Table Of Content

Astronomy&Astrophysicsmanuscriptno.oscillatory˙1712 (cid:13)c ESO2011 January11,2011 Oscillatory dynamos and their induction mechanisms M.Schrinner1,L.Petitdemange2,andE.Dormy1 1 MAG(ENS/IPGP),LRA,EcoleNormaleSupe´rieure,24rueLhomond,75252ParisCedex05,France e-mail:[email protected] 2 Max-Planck-Institutfu¨rAstronomie,Ko¨nigstuhl17,69117Heidelberg Received;accepted ABSTRACT 1 1 Context.Large-scalemagneticfieldsresultingfromhydromagneticdynamoactionmaydiffersubstantiallyintheirtimedependence. 0 Cyclicfieldvariations,characteristicforthesolarmagneticfield,areoftenexplainedbyanimportantΩ-effect,i.e.bythestretching 2 offieldlinesduetostrongdifferentialrotation. Aims.Thedynamomechanismofaconvective,oscillatorydynamomodelisinvestigated. n Methods.Wesolvethe MHD-equations for a conducting Boussinesq fluidinarotating spherical shell. Foraresulting oscillatory a J model,dynamocoefficientshavebeencomputedwiththehelpoftheso-calledtest-fieldmethod.Subsequently,thesecoefficientshave beenusedinamean-fieldcalculationinordertoexploretheunderlyingdynamomechanism. 0 Results.Theoscillatorydynamomodelunderconsiderationisofα2Ω-type.Althoughtheratherstrongdifferentialrotationpresentin 1 thismodelinfluencesthemagneticfield,theΩ-effectaloneisnotresponsibleforitscyclictimevariation.IftheΩ-effectissuppressed theresultingα2-dynamoremainsoscillatory.Surprisingly,thecorrespondingαΩ-dynamoleadstoanon-oscillatorymagneticfield. ] R Conclusions.TheassumptionofanαΩ-mechanismdoesnotexplaintheoccurrenceofmagneticcyclessatisfactorily. S . h p 1. Introduction implemented in a large number of solar dynamo models (e.g. - Steenbeck&Krause1969;Roberts1972;Roberts&Stix1972; o The study of the solar cycle has motivated dynamo theory for Stix 1976; Ossendrijver 2003; Brandenburg&Subramanian r many decades. Hence, the solar dynamo has become the pro- t 2005; Chanetal. 2008) corresponds to the isotropic compo- s totype of oscillatory dynamos.However,its explanationis still nent of a in relation (2), while the Ω-effectresults from the φ- a controversial (Jonesetal. 2010). Most solar dynamo models [ componentofthe∇×(V×B)terminequation(1). have been built on the assumption of an αΩ-dynamo mecha- However, a strong differential rotation is not a neces- 1 nism(Ossendrijver2003);thatis,thepoloidalfieldresultsfrom sary condition for oscillatory solutions of the dynamo equa- v the interaction of helical turbulence with the toroidal field (α- 7 effect)whereasthetoroidalfieldisthoughttooriginatefromthe tion (1), as has been demonstrated in several papers (see e.g. 3 shearingofpoloidalfieldlinesbystrongdifferentialrotation(Ω- Ra¨dler&Braüer 1987; Schubert&Zhang 2000; Ru¨digeretal. 8 effect).Thisattemptisattractiveformainlytworeasons: 2003;Stefani&Gerbeth2003).Theseauthorsconstructmodels 1 inwhichthetoroidalfieldislikewisegeneratedfromthepoloidal First, the existence of a strong shear layer at the bottom of . field by anα-effect(α2-models)andinvestigatenecessarycon- 1 thesolarconvectionzoneisobservationallywellestablished,and 0 theimportanceofaresultingΩ-effectisnon-controversial. straintsona,theboundaryconditionsforthemagneticfieldand 1 Second,Parker’splanelayermodel(Parker1955)andinpar- the geometry of the dynamo region in order to obtain oscilla- 1 ticular mean-fieldelectrodynamics(Steenbecketal. 