Simulations of small-scale turbulent dynamo

1.7 MB·English

by  A. A. Schekochihin

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 mag-
netic 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 mag-
neticfields. 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,turbu-
lent. 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.