Table Of Content

DRAFTVERSIONFEBRUARY2,2008 PreprinttypesetusingLATEXstyleemulateapjv.10/09/06 DRAPINGOFCLUSTERMAGNETICFIELDSOVERBULLETSANDBUBBLES–MORPHOLOGYANDDYNAMIC EFFECTS L.J.DURSIANDC.PFROMMER CanadianInstituteforTheoreticalAstrophysics,UniversityofToronto,Toronto,ON,M5S3H8,Canada DraftversionFebruary2,2008 ABSTRACT High-resolution X-ray observations have revealed cavities and ‘cold fronts’ with sharp edges in tempera- ture and density within galaxy clusters. Their presence poses a puzzle since these featuresare not expected to be hydrodynamicallystable, or to remain sharp in the presence of diffusion. However, a moving core or 8 0 bubblein evena veryweakly magnetizedplasma necessarilysweeps upenoughmagneticfield to buildup a 0 dynamicallyimportantsheath;thelayer’sstrengthissetbyacompetitionbetween‘plowingup’andslipping 2 aroundoffieldlines,anddependsprimarilyontherampressureseenbythemovingobject. Inthisinherently three dimensionalproblem, our analytic argumentsand numericalexperimentsshow that this layer modifies n thedynamicsofa plungingcore, greatlymodifyingthehydrodynamicinstabilitiesandmixing,changingthe a geometryofstrippedmaterial,andslowingthecorethroughmagnetictension.Wederiveanexpressionforthe J maximummagneticfieldstrengthandthicknessofthelayer,aswellasfortheopeningangleofthemagnetic 8 wake.Themorphologyofthemagneticdrapinglayerimpliesthesuppressionofthermalconductionacrossthe layer,thusconservingstrongtemperaturegradients.Theintermittentamplificationofthemagneticfieldaswell ] h astheinjectionofMHDturbulenceinthewakeofthecoreisidentifiedtobeduetovorticitygenerationwithin p themagneticdrapinglayer. Theseresultshaveimportantconsequencesforunderstandingthecomplexgasdy- - namicalprocessesoftheintra-clustermedium,andapplyquitegenerallytomotionsthroughothermagnetized o environments,e.g.theISM. r t Subjectheadings:hydrodynamics—magneticfields—MHD—turbulence—galaxies: clusters: general— s a diffusion [ 2 1. INTRODUCTION through the intercluster medium, as seen at the centers of v manycool-coreclusters(e.g.,McNamaraetal.2005;Bîrzan Recent observations of very sharp ‘cold fronts’ in galaxy 3 et al. 2004). In this case, the bubble would be quickly dis- clustersraiseunansweredquestionsinthehydrodynamicsof 1 rupted on rising absent some sort of support (e.g., Robin- galaxyclusters(seeforinstancethereviewofMarkevitchand 2 son et al. 2004). However, the draping of a pre-existing 0 Vikhlinin2007),forsuchabrupttransitionsarenotexpected magneticfield maystronglyalterthedynamicsandsuppress . to be stable against either hydrodynamicalmotionsor diffu- 1 hydrodynamic instabilities, as seen recently in simulations sionforextendedperiodsoftime. 1 (Ruszkowski et al. 2007b). The morphology of the draped It has been known for some decades in the space science 7 magnetic field may be able to suppress transport processes community that an object moving super-Alfvénically in a 0 acrossthe bubble interfacesuch as cosmic ray diffusionand magnetizedmedium can very rapidly sweep up a significant : v magneticlayerwhichisthen‘draped’overtheprojectile(e.g., heatconduction.Thishasimportantconsequencesforcosmic i BernikovandSemenov1980).Forconcretenessindiscussing ray confinement in these buoyantly rising bubbles and may X explainre-energizedradio‘relic’sources,broadcentralabun- the process, we show in Fig. 1 a picture of this mechanism r danceprofilesofclusters,andtheexcitationoftheHαlinein a takenfromoneofoursimulations,whichwillbedescribedin filaments trailing behind bubbles(Ruszkowski et al. 2007a). moredetailinlatersections. Although the analytics and simulations we discuss here fo- There has been significant interest in applying this idea cusonthecaseofanoverdense‘core’movinginanexternal of magnetic draping in galaxy clusters (e.g., Vikhlinin et