Current helicity and electromotive force of magnetoconvection influenced by helical background fields

0.39 MB·

by  Guenther Ruediger

Astronomy&Astrophysicsmanuscriptno.ruediger2 c ESO2016
(cid:13)
January18,2016
Current helicity and electromotive force of magnetoconvection
influenced by helical background fields
G.Ru¨digerandM.Ku¨ker
Leibniz-Institut fu¨r Astrophysik Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany, email: [email protected],
[email protected]
Received;accepted
6
ABSTRACT
1
0
Motivated by theempirical finding that theknown hemispheric rulesfor the current helicityat thesolar surfaceare not strict,the
2
excitationofsmall-scalecurrenthelicitybytheinfluenceofalarge-scalehelicalmagneticbackgroundfieldsonnonrotatingmagne-

n toconvectionisdemonstrated. Itisshownwithinaquasilinearanalytictheoryofdriventurbulenceandbynonlinearsimulationsof
a magnetoconvectionthattheresultingsmall-scalecurrenthelicityhasthesamesignasthelarge-scalecurrenthelicitywhiletheratioof
J bothpseudo-scalarsisoftheorderofthemagneticReynoldsnumberoftheconvection.Thesamemodelsdonotprovidefinitevalues

5 ofthesmall-scalekinetichelicity.Ontheotherhand,aturbulence-inducedelectromotiveforceisproducedincludingthediamagnetic
1 pumpingtermaswellastheeddydiffusivitybutnoαeffect.ItisthusarguedthattherelationsbyPouquet&Patterson(1978)and
Keinigs(1983) forthesimultaneousexistenceofsmall-scalecurrenthelicityandαeffectdonot holdfortheconsidered modelof

