Astron. Nachr./AN000,No.00,1–8(0000)/DOIpleasesetDOI! Modeling a Maunder Minimum A.Brandenburg1andE.A.Spiegel2 1 NORDITA,Roslagstullsbacken23,SE-10691Stockholm,Sweden 2 AstronomyDepartment,ColumbiaUniversity,NewYork10027,USA Thedatesofreceiptandacceptanceshouldbeinsertedlater Weintroduceon/offintermittencyintoameanfielddynamomodelbyimposingstochasticfluctuationsineitherthealpha effectorthroughtheinclusionofafluctuatingelectromotiveforce.Sufficientlystrongsmallscalefluctuationswithtime 8 scalesoftheorderof0.3-3yearscanproducelongtermvariationsinthesystemontimescalesoftheorderofhundredsof 0 years.However,globalsuppressionofmagneticactivityinbothhemispheresatoncewasnotobserved.Thevariationofthe 0 magneticfielddoesnotresemblethatofthesunspotnumber,butismorereminiscentofthe10Berecord.Theinterpretation 2 ofourresultsfocusesattentionontheconnectionbetweenthelevelofmagneticactivityandthesunspotnumber,anissue thatmustbeelucidatediflongtermsolareffectsaretobewellunderstood. n a J (cid:13)c 0000WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim 5 1 1 Introduction In modeling the solar magnetic variability with simple ] oscillators, it is not obvious how to connect the output of h the models with the sunspot number. It may therefore not The sun showsvariabilityon a broadrangeof time scales, p bedamningifthevariationsproducedbyamodeldonotre- - frommillisecondstomillenniaandontoevenlongerscales. o produceevenqualitativelythevariationsseeninthesunspot Hereweareinterestedinthetimescalesassociatedwiththe r number. Nevertheless, in terms of nonlinear lumped mod- t solar cycle and its grand minima. A salient manifestation s els, whose behavior is purely temporal, it has been pos- a ofthecyclicbehaviorisseeninthesunspotnumber,whose [ annualmeanoscillatesonascaleofelevenyears(ortwenty- sible to produce variations of the cyclic behavior that re- twoyears,ifonegoesbymagneticpolarityvariations).The semble those seen in the grandminimaqualitativelyeither 1 by modulation (Weiss et al. 1984) or intermittency (Platt v term“cycle”isusedtodescribethisoscillationinthesense etal.1993b).Butthesolarvariabilityismanifestlyspatio- 6 that the sunspot number qualitatively performs the same 5 kind of oscillation approximatelyevery eleven years, with temporal and lumped models can at best provide clues to 1 theactualprocessesinvolvedinthesolarcycle.Herewede- anamplitudethatvariesinawaythatisreminiscentofsome 2 scribeanattemptto gobeyondmodelswithpurelytempo- chaotic oscillators. However,it has been foundthat we do . 1 ralvariationstomodelsshowingspatio-temporalvariations not have sufficient data to decide whether the global solar 0 thatalsoproduceanaloguesofthegrandsolarminima. magneticvariationischaotic,inthesensethatitstemporal 8 0 behaviormaybethatofalow-orderdeterministicdynami- An earlier version of this paper was composed in the : cal system (Spiegel& Wolf 1987).Nevertheless,a chaotic v early nineties and some added remarks were inspired by oscillatoroffersanaturalwaytomodelvariousirregularities i discussions at the Enrico Fermi School in Varenna (Cini X ofthesolarcycle(Spiegel1977;Tavakol1978;Ruzmaikin Castagnoli & Provenzale 1997). In the meantime a lot of r 1981). newworkhasemerged,butwefeelthattheideaspresented a Likecertainsimple chaoticsystems, thesun repeatsit- here are still relevant. Particularly important has been the self magnetically over and over again, but never quite the workofBeeretal.