al. field,weexpectthebasicmagneticdynamicstoalsoextendto 2001; Lyutikov 2006; Asai et al. 2004, 2005, 2007) as such thecaseofanunderdensebubbleprobablydependingonthe a magnetic field could naturally inhibit thermal conduction magneticenergydensityoftheplasma. across a front (e.g., Ettori and Fabian 2000) allowing it to Becauseweareinterestedhereinthefundamentalsofaba- remain sharp over dynamically long times. Although such sic process—thatofthedrapingofa fieldaroundanobject draping has been explored in the past, in the space sciences andtheresultinghydrodynamicaleffectontheobjectandits the resulting dynamicsis relatively simpler, as generally the interactionwiththeexternalmedium—weconsiderforthis objectbeingdrapedisasolidbody,withlittleinteriordynam- paper,inbothouranalyticandcomputationalwork,thesim- icsofitsown. However,inthecaseofforinstanceamerger plestpossiblecase; anoverdense,non-self-gravitating‘blob’ ofgas-richclusters, thehydrodynamicsofthe drapedplung- moving through a quiescent medium with a magnetic field ingcorecanalsobemodified,withthestrongmagneticlayer uniformonthe scalesconsidered. (Thetermwe willuse for providing some stabilization against instabilities that would thisblobwilldependonthesituation;whendiscussingastro- otherwiseoccur(Dursi2007). physical implications, we will speak of ‘cores’ or ‘bullets’, Theeffectofastrongdrapedmagneticlayercouldbeeven dependingonthecircumstances,or‘bubbles’forunderdense greater for underdense objects, such as for bubbles moving regions;forthecaseofournumericalsimulations,wewillre- Electronicaddress:[email protected],[email protected] fer to ‘projectiles’, as the overdense fluid in the simulations 2 Dursi&Pfrommer differsinstructurefromcoresorbulletsinlackingselfgrav- Because in this case the flow falls quickly to zero at the ity;inouranalyticworkwheretheblobisarigidsphere,we surface of the moving sphere, magnetic field rapidly ‘builds willrefertotheblobasasphereorsphericalbody.) up’ around the projectile, and in the kinematic limit eventu- We furtherconsiderthecaseoftheobjectmovingsubson- allybecomesinfinite.Thehighdegreeofsymmetryalongthe ically;whilethecaseofsupersonicmotionisinterestingand stagnation line (the axis of symmetry of the object pointing highly relevant, we anticipate that in the usual case where inthedirectionofmotion)greatlysimplifiesthemathematics, the bow shock is well separated from the magnetic layer – and as shown in for instance Lyutikov (2006), the magnetic that is where the standoff distance ∆ ρ0R R, (where R fieldstrengthdirectlyalongthestagnationlineisgivenby ≈ ρs ∼ tihsethsehoracdkieudsdoefntshitey)coisrem, ρu0chisgtrheeataemrtbhiaenntthdeenmsaitgyn,eatnicdlρasyeisr |B| = B0 1 (1) thicknessl - 2R R, (wherel is the approximatemag- ρ ρ0 1- R 3 ≈MA ≪ R+s netized layer thickness and the alfvénic Mach number, asdiscussedinmoredetailinMthAenextsection)thattheargu- whereB0istheambientmagnqeticfi(cid:0)eld,(cid:1)ρ0istheambientden- sity, R is the radius of the solid sphere projectile, and s is ments here will also hold, so we save the more complicated thedistancealongthestagnationlinefromthesurfaceofthe geometryandlargerparameterspaceofthecompressiblecase sphere. forfuturework. The analytic works cited, and presented here, considered purelyincompressibleflow;foroursimulations,weconsider only very modest compressibility, with projectile motions throughtheambientfluidquitesubsonic,soitsufficesforthe momenttoconsiderintheexternalmediumρ=ρ . Thefluid 0 here is further considered to be infinitely conducting; how- ever, the buildup of magnetic field without a corresponding buildupofmassdoesnotviolatethe‘flux-freezing’condition, asshowninthecartoonFig.2asincomingfluidelementsare ‘squished’alongthesidesoftheincomingsphere,sothatthe magnetic flux coming out the sides of the fluid element re- mainsconstant,evenastheconcentrationoffieldlinesbuilds upalongthestagnationline. Furtherincreaseinmagneticen- ergycomesfromthe stretchingoffield linesin the direction FIG.1.