1966) pro- torysolutionsof(1).Recently,oscillatorydynamomodelshave : alsobeeninvestigatedbymeansofdirectnumericalsimulations. v vide a very elegant theoretical framework for this approach. Mitraetal. (2010) performed dynamo simulations in a wedge- i Within mean-field theory, attention is focused on large scale, X shapedsphericalshellwith anappliedforcinganddemonstrate i.e. averaged fields, only, and the induction equation may be r replaced by a mean-field dynamo equation (Krause&Ra¨dler againtheexistenceofoscillatoryα2-dynamomodels. a 1980) However, the success of mean-field models in reproducing solar-like variations of the magnetic field relies partly on the ∂B =∇×(E+V×B−η∇×B), (1) largenumberoffreeparameters,i.e.onthearbitrarydetermina- ∂t tionofthedynamocoefficientsaandb.Analternativeapproach inwhich BandV denotetheaveragemagneticandtheaverage is presented by Pe´tre´lisetal. (2009). They constructamplitude velocityfield,ηstandsforthemagneticdiffusivityandEisthe equationsguidedfromsymmetryconsiderationsandanalysepo- meanelectromotiveforce.Moreover,itisassumedthatEisho- larity reversals and oscillatons of the magnetic field resulting mogeneousinthemeanmagneticfieldandmaybereplacedby fromtheinteractionbetweentwodynamomodes. aparameterisationintermsofBanditsfirstderivatives Self-consistent, global, convective dynamo models with cyclic magnetic field variations have been bublished by E=aB+b∇B. (2) Busse&Simitev (2006), and Goudard&Dormy (2008). In(2), the so-calleddynamocoefficientsa andb are tensorsof Convective dynamo simulations with stress-free mechanical secondandthirdrank,respectively,anddependonlyontheve- boundary conditions (Busse&Simitev 2006) exhibit a strong locityfieldandthemagneticdiffusivity.Thetraditionalα-effect andaweakfieldbranch,dependingontheinitialconditionsfor 2 M.Schrinneretal.:Oscillatorydynamosandtheirinductionmechanisms themagneticfield.Ifthemagneticfieldisinitiallyweak,stress inwhichthelinearoperatorDisdefinedas free boundary conditions enable the development of a strong zonal flow carrying most of the kinetic energy and rendering 1 DB=V×B+aB+b∇B− ∇×B. (7) convectionineffective. The magnetic field resulting from these Pm dynamosisrathersmallscaled,oftenofquadrupolarsymmetry Thetimeevolutionofeachmodeisdeterminedbyitseigenvalue and weak. Oscillatory solutions of the induction equation are σandproportionaltoexp(σt).Formoredetailsconcerningthe typicalforthisdynamobranch. eigenvaluecalculation,werefertoSchrinneretal.(2010b). A transition from steady to oscillatory dynamos may Wealsoconsidertheevolutionofakinematicallyadvanced also be governed by the width of the convection zone; magneticfield,B ,governedbyasecondinductionequation Goudard&Dormy(2008)foundoscillatorymodelsbydecreas- Tr ingthe shellwidth.Inthisstudy,we followtheirapproachand ∂B 1 analysethedynamomechanismfortheseoscillatorymodels.In ∂tTr =∇×(v×BTr)+ Pm∇2BTr. (8) particular, we address the question whether an Ω-effect is re- sponsibleforthecyclicvariationofthemagneticfield.Different ThetracerfieldB experiencestheself-consistentvelocityfield Tr frompreviouswork,wedeterminethedynamocoefficientsaand at each time step but does not contribute to the Lorentz force bfromdirectnumericalsimulationswiththehelpofthetest-field andispassiveinthissense(seealsoSchrinneretal.2010a).Its method(Schrinneretal.2005,2007).Theapplicationofaandb evolution will be compared with mean-field results originating inamean-fieldcalculationrevealstheirimportanceforthegen- likewisefromakinematicapproach.Moreover,akinematically erationofthemagneticfield. advanced tracer field allows us to test for the influence of the Ω−effectindirectnumericalsimulations.Inanumericalexperi- ment,wesubtractthecontributionoftheΩ−effectandthemean 2. Dynamocalculations meridionalflowintheequationforthetracerfield, WeconsideraconductingBoussinesqfluidinarotatingspheri- ∂B 1 calshellandsolvetheequationsofmagnetohydrodynamicsfor Tr =∇×(v×B )+ ∇2B −∇×(V×B ), (9) ∂t Tr Pm Tr Tr the velocityv, magneticfield B andtemperatureT as givenby Goudard&Dormy (2008) with the help of the code PaRoDy andstudyinthiswaytheoutcomeofakinematicα2-dynamo. (Dormyetal.