Helicity and alpha-effect by current-driven instabilities of helical magnetic fields

0.93 MB·English

by  M. Gellert

Mon.Not.R.Astron.Soc.000,1–??(2011) Printed5January2011 (MNLATEXstylefilev2.2)
Helicity and alpha-effect by current-driven instabilities of helical
magnetic fields
M. Gellert1(cid:63), G. Ru¨diger1,2, R. Hollerbach3
1AstrophysikalischesInstitutPotsdam,AnderSternwarte16,D-14482Potsdam,Germany
1 2ForschungszentrumDresdenRossendorf,P.O.Box510119,D-01314Dresden,Germany
1 3DepartmentofAppliedMathematics,UniversityofLeeds,Leeds,LS29JT,UK
0
2

n Accepted.Received;inoriginalform
a
J

4 ABSTRACT
Helicalmagneticbackgroundfieldswithadjustablepitchangleareimposedonaconducting

] fluidinadifferentiallyrotatingcylindricalcontainer.Thesmall-scalekineticandcurrenthe-
R
licitiesarecalculatedforvariousfieldgeometries,andshowntohavetheoppositesignasthe
S helicityofthelarge-scalefield.Thesehelicitiesandalsothecorrespondingα-effectscalewith
h. thecurrenthelicityofthebackgroundfield.Theα-tensorishighlyanisotropicasthecompo-
p nentsαφφ andαzz haveoppositesigns.Theamplitudesoftheazimuthalα-effectcomputed
- with the cylindrical 3D MHDcode are so small that the operation of an αΩ dynamo on the
o
basisofthecurrent-driven,kink-typeinstabilitiesoftoroidalfieldsishighlyquestionable.In
r
t anycasethelowvalueoftheα-effectwouldleadtoverylonggrowthtimesofadynamoin
s
theradiationzoneoftheSunandearly-typestarsoftheorderofmega-years.
a
[
Keywords: magneticfields-instabilities-stars:magneticfield-dynamo

2
v
5
3
5 1 INTRODUCTION (thedotsrepresenthigherderivativesofB0)theabove-mentioned
0 signrulesarealsothesignrulesoftheα-effecti.e.α∝g·Ω.
Nohydromagneticdynamocanexistdrivenonlybydifferentialro-
. This sort of α-effect only exists for inhomogeneous turbu-
1 tation(Elsasser1946),butitisknownthatsuchdynamoscanexist
lence. In planetary cores, however, and also in laboratory experi-
0 iftheturbulenceishelicalinthesensethatitskinetichelicity
mentstheonlyinhomogeneitiesresultfromboundaryconditions,
1
1 Hkin =(cid:104)u·curlu(cid:105) (1) as the density gradients are negligible. One can show that under
: thepresenceofmagneticbackgroundfieldsotherinhomogeneities
v and/oritscurrenthelicity alsoformpseudo-scalarsand,asaconsequence,leadtonewmech-
i
X 1 anisms for an α-effect (e.g. Gellert, Ru¨diger & Elstner 2008). In
H = (cid:104)b·curlb(cid:105) (2)
r curr µ0ρ thepresentstudywedemonstratethatinstabilitiesduetoinhomo-
a geneousmagneticfieldsalsoleadtofinitevaluesofthehelicities
donotvanish.HereuandbarethefluctuatingpartsoftheflowU
(1)and(2),andinaccordwith(3)alsotofinitevaluesofα.Indeed,
andmagneticfieldB.Thisconditionofnon-vanishinghelicityis
in the presence of electric currents the simplest existing pseudo-
clearlyfulfillediftheturbulenceisrotatingandstratified.Insuch
scalarisB ·curlB whichdoesnotvanishforhelicalfieldge-
0 0
turbulenceapseudo-scalarexistswhichallowsthepseudo-scalars
ometries.Weshowthatforsuchbackgroundfieldsthesmall-scale
(1)and(2)totakefinitevalues.Thesameistrueforlinearshear
helicitiesobtainfinalvalueswiththeoppositesignasthehelicity
flows where the stratified turbulence in the presence of the shear
ofthebackgroundfield.
alsocanformakinetichelicity(seeRu¨diger&Kitchatinov2006).
