Suppression of the large-scale Lorentz force by turbulence

0.16 MB·English

by  G. Ruediger

Astron. Nachr./AN000,No.00,1–8(0000)/DOIpleasesetDOI!
Suppression of the large-scale Lorentz force by turbulence
G.Ru¨diger1,⋆,L.L.Kitchatinov2,3,andM.Schultz1
1 Leibniz-Institutfu¨rAstrophysikPotsdam,AnderSternwarte16,D-14482Potsdam,Germany
2 InstituteforSolar-TerrestrialPhysics,P.O.Box291,Irkutsk664033,Russia
3 PulkovoAstronomicalObservatory,St.Petersburg196140,Russia
Received2011Nov17,accepted2011Dec5
Publishedonline2012Jan23
Keywords magnetohydrodynamics(MHD)–magneticfields–turbulence
2 The components of the total stress tensor (Reynolds stress plus Maxwell stress) are computed within the quasilinear
1 approximation foradriven turbulenceinfluencedbyalarge-scalemagneticbackground field.Theconducting fluidhas
0 anarbitrarymagneticPrandtlnumber andtheturbulencewithoutthebackground fieldisassumedashomogeneous and
2 isotropicwithafreeStrouhalnumberSt.Thetotallarge-scalemagnetictensionisalwaysreducedbytheturbulencewith
n thepossibilityofa‘catastrophicquenching’forlargemagneticReynoldsnumberRmsothatevenitssignisreversed.The
a totalmagneticpressureisenhancedbyturbulenceinthehigh-conductivitylimitbutitisreducedinthelow-conductivity
J limit.Alsointhiscasethesignofthetotalpressuremayreversebutonlyforspecialturbulenceswithsufficientlylarge
St>1.Theturbulence-inducedtermsofthestresstensoraresuppressedbystrongmagneticfields.Forthetensionterm
3
thisquenchinggrowswiththesquareoftheHartmannnumberofthemagneticfield.Formicroscopic(i.e.small)diffu-

sivityvaluesthemagnetictensiontermbecomesthushighlyquenchedevenforfieldamplitudesmuchsmallerthantheir
]
R equipartitionvalue.Intheoppositecaseoflarge-eddysimulationsthemagneticquenchingisonlymildbutthenalsothe
turbulence-inducedMaxwelltensorcomponentsforweakfieldsremainrathersmall.
S
.
h (cid:13)c 0000WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
p
-
o 1 Introduction notedbyuandthefluctuatingfieldcomponentsaredenoted
r
t byb.ThestandardMaxwelltensor
s
a Differential rotation and fossil fields do not coexist. A 1 1
[ nonuniformrotationlawinducesazimuthalfieldsδBφfrom Mij = µ0BiBj − 2µ0B2δij (2)
2 an original poloidal field BR which together transport an- forthe consideredMHD turbulenceturnsintothe general-
v gular momentumin radialdirectionreducingthe shear δΩ
izedstresstensor
5 viathelarge-scaleLorentzforceJ B,i.e.
4 × Mtot =M ρQ +MT, (3)
3 RδΩ BRδBφ. (1) wiitjhtheoniej-p−ointcijorrelatiijontensor
3 δt ≃ µ ρ∆R
0
9. Qij = ui(x,t)uj(x,t) (4)
AstheinducedB resultsasδB ∆ΩB δttheduration h i
0 φ φ ≃ R oftheflowandtheturbulence-inducedMaxwelltensor
1 ofthecompletedecayoftheshear(i.e.δΩ = ∆Ω)isδt
≃ 1 1
v:1 f√osµs0ilρRfie/lBdRof. 1ThGisauisssacsohmoprtartiemdewoitfhotrhdeertim10e0s0c0alyeroffotrhea MiTj = µ0hbi(x,t)bj(x,t)i− 2µ0hb2(x,t)iδij. (5)
i starformation.Allprotostarsshouldthusrotaterigidly. ThegeneralizedLorentzforceF isthen
X
Equation(1)isalsousedfortheexplanationoftheob- F =Mtot. (6)
r i ij,j
servedtorsionaloscillationsoftheSun.WithB 5Gauss
a R ≃ If the onlypreferreddirectionin the turbulenceis the uni-
and B 10000Gaussthe estimationforRδΩ is10 m/s
whichφis≃closetotheobservedvalueof5ms−1.Theresult formbackgroundfieldBboththetensorsQijandMiTj have
the same form as the Maxwelltensor (2) butwith two un-
isthat–ifEq.(1)iscorrect–themaximalfieldstrengthof
knownscalarparameters.Itmakesthussensetowrite
theinvisibletoroidalfieldsshouldnotbemuchhigherthan
1 1
10000Gauss.However,thesolarconvectionzoneisturbu- Mtot = (1 κ)B B (1 κ )B2δ (7)
lent and it is not yet clear whether Eq. (1) is also true for ij µ0 − i j − 2µ0 − p ij
conductingfluidswithfluctuatingflowsandfields. forthetotalstresstensor(3).ThefirsttermoftheRHSde-
In this paper the total Maxwell stress is thus derived scribes a tension along the magnetic field lines while the