(1998)andfurtherdevelopments), ∂v E +v·∇v−∇2v +2z×v+∇P= 3. Results ∂t ! The modelunderconsiderationhas been previouslystudied by r 1 Ra T + (∇×B)×B, (3) Goudard&Dormy (2008). It is defined by E = 10−3, Ra = ro Pm 100(= 2.8Rac), Pm = 5, Pr = 1 and an aspect ratio of 0.65. ∂B 1 Except for the stress-free mechanical boundary condition ap- =∇×(v×B)+ ∇2B, (4) ∂t Pm pliedatr = ro andanincreasedaspectratio,thegoverningpa- ∂T 1 rameters are those of a rather simple, quasi-steady benchmark ∂t +(v·∇)(T +Ts)= Pr∇2T. (5) dynamo(Christensenetal.2001).However,Goudard&Dormy (2008) report a transition from steady, dipolar to oscillatory Governingparameters are the Ekman number E = ν/ΩL2, the models for these parameter values. Note that the model re- (modified) Rayleigh number Ra = α g ∆TL/νΩ, the Prandtl quires a rather high angular resolution up to harmonic degree T 0 numberPr = ν/κ andthemagneticPrandtlnumber Pm = ν/η. l =112. max In these expressions, ν denotes the kinematic viscosity, Ω the Figure1 displaystheradialcomponentofthe velocityfield rotationrate, L theshellwidth,α the thermalexpansioncoef- at a given radial level. A typical columnar convection pattern T ficient,g isthegravitationalaccelerationattheouterboundary, is visible, even though the convection columns are noticeably 0 ∆T standsforthe temperaturedifferencebetweenthe spherical disturbedbytheinfluenceofthecurvedboundariesandastrong boundaries,κisthethermalandη=1/µσthemagneticdiffusiv- zonalflowcarryingabout50percentofthekineticenergy.The ity withthe magneticpermeabilityµandthe electricalconduc- magnetic Reynolds number based on the rms-velocity and the tivity σ. Furthermore,the aspectratio is definedas the ratio of shellwidth,Rm = v L/η,isabout90.Theflowissymmetric rms the innerto the outershell radius, r/r ; it determinesthe shell with respect to the equatorialplane and convectiontakes place i o width. onlyoutsidetheinnercoretangentcylinder. In our models,convectionis drivenby an imposed temper- Theevolutionofthemagneticfieldiscyclic.Infigure2(top), ature gradient between the inner and the outer shell boundary. contours of the azimuthally averaged radial magnetic field at Themechanicalboundaryconditionsarenoslipattheinnerand the outer shell boundaryvarying with time are plotted in a so- stressfreeattheouterboundary.Moreover,themagneticfieldis called butterfly diagram. A dynamo wave migrates away from assumedtocontinueasapotentialfieldoutsidethefluidshell. the equator until it reaches mid-latitudes where the inner core Time-averaged dynamo coefficients for an axisymmetric tangent cylinder intersects the outer shell boundary. The mag- mean magnetic field have been determined from direct numer- neticfieldlooksrathersmallscaledandmultipolar.Thisiscon- icalsimulationsasdescribedindetailbySchrinneretal.