(1998)indemonstratingthepersistence samewaytwice,muchinthemannerofsimplechaoticos- ofasolaractivitycyclethroughoutthetimeoftheMaunder cillators. On the otherhand, the simplest chaotic dynamos Minimum.Suchbehavioremergedfroma purelytemporal (Allan 1962; Robbins 1979) do not exhibit the kind of model(Pasquero1996)as well as froma one-dimensional strongintermittencysuchaswasseenintheMaunderMin- dynamomodel(Tobias1996)withsmallturbulentmagnetic imum (Eddy 1978) when the amplitude of the oscillation Prandtl number (so that the viscous diffusion timescale is in sunspot numbers went nearly to zero for seventy-five muchlongerthanthatformagneticdiffusiontime).Similar years in the time of Newton. This behavior is suggestive results — also for small turbulent magnetic Prandtl num- of the possibility that a stronglyintermittentchaotic oscil- bers—havebeenobtainedfortwo-dimensionalmodelsby lator needs to be invoked in trying to understand the so- incorporatingquenchingintheΛ-effectthatdrivesthedif- lar oscillation. Oscillators that behave this way are known ferentialrotation(Ku¨keretal.1999).Intermittentbehavior (Spiegel1981;Fautrelle&Childress1982)but,whenthey indynamomodelshasbeenstudiedfurtherbyTworkowski areintheactivephase,theirvariationsdonotnormallyre- etal.(1998)usingtime-dependentalpha-quenching,andby semblethoseofthesunspotnumber. Charbonneauetal.(2005)usingalgebraicalpha-quenching. (cid:13)c 0000WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim A.Brandenburg&E.A.Spiegel:ModelingaMaunderMinimum Astron.Nachr./AN(0000) However, intermittency can equally well occur in stochas- chaotic or stochastic. The role of the driveris to movethe tically forced models (e.g. John et al. 2002). In the fol- systemintoandoutoftheunstablestate.Observationsofthe lowing, we discuss the spatio-temporal variability of sim- bursts cannot readily distinguish the chaotic driver from a ilarly forced dynamo models in two dimensions assuming stochasticalternativewhichworksequallywell(vonHard- axisymmetry. enbergetal.1997),andthenumberofdegreesof freedom involvedin suchan objectin thesolar case willbehardto determine(butseeHeagyetal.1994).Fortunately,thequal- 2 Spatio-temporal variability itativefeaturesoftheprocessdonotdependsensitivelyon thisdifference,andweshalluseastochasticdriverhere.In Thespatio-temporaldynamicsofthesolarcycleasseenin ourpresentconsiderations,theactionofthedriverismeant spacetimediagramsliketheMaunderbutterflydiagramsug- to modeltheinfluenceoftheconvectivesolardynamoand gestthatsolitarywavesmayplayanactivepartinthesolar the on/offoscillator representsthe tachocline.Here we in- activitycycle(Proctor&Spiegel1991).Anonlinearversion troduceon/offintermittencyintoaworkingmodeloftheso- ofParker’s(1955)dynamowaves(Worledgeetal.1997)or larcycle(Ru¨diger&Brandenburg1995)thathasnotprevi- solitary waves arising from another overstability could be ouslyproducedgrandminima.Modulationalgrandminima the mechanism of the drifting of the center of activity in indynamomodelshavebeenfoundbyTobias.Inthecaseof latitude through the cycle. The finite width of the activity a distributeddynamo(Brandenburg2005),on/offbehavior zoneatanygiventime suggeststhatthe wavesinquestion maysimilarlybeproduced. are themselvesconfinedor guidedby a layer of some cor- respondingthickness.Thepicturethatweadopthereisthe The effects of stochastic noise on mean field αΩ dy- nowconventionalonethatthelayeristhetachocline,though namos have been studied previously (Choudhuri 1992; such a layer has been variously considered to lie deep in Mossetal.1992;Hoyngetal.1994),mainlytomodelirreg- the convectionzone (DeLuca1986),justbelow it (Spiegel ularitiesofthesolarcycleontimescalescomparabletothe & Weiss 1980) or occupy the full convective zone (Bran- solarcycleitselforshorter.Mossetal.