fora turbulentfluidunderthe presenceofa uniformback- second term is the sum of the magnetic-inducedpressures
ground field B. The fluctuating flow components are de- transverse to the lines of force. The main role of the first
term in stellar physics is an outward-directedangular mo-
⋆ Correspondingauthor:[email protected] mentum transport if BRBφ <0. If its coefficient 1 κ
−
(cid:13)c 0000WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
2 G.Ru¨diger,L.L.Kitchatinov&M.Schultz:Suppressionofthelarge-scaleLorentzforcebyturbulence
wouldchangeitssignunderthepresenceofturbulencethen and the same for the velocity field. The influence of mean
for the same magnetic geometry the angular momentum magneticfieldonturbulenceisdescribedbytherelation
transportwouldbeinwardlydirected.TheLorentzforce(6)
uˆ(0)(k,ω)
with(7)becomes uˆ(k,ω)= , (11)
1+ (k·V)2
1 (−iω+ηk2)(−iω+νk2)
F =(1−κ)J ×B− 2µ0(κ−κp)∇B2, (8) where V = B/√µ0ρ is the Alfve´n velocity of the large-
scale backgroundfield, η is the microscopicmagnetic dif-
sothatthe‘laminar’LorentzforceJ B hastobemulti-
× fusivity, and ν is the microscopic viscosity. In Eq. (11) uˆ
plied with the factor1 κ andan extramagneticpressure
− is the (Fourier-transformed)velocity field modified by the
appearsif theκ’sareunequal(andtheyare)duetotheac-
meanmagneticfieldanduˆ(0) standsforthevelocityofthe
tion of the turbulence. If the κ is positive then its ampli-
‘original’turbulencewhichisassumedtoexistforB = 0.
tude should not exceed unity as otherwise the direction of
Theoriginalturbulenceisassumedasstatisticallyhomoge-
the Lorentzforcereversed.Roberts& Soward(1975)con-
neousandisotropic,i.e.
sideringonlythetermsoftheMaxwellstressfound(large)
pthoesiwtivelel-κknaonwdnneedgdatyivdeifκfups,iiv.iet.yκ(s=ee−Eqκ.p(4=2)η,Tbe/lηoww)i.thηT huˆ(i0)(k,ω)uˆ(j0)(k′,ω′)i= E1(6kπ,kω2) δij − kkik2j
(cid:18) (cid:19)
Ru¨diger et al. (1986) found κ also as positive and as δ(k+k′)δ(ω+ω′), (12)
×
runningwiththemagneticReynoldsnumberRmofthetur-
whereE(k,ω)isthepositive-definitespectrumfunctionof
bulence even for Rm > 1. Kleeorin et al. (1989) suggest
theturbulence.Here
thatκp >0andevenlargerthanunitysothatthetotalmag- ∞ ∞
netic pressure changesits sign and becomesnegative.The
u2 = E(k,ω)dkdω (13)
resultinginstabilitymayproducestructuresofconcentrated
Z Z
magneticfieldandmaybeimportantforsunspotformation 0 0
definesthermsvelocityuoftheoriginalturbulence.Equa-
(Kleeorinetal.1990;Brandenburgetal.2010,2011).
tions(9)to(12)sufficetoderivethevaluesoftheκ’s.
Hence,the κ’s haveanimportantphysicalmeaning.In
the simplest case they both wouldresult as negative.Then
the effective pressure is increased by the magnetic terms 3 Weakfield
and also the tension term 1 κ remains positive so that
−
the Lorentz force in turbulent media is simply amplified. We proceed by considering special cases. For weak mean
Many more serious consequences would result from posi- magnetic field one finds from the expressions given by
tive κ and κ if exceeding unity. In this case the Lorentz Ru¨diger&Kitchatinov(1990)therelation
p
force changes its sign with dramatic consequencesfor the ∞∞
1 Ek2(ν(2η+ν)k4 ω2)
theoryoftorsionaloscillationsandsolaroscillations(Klee-
κ= − dkdω,
orin&Rogachevskii1994:Kleeorinetal.1996). 15 (ν2k4+ω2)(η2k4+ω2)
ZZ
0 0
The turbulence-induced modification of the Maxwell ∞∞
stress can also be important in other constellations where 1 Ek2(ν(8η ν)k4 9ω2)
κ = − − dkdω. (14)
large-scale fields and turbulence simultaneously exist. p 15 (ν2k4+ω2)(η2k4+ω2)
ZZ
Tachocline theory, jet theory, the structure of magnetized 0 0
Theexpressionsdonothavedefinitesignssothatitremains
galactic disks (Battaner & Florido 1995)or oscillations of
unclearwhetherthelarge-scaleMaxwellstressisincreased
convectivestarswithmagneticfieldscouldbementioned.
ordecreasedbytheturbulence.Eventhesignsofκandκ
p
maydependonthespectrumoftheturbulence.
