnose the folded structure. Ott kB×J=(cid:20)h| ×B4 | i(cid:21) , (14) and coworkers studied the field reversals in map dynamos in h i whereJ= B. Theoverallfieldvariationismeasuredbythe termsof magnetic-fluxcancellation: see review byOtt (1998) ∇× rmswavenumber andnumericalresultsbyCattaneoetal.(1995).Chertkovetal. (1999)consideredtwo-pointcorrelationfunctionsof themag- 1 ∞ 1/2 B2 1/2 k = dkk2M(k) = h|∇ | i . (15) neticfieldandfoundlarge-scalecorrelationsalongthefieldand rms (cid:20) B2 /2Z (cid:21) (cid:20) B2 (cid:21) h i 0 h i 6Theseinclude,e.g,requiringthatthemagnetic-fieldsecond-ordercorrelationfunctiondecayexponentiallyatlargedistances(Artamonova&Sokoloff1986;Sub- ramanian1997)orsettingM(k∗)=0(Schekochihinetal.2002a). Kulsrud&Anderson(1992)solvedtheintegrodifferentialequationforthespectrumnumerically, usingaKolmogorovspectrumforvelocity,andalsoobtainedλ=3/4.SeealsoGruzinovetal.(1996)foryetanotherargumentleadingtothesameresult. 7WehesitatetoclaimthatthisisacorroborationofthespecificnumberthatobtainsintheKazantsevmodelratherthanamerecoincidence. TheKazantsevvelocity isδ-correlatedintime,whichcannotbeagoodquantitativedescriptionofthevelocityfieldinoursimulations(see§2.3).Itiswellknownthatfinite-correlation-time effectscanresultinorder-onecorrectionstothedynamogrowthrate(Chandran1997;Kinneyetal.2000;Schekochihin&Kulsrud2001;Chou2001;Kleeorinetal. 2002). WhilethescalingsintheKazantsevmodelmaybeuniversal(e.g.,thepowertailofthecurvaturepdf,§3.2),thevaluesofgrowthratescanonlyaprioribe consideredasqualitativepredictions. 8 SCHEKOCHIHINETAL. FIG.6.—(a)Evolutionofcharacteristicwavenumbers(definedin§3.2.1)andofKrms=h|ˆb·∇ˆb|2i1/2forrunS4.(b)Averagedvaluesofthesamewavenumbers vs.PrmduringthekinematicstageofrunsS0-S4.NotethatforrunS0dynamoisresistivelysuppressed.ThevaluesplottedarelistedinTable1. While both k and k grow exponentially during the Wefind(Fig.7)thatallthesespectragrowself-similarlyand B×J rms diffusion-freeregimeand saturate at valuesthat scale as k k Pr1/2 (cf. Brummell et al. 2001), the parallel wavenuηm∼- M4(k)∼k0, MB×J(k)≃ 41k2M4(k), (22) bνer kmk remains approximately constant and comparable to kν MB·J(k)≃kB2·JM4(k), T(k)≃kk2M4(k), (23) (Schekochihin et al. 2002b): see Fig. 6. For comparison, we withk andk definedbyeqs.(13)and(17). alsoplot(Fig.6)thermswavenumberofthevelocityfield, k B·J This simple behavior is due to the folded structure. If the 1 ∞ 1/2 u2 1/2 √5 magnetic-field variation is dominated by direction reversals, k = dkk2E(k) = h|∇ | i = ,(16) λ (cid:20)hu2i/2Z0 (cid:21) (cid:20) hu2i (cid:21) λ then,forkν ≪k≪kη,themaincontributiontotheintegral where E(k) is the angle-integrated kinetic-energy spectrum [B2](k)2 = and λ is the Taylor microscale of the flow defined in the h| | i Rstean- 1d/a2rld waRye1(/F4rlis,csho1k995).ReI-n1/K4kolm. ogorov turbulence, λ∼ Z d3k′Z d3k′′ hB(k′)·B(k- k′)B(k′′)·B(- k- k′′)i(24) 0 ν λ ν ∼ ∼ Inordertoestimatethedegreeofmisalignmentbetweenthe isfromk′,k′′ k withtheprovisothatk′,k′′,andkareallin direction-reversingfields,itisinstructivetolookatacharacter- ∼ η the direction of the reversal(i.e. transverse to the flux sheet). istic wavenumberoffieldvariationin thedirectionorthogonal Expandingin k, we getto zerothorder, [B2](k)2 B4 tobothBandB×J: k0. Sincethespectrumisessentiallyoneh-|dimensi|onia≃l(pheaike∝d B J2 1/2 forktransversetothefluxsheet),wegetM (k) k0. Therest kB·J=(cid:20)h| B·4| i(cid:21) . (17) ofthefourth-orderspectracanbeworkedou4tin∼thesameway Figure6andTable1showthahtk i k anddoesnotincrease andeqs.