AccordingtotheRayleighcriterion,intheabsenceofMHD
Thesimplestpseudo-scalaristhescalarproductg·Ωwithgasthe
effects an ideal flow is stable against axisymmetric perturbations
gradient vector of the turbulence (or the fluid density) and Ω the
wheneverthespecificangularmomentumincreasesoutward
rotationvector.Inspheresthegradientvectorgismainlyradialso
thepseudo-scalarg·Ωhasoppositesignsinthetwohemispheres, d
(R2Ω)2 >0, (4)
and vanishes at the equator. Because of the close relationship of dR
the helicity to the α-effect in the mean-field electrodynamics of
turbulentmedia, whereΩistheangularvelocity,and(R,φ,z)arecylindricalcoor-
dinatesinaright-handedsystem.Inthepresenceofanazimuthal
(cid:104)u×b(cid:105)=αB0+... (3) magneticfieldBφ,thiscriterionismodifiedto
1 d R d (cid:18)B (cid:19)2
(R2Ω)2− 0,φ >0, (5)
(cid:63) E-mail:[email protected] R3dR µ ρdR R
0
(cid:13)c 2011RAS
2 M.Gellert,G.Ru¨digerandR.Hollerbach
where µ0 is the permeability and ρ the density (Michael 1954). β = Bin. (13)
Notealsothatthiscriterionisbothnecessaryandsufficientfor(ax- B0
isymmetric)stability.Inparticular,allidealflowscanthusbedesta- For our standard profile µ = 1 we have B · curlB =
B 0 0
bilized,byazimuthalmagneticfieldswiththerightprofiles(steeply 2β B2/ρR for the helicity of the background field. For fixed
0 in
increasingoutwards)andamplitudes. toroidalfieldamplitudethisquantityscalesasβ−1:
Ontheotherhand,fornonaxisymmetricmodesonehas
2 B2
d (RB2 )<0 (6) B0·curlB0 = 3β Riinn. (14)
dR 0,φ
The sign of β determines the sign of the current helicity. If the
as the necessary and sufficient condition for stability of an ideal
toroidalfieldisduetotheinteractionofapoloidalfieldwithadif-
fluidatrest(Tayler1973).Again,outwardlyincreasingfieldsare
ferential rotation with negative shear then β is negative and vice
thus unstable. If (6) is violated, the most unstable mode has az-
versa.Interchanging±βsimplyinterchangesleftandrightspirals,
imuthalwavenumbersofm=±1.
m→−m.
Asusual,thetoroidalfieldamplitudeismeasuredbytheHart-
mannnumberandtheglobalrotationbytheReynoldsnumber,i.e.
2 EQUATIONS B D Ω D2
Ha= √ in , Re= in . (15)
µ ρνη ν
We are interested in the stability of the background field B = 0
0
(0,Bφ(R),B0), with B0 = const, and the flow u0 = D = Rout −Rin is used as the unit of length, ν/D as the unit
(0,RΩ(R),0). The governing equations for the flow U and the ofvelocityandB astheunitoftheazimuthalfields.Frequencies,
in
fieldBare includingtherotationΩ,arenormalizedwiththeinnerrotationrate
√
∂U 1 Ωin.TheLundquistnumberSisdefinedbyS = PmHa.The
∂t +(U ·∇)U =−ρ∇p+ν∆U + magnetic-diffusionfrequencyisωη = η/D2 andthentheAlfve´n
1 frequencyΩA =Sωηis
+ curlB×B, (7)
µ0ρ B2
Ω = in . (16)
A µ ρD
0
∂B
=curl(U ×B)+η∆B, (8) Throughout the whole paper numerical values of helicities are
∂t
giveninunitsofΩ2D.Inthisnotationthehelicity(14)oftheback-
anddivU =divB =0,wherepisthepressure,νthekinematic A
groundfieldcanbewrittenas
viscosityandηthemagneticdiffusivity.Theirratioisthemagnetic
Prandtlnumber 1 B ·curlB = 2 Ω2D D (cid:39) Ω2AD, (17)
ν µ0ρ 0 0 3β A Rin β
Pm= . (9)
η The boundary conditions associated with the perturbation
From now on we drop the subscripts from the large-scale values equationsareno-slipforuandperfectlyconductingforb,atboth
sothatthetotalflowisU +uandthetotalfieldisB +b.The R=RinandR=Rout,wherewefixRout =2Rin,i.e.ηˆ=0.5.