] nonrotatingmagnetoconvection.CalculationsforvariousvaluesofthemagneticPrandtlnumberdemonstratethatfortheconsidered
R diffusivitiesthecurrenthelicitygrowsforgrowingmagneticReynoldsnumberwhichisnottrueforthevelocityofthediamagnetic
S pumping–inagreementwiththeresultsofthequasilinearanalyticalapproximation.
.
h Keywords.Magnetohydrodynamics(MHD)–Magneticfields–convection–Sun:activity
p
-
o
r 1. Introduction electromotiveforceviaanαeffectassuggestedbytherelation
t
s
a Anincreasingnumberofobservationsconcernsthesmall-scale
[ currenthelicity b(x,t) curlb(x,t)
1 α=−ηh ·B¯2 i (2)
v curr = b(x,t) curlb(x,t) (1)
H h · i
7
7 ofKeinigs(1983),Itisbasedontheexistenceofhomogeneous
atthesolarsurface,allshowingthatitisnegative(positive)inthe
8 and stationary turbulence with finite kinetic helicity (Seehafer
northern(southern)hemisphere(Hale1927;Seehafer1990;Rust
3
1996) which conditions are obviously not fulfilled if one of
& Kumar 1994; Abramenko et al. 1996: Bao & Zhang 1998;
0
the magnetic field components is inhomogeneous. Pouquet &
. Pevtsov 2001; Kleeorin et al. 2003; Zhang et al. 2010: Zhang
1 Patterson(1978)presentedtherelation
2012).Herethenotationbdenotesthefluctuatingpartsoftheto-
0 talmagneticfieldintheformB =B¯+b.Itbecameclearthatthe
6
scalar quantity (1) shows a strict equatorialantisymmetry with
1 1
: signswhichdonotchangefromcycletocycle.Notethatforthe α b curlb u curlu , (3)
v linearα2 dynamoaswellasthesimpleαΩ dynamosthelarge- ∝ µ0ρh · i−h · i
Xi scale helicity B¯ curlB¯ similar to (1) also shows equatorial
antisymmetryin a·ccordancewith sign(B¯ curlB¯) = sign(α)
ar (Steenbeck & Krause 1969). That for flu·x transport αΩ dy- whichalsomeansthateventurbulentfluidswithvanishingkine-
namostherelationismuchmorecomplicated(positivecorrela- matichelicitybutfinitesmall-scalecurrenthelicityshouldpos-
tiononlyduringthecycleminima)indicatesthatadirectrelation sessanαeffectbutwithoppositesignasin(2).
α B¯ curlB¯ notnecessarilyexists.Itisneverthelessimpor- Studies by Yousef et al. (2003),Blackman & Subramanian
∝ ·
tantthatallthementionedscalarssuchashelicitiesandαeffect (2013)andBhatetal.(2014)concernthedissipationofhelical
arepseudoscalarswhichmightberelatedtoeachother. and nonhelical large-scale background fields under the influ-
There are many theoretical studies where both the current enceofanonhelicalturbulentforcing.Itisshowninthesepapers
helicityandtheαeffectarederivedasconsequencesoftheexis- that the kinematichelicity is dominatedby the magnetichelic-
tenceofthepseudoscalarg Ω inrotatingconvectionzoneswith itywhichisconfirmedbyourquasilinearcalculationsforforced
·
gasthe(radial)directionofstratification.Yetitisshowninthe turbulenceandthenonlinearsimulationsofmagnetoconvection
presentpaperthatthecurrenthelicitycanalsobeproducedeven under the influence of a large-scale helical field. In contradic-
without rotation if the convection is influenced by a magnetic tion to the relation (3) both the quasilinear approximation and
backgroundfieldwhichishelical.Thenthequestioniswhether nonlinearsimulationsprovidereasonablevaluesforthecurrent