—Arenderingofoneofthethree-dimensionalsimulations(referred toasRunFlaterinthiswork)performedforthiswork,discussedinmorede- ofmotionofthecore. tailinlatersectionsbutincludedheretoillustratethephysicalpicture. An overdenseprojectileissentthroughauniformlymagnetizedmedium,sweep- ingupmagneticfieldaheadofit.Plottedisadensityisosurface,correspond- ingtothemeandensityofthebullet,andsomefiducialmagneticfieldlines. Thecut-planeiscolouredbymagneticenergydensity,asarethefieldlines. Theminimumofthecolourmapforthemagneticenergydensityissettobe themagneticenergydensityintheambientmedium;energieslessthanthat (whichoccurinthewake,duetoentrainmentofunmagnetizedmaterialasthe bubblerises)issettothecoloroftheambientenergydensity. Themagnetic fieldis‘draped’intoathinlayerformingabowwave,leavingturbulenceina FIG.2.—Acartoonshowingthedistortionofincomingfluidelementsand wakebehindthebullet.Magneticfieldlinespileupalongthestagnationline stretchingoffieldlinesasaredsphericalprojectilemovesupwardsthrough ofthisinitiallyaxisymmetricbullet,whileintheplaneperpendiculartothe theambientmedium. initialfield,thefieldlinescansliparoundthebullet. APDFversionofthis manuscriptwithaninteractive3dversionofthisfigure,followingBarnesand Inreality,ofcourse,themagneticfielddoesback-reactonto Fluke(2007),isavailableathttp://www.cita.utoronto.ca/∼ljdursi/draping/. the flow, and the kinematic potential flow solution fails for tworeasons–buildupofastrongmagneticfieldlayer(which In§2wegiveanoverviewofthephysicsofdraping,putting violates the kinematic assumption) and creation of vorticity ourworkinthecontextofpreviousresults;in§3wedescribe (inconflictwiththepotentialflowassumptions). ouranalyticandcomputationalapproaches;in§4wecompare Themagneticfieldshouldexertasignificantback-reaction theresultsofourtwoapproaches,andfromtheunderstanding whentheresultingmagneticpressureiscomparabletotheram gainedtherein§5wedescribecharacteristicsofdraping;we pressureoftheincomingmaterial: B2/8π ρ u2,whereuis discusstheeffectofinstabilitiesin§6,considerthelimitations ∼ 0 thespeedofthecorethroughthequiescentambientfluid.The ofourresultsandconsiderapplyingthemtolatertimesin§7, firstplacethiswillhappenisalongthestagnationline,which andfinallyconcludein§8. bysymmetrywillbethelocationofthelargestmagneticen- ergydensity. Thelayerofmagneticfieldwiththismagnitude 2. GENERALPHYSICSOFMAGNETICDRAPING isexpected(fromEq.1 andassumingl R)to beofthick- Previouswork(e.g.,BernikovandSemenov1980;Lyutikov ness ≪ 2006) has looked at the basic picture of magnetic draping 1 l= R (2) in a simplified way in some detail; we summarize some of 6α 2 theirkeyresultsaswellasournewinsightintothisproblem MA here. Intheseworks,theknownpotentialflowaroundasolid where A=u/υA istheAlfvénicMachnumberofthecore, M sphereistakenasaninput,andapurelykinematicmagnetic υA2 =B20/4πρ0 istheambientAlfvénspeed,andαisthecon- field, uniform and perpendicular to the direction of motion, stant of proportionality describing the maximum magnetic is added. The derivation of Bernikov and Semenov (1980) pressure in units of the incoming ram pressure, B2 /8π = max is clarified, and a novelset of useful approximationsfor the αρ u2. Wewillseethatα 2andfiducialvaluesforthesitu- 0 resultingfieldnearthesolidspherearegiven,inAppendixA. ationsconsideredherewil≈linvolve 2 3,sothatatypical MA≈ MagneticDrapinginClusters 3 FIG. 3.—Acartoonshowingtheexpectedgeometryofthedrapedmag- neticfield(blue)overtheobject(red). Seenintheplaneofthedirectionof theambientfield,panel(a),withthedirectionoftheambientfieldshown,a cleanbowwaveispresentedwithawell-definedopeningangle.Intheplane perpendiculartotheambientfield,panel(b),thefieldlinescansliparound theprojectile, andtheflowwouldclosebackinonastagnationlineonthe FIG. 4.