stationarybackgroundsolutionis The computational domain is periodic in z. The nonlinear MHD
codeusedforthesolutionofEqs.(7)and(8)hasbeendescribedin
b b
Ω=a + Ω, B =a R+ B, (10) detailbyGellert,Ru¨diger&Fournier(2007)(seealsoFournieret
Ω R2 φ B R al.2004,2005).
wherea ,b ,a andb areconstantsdefinedby
Ω Ω B B
µ −ηˆ2 1−µ
aΩ =Ωin 1Ω−ηˆ2 , bΩ =ΩinRi2n1−ηˆΩ2, 3 RESULTS
a = Binηˆ(µB−ηˆ), b =B R 1−µBηˆ, (11) Fig.1showsthegrowthratesforapurelytoroidalfield(β = ∞),
B R 1−ηˆ2 B in in 1−ηˆ2 andnorotation.WeseethatbeyondthecriticalLundquistnumber,
in
thegrowthrateisessentiallylinear,i.e.
with
ηˆ= Rin , µ = Ωout, µ = Bout. (12) ωgr ∝ΩA, (18)
R Ω Ω B B
out in in wherePm = 1hasthesteepestslope,andisthusmoreunstable
R andR aretheradiioftheinnerandoutercylinders,Ω and thanbothPm<1andPm>1.
in out in
Ω aretheirrotationrates,andB andB aretheazimuthal TheazimuthalwavenumberofthemodesshowninFig.1is
out in out
magneticfieldsattheinnerandoutercylinders.Afieldoftheform m = ±1.Forapurelytoroidalfield,±m,correspondingtoleft-
b /R is generated by running an axial current only through the andright-handedspirals,aredegenerate,andnecessarilyhaveex-
B
innerregionR < R ,whereasafieldoftheforma Risgener- actlythesamegrowthratecurves.SeealsoHollerbach,Teeluck&
in B
atedbyrunningauniformaxialcurrentthroughtheentireregion Ru¨diger(2009),whoobtainedthesameeffectinmagnetorotational
R<R ,includingthefluid. instabilities,andRu¨diger,Kitchatinov&Elstner(2011a),whocon-
out
Given the z-component of the electric current, curl B = siderinstabilitiesoftoroidalfieldsinspheres.
z
2a , one finds for the current helicity of the background field Wenextconsiderthenonlinearequilibrationofthesemodes.
B
B·curlB=2a B ,whichmaybepositive,negative,orzero. AsFig.2shows,eventhoughm = ±1aredegenerate,theequi-
B 0
The inner value B is normalized by the uniform vertical libratedsolutionsdonotconsistofequalmixturesofbothmodes.
in
field,i.e. Instead,eithertheleftortherightmodewinsout,andcompletely
(cid:13)c 2011RAS,MNRAS000,1–??
Helicityandalpha-effectbycurrent-driveninstabilitiesofhelicalmagneticfields 3
Figure1. GrowthratecurvesforvariousmagneticPrandtlnumbers,for
stationarycylindersandapurelytoroidalfield,withµB =1.Thegrowth
rateandthemagneticfieldarenormalizedwiththemagnetic-dissipation
frequencyωη.