(1992)suggestedthat denburg2005).Itmightoperateonthestandardingredients amodulationofthesolarcycleonlongertimescalesofthe of a dynamo — differential rotation and cyclonic convec- order of centuries should also be possible. More recently, tion — from which we here make a model. Since we first workalongthoselines(Schmittetal.1996)hasintroduced wrote those lines, the layerhasbeenwell studiedbothob- theon/offintermittencymechanismintoa mean-fieldstyle servationallyandtheoretically(Hughesetal.2006).Itme- of dynamo, as had been used already in a lumped model diates the transition between the outer differentially rotat- of the solar cycle(Platt etal. 1993b).The presentpaperis ing layers of the convection zone and the inner core with similarly based on a relatively realistic model of the solar itsnearlyconstantangularvelocity.Ithasbeenrenamedthe dynamoin beingspatio-temporalandinto whichwe intro- tachocline(Spiegel& Zahn1992)in keepingwith its gain ducetheon/offintermittencymechanism. inrespectability. The main difference between our work and that of Hereweusemeanfieldtheory(Moffatt1978;Krause& Schmittetal. isthatwe considera standardα-effectwhile Ra¨dler 1980) with the effects of small scale motions sub- they introduced a lower cutoff excluding field generation sumed into turbulent diffusivity and the α-effect. Though belowafieldstrengthof1kGatthebaseoftheconvection mean field dynamos may not tell the whole story of the zone.ThiskindoffilteringisanalogoustoDurney’s(1995) solar magnetic fluctuations (e.g. Hoyng 1987; 1988), they introductionofacriticalfieldvalueinthetachoclineabove willsuitourpurposeofmodelingtheintermittencysignaled whichamagnetictubeisejected.Thisalsoproducesgrand bytheMaunderMinimum.Inmodelingsolarintermittency, minimainspatio-temporalmodels. we need to be aware that there are several forms of inter- mittencythathavebeenisolatedindynamicalsystemsthe- AnotherdifferencewiththecalculationofSchmittetal. ory.However,thereisaparticularonethatmodelsthesolar is that we consider the full sun whereas they studied only grandminimaquitewell(Pasquero1996)andthathascome a single hemisphere and produced one-winged butterflies. tobecalledon/offintermittency(Plattetal.1993a;seealso Thisrelatestoanotheraspectoftheproblemthatourmodel Spiegel1994).[Infact,thisformofintermittencywasfash- isintendedtobringout.Itisverydifficulttohavebothhemi- ioned(Spiegel1981)withgrandminimainmind.] spheres go completely inactive for appreciable times. The On/offintermittencyisliketheintermittencydetectedin probabilityofa completeturnoffis exceedinglylowandit theoutputofaprobeina turbulentfluidregisteringabrupt seems unlikely that any propagative intermittency mecha- changesaslaminarandturbulentfluidregionsflowpastit. nism with retardation effects could turn off the magnetic Onemaythinkofthisasaseriesofburstsorchaoticrelax- activity globally in models with large spatial extent. The ationoscillations.Whatcharacterizesmodelsthathavebeen modeldescribedhere doeshoweverproduceloweredsolar madeofthisprocess(e.g.Spiegel1981;Ott&Chen1990; activityoverlargeportionsofitscomputationaldomainand Pikovsky& Grassberger1991)is thatthe (potentially)un- thereforegivesthekindofreductionintotaloverallactivity stableoscillatorperformingthecycleisdriventoinstability thatisconsistentwithwhatwasseenintheMaunderMini- throughcouplingtoanaperiodicdriverthatiscontinuously mum. 2 A.Brandenburg&E.A.Spiegel:ModelingaMaunderMinimum Astron.Nachr./AN(0000) The hemispheric asymmetry has been modeled by wouldneedverylargefluctuationsatthatvalue.Inthisex- Knoblochetal. (1998;see also Weiss 1993)andthiswork ploratory study, we prefer to operate at more modest pa- is related to the problem of making an extended system rametervaluestoavoidtheneedforlargefluctuationsinthe demonstrate on/off intermittency. The problem has some- driver. Therefore we choose a slightly subcritical value of thing in common with that involvedin laying a rug. Once α and introduceonly modest fluctuations in its magnitude therugisnaileddown,ifthereisabumpsomewhere,there so that the driver can move it into and out of the unstable