the same constellationwould also lead to a turbulence-induced helicitybuttheydonotleadtoafiniteαeffect.
1
G.Ru¨digerandM.Ku¨ker:Currenthelicityandelectromotiveforce
2. Thecurrenthelicity Withourapproximations(quasilinearequations,forcedtur-
bulence,norotation)the ratio ofthe two helicitiescorresponds
In order to find the current helicity due to the interaction of a
totheratioofthebothmagneticenergies.Equation(11)requires
prescribedstochasticvelocityu(x,t)andalargescalemagnetic
thesamesignforthesmall-scaleandthelarge-scalecurrenthe-
fieldB¯(x)itissufficienttosolvetheinductionequation licities.With(2)onefindssign(α)= sign(B¯ curlB¯)which
− ·
∂B contradictstheabovementionedobservationsthatinthesimple
=curl(u B)+η∆B, (4) α2 dynamossign(B¯ curlB¯) = sign(α).Theonlysolutionof
∂t × ·
thisdilemmaisthatnonrotatingconvectionundertheinfluence
wherethefluctuatingmagneticfieldmaybewrittenasB =B¯+ ofhelicallarge-scalefieldsmayproducesmall-scalecurrenthe-
b. Theinfluenceof field gradientson the mean currenthelicity licitybutitdoesnotgenerateanαeffect.
(1)atlinearorderisgovernedby If calculated under the same analytical assumptions as the
currenthelicity,thekinetichelicity u curlu identicallyvan-
∂b h · i
i η∆b =x B u u B , (5) ishes.Blackman&Subramanian(2013)demonstratedthatdur-
i p jp i,j j ij
∂t − − ingtheturbulence-induceddecayofhelicallarge-scalefieldsthe
kinetic helicity remains unchanged.In the quasilinear approxi-
where the prescribed inhomogeneous mean magnetic field B¯
mationits partdueto the backgroundfieldsis identically zero.
has been introduced in the form B¯ = B x with the nota-
j jp p MHDturbulenceundertheinfluenceofahelicallarge-scalefield
tion B¯ = B and with B = 0. It follows B¯ curlB¯ =
j,p jp jj · isthussuspectedtogeneratecurrenthelicitybutnokinetichelic-
ǫ B B x which is here considered to be the only pseudo-
ilk ij kl j ity.Thequestioniswhetheronthisbasisandaftertherelations
scalarinthesystem.
(2)and/or(3)alsononlinearsimulationsofconvectioninastrat-
To solve the equation (5) the use of the inhomogeneous
ifiedplasmaprovidecurrenthelicitybutnokinetichelicityand
Fouriermodes
noαeffect.
Inapreviouspaperwesimulatedmagnetoconvectionunder
u (x,t) = uˆ (k,ω)ei(kx−ωt)dkdω,
i i theinfluenceofauniformverticalmagneticfieldinordertopro-
ZZ
duce finite values of cross helicity (Ru¨diger et al. 2012, from
b (x,t) = (ˆb (k,ω)+xˆb (k,ω))ei(kx−ωt)dkdω (6) nowonreferredtoaspaperI).Inthepresentpaperthehelical
i i l il
ZZ magneticbackgroundfield
issuggestive.Itisastandardprocedurewhichyields z
B =B¯ B =B¯ (13)
x x y y
H
ˆb = Bij + 2ηk−lkiωm+Bηlkm2δij uˆ , ˆb = ikjBjl uˆ (7) with uniform values of B¯ and B¯ has been applied to probe
i − iω+ηk2 j il iω+ηk2 i x y
the possible productionof both small-scale kinetic and current
− −
(Ru¨diger1975)fromwhichthesmall-scalecurrenthelicity helicity.Inthecomputationallayertheverticalcoordinatezruns
from 0 to 2H while the convection zone is located in [H,2H].
Hcurr = ZZ hbˆ(k,ω)·curlbˆ(k′,ω′)iei((k+k′)x−(ω+ω′)t) cTuhrelBv¯alueB¯oyfBt¯hxebeyledcetrfiicn-ictiuornr.entis Jy = B¯x/(µ0H). It is B¯ ·
∝
dkdk′dωdω′ (8)