(2007) firmed by the magnetic energy spectrum which is essentially andasrecentlydiscussedfortime-dependentdynamomodelsby white,exceptforanegligibledipolecontribution.Furthermore, Schrinner(2011).Inasecondstep,thesecoefficientshavebeen themagneticfieldisweak,asexpressedbyanElsassernumber appliedin a mean-fieldmodelbased on equation(1) written as ofΛ= B2 /(µρηΩ)=0.13. rms aneigenvalueproblem, The kinematically advanced tracer field grows slowly in time,i.e.themodelunderconsiderationiskinematicallyunsta- σB=∇×DB, (6) ble according to the classification by Schrinneretal. (2010a). M.Schrinneretal.:Oscillatorydynamosandtheirinductionmechanisms 3 Fig.1.Snapshotoftheradialvelocityoftheconsidereddynamo model at r = 0.79r . The velocity component has been nor- o malised by its maximum absolute value, v = 24.46ν/L. r,max Hence, the colour coding ranges from −1, white, to +1, black. Contourlinescorrespondto±0.2and±0.6. But,deviationsofthetracerfieldfromtheactualfieldarehardly noticeable in the field morphology. Moreover, the very same dynamo wave persists in the kinematic calculation (see also Goudard&Dormy(2008)),asvisibleinfigure2(middle).Note thatthe tracerfield in figure 2 hasevolvedfromrandominitial conditions. A mean-field calculation based on the dynamo coefficients a, b and the mean flow V determined from the self-consistent model is presented in the bottom line of figure 2. The fastest growing eigenmodes form a conjugate complex pair and give rise to a dynamo wave which compares nicely with the direct numericalsimulations.Since thismodeldependsonthe fulla- tensorandthemeanflow,werefertoitasanα2Ω-dynamo. Theinfluenceofthedifferentialrotationmaybesuppressed inthekinematiccalculationofthetracerfieldwithoutchanging anyother componentof the flow. A butterflydiagramresulting fromakinematicallyadvancedfieldaccordingtoequation(9)is presentedinfigure3(top).Theevolutionofthemagneticfieldis again cyclic. Apart from small-scale variationson shorter time scales,adynamowavemigratesfrommid-latitudestowardsthe equator. This is in agreement with a correspondingmean-field calculationinwhichthemeanflowV in(7)hasbeencanceled: Thebottomchartoffigure3providesthebutterflydiagramstem- ming fromthe fastest growingeigenmodesof the resulting α2- dynamo.An explanationwhy directnumericalsimulationsand meanfieldcalculationscomparesomewhatbetterinfigure2than infigure3isprovidedinappendixA. Fig.2. Azimuthally averaged radial magnetic field at the outer ThetimeevolutionoftherelatedαΩ-dynamoisoffurtherin- shell boundary varying with time (butterfly diagram) resulting terest.Astheα-effectisnotdirectlyaccessibleindirectnumer- from a self-consistent calculation (top), kinematic calculation icalsimulations,thecorrespondingαΩ-dynamocanberealised according to (8) (middle) and mean-field calculation (bottom). ina mean-fieldcalculation,only.Ina firstattempt,we haveset Thecontourplotshavebeennormalisedbytheirmaximumabso- arr = aθθ = 0 in order to suppress the generation of toroidal lutevalueateachtimestepconsidered.Thecolourcodingranges fieldfrompoloidalfieldbyanα-effect.Bothcomponentsmake from−1,white,to+1,black. major contributionsto this process. The leading eigenmodere- sulting fromthis calculationis shown in figure 4; it is real, i.e. weintroduceaCartesiancoordinatesystem(x,y,z)correspond- non-oscillatory, and close to marginal stability. The results re- ingtothe(φ,θ,r)directionsanddefinemeanquantitiestobex- mainsimilarifweneglectfurther,non-diagonalcomponentsof a. independent.Moreover,wewriteB= Bex+Bp= Bex+∇×Aex andreformulate(1)inthefollowingsimplifiedmanner ∂A 1 4. Discussion = α B+ ∇2A, (10) ∂t xx Pm The Frequency and the propagation direction of the dynamo ∂B ∂ ∂A dV ∂A 1 wave visible in figure 2 depend strongly on the differential ro- = − (α )+ x + ∇2B. (11) ∂t ∂y zz∂y dz ∂y Pm tation, in agreement with Busse&Simitev (2006). We follow