seemstobenowaytosqueezeitoutofexistence.Wehave state easily. For this purposewe introducea scaling factor observedthesamebehaviorwiththedynamomodel.Ifwe cα in front of certain components of the α-tensor (those produceagrandminimumlocallysomewhereinspace,we components which result from the interaction of rotation find invariably that there is usually some excitation else- andstratification).Fordetailsseetheoriginaldiscussionof where. While we can get a whole hemisphere to be quiet themodel(Ru¨diger&Brandenburg1995)where,withcα= atonetime,theotheronemaystillshowactivity.Theout- 1,theresultingtoroidalmagneticfieldisa fewkgaussand putofthedynamomodelisthereforesomewhatlikethesun thepoloidalfieldatthesurfaceabout10gauss.Inthecases inintermission—ontheglobalscalethereisusuallysome studiedhere,however,wherethedynamoisjustmarginally weakactivitysomewhereanditdoeshavecycles.[Pasquero excited,thegeneratedmagneticfieldisweak,andcouldnot (1996) has shown how this may be achieved in the purely explainthefieldstrengthobservedinsunspots.Thisfeature temporal oscillators as well.] As we have learned at the ofthemodelcanbecorrected,astheworkofSchmittetal. Fermi School on the Interaction of the Solar Cycle with (1996)shows. TerrestrialActivity in June of 1996,certain terrestrialdata We consider separately two kinds of fluctuation in the havemuchmoreincommonwiththeoutputofthedynamo model: fluctuations in α or in an imposed electromotive modelsthan the sunspotnumber.Moreover,some of these force E representing the effect of the fluctuations in the terrestrialdataareveryconvincingproxydataforthesolar main solar dynamo. The latter is a more elementary pro- activity(Beeretal.1990;1994;1996;Solankietal.2004), cessthatdoesnotrelymuchondynamotheory.Thepicture whileothersmaymimicthesunspotnumber.[Therearealso hereisthatinthebulkoftheconvectionzoneasmallscale auroralindicatorsofsolaractivityrecordedinclassicalan- dynamo operates(e.g. Meneguzzi& Pouquet1989;Nord- tiquity(Stothers1979,Solow2005).]Toagreatextent,the lund et al. 1992) producinga magnetic field that is highly issueisatheartoneofknowinghowtocomparetheoutput variableinspaceandtimeasrepresentedbyE.Inthebulk ofdynamomodelstothevariousmeasuresofsolaractivity. oftheconvectionzonethefluctuationsareimmense,hence We describe next some particulars of the model itself, even spatial and temporal averages (E = hu′ × B′i) re- including the manner in which the fluctuations are intro- main fluctuating, albeit on longer scales (Brandenburg et duced.Readerswhoarenotinterestedinsuchdetailedinfor- al. 2008).Theangularbracketsreferto ensembleaverages mationshouldskipdirectlytoSect.4whereweoutlinethe inprinciplebut,inpractice,theyareapproximatedbyspa- mainresults.Inthese,wefocusonmodelswitharatherhigh tialandtemporalcoarsegrainingaverages.Theerrorresult- noise level exceeding the electromotive force from the α- ing from this approximationis sometimes interpreted as a effectbyanorderofmagnitudesinceweexpecttheglobal sourceofstochasticnoise(Hoyng1987;1988;1993;Moss convectivedynamoactiontobemorevigorousthanthatof et al. 1992; Brandenburg et al. 2008). As we have men- thetachocline. tioned, there is ample reason for expecting fluctuations to appearinanyrealisticmodel. Weadoptwhitenoisewithvanishingmeanvalueanda 3 The model rootmeansquarevalueofunity.Thetemporalpowerspec- trumofthenoiseisflatforfrequenciessmallerthan2π/τ. We use a mean-field model of the dynamo action in The value of τ determines the time span over which long the tachocline (Ru¨diger & Brandenburg 1995) with an termvariabilityoftheresultingmeanmagneticfieldispos- anisotropic α-effect and a turbulent magnetic diffusivity. sible.Theamplitudeofthenoise,nE,ismeasuredinterms Magnetic buoyancy is also included as it is in the more of the rms velocity of the turbulent motions and the lo- elaboratemodelofJiangetal.