2 Equations The simplest case is a turbulence with a white-noise
spectrum containing all frequencies with the same ampli-
We applythequasilinearapproximationknownalso asthe tude. Here and in the following we shall use the Strouhal
second order correlation approximation (SOCA). All pre- numberStandthenormalizedcharacteristicfrequencyw∗
liminary steps to derive main equationsfor the problemat u wl2
St= τ , w∗ = c (15)
handweredescribedbyRu¨diger&Kitchatinov(1990).The l c η
c
fluctuating magnetic and velocity fields are related by the
(l correlation length, τ correlation time). The turbulence
c c
equation frequencyw∗ measures the characteristic frequencyof the
i(k B) turbulencespectruminrelationtothediffusionfrequency.It
bˆ(k,ω)= · uˆ(k,ω), (9)
iω+ηk2 islargeforflatspectrasuchaswhitenoiseanditissmallfor
− verysteepspectralikeδ functions.Ontheotherhand,itis
wherethehatnotationmarksFourieramplitudes,e.g.
largeinthehigh-conductivitylimitanditissmallinthelow-
b(r,t)= ei(kx−ωt)bˆ(k,ω)dkdω, (10) conductivity limit. E.g., it is much larger than unity if the
microscopic(Spitzer)diffusivityisused(high-conductivity
Z
(cid:13)c 0000WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim www.an-journal.org
Astron.Nachr./AN(0000) 3
∞∞
limit).Itshouldbeunityif–asitisusedinlarge-eddysim- 1 ωk2 ∂E
ulations – η = η wl2. In the numerical integrations dkdω, (20)
presented below tThe≃limitcw∗ 0 (i.e. low-conductivity − 15Z0 Z0 ω2+η2k4 ∂ω
→
limit) applies to the case of the frequency spectrum as a fromwhichκ provestobepositive-definiteforallspectral
Diracdeltafunctionδ(ω). functions E which do not increase for increasing ω. We
TheproductofStandw∗givingthemagneticReynolds shall see that the positivity of κ which reduces the effec-
number tivityoftheangularmomentumtransportisageneralresult
Rm= ulc, (16) oftheSOCAtheory.
η Furthersimplificationscanbeachievedbyapplyingthe
wherewehaveusedtherelationτ =1/wasadefinitionof modelspectrum
c
thecorrelationtime.Thenitisw∗ =Rm/St. 2w
E(k,ω)=q(k) (21)
In the high-conductivitylimit (‘whitenoise’)one finds π(w2+ω2)
thesimpleresults
with
1 1 ∞
κ= RmSt, κ = RmSt, (17)
p
15 −15 q(k)dk =u2, (22)
so that the κ is positive and runs with RmSt = u2τ /η
c Z
0
whichisthe(large)ratiooftheeddydiffusivityandthemi-
wherewisacharacteristicfrequencyoftheturbulencespec-
croscopic diffusivityhence the magnetic tension is always
trum.For w 0 (21) representsa Dirac δ-functionwhile
(strongly) reduced. On the other hand, the magnetic pres- →
w gives‘whitenoise’.Theresultsare
sureisincreased(seeEq.7).Thenegativesignofthevalue →∞
∞
of κp excludes the possibility that the effective pressure 1 w+3ηk2
term in Eq. (7) changes its sign so that the total magnetic κ= qdk (23)
15η (w+ηk2)2
pressure becomes negative. This is formally possible after Z
0
(17)forthemagnetictensionparameter1 κ. and
−
Thewhite-noiseapproximation,however,isnotperfect. ∞
If,forexample,theoppositefrequencyprofileforverylong 1 7ηk2 w
κ = − qdk. (24)
correlation times, i.e. E δ(ω), is applied to (14) then p 15η (w+ηk2)2
again the κ is positive bu∝t the sign of κ depends on the Z0
p
Againthe κ is positive-definite.Note thatfor w the
magneticPrandtlnumber
→ ∞
high-conductivityresults (17) are reproduced. In this case
ν
Pm= . (18) the κ’s are running with 1/η while for w 0 the κ’s
η →
are runningwith 1/η2. This is a basic result: for low con-
Itis thusnecessaryto discusstheintegralsin (14) in more
ductivity and for high conductivity the dependence of the
detail.