theirapproachandgiveanestimateforthecyclefrequencyap- In the above equations, we have considered only the dominant plyingParker’splanelayerformalism(Parker1955).Tothisend, diagonalcomponentsofa,a anda correspondingtoα and rr φφ zz 4 M.Schrinneretal.:Oscillatorydynamosandtheirinductionmechanisms Fig.4. The leading dipolar eigenmode resulting from a mean- field calculation with a = a = 0. Contour plots of all three rr θθ componentsare presented, each normalised separately by their maximumabsolutevalues.Maximaandminimaarewrittennext to each plot. The colour coding ranges from −1, white, to +1, black, and contour lines correspond to ±0.1, ±0.3, ±0.5, ±0.7 and±0.9. and ω = ℑ(σ)= Fig.3. Azimuthally averaged radial magnetic field at the outer 1/2 shell boundary varying with time (butterfly diagram) resulting 2 α dV ∂α apfrloocmtosrraaerskepipnorenemdsieanntgitcemdcaaelsacnuin-lfiafiteiglodunrcewa(li2cthu).lsautibotnrac(bteodttoΩm-e)ff.eTchte(tocopn)taonudr ±r 2xx vt(αzzky2)2+dz − ∂yzz ky2−αzzky2 . (17) Thesignin(17)isdeterminedbythesignofk (dV/dz−∂α /∂y). y zz Ifwefurtherassumethatthefrequencyisdominatedbydifferen- tialrotationandneglecttheα-termsin(17),weestimatesimilar α ;allcomponentsofbandthemeanmeridionalflowhavebeen xx toBusse&Simitev(2006) neglected.Furthermore,V hasbeenassumedtodependonlyon z.Then,theansatz π 1/2 ω=ℑ(σ)≈± α 2E . (18) (A,B)=(Aˆ,Bˆ)exp(ik·x+σt), (12) L2 xxq T! In(18),E denotesthethekineticenergydensityduetotheax- T leadsto isymmetrictoroidalvelocityfieldandk ≈ 2π/Lhasbeenused. y Approximating α by the rms-value of a , α = 11.50ν/L, pAˆ = αxxBˆ, (13) and with E = 7x1xν2/L2, we find ω ≈ ±φ1φ03.7xxη/L2 which is T dV ∂α surprisingly close to ω = ±100.95η/L2 in the full calculation pBˆ = α k2+ik − zz Aˆ, (14)  zz y ydz ∂y  presentedin figure 2. Note that αxx and dV/dzare of the same    orderofmagnitudeandcontributeequallytoω.Thesignin(18) with p = σ+|k|2/Pm.From(13),(14),we derivea dispersion is determined by the sign of the product aφφ∂Vφ/∂r which is relation positiveinthenorthernandnegativeinthesouthernhemisphere. Therefore,ourestimatein(18)predictsadynamowavemigrat- dV ∂α p2 =α α k2+ik − zz , (15) ingawayfromtheequator.Thisisinagreementwiththesimu- xx zz y ydz ∂y  lationsshowninfigure2.    However, the attempt to describe the model under consid- from which the real and the imaginarypart of σ can be calcu- erationasanαΩ-dynamofails.Anoscillatorymodewithafre- lated.Ifα ispositive(e.g.inthenorthernhemisphere),itfol- xx quencyclosetotheaboveestimateturnsouttobeclearlysubcrit- lows icalinamean-fieldcalcuation,ifa anda areomitted.Instead, rr θθ thismodelisgovernedbyareal,dipolarmodeclosetomarginal λ = ℜ(σ)=−|k|2/Pm stability(seefigure4).Hence,theΩ-effectisonlypartlyrespon- 1/2 2 sibleforthegenerationofthemeanazimuthalfield,asconfirmed α dV ∂α +r 2xx vt(αzzky2)2+dz − ∂yzz ky2+αzzky2 (16) rbByrfi∂g(ru−r1eV5φ.)/T∂hre+crh−1arstininθBthθe∂(msiindθd−le1Vcφo)m/∂pθarinesgrtheyescΩa-leeffweictth, M.Schrinneretal.:Oscillatorydynamosandtheirinductionmechanisms 5 the mean azimuthal field displayed by superimposed contour lines.Inparticular,theelongatedfluxpatchesclosetotheinner coretangentcylinderare,ifatall,negativelycorrelatedwiththe Ω-effect.Consistentwiththisfinding,thepoloidalaxisymmetric magneticenergydensityexceedsthetoroidaloneby20%. Differential rotation alone is not responsible for the cyclic timeevolutionofthemagneticfield,despiteitsinfluenceonthe frequency and the propagation direction of the dynamo wave. Thisismostclearlyvisibleinfigure3.Simulationswithoutdif- ferentialrotationstillleadtoadynamowaveeventhoughitsfre- quency