(2007;seefurtherreferences calequipartitionfieldstrength.Whicheverofthefluctuation therein). The prescribed angular velocity is taken from mechanismsapplies—intheemforintheαeffect—the theobservationalfindingsofhelioseismology(Christensen- procedureissimilar.Fortheαcase,forexample,weadda Dalsgaard&Schou1988).Toobtaina22yrmagneticcycle fractionnαofthenoisycomponenttotheoriginalα. period,weintroduceascalingfactorinthemagneticdiffu- sivity of 0.5 and, to get a butterfly diagramwith sufficient activityatlowlatitudes,weintroduceasuitablelatitudede- 4 Results pendenceinα. In the original model calculations, the value of α was 4.1 Fluctuatingα-effect typicallysetatavalueapproximatelytwentytimesthecrit- ical value for instability. To produce on/off intermittency Toregulatethestrengthoftheeffectofnoise,weintroduce (Platt et al. 1993a), in the highly supercritical case, we a factor cα in frontof the familiar α term, as described in 3 A.Brandenburg&E.A.Spiegel:ModelingaMaunderMinimum Astron.Nachr./AN(0000) 4.2 Magneticnoise If we introducean externalstochastic emf, as describedin Sect.3,wefindaqualitativelysimilarbehaviortothatwith fluctuatingα;see Fig. 4.Fluctuationsofvariouskindscan evidentlyproducelongtimeintermittency. The model shows two distinct activity waves, one mi- gratingequatorwardandtheotherpoleward.Thetwowaves seem to be modulated independently,in each hemisphere. So do the modulationsin the two hemispheresseem to be only weaklycoupled,with little tendencyto forma dipole structure. It may be that the approximate antisymmetry of the toroidalmagneticfield ofthe sun,suggestedby Hale’s polarity law, may not be a very stable feature, and that othertypesof(a)symmetrymighthaveoccurredinthepast. Fig.1 Butterflydiagramsofthetoroidalfieldatthebase Other, more regular parity variationsof the magnetic field oftheconvectionzoneforcα =0.02,nα =5andτ =3yr. have previously been seen in nonlinear models (Branden- burgetal.1989a,b;1990;Jennings&Weiss1991;Sokoloff &Nesme-Ribes1994). Sect.3.Witheverythingelsefixed,instabilityoccurswhen cα exceeds the critical value c(αcrit) ≈ 0.03. We begin by 5 Interpretation adoptingthesubcriticalvaluecα = 0.02andweadjustthe fluctuationsinαtohaveatimescaleτ =3yr. Inthemodeldescribedhere,theseatofthesolaractivitycy- cleisinthesolartachocline,thelayerthatmatchesthedif- Ifthenoiselevel,nα,istoolow,themagneticfieldde- ferentialrotationofthesolarconvectionzonetothe(nearly) caystozero,asitdoeswhennα=0.02.Shortnoisyburstsin rigid rotation of the inner sun (Hugheset al. 2006).While αareinsufficienttobringthemagneticfieldtoappreciable the precise hydromagnetic process that drives the activity strength.Ontheotherhand,ifnα isratherlargerthanthis, hasnotyetbeensecurelyidentified,we haveassumedthat themagneticfieldisalmostentirelydominatedbythenoise theprocessmaybemodeledasanαω dynamoforthepur- andcyclicbehaviordoesnotoccur,exceptforirregularre- pose of exploring the cause of the grand minima of solar versalsonthetimescaleofcenturies.Suchacaseisdepicted activity.However,as we have alreadyimplied,anyof sev- inFig.1,whereweshowacolor-codedrepresentationofthe eraloverstabilitiesmightequallyserveourpurposes. toroidalmagneticfieldatthebottomoftheconvectionzone Aswe havetacitly assumed,the differentialrotationin asafunctionoftimeandlatitude.Inthefollowingwerefer thetachoclineislikelytobesignificantinsuchinstabilities. to such representationsasbutterflydiagrams.A verysimi- Inparticularitislikelytogiverisetoatoroidalfield.Given larresult,withreversalsonalongtimescale,isfoundeven a suitable depth dependence of the strength of this field, whenthenon-randomcomponentofαisabsentaltogether an instability driven by magnetic buoyancy (Parker 