(22-23)arereadilyobtained. B·J ν withPr inthekinematicregime. T∼hus,thereversingstraight Theseresultsproveusefulin§4.2and§5.3.4. m fieldsarefairlywellalignedandthefluxsurfacesaresheets. 3.2.3. Magnetic-FieldStrengthandCurvature 3.2.2. Fourth-OrderSpectra Thefieldgeometrycanbestudiedinamoredetailedwayin One can probe further into the behavior of the quantities ˆ ˆ ˆ termsof thecurvatureK=b b ofthe field lines(b=B/B). B B, B J, and B J by lookingat their spectra. All these ·∇ sp·e∇ctra are×, in fact, c·ontrolled by the spectrum of B2, which K= K satisfies(cf.Drummond&Münch1991) | | turnsouttobeflat. Defi1neangle-averagedspectra d K= nˆnˆ: u- 2ˆbˆb: u K+ˆbˆb: u nˆ, (25) M (k)= dΩ k2 [B2](k)2 , (18) dt ∇ ∇ ∇∇ · 4 2Z k h| | i wherenˆ=K/(cid:0)K,andresistiveterm(cid:1)shavebe(cid:0)endrop(cid:1)ped. Equa- 1 M (k)= dΩ k2 [B J](k)2 , (19) tion (25) is a straightforwardconsequenceof eq. (2). A stan- B×J 2Z k h| × | i dardkinematiccalculationofthermscurvature,K = K2 1/2, rms 1 h i M (k)= dΩ k2 [B J](k)2 , (20) shows that it growsexponentiallyin the diffusion-freeregime B·J 2Z k h| · | i (Malyshkin2001;Schekochihinetal.2002b):seeFig.6. Since 1 curvature is an inverse scale, it is clear that in the diffusive T(k)= dΩ k2 [B B](k)2 . (21) k regime,K mustsaturateatsomePr -dependentvalue. 2Z h| ·∇ | i rms m SIMULATIONSOFTHESMALL-SCALETURBULENTDYNAMO 9 FIG.7.—(a)TensionspectraforrunS4for8≤t≤40atintervals∆t=2andinthesaturatedstate.(b)Fourth-orderspectradefinedineqs.(18-21)(normalized byhB4i/2andaveraged)duringthekinematicstageofrunS4.TheevolutionofallthesespectraisanalogoustothatofT(k)shownin(a). The growth of K does not contradict our previous claim is quite wide. It is clear, however, that magnetic fields with rms thatmagneticfieldlinesarestraightuptotheflowscale.Asthe B&B havecurvatureswellbelowK whilethefieldswith rms rms field is stretchedandfolded,itis organizedin longthinstruc- curvaturesK&K arequiteweak. rms tures (folds)where it is only significantlycurvedin the bends Ifweconsidereq.(25)togetherwiththeevolutionequation (turning points). The flow stretches the straight segments of forthemagnetic-fieldstrength, thefield,whilethecurvedfieldsinthebendsremainweak(this issimplya consequenceoffluxandvolumeconservation: see d B= ˆbˆb: u B+η∆B- η ˆb2B, (28) dt ∇ |∇ | Schekochihinetal.2002b,c). Thus,thereisanticorrelationbe- (cid:0) (cid:1) tweenthefield-linecurvatureandthefieldstrength. anddropboththeresistivetermsandthesecondderivativesof Themoststraightforwardstatisticalmeasureofthisanticor- thevelocityfield(bendingterms),wemightobservethat relationisthecorrelationcoefficientdefinedasfollows d 1 K2B2 - K2 B2 dt ln BK1/2 = 2 nˆnˆ:∇u . (29) CK,B= h Ki2 hB2ih i. (26) Theformalsolutionin(cid:0)theco(cid:1)movin(cid:0)gframe(cid:1)is h ih i Inallruns,itisfoundtobewithin6%ofitsminimumpossible 1 t 1 value of - 1 (Table 1). Furthermore, we notice that the mean ln BK1/2 (t)= dt′nˆnˆ: u(t′) ζ (t), (30) n 2Z ∇ ≡ 4 squaretensionforce[seeeq.