Figure3. Theturbulenceintensitiesforflow(top)andfield(bottom)fluc-
tuationsinunitsofD2Ω2.Notethatonceβexceeds∼100ithasvirtually
A
nofurtherinfluence.µB =1,µΩ=0.5,Pm=1.
axial component (β < ∞). In contrast to Figs. 1 and 2, we also
includeadifferentialrotationhere.Theprofilesofthebasicstate
Figure2. Theleftandrightspiralsthatcanbeexcitedbyapurelytoroidal fieldandflowarefixedatµ =1andµ =0.5.Theiramplitudes
B Ω
basicstate.Apartfromhavingtheoppositehandedness,thetwomodesare areHa = 100andRe = 200forthefirstseries,andHa = 200,
exactlyequivalent;theirkinetichelicityis±6.0·10−4 andtheircurrent
Re = 20forthesecond.Thefirstseriesisthusrotationallydom-
helicityis±3.5·10−3(bothinunitsofΩ2D),wheretheleftspiralispos-
A inated(Ω > Ω ),whereasthesecondismagneticallydominated
itiveandtherightspiralnegative.Thesemodesdonotdriftintheazimuthal A
(Ω > Ω).Theastrophysicallyrelevantcaseisrotationallydomi-
direction.Theparametervaluesareβ=∞,µB =1,Re=0,Ha=200, A
andPm=1. nated,whichisnottheclassicalrealizationoftheTaylerinstability.
WehavecalledthisinstabilitytheAzimuthalMagnetorotationalIn-
stability(AMRI,seeHollerbach,Teeluck&Ru¨diger2009).
suppressestheother.Whichmodeoneobtainsdependsonthepre- ForbothseriesofrunsFig.3showsthekineticandmagnetic
cise initial conditions. If these already favor one mode, then (not turbulence intensities (cid:104)u2(cid:105) and (cid:104)b2(cid:105). For sufficiently large β its
surprisingly)thatonewins,butiftheinitialconditionisevenlybal- influenceisverysmall;theaxialcomponentofB isthensoweak
0
ancedbetweenthetwomodes,itisultimatelyjustnumericalnoise thatithasnofurtherinfluence.Thisisnottrueforsmallβ,where
thatdetermineswhichmodewins.Eventuallythoughonemodeal- the axial field starts to dominate. For β < 1 the kink-instability
wayswins;thesolutionconsistingofanequalmixtureofbothis is strongly stabilized Ru¨diger, Schultz & Elstner(2011b) and the
unstable. resultingenergiesoftheperturbationsarereduced.
Spontaneousparity-breakingbifurcationsofthistypearewell Fig.4showsthekinetic(1)andcurrent(2)helicitiesforthe
knowninclassical,non-magneticTaylor-Couetteflow(e.g.Hoff- twoseriesofruns.Forbothseries,bothhelicitieshavetheopposite
mannetal.2009,Altmeieretal.2010andreferencetherein),but signasβ(seealsoKa¨pyla¨&Brandenburg2009)forcomparison).
are almost unknown in magnetohydrodynamic problems. To the Instatingthisresult,itisimportantthoughtospecifycarefullythe
bestofourknowledge,theonlyotherexampleisintheveryrecent nature of the initial conditions used in each run. For β = O(1),
workbyChatterjeeetal.(2010).Giventheimportanceofhelicity thebasicstatehasasufficientlystronghandednessthatitforcesthe
inmean-fielddynamics,anyeffectthatgenerateshelicityfroman instabilitiestohaveaparticularparityaswell,whichasindicated
underlyingbasicstatewithouthelicitycouldbesignificant. turnsouttobeoppositetothatofthebasicstate.Ifonethengrad-
WenextpresenttwoseriesofsolutionswhereB includesan uallyincreasesβ,eachtimeusingtheprevioussolutionasthenew
0
(cid:13)c 2011RAS,MNRAS000,1–??