1979) providedthereis globalshear in the tachocline,as thereis mayariseand,especiallyinthepresenceofastabledensity inaccretiondisks(Vishniac&Brandenburg1997). stratification, it may drive waves of excitation (Proctor & Spiegel1991).Anotherpossibledriverforsuchwavesmay Togetsufficientlylargemagneticfieldstrengths,wefo- be magnetorotationalinstability (Balbus & Hawley 1998). cus attention on models with cα = 1. With τ = 3yr we Parfrey & Menou (2007) have studied the local instabil- findlongtermvariabilityonatimescaleof200–500yr;see ity of the tachoclineand foundinstability at high latitudes Fig.2.Weassociatethisvariabilitywithgrandminima(and andstabilityatlowlatitudes.However,theircalculationdid maxima)asdiscussedbelowinSect.5althoughasnotedal- not include a toroidal field and, as Knobloch (1992) has ready, there is nevera complete turningoff of the cycle at observed, an azimuthal field can cause a Hopf bifurcation alllocationsatonce. in MRI. As we may reasonably expect to find a toroidal field in the tachocline,this overstability is another mecha- Even if τ is decreased to a value of 0.3yr, say, long nismthatcouldperhapsengenderwaveswhosedescription termvariabilitystilloccurs,butindividualcyclesmayvary wouldtypicallybebythecomplexGinzburg-Landauequa- significantly in amplitudeas seen in Fig. 3. Such behavior tion(Aranson&Kramer2002).Arelatedcalculationisde- is foundonlyifthefluctuationsofαaresufficientlylarger scribed by Kitchatinovand Ru¨diger (2007);see also Cally thantheaveragevalue;hereafactoroffiveisneeded.Indi- (2003),Gilmanetal.(2007),Ru¨diger&Kitchatinov(2007), vidualfluctuationsofαcanstillbemuchlargerbecausewe andZahnetal.(2007). haveassumedanexponentialdistributionfortheprobability Given that the tachocline is poised to drivehydromag- densityofαfluctuations. netic activity, by dynamo processes or otherwise, we have 4 A.Brandenburg&E.A.Spiegel:ModelingaMaunderMinimum Astron.Nachr./AN(0000) Fig.2 Butterflydiagramsofthetoroidalfieldatthebaseoftheconvectionzoneforcα = 0.02,nα = 5andτ = 3yr. Butterflydiagramsofthetoroidalfieldatthebaseoftheconvectionzoneforcα=1,nα =0.2andτ =3yr. convectivedynamo.Itisthisprocessthatispresumablyre- sponsible for the magnetic carpet (Simon et al. 2001) in a continuousprocessthatis thoughtto producerapidlyfluc- tuating fields of moderate strength (Priest et al. 2002). In viewofthecomplicationsinthetheoryofthisprocess,we havesimplymodeledthehighlyfluctuatingfieldsinofthe solarconvectionzoneasastochasticprocess.Inthemech- anismofon/offintermittencythatwehaveintroducedhere, the fluctuations produced by the main dynamo move the tachocline into and out of states of instantaneous oversta- bility.Thismechanismproduces,aswe havenoted,longer responsetimesinthemodeltachoclinethanthetimescales of the fluctuationsthat give rise to more intense field con- centrations. [We have seen a similar kind of symbiosis in thesimulationsofthegeodynamobyGlatzmaier&Roberts Fig.3 Butterflydiagramsofthetoroidalfieldatthebase (1995)].The details of the processare complicated but, in oftheconvectionzoneforcα =1,nα =5andτ =0.3yr. gross, when the local field fluctuations produced are too weak,thegroupingoffluxropesintoasunspotfieldisnot achieved,eventhoughmuchofthe normalactivitycontin- ues in the convection zone. To illustrate how such details mayrelatetotheobservedspotnumbers,wehavemapped the activityto the butterflydiagramwith a particularfunc- tionalofthefieldinthetachoclineforthepresentpurposes. Like the sun, the model we have studied here mani- fests spatio-temporalintermittencybut the magnetic activ- ity almost never turns off everywhere. That is, our results suggestthatsimpledynamomodels,witheitherfluctuating emfsand/orwithmagneticnoiseinjectedintothebulkofthe convectionzone,can produceintermittencyon sufficiently