κ’sonthemagneticReynoldsnumberRmdiffers.Forhigh
conductivity(whitenoise)theκ’sareproportionatetoRm
3.1 Pm≥ 1 while for low conductivity (steep spectra) the factor Rm2
appears.Notethatforδ-likespectralfunctionsthenumeri-
For Pm > 8 one finds that κ is negative-definite for all
p calcoefficientforκ is 0.2while forκ thisfactoris about
possible spectral functions. The coefficient 1 κ of the p
− p 0.5.OnecanalsofindthesevaluesattheordinateofFig.2.
magnetic pressure is thus positive-definite and cannot be-
Abasicdifferenceexistsforκandκ ,too.Whiletheκ
comenegative.Thisisnottrueforκ.Weshallshowthatthe p
ispositive-definite,theκ canchangeitssign.Generallyit
κwill‘almostalways’bepositivesothattheLorentzforce p
willbepositiveonlyforsmallw∗ butitshouldbenegative
terminthegeneralizedLorentzforceexpression(8)is‘al-
for large w∗. Already from these arguments one finds the
mostalways’quenchedbytheexistenceoftheturbulence.
maincomplicationoftheproblem.Theshapeoftheturbu-
Forν =ηtheexpressions(14)turninto
lencespectrumhasafundamentalmeaningfortheresults.
∞∞
1 Ek2(3η2k4 ω2) Toprobetheseresultsindetailaspectralfunctionq(k)
κ= − dkdω,
15 (ω2+η2k4)2 2l u2
Z0 Z0 q c (25)
∞∞ ≃ π 1+k2l2
1 Ek2(7η2k4 9ω2) c
κ = − dkdω, (19) isused.Theintegrationyields
p 15 (ω2+η2k4)2
Z0 Z0 1 StRm√w∗(2+√w∗)
κ= , (26)
whichagaindonothavedefinitesigns.Onecanwrite,how- 15 (1+√w∗)2
ever,theexpressionforκas
sothat
∞∞
κ= 115Z0 Z0 (ω22η+2kη62Ek4)2dkdω κ≃( 125R115mRmSt√Stw∗ ) for w∗(cid:26) ><41. (27)
www.an-journal.org (cid:13)c 0000WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
4 G.Ru¨diger,L.L.Kitchatinov&M.Schultz:Suppressionofthelarge-scaleLorentzforcebyturbulence
spectrawithfinitecorrelationtime.Stationarypatternswith
E δ(ω)areexcluded.In thelimitofverysmallPmthe
∝
Eqs.(14)reduceto
∞ ∞∞
π 1 E(k,ω)k2
κ= E(k,0)dk dkdω,
15η − 15 ω2+η2k4
Z ZZ
0 0 0
∞ ∞∞
4π 9 E(k,ω)k2
κ = E(k,0)dk dkdω. (30)
p 15η − 15 ω2+η2k4
Z ZZ
0 0 0
Thespectrum(21)leadsto
∞
1 2ηk2+w
κ= q(k)dk,
15ηw ηk2+w
Z
0
Fig.1 Theκ vs.RmafterEq.(28).Fromtoptobottom: ∞
p 1 8ηk2 w
St = 7,St = 4,andSt = 1.Allcurveshaveamaximum. κ = − q(k)dk. (31)
Notethatallκ becomenegativeforsufficientlylargeRm; p 15ηw ηk2+w
p Z
0
Pm=1.
Again κ is positive-definite. If the wave number spectrum
hasonlyasinglevaluethen
Hence, for small w∗ (low conductivity) the κ runs with
1 Rm22+w∗
St0.5Rm1.5whileforhighconductivitytherelationissim- κ= ,
plyStRm.Onefindsagainthedifferencesbetweenthetwo 15 w∗ 1+w∗
limits.Inlarge-eddysimulationsfortheeffectivediffusivity 1 Rm28 w∗
κ = − . (32)
therelationη ul isusedsothatRm Pm 1.Asin p 15 w∗ 1+w∗
c
≃ ≃ ≃
themajorityoftheapplicationsalsotheStrouhalnumberSt Thelimitw∗ 0isherenotallowed.Againtheκ ispos-
p
isofthesameorderthecoefficient(27)isasmallnumber. itive(negative→)forsmall(large)w∗.Formally,theStrouhal
If for direct numerical simulations the numerical value of numberStdoesnotappear.Replacingthew∗ byRm/Stin
Rmbecomeslargethenthereisnoreasonthat(26)remains bothlimitsthe κ runslinearlywithStRm,i.e.with1/η.
p
smallerthanunity. | |
Theκalsorunswith1/ηinthehigh-conductivitylimit,
Forκpthereisanothersituation.Oneobtains i.e.