caneasilybeformed.A homogeneousandisotropicturbulence
fieldwiththespectraltensor
4
E(k,ω) k k
Qˆ(0)(k,ω)= δ i j (9)
ij 16πk2 (cid:18) ij − k2 (cid:19) 3
may be postulated where the positive spectrumE givesthe in-
2
tensityofthefluctuations.Aftermanipulationsonederivesfrom
(8)and(9)thefinalexpression
1
∞ ∞ 0.0 0.5 1.0 1.5 2.0
1 k2E(k,ω)
Hcurr = 6Z Z ω2+η2k4dkdω B¯ ·curlB¯. (10) Fig.1. Snapshots of temperature (red lines) and density (blue
0 −∞ lines)duringtherunforB¯x =10,B¯y =10.Thesurfacevalues
arefixedtounity.
The integral in (10) does only exist for the high-conductivity
limitη 0aftermultiplicationwithηsothat
→ The compressible MHD equations have been solved with
Rm B¯ curlB¯ (11) the NIRVANA code (Ziegler 2004) in a Cartesian box with
Hcurr ≃ 3 · gravitation along the negative z-axis. The box is periodic in
the horizontaldirectionsand all mean-field quantitiesare aver-
scaleswiththemagneticReynoldsnumberRm=u ℓ /η.
rms corr agedoverthe horizontalplane.Forthe Prandtlnumberandthe
Equations(3)and(11)leadto
magneticPrandtlnumberacommonvalueisfixedto0.1.Asin
α u curlu + Rm B¯ curlB¯ (12) tPhaepesrpeIetdheofveslooucnitdiesatatrheeustoepd ionfuthneitscoonfvceacct/io1n00bwoxhe(rseeecaFcigis.
∝−h · i 3µ ρ ·
0 1). Correspondingly, the magnetic fields are given in units of
for the α effect for weak magnetic fields ( see Blackman & √µ0ρcac/100 with µ0 = 4π. We again assume an ideal, fully
Brandenburg2002,Brandenburg&Subramanian2005). ionized gas heated from below and kept at a fixed temperature
2
G.Ru¨digerandM.Ku¨ker:Currenthelicityandelectromotiveforce
atthetopofthesimulationbox.AsispaperIthedimensionless
Rayleigh numberis fixed to 107. Periodic boundaryconditions 1.0 1.0
applyatthehorizontalboundaries.Theupperandlowerbound-
0.5 0.5
ary are perfect conductors so that there for the total magnetic
fields 0.0 0.0
dB dB
x = y =B =0, (14) -0.5 -0.5
z
dz dz
-1.0 -1.0
similar to the stress-free conditionsof the velocity, dux/dz = 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
du /dz =u =0.Attheinnerboundarytheheat-fluxisfixed. <> = -0.06 <> = 0.42
y z
One finds in paper I that for very weak magnetic fields the re-
sultingrms-valueoftheconvectionvelocityintheusedunitsis Fig.3.ThesameasinFig.2butforthecorrelationcoefficients
about8(seealsotheleftpanelofFig.2below)sothatwiththe ckin(left)andccurr(right).Thenumbersbelowthegraphsarethe
dimensionlwssη = 0.06theglobalmagneticReynoldsslightly volume averages of the coefficients over the whole convective
exceeds100whichislargerbydefinitionthanthelocalmagnetic domain.Theaverageofallsnapshotsinbothcasesisbelowunity.
ReynoldsnumberRm=u ℓ /ηintroducedabove.
rms corr
We start to compute the intensities of the magnetoconvec-
small-scale kinetic helicity but it generates small-scale current
tion u2 = u2 and b2 = b2 (Fig. 2). The velocity at
the toprmosf thehconivectivermbosx reshultsias <∼10% of the surface helicity.
value of the sound speed which for the Sun is about 10 km/s.
The giventime averagesandverticalaveragesin z=[1,2]of all
snapshots are taken as the characteristic values of the quanti- 1000
ties.Thedifferencesofthevelocitydispersiontothosegivenin 400
500
paper I, where much weaker vertical fields have been applied, 200
are very small. The magnetic quenching of the velocity fields 0 0
does exist but it is weak. However, the ratio q = b2 /B¯2 is -200
h i -500
strongly affected by the mean magnetic field. While its value
is about 30 for the vertical field B¯z = 1 in paper I the helical -400 <> = 186.05 -1000 <> = 351.41
field B¯x = B¯y = 10onlyallowsq valuesoforderunity.Note 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
also thatu2rms >∼b2rms/4πρ, i.e. the convectionis notmagnetic-
dominated.Contrarytothat,weshalldemonstratethatneverthe-
lessthemagnetichelicitystronglydominatesthekinetichelicity. Fig.4.Thesmall-scalecurrenthelicity b curlb asduetothe
large-scalecurrenthelicity.Leftpanel:Bh¯ ·=10,Bi¯ =10,right
x y
panel:B¯ = 10,B¯ = 20.The numbersin the plots markthe
x y
averages of all given snapshots. The stratified layer is convec-