—Wecomparethedrapedmagneticpressureandrampressurein othersideoftheobjectexceptforlargely-2dvorticalmotionsinducedbyin- theplanethatisparallel (perpendicular) totheinitial magnetic fieldinthe stabilitiesatthemagneticinterface. Thegeometryoftheflowintheregion top(bottom)panelswithalogarithmiccolorscale. Intheparallelplane,the indicatedbydashedboxdependsheavilyonthefinalshape,andthusinternal overpressure of the magnetic draping layer is only partly compensated by structure,ofthemovingobject. adeficit inrampressure andeventually isresponsible fordecelerating the coreduetomagnetictension.Intheperpendicularplane,therampressurein valueforlwillbeapproximatelyR/36.Evensuchaverythin thewakeofthecoreattainsmuchhighervaluesandthedrapinglayercloses towards thesymmetry axis unlike theparallel plane where anice opening layer can have important effects, both in terms of suppress- coneforms.Shownhereisazoom-inonasmallregionofourcomputational ing thermalconduction(Ettoriand Fabian 2000)and hydro- domain.Thisfigurehasbeensomewhatdegraded. ForaPDFversionofthis dynamicinstabilities(Dursi2007). manuscriptwithhigherresolutionfiguresandinteractive3dgraphics,please seehttp://www.cita.utoronto.ca/∼ljdursi/draping/. ItshouldbenotedherethatwhenweusetheAlfvénicMach number throughthisworkitshouldreallybeconsidered A in Fig. 3 and for our simulation in Fig. 4. This shows that M a dimensionless ratio of ram pressure to magnetic pressure thedrapinglayerbecomesdynamicallyimportantandfillsin (uMsed2Aw=hρe0nui2n/t(e2rPpBre0)t)in,goritaatsthaeralteiaosto,fsvoemloecictaieusti(oun/υshAo)ualsdthbee tphreesdseufirecisthoofwrasmanproevsesru-rper.esTshueresuamheaodftohfetmheagcnoereticthaantdleraadms velocitiesareorientedindifferentdirections;intheworkpre- toadecelerationoftheprojectile. sented here, the velocityof the drapedobjectwill always be completelyorthogonaltotheambientdirectionofpropagation 3. METHODOLOGY of Alfvén waves. Thusthere is an importantsense in which In order to understandthe full non-linearphysicsof mag- our projectiles are always (infinitely) super-Alfvénic, which neticdrapingaroundadynamicallyevolvingdenseprojectile isnotcapturedintheratio . A moving in a magnetized plasma we perform our analysis in M Sweepingupsuchamagneticfieldwilloccuronatimescale two steps. First, we analytically study the properties of the t/t √α(l/R) (√α )- 1,wheret =2R/uisthepro- c A A c flow of an ideally conducting plasma with a frozen-in mag- ∼ M ∼ M jectile’s own crossing time. This result means that, because neticfieldaroundaspheretoexplorethecharacteristicsofthe the magnetic layer is very thin, a strong field can be built magneticfield near the surface of the body. To this end, we up extremely quickly. Crucially, particularly for the prop- disregardanypossiblechangeintheflowpatternbymeansof agation of bubbles, the sweep-up time can be significantly theback-reactionofthemagneticfield. Whilethederivation smaller than a single crossing time; this is relevant because ofthisproblemcanbefoundinAppendixA, we summarize apurelyhydrodynamicbubblewillgenerallyself-disruptinto thekeyresultsinthissection.Inthesecondstep,wecompare a torus, or smaller fragments in a turbulent medium, in on thisanalyticalsolutiontoanMHDadaptivemeshrefinement ordera crossing time (Robinsonet al. 2004;Pavlovskiet al. simulationandexploreitquantitatively. 