1 RmSt√w∗(3 √w∗) 1
κ = − , (28) κ StRm, (33)
p 15 (1+√w∗)2 ≃ 15
whileforlowconductivitythevalueisκ (2/15)StRm.
hence, Forlargew∗,i.e.forRm > St,andPm <≃1thereisprac-
κ 51Rm1.5 St0.5 for w∗ <9 (29) ticallynoinfluenceofthenumericalvalu∼eofthemagnetic
p ≃( −115RmSt ) (cid:26) >9. Prandtl number (see Eq. 27). Below we shall also demon-
strate by numericalsolutions of the integrals that Eq. (33)
For w∗ > 9 the κ is negativeso thatthe total pressureis
p forms the main result of the presentanalysis. Whether the
alwayspositive.Forw∗ <9,however,theκ becomespos-
p κ-coefficientmaybecomelargerthanunityonlydependson
itive. In this case for largeStrouhalnumberthe total mag-
thenumericalvaluesofStandRm.Forlarge-eddysimula-
netic pressure 1 κ becomes negative. Figure 1 demon-
− p tionswithSt = Rm = Pm = 1theκisbasicallyonlyof
strates that κ exceeds unity for St>4. For St>4 one
p order0.1.
findsκ >1forRm=14,i.e.w∗ =3.5.Wehavetostress,
p
Withthespectralfunction(25)theresultsareverysim-
however, that the SOCA approximation only holds if not
ilar,i.e.
boththequantitiesStandRmsimultaneouslyexceedunity.
For turbulences in liquid metals in the MHD laboratory 1 Rm28 √w∗
κ = − . (34)
Rm 1isatypicalvalue.Kemeletal.(2012)reportanin- p 15 w∗ 1+√w∗
≃
creaseofκ withRm2 (theirFig.9,bottom).Thenegative
p ThisexpressiononlyexceedunityforSt 1.ForsmallSt
branchof(29)doesnotexistinthesimulations(Pm=0.5). ≫
the sum 1 κ is thusalways positive independentof the
p
−
actualvalueofRmcontribution.
3.2 Pm≪ 1
ThesituationismoreclearforsmallmagneticPrandtlnum- 4 Strong fields
bers, which exist, e.g., in stellar interiors, protoplanetary
disksandalsointheMHDlaboratory.Itispossibletocon- Sofaronlytheinfluenceofweakmagneticfieldshasbeen
siderthelimitν 0intheEqs.(14)butonlyforturbulence considered.The influenceof strong magneticfields is also
→
(cid:13)c 0000WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim www.an-journal.org
Astron.Nachr./AN(0000) 5
Fig.3 ThesameasinFig.2forPm = 0.1.From topto
bottom:S=0.01,S=1,andS=10.Thequantitiesvanish
as1/w∗ forw∗ .
→∞
the frequency integration so that the turbulence quantities
κ/Rm2andκ /Rm2onlydependontheLundquistnumber
p
Bl
S= c (36)
√µ0ρη
Fig.2 The κ/Rm2 (top) and κp/Rm2 (middle, bottom) of the magnetic field, the frequency w∗ and the magnetic
vs.w∗fortheone-modemodel(35).Thecurvesintheplots
PrandtlnumberPm.
(from top to bottom) are for S = 0.01, S = 1, S = 3,
Intheweak-fieldlimit,S 1,onefindstheoverallre-
and S = 10. At the left vertical axis the values are valid sult that κ/Rm2 runs as 1/1≪5w∗ (Figs. 2 and 3, top) so
forthe deltafunctionspectra(low-conductivitylimit).The
that again the general result (33) is reproduced. For very
quantitiesvanishas1/w∗ forw∗ → ∞(high-conductivity smallw∗,i.e.fordeltafunctionfrequencyspectra(or,what
limit,rightverticalaxis)leadingtotheresult(33).Bottom:
isthesame,forverylongcorrelationtimes),theκ’srunwith
detailsforκ /Rm2;Pm=1.
p 1/Rm2–asalreadyshownabove.