tivelyunstableattheright-handsideoftheverticaldashedline
100 600
wherez >1.
80 500
400
60 The behavior of the magnetoconvection complies with the
300 aboveanalyticalresultof the incapability of large-scalehelical
40 fieldstoproducesmall-scalekinetichelicityandtheyareincor-
200
respondencewiththenumericalresultofBhatetal.(2014)who
20 100
foundthe helicitiesasmagnetic-dominatedin theirsimulations
<> = 72.76 <> = 383.55
0 0 ofdecayinghelicalbackgroundfieldsundertheinfluenceofnon-
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
helicallydriventurbulence.

Fig.2.Theturbulence-intensitiesu2 (left)andb2 (right)for After(11)oneexpectsforthecurrenthelicitynumericalval-
B¯x = 10,B¯y = 10.Thenumbersrimnsthegraphsarrmesthevolume uesexceedingB¯xB¯y whichindeedisconfirmedbythesimula-
averagesof the coefficients overthe whole convectivedomain. tions(Fig.4).Theaveragenumberoftherightplotisjusttwice
Theaveragesoverallblue-linesnapshotshereandinallthebe- the number in the left plot demonstrating the linear run of the
lowplotsaregivenbysolidredlines. correlations with the the large-scale helicity. The relation (11)
whichlinearlyconnectsthesmall-scalecurrenthelicitywiththe
large-scalecurrenthelicitycanthusbeconsideredasrealizedby
Forthesamemodelthecorrelationcoefficients thenumericalsimulations.Thenumericalvaluebetweenthetwo
helicitiesanditsrelationtothemagneticReynoldsnumberwill
H u curlu H b curlb
c = h · i, c = h · i (15) bediscussedbelowinthesection3.2.
kin u2 curr b2
rms rms
for the kinetic helicity and the currenthelicity have been com-
3. Theelectromotiveforce
puted (Fig. 3). The numericalvalues strongly differ. While the
fluctuationsofbandcurlbarewellcorrelatedthisisnottruefor The same magnetoconvection model is used to calculate the
theflowfield. turbulence-inducedelectromotiveforce.Aselectric-currentsex-
Asystematicbehaviorofthez-profiledoesalsonotexist.In istintheconvectionzoneitcertainlywillcontaintensorialcom-
accordance with the analytic quasilinear calculations magneto- ponentsofmagneticdiffusion.The questioniswhetheralso an
convectionsubjecttohelicalbackgroundfieldsdoesnotgenerate electromotiveforcealongthemagneticfieldoccurswhichmust
3
G.Ru¨digerandM.Ku¨ker:Currenthelicityandelectromotiveforce
beinterpretedasanαeffect.Surprisingly,theanswerwillbethat 3.1. Numericalsimulations
noαeffectisgeneratedbytheconvectioninfluencedbythehe-
The two horizontal components of the electromotive force are
licalmagneticbackgroundfield(15).Assimultaneouslycurrent
calculated for two different cases. In the first simulations the
helicitywithoutkinetichelicityexists,Eq.(3)cannotbecorrect
applied magnetic fields are assumed as free of helicity, i.e.
forallpossibleturbulencefields.
TheelectromotiveforceE= u b inturbulentplasmaisa B¯ ·curlB¯ =0.TheotherpossibilityworkswithB¯ ·curlB¯ 6=0.
h × i Iftheresultsofbothsetupsareidenticalthentheglobalhelicity
polarvector.Ifonewantstoexpressitlinearlybytheaxialmean
doesnotcontributetotheelectromotiveforceasissuggestedby
magneticfieldvectorB¯ thenonlytheformulation
Eq.(17).
E= η curlB¯ γ curlB¯ +.... (16) Consider first the two differenthelicity-free cases where in
− T − × they directioneitheronlyafieldoronlyanelectric-currentex-
ispossibleifpseudoscalarssuchasg Ω orB¯ curlB¯ arenot ists.Hence,forB¯x =0
· ·
available.Withoutglobalrotationanαeffectintheformulation =γ B¯ , =0. (20)
E = αB¯ ...doesnotoccurinthelinearrelation(16).Thepa- Ex z y Ey
rameterηT describestheturbulence-inducededdydiffusivityin Figure5(left)displaystheresultingγz aspositive(i.e.theflow
the fluid while the vector γ gives a velocity which transports isdownwards)andoforderγ 0.24inunitsofc /100which
z ac
(‘pumps’)the large-scalemagneticfield downwardforpositive withc 10km/sapproxima≃tely1leadstoadownwardpump-
ac
γz.If,however,termsnonlinearinB¯ areincludedintheheuris- ingof20≃m/s.Thesimulatedelectromotiveforce y vanishesas
ticexpression(16)thentheextraterms requiredby(20). E
ThealternativeexamplewithB¯ =0yields =0and
y x
1 E
E=....+ηˆ (cid:18)B¯2curlB¯ + 3(cid:20)∇B¯2+(B¯ ·∇)B¯(cid:21)×B¯(cid:19) (17) = (η ηˆB¯x2 )+ γ 2 z ηˆB¯x2 z B¯x. (21)
Ey −(cid:18) T− µ ρ z− 3H2 µ ρ (cid:19) H
0 (cid:0) 0 (cid:1)
occur within the second-order-correlation approximation
(Kitchatinovet al. 1994). The first term of the RHS represents Indeed,thenumericallysimulated xvanisheswhile y isnega-
E E
theknownmagneticinfluenceontheeddydiffusivitywhilethe tive(seeFig.5,right).
second one forms a magnetic-induced pumping effect. There
is no term (B¯ curlB¯)B¯ representing an α effect with the
large-scale curre·nt helicity B¯ curlB¯ of the background field
as(pseudoscalar)coefficient.F·ornonrotatingandnon-helically 5 5
driventurbulencethelarge-scalecurrenthelicitydoesnotcreate
0 0
an α effect in this approximation. Indeed, we do not find an
α effect in the numerical simulations below. There is instead
-5 -5
the – sometimes ignored – magnetic-induced new pumping <> = 2.09 <> = -0.19
term which for positive ηˆ advects the magnetic field towards 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

the maximum magnetic field. This effect should appear in all
numericalmodelswithhelicallarge-scalefieldandnonhelically
5 5
drivenflow(Yousefetal.2003;Bhatetal.2014).
Equations(16)and(17)yield
0 0
B¯2 ηˆ B¯2 -5 -5
E= (η ηˆ )curlB¯ (γ ∇ ) B¯. (18) <> = 0.02 <> = -4.78
T
− − µ0ρ − − 3 µ0ρ × 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