and propagation direction have changed. In the framework of Parker’s plane layer formalism, the frequency of this oscillatoryα2-dynamodependscruciallyon−∂α /∂yinsteadof zz dV /dz. Note the additional minus sign, which might explain x thereversedpropagationdirectioniftheassumethat∂α /∂yis zz predominantlypositive. But differentfromthe radialderivative of the mean azimuthal flow, (1/r)∂arr/∂θ is highly structured, Fig.5. Left:∂Vφ/∂rinunitsofν/L2.Middle:Ω-effectasgiven changessigninradialdirectionandexhibitslocalisedpatchesof by rB ∂(r−1V )/∂r + r−1sinθB ∂(sinθ−1V )/∂θ (greyscale) lownegativevalues(seefigure5).Therefore,wedonotattempt r φ θ φ and B (superimposed contour lines, solid [dashed] lines indi- togiveanestimateforthefrequencysimilarto(18). φ cate positive[negative]values).Right:(1/r)∂a /∂θ in unitsof rr ν/L2.Thecontourplotsarepresentedinthesamestyleasinfig- In order to better understand the influence of the mean ure4. flow on the frequency of the dynamo wave, we have gradu- ally changed the amplitude of V in a series of kinematic calculations. Results are presented in figure 6. Stars denote fre- quenciesobtainedfromeigenvaluecalculationsaccordingto(6), wherastrianglesstandforfrequenciesestimatedfromkinematic results due to equation (8). In both cases, the amplitude of the meanflowV hasbeenvariedbymultiplicationwithascalefac- tor f. For f = 1, the originalcalculation is retained, while for f = 0, we reproduce the α2-dynamo already discussed above. Frequencies of dynamo waves resulting from direct numerical simulationsaccordingto(8)havebeenmeassuredfor f =1, 0.7, and0.5.Owingtotheturbulencepresentinthesimulations,these are rather rough estimates and error bars have been included. Nevertheless,the resultsobtainedare in satisfactory agreement with the eigenvalue calculations. The frequencies in figure 6 decrease continously with decreasing scale factors. If the amplitude of V is reduced to 25 per cent of its original value, ω changessignandthepropagationdirectionofthedynamowave isreversed.Thedashed-dottedlineinfigure6givesωaccording to (18) as predicted for an αω-dynamo. It matches the numer- Fig.6. Frequencies resulting from kinematic calculations in ical results if dV/dz dominates in (17) but deviates clearly for whichtheamplitudeofthemeanflowhasbeenchangedbymul- smaller amplitudes. On the other hand, it is illustrative to use tiplicationwithascalefactor, f.Starsdenotefrequenciesstem- relation (17) to model the dependence of ω on the mean flow. ming from an eigenvaluecalculation accordingto (6), whereas triangles are estimates obtained from kinematic results due to If we set ∂α /dy = 0.25dV/dz anddeterminea representative zz (8). The dashed-dotted line gives frequencies as predicted for valueforα inverting(17)fordV/dz=0andω=−29.15η/L2, zz anαω-dynamoby(18),whilethedashedlinerepresentsωasa thedashedlineinfigure6resultsfrom(17).Itfitsthenumerical functionofVforanα2ω-dynamoaccordingto(17). dataratherwellandconvergestowardsthefrequenciespredicted foranαω-dynamo,iftheamplitudeofVissufficientlyhigh. Let us stress againthat some cautionis neededin applying 5. Conclusions the present mean-field analysis to non-linear direct numerical simulations, as the the dynamo modelconsidered here is kine- A particulardynamomechanismdoesnotseemto beresponsi- maticallyunstable.Strictlyspeaking,ourmean-fieldresultsare blefortheoccurrenceofperiodicallytime-dependentmagnetic only relevant for the kinematically advanced tracer field. But, fields. It turns out, that the