Whenthefieldisnotweak,thestressparametersrapidly
decreasewith S. Figures2 and3 also demonstratethatthe
importanttoknow.TherathercomplexresultsoftheSOCA
magneticquenchingcanbewrittenas
theory with arbitrary magnetic field amplitudes and with
κ
free valuesofbothdiffusivitiesare givenin theAppendix. κ 0 (37)
≃ 1+ǫS2
Theseexpressionscanbediscussedbyapplyingthesingle-
(see Fig. 4), in confirmation to Brandenburg et al. (2010)
scalewavenumberspectrum
whofoundthemagneticquenchingintermsof1/B2.From
q(k)=2u2δ(k l−1) (35) theFiguresonefindsthatǫ<∼1forPm < 1.ForlargePm
− c
the ǫ is even smaller. The magnetic quenching of the κ-
and the frequency spectrum (21). Such an approximation parameteristhusstrongerforsmallmagneticPrandtlnum-
allows to solve the Eqs. (A1)...(A4)numericallyincluding ber than for largePm. While a magneticfield with S = 1
www.an-journal.org (cid:13)c 0000WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
6 G.Ru¨diger,L.L.Kitchatinov&M.Schultz:Suppressionofthelarge-scaleLorentzforcebyturbulence
reduces the κ remarkablyif Pm < 1 in the opposite case
Pm > 1 the κ is almost uninfluenced by S = 1. Figure
5demonstratestheinversedependenceoftheǫonthemag-
neticPrandtlnumber.Onefindsǫ 0.75/Pm.Thequench-
≃
ingexpression,therefore,turnsforPm=1into
6
κ
κ 0 , (38)
≃ 1+0.75Ha2
with the Hartmann number Ha=S/√Pm instead of the
LundquistnumberS.Forthemagneticquenchingitisthus
notimportantwhichofthediffusivitiesislargeandwhichis
small.Thequenchingisverystrongifoneofthemissmall
(see Roberts & Soward 1975). For the high-conductivity
limit(η 0)orforinviscidfluids(ν 0)the Hartmann
→ →
numberHa takesverylargevaluesso thatevenveryweak
Fig.4 The verification of the relation (38) for the func-
fieldsstronglysuppresstheκ-effect.
tions w∗κ marked by their Lundquist numbers S. The re-
Notethat
sultingvalueforǫisabout0.75;Pm=1.
B
S=Rm , (39)
B
eq
with B = µ ρ u2 as the equilibrium field strength.
eq 0
h i
The magnetic quenching of the κ-term thus grows with
p
Rm2(Brandenburg&Subramanian2005)sothatforgrow-
ingRmtheκbecomessmallerandsmaller:
1 StB2
κ eq . (40)
≃ 15ǫRmB2
Itbecomesthusclearthatinthehigh-conductivitylimiteven
forrathersmallfieldstheκ-terminEq.(7)takesverysmall
valueswhichdonotplayanimportantroleinthemean-field
magnetohydrodynamics.
Ontheotherhand,forRm=1thewell-knownstandard
expression
κ
κ= 1+ǫ0BBe22q (41) Fthieg.m5agnTethiceP(wraenadkt)lnduepmebnedrePncme.of the quantityPm·ǫ on
for magneticquenchingappearswith ǫ of orderunityonly
slightlydifferingforsmallandlargew∗.
Because of St=Rm=1 in this case the κ’s always themodifiedmagnetictensiontermiswrittenas1 κ.The
−
remain smaller than unity in accordance to (33). Hence in quantities κ and κ have been computedwithin the quasi-
p
both the possible concepts, i.e. the use of the microscopic linearapproximation(SOCA)whichcanbeusedifthemin-
diffusivitiesandtheuseofthelarge-eddysimulationswith imum of both the numbers St and Rm is (much) smaller
subgrid diffusivities, the values of the turbulence-induced than unity. As almost all turbulences fulfill the condition
Maxwelltensorcoefficientsremainsmall. St 1, the validity of SOCA requires Rm=ul /η <1.
c
The numerical simulations by Kemel et al. (2012) in- Und≃er this restriction the resulting κ′s are always smaller
deed yield a magnetic quenching of the pressure term in thanunity.ForallmagneticPrandtlnumbersPmwefound
termsofRm2butonlyforRm<10. κ as positive so that the non-pressure force term (B )B
∇
isreducedundertheinfluenceofturbulence.Thisisinpar-
ticular true for the coefficients of the angular momentum
5 Catastrophic quenching?
transport terms B B and B B which, therefore, be-
φ R φ z
comemoreandmoreineffectiveinturbulentfluids.