longtimescales,buttheydonotswitchoffgloballyoverthe wholesun,inbothhemispheresatonce.Itseemslikelythat thisfeatureisinherentinallmodelswithpropagativebehav- Fig.4 Butterflydiagramsofthetoroidalfieldatthebase ior and thatit cannotbe expectedthatthe cycle-producing oftheconvectionzoneforcα =1,nE =1andτ =3yr. processesofthesunswitchesoffeverywhereforseveralcy- cles as might be imagined on the basis of certain lumped models(suchasPlattetal.1993b).Whenwefirstproduced also assumedthatthe ambienceis highlyfluctuating.Here thepresentresults,wethoughtthisfeaturewasadeficiency. wehaveinmindthatthemaindynamointhesunisaglobal But as Ribes & Nesme-Ribes (1993) have reported, even 5 A.Brandenburg&E.A.Spiegel:ModelingaMaunderMinimum Astron.Nachr./AN(0000) Fig.5 Timeseriesof|B|(dottedline)andtheactivitypa- Fig.6 10Be data (solid line) together with the sunspot rameterR(solidline)forthesamerunasinFig.2.Notethat number(dottedline),asprovidedbyDr.Ju¨rgBeer.Thedata thescalefor|B|increasesdownwards,inordertomimicthe arefromashallowcore(300m)drilledatDye3,Greenland, approximateanticorrelationbetweenthe10Be dataandthe in1986.Thedatawerefilteredusingaspectralfilterwitha sunspotnumber. cut-offof6yearsandinterpolatedusingacubicspline.The youngerpart(1783-1985)ispublishedin(Beeretal.1990), andthewholerecordappearedinBeeretal.(1994). during the Maunder Minimum, a weak solar cycle contin- ued.Thusagloballydepletedactivitylevelismorelikewhat iswantedandwehavegottenthatfromthemodelinkeep- netic field in the solar wind (Beer et al. 1996). In Fig. 6 ing with the historical recordsas interpreted by Ribes and we reproducedata kindly providedby Dr. Ju¨rg Beer com- Nesme-Ribes. paringthe10Berecordswiththesunspotnumberfornearly There still remains the need for an additional physical fourcenturies.Weseethatacyclicmodulationisverymuch feature that relates to the production of sunspots, or more in evidence during the Maunder Minimum, with a rather specifically, strong flux tubes of sufficiently large cross- modest reduction in its amplitude compared to that of the section. That is the message of the procedures of Durney sunspotnumber. (1995)andSchmittetal.(1996).Asimplewayofformulat- ThesituationasbroughtoutinthecitedpapersofBeer ingthisproblemistosaythatthesunspotnumber,whichis et al. is that variousmeasuresofsolar activityare notper- aglobalparameter,is,asalreadymentioned,afunctionalof fectly correlated. If we were to think of the output of a the variousfields producedin the model.To getthatfunc- modelsolar dynamo as we would one of these other mea- tional,weneedtooperatewithanexplicitsunspotproduc- sures,wewouldnotbesurprisedthatitdoesnotnecessarily tionmechanism. representallofthemfaithfully.Unfortunately,thereisasyet LetR(t)betheaveragemagneticfieldstrengthbetween no clear theoretical indication which one of them a model ±10◦ and±30◦ latitude,butallowonlythose areaswhere shouldmostcloselyrepresent.If,inagrandminimum,there thefieldexceedstherootmeansquarevalueby30percentto is high magnetic activity in only one hemisphere, it is not contributetothisaverage.ThisquantityisplottedinFig.5. unreasonablethatthemodulationsof10Beshouldcontinue Note that R(t) shows periods with almost vanishing mag- with reasonable strength. The question for the theory then netic activity as in a Maunder Minimum. The grand min- iswhatistherelationshipbetweenthevariationsproduced imaonthisinterpretationoftheoutputofthemodelsresult bythemodelsand(say)thesunspotnumber.Thisispartic- fromwhatmay beregardedaslean cyclesmagnetically.If ularlydifficultsincethesunspotsseemtobeproducedwell therearemagneticdroughts,theyareonlylocal,notglobal, within thesun andnotatthe surface,whereassomeof the though the magnetic means may be low. In that sense, the