2007). This buildup of magnetic field will greatly effect the flow 3.1. Analyticalsolution in the direction of the ambient field lines, and the projectile will leave a magnetic bow wave behind it; by analogy with The potential flow solution for an incompressible flow aroundasphericalbodyreadsas othersimilarbowwaves,weexpectittohaveanopeningan- gle of tanθ υA/u. In the plane perpendicular to the am- R3 R3 bientmagne≈ticfield, however,the magneticfieldwillhavea υ=e - 1 ucosθ+e +1 usinθ, (3) r r3 θ 2r3 much less direct effect as field lines can simply slip around (cid:18) (cid:19) (cid:18) (cid:19) theprojectileandinstabilitiescanoccur. Inthepotentialflow whereRdenotestheradiusofthesphereanduisthespeedof simulation,theflowsmoothlyreattachesattherearofthepro- thecorethroughthequiescentambientfluid. Usingthissolu- jectile;however,inthiscase,vorticalmotionsgeneratedatthe tion,wesolvefortheresultingfrozen-inmagneticfieldwhile magneticcontact(wherethemagneticpressureandmagnetic neglectingits back-reactiononto the flow. For convenience, tensionforceismisalignedwiththedensitygradient)andby weshowheretheapproximatesolutionwhichisvalidnearthe instabilitiesatthemagneticinterface(whicharenotstabilized sphere, inthisplane)detachthewakefromtheobject,leavinglargely two-dimensionalvorticalmotionsalongthefieldlinesinthis 2 3s sinθ B = B sinφ, (4) plane.Theresultingexpectedgeometryisshowninacartoon r 3 0 R 1+cosθ r 4 Dursi&Pfrommer theprojectileischosensothatthematerialisinitiallyinpres- sureequilibrium. Theprojectileinitialvelocityistypicallychosentobe1/4, for a Mach numberinto the the ambientmedium of 0.32. Thesimulationinthetransversedirectionsrangefrom≈[- 4,4], and in the direction of motion of the projectile ranges from [0,28] for an aspect ratio of 2:7; in most of the simulations with projectiles larger than the fiducial R = 1, the domain size is increased proportionately. The initial magnetic field strength can be defined in terms of α =ρu2/P ; a typical 0 B0 valueusedin these simulationsis25/4, orP =1/100. Pe- B0 riodicboundaryconditionsareusedinthedirectionsperpen- diculartothedirectionofmotion,andzero-gradient‘outflow’ conditionsareusedin thezdirection. Experimentswithdif- ferenthorizontalboundaryconditionsproducednomajordif- ferencesinresults. Theprojectilefluidisinitiallytaggedwithapassivescalar, so that the material corresponding to the projectile can be tracedthroughoutthesimulation. FIG.5.—Diagramshowingthegeometryofthesimulationspresentedhere. Asphericalprojectilewithasmoothdensityprofileρmax(1+cos(πr/R))/2is 3.2.2. CodeChoice sentinthe+zdirectionwithaninitialvelocityυthroughanambientmedium with density ρ0 and a uniform magnetic field pointed in the +y direction. As can be seen from analytic arguments, and is shown Themagneticfieldstrength‘turnson’throughthedomainwithatanh-profile in some detail in §2, two features characterize the problem inthedirection ofmotionoftheprojectile, asindicated bytheshading of of magnetic draping: the formation of a narrow strongly- thebox;thisallowsustostartthecoreinanessentiallyfield-freeregionand smoothlyenterthemagnetizedregion.Periodicboundaryconditionsareused magnetized layer, and the relative simplicity of the dynam- inthedirectionsperpendiculartothedirectionofmotion,andzero-gradient ics, inthata potentialflowsolutionwithonlymagneticfield ‘outflow’conditionsareusedinthezdirection. kinematics captures much of the problem, lacking only the R magneticfieldback-reaction. B =B sinφ , (5) θ 0 3s Becauseoftheseparationofscales(arelativelylargeobject r moving through an ambient medium and a relatively small R B =B cosφ , (6) layerformingaroundit), thehighestresolutionrequirements φ 0 3s r wouldimposealargecostonthesimulationsiftheresolution where we introduceda radialcoordinatefrom the surface of hadtobeeverywhereuniform;indeed,itisonlyasmallpor- thesphere,namelys=r- R.Theseapproximatesolutionsuni- tion of the simulation domainwhich needsto be resolvedat formly describe the field near the sphere with respect to the thehighestlevel. Thisisespeciallytruesincethesimulations angle θ. As describedin AppendixA, the energydensity of we will need to perform are three-dimensional (as we will