(1)]is (cid:0) (cid:1) B B2 = B4K2 + B4 B/B2 , (27) whileforthefield-strength,wehave k sPorth)ewfhaicltethhk|2at=·K∇r2mBs|∼ihB|Bh2·∇/BB|i42/Bh4ik|2i∇sallasrogein|(dainicdatgersowthsatwtihthe lnB(t)=Z tdt′ˆbˆb:∇u(t′)≡ 21ζb(t). (31) m k h| ·∇ | i h i∼ ν curvature in the regions of growing field remains comparable It can be shown (A. A. Schekochihin 2002, unpublished), for totheinversescaleoftheflow.8 theKazantsevmodelvelocity(7)andinneglectofbendingand Let us give an illustration of the anticorrelation between B diffusion,thatthejointpdfofζ (t)/t andζ (t)/t isexactlythe b n and K for a typical snapshot of the field. In Schekochihin et sameasthejointpdfofζ (t)/t andζ (t)/t,thefinite-timeLya- 1 2 al. (2002b), we notedthat this anticorrelationwas manifestin punovexponentscorrespondingtothestretchingandthe“null” cross sections of field-strength and curvature. Figure 5 does, directionoftheflow(see§3.3foraquickoverviewoftherele- indeed, showthatfield linesarestraight(anddirectionrevers- vantdefinitions).Thismeansthat,whileζ (t)increaseslinearly b ing) in the areas of strong field. The same pointcan be made withtimeandisresponsibleforthefieldstretching,ζ (t)fluc- n by scatter plots of B versus K during the kinematic stage of tuates around zero. Equation (30) then suggests that BK1/2 is our run S4 (Fig. 8). Since the distributions of both magnetic a special combinationin whichthe effectof stretchingis can- field(§3.4)andcurvature(§3.2.4)areintermittent,thescatter celled.9 Least-squaresfits performedon log-scatterplotsof B 8Thesecondtermineq.(27)representsthecontributionfromthemirrorforce∇ B/B,whichisalsolargeonlyinthebends,whereitsrmsvaluegrowsatthesame k rateasKrms(Schekochihinetal.2002b). Thepdfofthemirrorforcehasapowertailwiththesamescalingasthepdfofcurvature(§3.2.4)andis,therefore,also dominatedbytheouterscales(A.A.Schekochihin2003,unpublished). 9InSchekochihinetal.(2002c),wearguedthatBK∼constbasedonaflux-conservationargument,which,however,involvedsomeadhocassumptionsaboutthe foldgeometry.Constantinetal.(1995)andBrandenburgetal.(1995)alsoarguedinfavorofBK∼constonthebasisofanalternativeformofeq.(25).Thenumerical evidencepresentedhereandin§4.1appearsrathertosupportBK1/2=const.Itis,however,possiblethatintheregionsofstrongfieldandlowcurvature,therelation betweenBandKisclosertoB∼1/K. 10 SCHEKOCHIHINETAL. versusKsuchasFig.8giveBKα=constwithα 0.47,which being a geometricalproperty of the field lines independentof ≃ givesameasureofboththedetailedanticorrelationbetweenB thefluctuatingstretchingrates(seefootnote10). and K andtheextentto whichit canbe understoodviaessen- A stationarypower-likepdfofK ispossible becausecurva- tiallygeometricalargumentssuchastheonewehavejustpre- ture,unlikefieldstrength,hasanexplicitdependencenotonly sented.10 Clearly,theseshouldbreakdownforcurvaturesclose on ubutalsoon u. Thesecondderivativesofuenteras ∇ ∇∇ toeithertheinverseflowscale(K k )ortheinverseresistive a source term in eq. (25) and are responsible for bending the ν ∼ scale(K k ),butitisnontrivialthattheyappeartoworkquite fields. Thus,theflowscaleisexplicitlypresentinthecurvature η ∼ wellawayfromthecutoffs. equationand,consequently,inthecurvaturestatistics. Formallyspeaking,integermomentsofthedistribution(32) diverge:inthediffusion-freecalculationofSchekochihinetal. (2002b),whilethepdfconvergestothestationaryprofile(32), all moments Kn grow exponentially without bound. In the h i