We have computed the stress tensor which is formed by
large-scale background fields, by the Reynolds stress of a Thesignofκ stronglydependsonthemagneticPrandtl
p
turbulencefieldundertheinfluenceofthefieldandthetur- number.Itprovestobenegative-definiteforlargePm.For
bulentMaxwellstressofthefieldfluctuations.Allcontribu- smallerPmthesignofκ dependsontheshapeofthefre-
p
tions can be summarizedin form of the classical Maxwell quency spectrum of the turbulence. For steep profiles, i.e.
stresstensorbutwithturbulence-modifiedcoefficients(see verylongcorrelationtimes,theκ becomespositivewhile
p
Eq. 7). The modified pressure term is now 1 κ while forflatfrequency-spectraoftheturbulencewhichareasflat
p
−
(cid:13)c 0000WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim www.an-journal.org
Astron.Nachr./AN(0000) 7
asthespectrumofwhitenoise(veryshortcorrelationtimes) 1 S2 forsmallSandlike S−3 forlargeS. Of course,by
−
theκ forRm<1becomesnegative. this procedure the η cannot become negative. We know,
p T
OnecouldbelievethatrelationsvalidforsmallRmlike ontheotherhand,thatthemagneticquenchingoftheeddy
κ St RmcanbealsousedforRm>1sothatfinallythe diffusivityinsunspotsreducesitsvalue(only)from5 1012
efpfe∝ctive·magnetic pressure becomes negative. This, how- cm2s−1 to about 1011 cm2s−1 what – together wi×th the
ever,isnottrue.Theκ changesitssignforRm Stand time decay law of the sunspots – can be understood with
p
becomes negative. Hence, the total magnetic pre≫ssure re- quenching expressions like (42) for Rm = 1 (Ru¨diger &
sultsasmostlypositive.Theonlyexceptionexistsforsuffi- Kitchatinov2000).Itisthussuggestedtoworkwiththesim-
cientlylargeStandsufficientlysmallRm(seeFig.1). plerelationsRm = 1andS B/Beq inapplicationswith
≃
turbulentconvection.
More dramatic is the situation with the magnetic ten-
sionanditscoefficient1 κwhichisalsothecoefficientof Similarly, also the κ increases for vanishing η. There
thevectorJ Bintheg−eneralizedLorentzforceinturbu- is, however, no nonmagnetic term against which the mag-
×
lent media. This coefficientis positive for small κ, i.e. for netic influence can be neglected as it must be compared
sufficiently small Rm if St = 1. It is positive and smaller with the large-scaleLorentz force J B which is also of
than unity for the large-eddy simulations (‘mixing-length thesecondorderinB.Theonlypossi×bilitytokeepthetur-
model’)consideredattheendofSect.3.2withRm=St= bulence contributionsmall for large Rm is to put St 1.
≪
Pm=1(seeFig.2). However,ifthemagneticfieldissuper-equipartitionedthen
Thequestion,however,whethertheκcanexceedunity the κ is magnetically quenched which introduces a new
(sothat1 κbecomesnegative)cannotfinallybeanswered factor Rm−2. Then the magnetic-induced κ-effect finally
withinthe−quasilinearapproximation.Itisκ 0.1St Rm runswith1/Rmsothatitvanishesinthehigh-conductivity
where one of the factors St and Rm must b≃e smaller·than limit. In summary, for large Rm and for very weak mag-
unitybuttheproductSt Rmisformally notrestrictedby netic field the κ can exceedunity(so thatthe stress tensor
the SOCA. It is thus a c·lear and surprising result also in reverses sign) but this phenomenondisappears already for
the frame of SOCA that the angular momentum transport ratherweakfields.
by large-scale magnetic fields can strongly be suppressed
undertheinfluenceofturbulence.Thepossibleexistenceof Acknowledgements. This work was supported by the Deutsche
Forschungsgemeinschaft and by the Russian Foundation for Ba-
an instability resultingfromκ > 1 hasbeenconfirmedby
sicResearch(projects10-02-00148,10-02-00391).
thenumericalsimulationsbyBrandenburgetal.(2011).
The formal background of this phenomenon is that
the integrals defining κ and κp do not exist in the high- References
conductivitylimit or, what is the same, in the ideal MHD.