themagneticfieldforminginthewakebehindthebodyispre- seeinthenextsub-section,itisimpossibletodomeaningful dictedtodivergelogarithmically.Wepointoutthatthevalid- simulationsofmagneticdrapingintwodimensions). Thus,a ityofthepotentialflowsolutionheavilyreliesonthesmooth simulationcodewhichallowssomeadaptivityofmeshingis irrotationalfluidsolutionwherethemagneticback-reactionis extremelyhelpfulforapproachingthisproblem. negligible. We will see that these assumptions are naturally The relatively straightforward magnetic field dynamics violatedinthewake. meansthat,unlikeinproblemsof(forinstance)studyingthe details of MHD turbulence, we do not require high-orderfi- 3.2. Numericalsolution nitedifferencemethods;thisisparticularlytruebecauseofthe 3.2.1. Setup sharpnessofthethinlayersandthelargedensitygradientsin this problem. Instead, an MHD solverwhich can accurately Thesimulationspresentedinthispaperaresetupasshown dealwithsharpgradientsisvalued. in Fig. 5. For clarity of understanding the physical pic- AsaresultoftheimportanceofAMRforthesesimulations, ture, we consider only the magnetohydrodynamics(MHD); thecodewe’vechosentoperformthesesimulationswithisthe noexternal-orself-gravityisconsidered,andwedeferother FLASHcode(Fryxelletal.2000;Calderetal.2002). FLASH physics such as self-consistent inclusion of thermal conduc- is an adaptive-mesh general purpose astrophysical hydrody- tivitytofuturework. Inthisreport,wealsoconsideronlythe namics code which is publicly available1. The MHD solver magneticfield of the externalmedium, and assume that it is we use here is a dimensionally-split second-order accurate uniformoverthescalesofinteresthere. 8-waveGodunov-typesolverwhichisdescribedinmorede- In the code units we consider here, the ambient material tail in Powell et al. (1999). The smallness of spuriousmag- has a density of ρ = 1, and a gas pressure P = 1. The 0 neticmonopolesisensuredbyadiffusion-type‘div-B’clean (unmagnetized) fiducial projectile has a radius that we vary operation. This diffusive cleaning approach can be prob- in our runs between R = 0.5 and 2, and a maximum den- lematic near strong shocks, where diffusion cannot operate sity of ρ = 750. With the density profile chosen ρ(r) = max quicklyenough;however,nosuchshocksoccurinthesesimu- ρ (1+cos(πr/R))/2, the mean density of the projectile is max lations. FLASHhasbeenoftenusedforrelatedproblemssuch (1- 6/π2)ρ /2 0.2ρ . Both the ambient and projec- max max as hot and magnetized bubbles in the intercluster medium ≈ tile material are treated as ideal, perfectly conducting fluids (e.g.,Robinsonetal.2004;Pavlovskietal.2007;Heinzetal. withratioofspecificheatsγ=5/3,andsotheadiabaticsound speed in the ambientmedium is 5/3. The pressure inside 1http://flash.uchicago.edu p MagneticDrapinginClusters 5 2006;Roediger et al. 2006;Gardini 2006; Pope et al. 2005; field,sothedoubletransientofbeingshockedandexposedto Dalla Vecchia et al. 2004;Heinz et al. 2003;Brüggen2003; magneticfieldatthesametimewouldnotbeappropriatefor BrüggenandKaiser2002). thecasewewishtoconsider. Othermorerecentwork,whichinlargepartmotivatedthe 3.2.3. Parameters research presented here, considers squarely the case we are interestedin–magneticdrapinginclusterenvironmentsAsai Performingsimulationsofdrapingoveraprojectilewithan et al. (2004, 2005, 2007). In these works, the authors simi- explicit hydrodynamicscode (so that compressibility effects larly consider a spherical projectile moving through an am- will be included, for ease of comparison with later, super- bient magnetized medium. These simulations include more sonic, work)with a finite resolutionplacessomerestrictions physics (e.g. anisotropic thermal conduction, tangled mag- ontherangeofparameterspacewhichcanbeexplored. neticfields)thanweconsiderhere,asthegoaloftheseworks For simulating these cases with no leading shock, we re- wastofindoutifthedrapingofthemagneticfieldcouldsig- quire that the velocity of the projectile, u, be less than the nificantlysuppressthermalconductionintoacoldcore.These sound speed in the ambient fluid – but to take a reasonable simulationsshowthatmagneticdrapingispotentiallyimpor- number of timesteps (avoiding computational expense and tantfor suppressingthermalconductionacrossthe interface, spuriousdiffusion)requiresthattheprojectilevelocityremain of the order of the sound speed; thus u.c , or ρ u2 .