influence of the large-scale radial because the model is close to dynamo onset and only weakly shear(the Ω-effect),isnotnecessaryforcyclic fieldvariations. non-linear,webelievethatourinterpretationisalsovalidforthe Instead, the action of small-scale convection, represented by a fullyself-consistentfield.Thisisinparticularconfirmedbythe spatiallystructureddynamocoefficienta ,happenstobeessen- rr rathergoodagreementofthethreebutterflydiagramspresented tial.Forthemodelpresentedhere,smallconvectivelengthscales infigure2. areforcedbya thinconvectionzone.Furtherinvestigationsare 6 M.Schrinneretal.:Oscillatorydynamosandtheirinductionmechanisms neededtoassesswhetherourfindingisrepresentativeforawider classofoscillatorymodels. Acknowledgements. MSisgratefulforfinancialsupportfromtheANRMagnet project.ThecomputationshavebeencarriedoutattheFrenchnationalcomput- ingcenterCINES. AppendixA: Theuseoftimeaverageddynamo coefficients In the following, azimuthal averages are, as throughout in the paper, denoted by an overbar, time averages are expressed by brackets, < ··· >. Initially, dynamo coefficients have been determined for an azimuthally averaged, mean magnetic field B. Hence,theevolutionofthelatterisgivenby ∂B 1 =∇×(aB+b∇B+V×B− ∇×B). (A.1) ∂t Pm But, the dynamo coefficients a, b and the mean flow V vary stochasticallyin time. In orderto describethe averagedynamo action, we take in addition the time averageof these quantities andwriteapproximatively, ∂B 1 ≈∇×(<a> B+<b>∇B+<V>×B− ∇×B). (A.2) ∂t Pm We emphasisethatthereis noa priorirelationbetweenthe left handsideandtherighthandsideofequation(A.2).Theactual, azimuthallyaveragedmagneticfieldwilldeviatefromourmean- field description the stronger, the more a, b and V fluctuate in time. Amongthese three quantities, the mean flow V is almost time independent, whereas a and b vary considerably. This is the reason, why the butterfly diagramsin figure 2 are in better agreementthaninfigure3,forwhichthestabilizinginfluenceof themeanflowhasbeenomitted. References Brandenburg,A.&Subramanian,K.2005,Phys.Rep.,417,1 Busse,F.H.&Simitev,R.D.2006,Geophys.Astrophys.FluidDyn.,100,341 Chan,K.H.,Liao,X.,&Zhang,K.2008,ApJ,682,1392 Christensen,U.R.,Aubert,J.,Cardin,P.,etal.2001,Phys.EarthPlanet.Inter., 128,25 Dormy,E.,Cardin,P.,&Jault,D.1998,EarthPlanet.Sci.Lett.,160,15 Goudard,L.&Dormy,E.2008,Europhys.Lett.,83,59001 Jones,C.A.,Thompson,M.J.,&Tobias,S.M.2010,SpaceSci.Rev.,152,591 Krause,F.&Ra¨dler,K.1980,Mean-fieldmagnetohydrodynamicsanddynamo theory, ed. Goodman,L.J.&Love,R.N.(Oxford,PergamonPress,Ltd., 1980.271p.) Mitra,D.,Tavakol,R.,Ka¨pyla¨,P.J.,&Brandenburg,A.2010,ApJ,719,L1 Ossendrijver,M.2003,A&ARev.,11,287 Parker,E.N.1955,ApJ,122,293 Pe´tre´lis,F.,Fauve,S.,Dormy,E.,&Valet,J.2009,Phys.Rev.Lett.,102,144503 Ra¨dler,K.&Braüer,H.1987,Astron.Nachr.,308,101 Roberts,P.H.1972,Phil.Trans.RSoc.Lond.A,272,663 Roberts,P.H.&Stix,M.1972,A&A,18,453 Ru¨diger,G.,Elstner,D.,&Ossendrijver,M.2003,A&A,406,15 Schrinner,M.2011,inpreparation Schrinner,M.,Ra¨dler,K.,Schmitt,D.,Rheinhardt,M.,&Christensen,U.2005, Astron.Nachr.,326,245 Schrinner, M.,Ra¨dler, K.,Schmitt, D.,Rheinhardt, M.,&Christensen, U.R. 2007,Geophys.Astrophys.FluidDyn.,101,81 Schrinner,M.,Schmitt,D.,Cameron,R.,&Hoyng,P.2010a,Geophys.J.Int., 182,675 Schrinner,M.,Schmitt,D.,Jiang,J.,&Hoyng,P.2010b,A&A,519,A80+ Schubert,G.&Zhang,K.2000,ApJ,532,L149 Steenbeck,M.&Krause,F.1969,Astron.Nachr.,291,49 Steenbeck,M.,Krause,F.,&Ra¨dler,K.1966,Zeit.Nat.A,21,369 Stefani,F.&Gerbeth,G.2003,Phys.Rev.E,67,027302 Stix,M.1976,A&A,47,243