Thesameistrueforthemuchsimplermagnetic-suppression Battaner,E.,Florido,E.:1995,MNRAS277,1129
problemoftheeddydiffusivity.Wetaketheexpression Blackman,E.G.,Field,G.B.:2000,ApJ534,984
Blackman,E.G.,Brandenburg, A.:2002,ApJ579,359
∞∞
1 ηk2E Brandenburg,A.,Subramanian,K.:2005,Phys.Rep.417,1
ηT = 3 ω2+η2k4 Brandenburg,A.,Kleeorin,N.,Rogachevskii,I.:2010,AN331,5
ZZ Brandenburg,A., Kemel, K., Kleeorin,N., Mitra, D., Ro-
0 0
gachevskii,I.:2011,ApJ740,L50
6 η2k4 ω2 B2
1 − dkdω (42) Kemel, K., Brandenburg,A., Kleeorin,N., Mitra, D., Ro-
(cid:18) − 5(ω2+η2k4)2 µ0ρ(cid:19) gachevskii,I.:2012,Sol.Phys.(subm.),arXiv:1112.0279
Kitchatinov,L.L.:1991,A&A243,483
(Kitchatinov et al. 1994) for the SOCA expression of the
Kitchatinov,L.L.,Pipin,V.V.,Ru¨diger,G.:1994,AN315,157
eddy diffusivityunderthe presence of a uniformmagnetic
Kleeorin,N.I.,Rogachevskii,I.:1994,Phys.Rev.E50,493
backgroundfield(Pm= 1).Theexpressionispartofase-
Kleeorin,N.I., Rogachevskii, I.V., Ruzmaikin,A.A.: 1989, SvA
riesexpansionwhichconvergesifthesecondtermissmaller Lett.15,274
thanthefirstterm.ThesecondtermoftheRHSofthisex- Kleeorin,N.I., Rogachevskii,I.V., Ruzmaikin,A.A.: 1990, JETP
pressionhastwoimportantproperties:i)itispositiveforall 70,878
spectralfunctionsE with∂E/∂ω < 0sothattheη isal- Kleeorin,N.,Mond,M.,Rogachevskii,I.:1996,A&A307,293
T
waysreducedbythemagneticfields,andii)itdoesnotexist Roberts,P.H.,Soward,A.M.:1975,AN296,49
Ru¨diger,G.,Kitchatinov,L.L.:1990,A&A236,503
forthelimitη 0.Inotherwords,forrathersmallηthein-
→ Ru¨diger,G.,Kitchatinov,L.L.:2000,AN321,75
tegralbecomeslargesothatthemagneticquenchingwould
Ru¨diger, G., Tuominen, I., Krause, F.,Virtanen, H.: 1986, A&A
be extremely effective for large Rm. This is why such a
166,306
seriesexpansiononlyholdsforveryweakfields.Thisphe-
nomenon has been called a ‘catastrophic’ quenching (see
Blackman&Field2000;Blackman&Brandenburg2002). A SOCA expressions ofthe κ’s
It exists within the SOCA theory for the eddy diffusivity
andalsofortheeddyviscosity.OnefindsfromEq.(42)that Theexpressionsforthemean-fieldLorentzforceparametersκand
the mentioneddiffusivitiesare magneticallyquenchedlike κpofEq.(7)providedbythequasilineartheoryforarbitrarymag-
www.an-journal.org (cid:13)c 0000WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
8 G.Ru¨diger,L.L.Kitchatinov&M.Schultz:Suppressionofthelarge-scaleLorentzforcebyturbulence
neticamplitudescanbewrittenas
κ= ∞ ∞ωE2(k+,ωη2)kk24 K(B,k,ω)dkdω, Kp = ωω22++νη22kk44 1/2 4β14 38β2+(β2−1)4βLsNinφ −
Z Z (cid:18) (cid:19) (cid:18) 2
0 0
AR 1 LN
∞ ∞ (β2+1) + 2−(β2−1+2cosφ)
E(k,ω)k2 2βcosφ 4β4 4βsinφ
κp = ω2+η2k4 Kp(B,k,ω)dkdω. (A1) 2(cid:19) (cid:18) 2
Z0 Z0 −(β2+1+2cosφ) AR . (A4)
2βcosφ
ThekernelfunctionsKandKpdependonthemagneticfieldand 2(cid:19)
thevariableskandωvia Thefirstpartsintheseexpressionsrepresentthecontributionofthe
kV Reynoldsstresswhilethefollowinglinesrepresentthesmall-scale
β = ,
(ω2+η2k4)1/4(ω2+ν2k4)1/4 Maxwellstress.
β2−2βsinφ +1
LN =log 2 ,
β2+2βsinφ +1
(cid:18) 2 (cid:19)
β−sinφ β+sinφ
AR=arctan 2 +arctan 2 . (A2)
cosφ cosφ
(cid:18) 2 (cid:19) (cid:18) 2 (cid:19)
Here, cosφ = ηνk4−ω2 / (ω2+η2k4)(ν2k4+ω2).
Thekernelsread
(cid:0) (cid:1) p
ω2+η2k4 1/2 1 LN
K = −(β2+3) +
ω2+ν2k4 8β4 4βsinφ
(cid:18) (cid:19) (cid:18) 2
AR 1 LN
(β2−3) + 6−(β2−3+6cosφ)
2βcosφ 8β4 4βsinφ
2(cid:19) (cid:18) 2
AR
−(β2+3+6cosφ) (A3)
2βcosφ
2(cid:19)
and
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