Thenonlineartermsmagneticallyaffectboththeeddydiffusiv-
ityandtheadvectionvelocitywhicharebasicallyreducedifthe
factor Fig.5.Thetwohorizontalcomponents(top: ,bottom: )of
x y
E E
the turbulence-inducedEMF for backgroundfields withouthe-
∞∞
1 ηk4(νηk4 ω2)E(k,ω)dkdω licity.Leftpanel:B¯x =0,B¯y =5(onlyhomogeneousmagnetic
ηˆ=5Z Z (ω2+η2−k4)2(ω2+ν2k4) (19) fieldinydirection);rightpanel:B¯x =10,B¯y =0(onlyhomo-
0−∞ geneous electric-current in y direction). The averages over all
snapshotsarenotedintheplots.
ispositive.Theηˆpossessesthedimensionofatimeandcanbe
estimated as ηˆ Rm2τ in the high-conductivitylimit. One
corr
easilyfindsηˆ>≃0forimportantcases.Forν =ηthecoefficient InFig.6thetwocomponentsoftheelectromotiveforceare
ηˆisalwayspositiveforallspectralfunctionswhichdonotgrow shownforafieldwithglobalhelicity,i.e.B¯xB¯y >0.From(18)
withgrowingfrequency.Moreover,forallPmthefullexpression onefinds
(19)ispositiveforE δ(ω)representinglongcorrelationtimes
aswellasforE con∝st,i.e.veryshortcorrelationtimes(‘white γ˜ = Ex, η˜= H(Ey + Ex z ), (22)
noise’).Ifcomp≃aredwiththemagnetic-diffusiontimeµ σℓ2 z B¯y − B¯x B¯y H
0 corr
thedelta-likespectrallinerepresentsthelow-conductivitylimit
where the tildes mark the magnetically quenchedvaluesof the
whilethewhite-noisespectrumrepresentsthehigh-conductivity
pumpingvelocityandtheeddydiffusivity.Thetoppanelsofthe
limit. In both limits the ηˆ is positive and, therefore, both eddy
diffusivityanddiamagneticpumpingaremagneticallyquenched 1 Numericalvaluesinphysicalunitsarealwaysgivenforthetopof
ratherthanamplified(‘anti-quenched’). theunstablezone.
4
G.Ru¨digerandM.Ku¨ker:Currenthelicityandelectromotiveforce
plots only concern the advection velocity γ˜ which is always tainlyonlytrueforlargevaluesofRm(high-conductivitylimit).
z
positive.Itsinksfromγ 0.42toγ˜ 0.25forB = 10by LargeRmrequirehighvaluesoftheReynoldsnumberReand/or
z z x
≃ ≃
themagneticsuppression.Fromthedefinitionofγ˜ oneimme- nottoosmallPm.Itwouldcertainlybemorerealistictoincrease
z
diatelyfindsηˆ 0.02forthequenchingcoefficient(19)inunits the Reynolds number rather than the magnetic Prandtl number
of100H/c .T≃hecodeworkswithρ =1. butourvalueReoforder103istheupperlimitdefinedbytech-
ac 0
Notethatinallcasesthenumericalvaluesof u b arevery nicalrestrictions.
h × i
smallcomparedwith thescalarproductu andb givenin Sofarthesimulationshavebeendonewithafixedmagnetic
rms rms
Fig.2.Theresultingcorrelationcoefficientsarehereverysmall. Prandtl number Pm = 0.1. For fixed molecular viscosity its
The existence of an α effect in the expression (17) for the increase (decrease) by one order of magnitude effectively in-
electromotive force would require a term (B¯ curlB¯)B¯. No creases (decreases) the magnetic Reynolds number by a factor
such term resulted from the quasilinear approx·imation for the of ten. We expect a similar reaction of the small-scale current
influenceofhelicallarge-scalefieldsonnonhelicallydriventur- helicity (11). The reasonis thatafter (11) it growsfor growing
bulence.Thisfindingcanbeprobedwiththenumericalsimula- magneticReynoldsnumbers.BecauseofRm=PmReforfixed
tions.NotethatthreemodelsinFigs.5and6arecomputedfor ReynoldsnumberthemagneticReynoldsnumberrunswithPm.
constant B and for growing helicity (B¯ B¯ =0, 50, 100). If TheresultsofthevariationofthemagneticPrandtlnumber
x x y
an α effect existed one should find that the values of grow are shown by the plots given in Fig. 7. From left to right the
y
E
forgrowinghelicity.Thenumericalresultsforthethreemodels, magnetic Reynolds number sinks by a factor of 33 while the
however,arealwaysidentical.Theydonotshowanynumerical current helicity sinks by a factor of about 17 (i.e. 50% of 33).
responsetothegrowinghelicitiesindicatingthenonexistenceof The current helicity indeed proves to be proportionate to Rm
an α termproportionalto B¯ curl B¯. The nonexistenceof the which, however,provesto be smaller than the globalmagnetic
α term (B¯ curl B¯)B¯ in th·e electromotive force (17) in the Reynoldsnumberoforder100.SinceinpaperIformuchweaker
·
quasilineartheoryisthus confirmedbynonlinearsimulations. fields the resulting Rm were much higher the suggestion may
Equation(22)allowstoderivethemagneticdiffusivityfrom be allowedthatfor the presentmodelsthe magneticquenching
the simulations. One finds by elimination of ηˆfrom the defini- effectisreducingthenumericalvalues.
tionstherelationη =η˜+(γ γ˜)H.Withthegivenmeanval-
T z
uesforallsnapshotsthemodel−withB¯ =5providesη˜ 0.11,
y
≃
so that η = 0.27 in units of Hc /100 for the unquenched
T ac 300
eddy diffusivity. If this result is compared with the traditional 1000
estimate η (1/3)u ℓ one finds ℓ /H 0.1. On 200
T ≃ rms corr corr ≃ 500
the other hand, if η˜ = 0 is used as the definition of a critical
100
magneticfield, i.e.B = η /ηˆ, thenB 3.6in unitsof 0
eq T eq
√4πρc /100sothatwithpρ 10−2 g/cm3 an≃dc = 10km/s -500 0
ac ac
≃
thefieldamplitudeBeq ≃13kGaussresultsinphysicalunits. -1000 <> = 416.13 -100 <> = 24.05
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