problem with diffusion, the power tail of the curvature pdf is cutoffattheresistivescale(K k )andthemomentssaturate η ∼ at values that scale with Pr : e.g., K k Pr2/7 asymptoti- m rms ∼ ν m cally with Pr . The numerical results demonstrate the m →∞ (nontrivial)factthatthecurvaturestatisticsabovetheresistive scale are not affected by the presence of diffusion. In other words,thereisnoresistiveanomalyforthecurvature: η +0 → and η=0 give the same result. Note thatthe use of hyperdif- fusion does not change the scaling of P (K) (Schekochihinet K al.2002c)—anotherindicationthatcurvaturestatisticsdonot depend on the dissipation mechanism. It can be seen in §3.4 that, unlike the curvature, the field strength has statistics that arecruciallyinfluencedbytheresistivecutoff. FIG. 8.—ScatterplotofBvs.Katt=8duringthekinematicstageofour runS4(the2563datawerethinnedoutbyafactorof1000). 3.2.4. TheCurvatureDistribution WehaveseenthatthelargevaluesofK areduetothelarge rms valuesofcurvatureinthebends,wherethefielditselfisweak. Figure8indicatesthattheseonlyoccupyasmallfractionofthe volume(inmostplaces,fieldsarestrongandstraight).Inorder to ascertain that this is true, as well as to get a more detailed statistical descriptionof the field-line geometry,Schekochihin etal.(2002b)foundthepdfofcurvatureanalytically: 6 K P (K)= , (32) K 7 [1+(K/K )2]10/7 ∗ where K = (2κ /7κ )1/2 [see eq. (8) for definitions of κ ∗ 4 2 2 and κ ]. This solution means that in most of the volume, 4 the field-line curvatureis comparableto the inverse eddy size FIG.9.—Curvaturepdf’sforrunS4duringthekinematicregime(withdiffu- (K K k ),whilethedistributionofcurvaturesinthebends sion)andinthesaturatedstate.Curvaturepdf’sforallotherrunsaresimilar. ∗ ν is c∼harac∼terizedby a powerlaw P (K) K- 13/7. This scaling K ∼ A note of caution is in order. We have found in the course is reproducedvery well in our simulations (see Fig. 9). Note ofournumericalinvestigationsthatthegeometricpropertiesof that we find all curvature-related quantities to be quite well the dynamo-generatedfield, while verywellbehavedin terms convergedalreadyaftera relativelyshortrunningtime (unlike of convergence in time, are quite sensitive to spatial resolu- the quantities containing field-strength, which fluctuate very tion. Specifically, with our spectral code, if the magnetic dif- strongly). We believethatthis is due to statistics of curvature 10 The underlying geometrical nature of the curvature-related statistics becomes especially clear in 2D. The curvature and field-strength in 2D are related by BK1/3 =const(again neglecting diffusion andbending). Thisfollows fromthefact that, in2D,nˆ⊥ˆb, so,ˆbbeing thestretching direction, nˆmustnecessarily bethecompressiveone:incompressibilitythenrequiresζn=- ζb,whencedln(BK1/3)/dt=0.AstatisticalcalculationfortheKazantsevmodelshowsthatthepdfof ln(BK1/3)isaδfunction(A.A.Schekochihin2002,unpublished). ApurelygeometricalconsiderationofthefieldlinesalsogivesBK1/3=const(Thiffeault2004). Furthermore,itturnsoutthatthepowertailofthecurvaturepdfin2DderivedbySchekochihinetal.(2002b)fortheKazantsevdynamomodel,PK(K)∼K- 5/3,can bereproducedforthecurvaturedistributionalongagenericparabola(J.-L.Thiffeault2002,privatecommunication). Itisanopenquestionwhethersimilarpurely geometrictreatmentispossiblein3D.