γP, despitethefactthattheirdrapinglayersareunder-resolvedby s 0 atleastafactorof100(cf.Eqns.2and7). where ρ and P are the unperturbeddensity and pressure of 0 Inaslightlydifferentclusterenvironment,MHDeffectscan the ambient medium. For the hydrodynamicsof draping to alsostabilizethemorphologiesofbuoyantlyrisingradiobub- berealistic,themagneticpressureinthefluidmustbesignifi- blesfromAGNinclustercenters. Ruszkowskietal.(2007b) cantlylessthanthegaspressure,P P. Finally,resolution B0 ≪ carefully set up magnetically isolated bubbles and tangled requirementsforresolvingthethicknessofthemagneticlayer clustermagneticfieldsintheambientmediumasrequiredfor willputsomeconstraintonthethicknessofthemagneticlayer causally or temporally disconnectedgenerating processes of l>R/NfromEq.2,withNbeingthenumberofpointswhich these fields. They showed that magnetic draping of cluster resolvetheradiusoftheprojectile;typicallythesizeofthedo- magnetic fields possibly supported by helicity of magnetic main(ifatfullresolution)willbe8N 8N 28N. Thiscon- × × fields internal to the bubble can strongly alter the dynamics straint,expressedintermsoftherelevantpressures(rampres- sureandinitialmagneticpressure)P &3α(ρ u2)/N. Com- andsuppresshydrodynamicinstabilities;thusresemblingthe B0 0 observedmorphologicallyintactX-raycavities. bined,theseconstraintsgive Knowing that under some circumstances thermal conduc- 3αρ u2 tion can be suppressed by magnetic draping and hydrody- γP&ρ0u2≫PB0 & N0 (7) namicstabilitycanbeenhanced,itbecomesworthwhiletoun- derstandtheprocessofdrapingitselfinsomedetail,inwhich For a given u – which is more or less arbitrary, fixed to be casetheadditionalmicro-physics(atthisstage)makesunder- near the (arbitrary) sound speed – there is thus a relatively standingthisbasicprocessmore,notless, difficult. Thuswe narrowrangeofinitialmagneticpressuresintermsoftheram considerhereacasewithnoexplicitmicro-physicalconduc- pressureoftheambientmaterialontotheprojectilewhichcan tionandsimplifiedmagneticfieldgeometry.Withthischoice, be efficiently simulated. As we will see, α 2, and for the our simulationsremain scale free and can be applied to var- simulationspresentedhere,N 32- 64,mea≈ningwearecon- ious astrophysical environments. Additionally, our simula- ∼ strainedto studyroughlythatpartofparameterspace where tions can be directly compared to analytics and we are able ρ0u2/PB0 ≈1- 10. to carefully examine the process of the draping itself. Fur- ther,themoresimplifiedphysicsalongwiththeuseofAMR 3.2.4. ComparisontoPreviousNumericalWork allowsustorunsimulationsforsignificantlylongertimesas Havingdescribedthe setup of the simulationswe perform themorecomplicatedearliersimulations. forthiswork,itisworthcomparingthisto previousnumeri- 3.2.5. TwoDimensionalResults cal work. One body of work (e.g., Miniati et al. 1999;Gre- goriet al. 1999, 2000)considereda very similar problemin In two dimensions, the imposition of a symmetry greatly thecontextoftheISM,whereadensecloudmaybemoving limitspossiblemagneticfieldgeometries.Inanaxisymmetric through(for instance) the magneticfield associated with the geometry,theonlymeaningfuluniformfieldgeometryispar- Galactic spiral arms. In this context, a cloud of cold-phase allel to the axis of symmetry, which in this case would also ISM gas may very well not be globally self-gravitating (or bethedirectionofmotionoftheprojectile;inthissomewhat onlyweaklyso)andsoaroughly‘top-hat’densityfunctionis artificialcasemagneticfieldcouldsomewhatconstrainapro- usedtodescribethecloud. Inaddition,asappropriateto the jectile(orabubble;Robinsonetal.