4 M.Gellert,G.Ru¨digerandR.Hollerbach
Figure4. Thekineticandcurrenthelicitiesofthenonaxisymmetricper- Figure5. Theα-effectintheslowlyrotatingcase,Ω < ΩA,Re = 20,
turbationsasfunctionsofβ.Top:forΩ > ΩA(Ha = 100,Re = 200). Ha=200.Thetopshowsαφφ,thebottomαzz.Theotherparametersare
Bottom:forΩA >Ω(Ha=200,Re=20).Thedash-dottedlinesindi- β=3,µΩ=0.5,µB =1,Pm=1.
catethelimits±6·10−4ofthekinetichelicityoftheleftandrightmodes
inFig.2.
4 ALPHA-EFFECTANDDYNAMOTHEORY
Wehavealsocalculatedtheα-effectin(3)withthesameaveraging
procedure over the azimuth. Because of the complex structure of
the background field it is even possible to determine parts of the
tensorialstructureoftheα-tensor.Inparticularweareinterestedin
thesignsandamplitudesoftheα-effectinbothazimuthalandaxial
initialcondition,thisparityoftheinstabilitiesispreservedallway directions.Accordingtothegeneralrulethattheazimuthalα-effect
toβ → ∞,wherethebasicstatenolongerhasahandedness,and isanticorrelatedwiththe(kinetic)helicityweexpecttheazimuthal
bothleftandrightinstabilitiescouldexistequallywell,asinFig. α-effecttobepositiveforβ >0.Theexpectedsignoftheaxialα-
2. effectisnotclear.Therearetheoriesandsimulationsleadingtoα
φφ
Thatis,bythetimeonereachesβ =500,say,thebasicstate andα withoppositesigns(seeRu¨diger&Hollerbach2004for
zz
makes sufficiently little distinction between left and right modes anoverview).Weshouldnotbesurprisedtofindasimilarbehavior
that both could exist, but because of the way we have reached inthepresentsimulations.Italsomeansthatanydynamowithvery
β = 500, we consistently obtain the right mode. However, sup- weakdifferentialrotationcannotbetreatedwithascalarα-effect.
poseonedoesthefollowingexperimentnow:Taketherightmode Figs.5and6givetheresultsforslowandrapidrotation.On
atβ = 500,swapitsparitytobeleft,andusethatasanewinitial thebasisofEq.(3)thedimensionlessα-effectintheform
condition for a series of runs where β is now gradually reduced.
αD
Eventuallytherecomesapointwherethebasicstate’shandedness C = (19)
α η
issufficientlygreatthatitnolongerallowstheinstabilitytohave
the‘wrong’parity,andthesolutionrevertsbacktotherightmode. is plotted for the components α and α . In both cases, these
φφ zz
This feature that both left and right modes are allowed for suffi- twocomponentshaveoppositesigns,withα > 0andα < 0
φφ zz
ciently large β (where the degeneracy between the two modes is almost everywhere in the meridional plane. This anti-correlation
onlyweaklybroken)butnotforsmallerβ(wherethedegeneracyis betweenthetwocomponentsisalsostrongestinthecenterofthe
stronglybroken)isinmanywaysanalogoustoanimperfectpitch- gap,andweakestneartheboundaries.Itisthereforenotcausedby
forkbifurcation. theboundaries.
(cid:13)c 2011RAS,MNRAS000,1–??
Helicityandalpha-effectbycurrent-driveninstabilitiesofhelicalmagneticfields 5
Figure7. ThedimensionlessdynamonumberCαoftheazimuthalα-effect
forΩ < ΩA andΩ > ΩA.Forhigherβ (smallerBz)theα-effectde-
creaseslikeC/βwithC(cid:39)0.05.µΩ=0.5,µB =1,Pm=1.