otheractivitymeasuresarenodoubtsuperficiallyproduced. sunspotnumber,thoughvaluableforhavingfocusedourat- Evenifwedohaveagoodmodelofthecyclicmechanism, tention on an interesting feature of the magnetodynamics, wealsoneedtounderstandhowsunspotsformandsurface mayinsomewaysbeamisleadingindicatorofwhatishap- beforewecanpredicttheirnumber. peningoverall. Norisitunimportanttotrytopredictthesunspotnum- Thisviewissupportedbystudiesofotherindicatorsof ber,orsomethingakintoit,foritisthisquantitythatseems solar activitythanthe sunspotnumber.The mostcomplete tobeconnectedtosomeclimatologicalvariations.Themost recordsofvariationsthatmayresultfromsolaractivityfluc- striking evidence of this is the discovery in tree ring data tuationsarefoundinthe10Berecordsfromicecores(Beer (Douglass 1927) that “the sunspot curve flattens out in a etal.1990;1994;1996).The10Bevariationsareplausibly striking manner ... from 1670 or 1680 to 1727.” This dis- attributedtomodulationofthecosmicrayfluxbythemag- coverywasmadebyA.E.Douglasbeforehehad“received 6 A.Brandenburg&E.A.Spiegel:ModelingaMaunderMinimum Astron.Nachr./AN(0000) a letter from Professor E. Maunder ... calling attention to Christensen-Dalsgaard, J., Schou, J.: 1988, in: E.J. Rolfe (ed.), theprolongeddearthofsunspotsbetween1645and1715.” Proc. Symp. Seismology of the Sun and Sun-like Stars, ESA AsDouglassandothershaveargued,variationsintreering SP-286,p.149 thicknessinturnareconnectedtorainfall,soitisnotanidle CiniCastagnoli,G.,Provenzale,A.(Eds.):1997,PastandPresent variabilityofthesolar-terrestrialsystem:Measurement,Data projecttotrytounderstandhowtogofromtheworkingsof Analysis and Theoretical Models, (IOS Press, Amsterdam), asolardynamotothemanufactureofsunspots. 311 In summary, solar activity waves at high and low lati- DeLuca,E.E.:1986,Dynamotheoryfortheinterfacebetweenthe tudesandinthetwohemispheres,leadsomewhatindepen- ConvectionZoneandtheRadiativeInteriorofaStar,Thesis, dent,weaklycorrelatedlivesandfluctuateseparatelyunder U.Colorado,NCARCT-104 the influence of noise. Spatial variationsof the solar cycle Douglass,A.E.:1927,SciLXV,220 during the Maunder Minimum are not an immediate indi- Durney,B.R.:1995,SoPh166,231 catorofthesunspotnumberand,ifthesunspotshavetheir Eddy, J.A.: 1978, in: J.A. Eddy (ed.), The New Solar Physics, AAAS Selected Symposium 17, Westview Press, Boulder, origin well beneath the solar surface, we must go another Colorado,p.11 stepinthediscussionbeforeweobtainresultsthatarefully Fautrelle,Y.,Childress,C.:1982,GApFD22,235 consistentwithhistoricrecordsofsunspotsassummarized Gilman,P.A.,Dikpati,M.,Miesch,M.S.:2007,ApJS170,203 by Ribes and Nesme-Ribes (1993). It is not at all obvious Glatzmaier,G.A.,Roberts,P.H.:1995,Nat377,203 what we may conclude about subconvective activity from Heagy,J.F.,Platt,N.,Hammel,S.M.:1994,PhRvE49,1140 observedsurfaceactivity. Hughes, D. W., Rosner, R., Weiss, N. O.: 2006, The Solar Tachocline,CambridgeUniversityPress,Cambridge Acknowledgements. WearegratefultoDr.Ju¨rgBeerforhishelp Jennings,R.,Weiss,N.O.:1991,MNRAS252,249 and advice and forproviding thedatafor Fig.6. Thiswasmade Jiang, J., Chatterjee, P., Choudhuri, A.R.: 2007, MNRAS, 381, possiblebyourparticipationintheE.FermiSchoolorganizedby 1527 Drs. J. Cini-Castagnoli and A. Provenzale. We thank Dr. David John,T.,Behn,U.,Stannarius,R.:2002,PhRvE65,046229 MossformakingusefulsuggestionsaboutthemanuscriptandDr. Hoyng,P.:1987,A&A171,357 A. Solow for a discussion of his work. A.B. is grateful for the Hoyng,P.:1988,ApJ332,857 hospitalityoftheAstronomyDepartmentatColumbiaUniversity, Hoyng,P.:1993,A&A272,321 where most of this work was carried out with support from the Hoyng,P.,Schmitt,D.,Teuben,L.J.W.:1994,A&A289,265 AFOSRunderundercontractnumberF49620-92-J-0061. 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