5 5
0 0 Fig.7. for B¯ = B¯ = 10. Left panel: Pm = 1, right
curr x y
H
panel:Pm=0.03.
-5 -5
<> = 1.27 <> = 1.68
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

It is easy to give the strength of this statement a test. It

is known that the advection term γ represented after (20) by
z
5 5 /B¯ does not vary with Rm (see Kitchatinov et al. 1994).
x y
E
Hence,we donotexpecta similar behavioras inFig. 7 forthe
0 0 electromotiveforce .Figure8indeedshowsthatforsmallRm
x
E
thereductionofthe magneticPrandtlnumberbyafactorof33
-5 -5
<> = -4.95 <> = -4.03 leavestheelectromotiveforce xbasicallyunchanged.
E
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 TheresultsofFigs.7and8canalsobeunderstoodasprob-

ingtheinnerconsistencyofthemodelsandthenumericalproce-
dures.Itisadditionallyshownthatbasicresultsofthequasilinear
approximationareconfirmedbysimulations.
Fig.6.ThesameasinFig.5butforthehelicalbackgroundfields
withB¯ =10,B¯ =5(leftpanel)andB¯ =10,B¯ =10(right
x y x y
panel).Theaveragesoverallsnapshotsarenotedintheplots. 4. Conclusions
By analytical (quasilinear) theory and numericalsimulations it
isshownthatinanonrotatingconvectivelayeralarge-scalehe-
lical magnetic backgroundfield producesa small-scale current
3.2. VariationofthemagneticPrandtlnumber
helicitywhilethekinetichelicityvanishes.Thecurrenthelicity
It remains to probe the analytical formulation(11) after which possesses the same sign as the helicity of the large-scale field.
the ratio of the small-scale current helicity and the large-scale TheratioofbothpseudoscalarsrunswiththemagneticReynolds
helicitygrowswiththemagneticReynoldsnumber.Thisiscer- number of the convection.The same magnetic model provides
5
G.Ru¨digerandM.Ku¨ker:Currenthelicityandelectromotiveforce

5 5
0 0
-5 -5
<> = 1.47 <> = 2.16
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

Fig.8.ThesameasinFig.7butfor .
x
E
finite values of the diamagnetic pumping term γ and the eddy
diffusivityη butanαeffectdoesnotoccurinthesameorder.
T
Ifitistruethattheαeffectinthestellarconvectionzonesis
duetotheactionoftheCoriolisforceontheturbulentconvection
then it is positive (negative) on the northern (southern) hemi-
spherewhich after Eq.(2) wouldlead to the oppositesigns for
thesmall-scalecurrenthelicity–asisindeedobservedasdom-
inating during the activity cycle. The new effect of the small-
scale currenthelicity due to the influence of a large-scale field
combinedwith a parallelelectric-currentyieldsopposite signs.
Duringtheminimumphaseofthesolaractivitycyclejustthese
signssometimesoccur.Zhangetal.(2010)givetheapproxima-
tive value 10−5G2/cm for the observed small-scale currenthe-
licity.Withtheradialscaleof100MmandwithB¯ B¯ 104G2
r φ
≃
oneonlyneedsamagneticReynoldsnumberRmofthefluctua-
tionsoforder10tofulfillthenumericalconstraints.
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