WeknowfromFig.7thatC (cid:39) C/β withC (cid:28) 1,sothat(22)
α
gives,atleastasanorder-of-magnitudeestimate,thecondition
|B |
C > R (23)
|B |
z
for self-excitation of a dynamo with differential rotation and
current-drivenα-effect.FordiskdynamosB dominatesB ,and
R z
forsphericaldynamosB iscomparabletoB .Inbothcasesthe
R z
conditionforself-excitationbecomesC >1,whichcannotbeful-
filled according to Fig. 7, which suggests instead that C <0.05.
∼
Theα-effectduetothecurrenthelicityofthebackgroundfieldap-
Figure6. Theα-effectintherapidlyrotatingcase,Ω>ΩA,Re=200, pearsasmuchtoosmalltoallowtheoperationofanαΩ-dynamo.
Ha=100.Thetopshowsαφφ,thebottomαzz.Theotherparametersare Anotherargumentconcernsthegrowthrateofsuchadynamo
β=3,µΩ=0.5,µB =1,Pm=1. (if it exists at all) in relation to the very long magnetic diffusion
timesinradiativezones.Assumethatforself-excitationC C >
α Ω
1,thenthegrowthrateω isgivenby
gr
ThatFigs.5and6showsuchsimilarresultsissurprising,and η √ η |B |
ω = C C (cid:39) C z . (24)
isoneofthebasicresultsofthispaper.Theinfluenceofrotation gr D2 α Ω D2 |B |
R
onαisevidentlyratherweak.Thefactthat–contrarytoprevious
Hence,onlyforC > 1thegrowthtimeofthedynamowouldbe
resultsforrotatingconvection–α isactuallysmallerforrapid
φφ shorterthanthemagneticdiffusiontimeD2/η,whichisknownto
rotation than for slow rotation illustrates just how different these
beoforderGyrfortheradiativeinteriorofstars.
magnetic-inducedhelicitiesarefromsomeofthepreviousresults.
Onecanalsoargueasfollows.Therelation(24)alsoreads
Finally,Fig.7showshowtheamplitudesofα varywithβ,being
φφ √
roughlyinverselyproportionalinbothcases. ω (cid:39) αΩ(cid:48) (25)
gr
Toconsidersomepossibleastrophysicalimplicationsofthese
independent of the magnetic diffusivity. On the other hand, for
results, imagine a disk dynamo with dominant field components
givenC (25)states
B andB .DynamowavesofαΩ-typerequireforself-excitation α
φ R
thattheproductof(19)and 1(cid:112)
ω (cid:39) C ηΩ, (26)
gr D α
D3 dΩ
CΩ =− η dR (20) whichforthecomputedvalueCα (cid:39) 0.01takenfromFig.7and
η (cid:39) 500cm2/sforthesolarcoreleadstovaluesoforder10−15
exceedsacriticalvalueoforderunity,i.e.C C (cid:62) 1.Theratio s−1,i.e.togrowthtimesoforder10Myr.AsitistypicalforαΩ-
α Ω
oftheamplitudesofthefieldcomponentsB andB followsthe dynamostheirgrowthtimesareonlyslightlyshorterthanthebasic
φ R
simplerule magneticdecaytime.
(cid:114)
|B | C
φ (cid:39) Ω (21)
|B | C
R α 5 SUMMARY
sothatdynamoexcitationrequires
Wehaveshownthatthecurrent-driveninstabilityofhelicallarge-
|B | scalefieldsdoesproducesmall-scalehelicity(kineticpluscurrent
φ C (cid:62)1. (22)
|B | α helicity) and even α-effects, but the resulting numerical values
R
(cid:13)c 2011RAS,MNRAS000,1–??
6 M.Gellert,G.Ru¨digerandR.Hollerbach
seem to be too small for the operation of large-scale dynamos in
radiativezonesofearly-typestars.
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