(2004))butdrapingcould ISM, supersonic motion was imposed on the ambient mate- notoccur. rial,withthecloudexperiencingahigh-Machnumbershock In planar symmetry, the field can have componentsout of and an ambient magnetic field at the same time; this might the plane, in the plane parallelto the directionof motion, or beappropriateforinstanceasa cold-phasecloudencounters intheplaneperpendiculartothedirectionofmotion. Acom- a hot-phase medium for the first time. In our work, we are ponent out of the plane will only have the dynamical effect consideringsubsonic motion initially, so that one significant of adding to an effective gas pressure ((e.g., Chandrasekhar differenceis the flow speed of the drapingmaterial, in addi- 1981)); the component along the direction of motion of the tion to the much flatter density profile used in the previous projectilecannotbedraped. work. Whilewedeferthesupersoniccasetofuturework,we Previouswork(e.g.,Asaietal.2005)hasexaminedthecase note that in our context, for example in the case of (for in- withtwodimensionalplanarsymmetrywithamagneticfield stance)acorefallingintoacluster,wewouldexpectthecore intheplaneofthesimulationandperpendiculartothedirec- toslowlyacceleratebeforeencounteringsignificantmagnetic tion of motion of the projectile. However, in this case, field 6 Dursi&Pfrommer Run R u Pb,0 ρu2/Pb,0 R/∆x A 1 1 1 6.25 64 2 4 100 B 1 1 1 6.25 64 4 100 C 2 1 1 6.25 32 4 100 D 1 1 1 3.125 64 4 50 E 1 1 1 1.5625 32 2 8 100 F 1 1 1 6.25 32 2 4 100 G 1 1 1 25 32 2 2 100 H 2 1 1 15.625 128 4 250 TABLE1 DETAILSOF3-DIMENSIONALSIMULATIONSRUNFORTHISWORK. SIMULATIONSWERERUNWITHANAMBIENTDENSITYANDPRESSURE OF1INCODEUNITS,ANDγ=5/3.SIMULATIONSWERERUNUNTIL 2 MAXIMUMMAGNETICFIELDONSTAGNATIONLINEWAS 16 1 3 APPROXIMATELYCONSTANT,TYPICALLY40-80TIMEUNITS. noitisoP telluB111024 8 6 0 20 40 60 80 100 120 140 Time FIG. 6.—Theaboveisaplotoftheprojectile position(calculated here bythemaximumheightatwhichthereissignificantprojectilematerialatany giventime)overtimeinatwodimensionaldrapingsimulation.Atabouttime 75,theprojectileisactuallybouncedbackundertheextremelystrongmag- netictensionwhichintwodimensionsmustgrowaheadoftheprojectile.The toppanelshowsB2inacloseupofthesimulationdomainatthreerepresen- tativetimesduringthebounce,withawhitecontourindicatingtheposition oftheoriginalprojectilematerial. Thisfigurehasbeensomewhatdegraded. ForaPDFversionofthismanuscriptwithhigherresolutionfiguresandinter- active3dgraphics,pleaseseehttp://www.cita.utoronto.ca/∼ljdursi/draping/. linescannotcannotsliparoundtheprojectile,andsoasmore andmoremagneticfieldgetssweptupbytheprojectile,mag- FIG.7.—Aplotofthemaximummagneticfieldstrengthalongthestagna- netic tension grows monotonicallyand linearly ahead of the tionlineforrunswiththreedifferentprojectilesizesandtwodifferentresolu- projectileuntiltheforcesbecomescomparablenotonlytothe tionsacrosstheprojectile,allwiththesamevelocityintotheambientmedium ram pressure seen by the projectile of the ambient medium, andthesameambientmagneticfield. Plottedisthemaximummagnitudeof themagneticfieldalongthestagnationlinevsthetime(scaledtothecrossing buttotherampressureoftheprojectileasseenbytheambient timeoftheprojectile).Theprojectileinitiallysitsinanunmagnetizedregion. medium. Atthispoint,theprojectiletrajectoryisreversed.A Themaximummagnetic fieldstrengthalongthestagnation lineisasensi- figuredescribingthisisshowninFig.6,wherethemagnetic tivemeasureofwhetherthestructureofthedrapedmagneticfieldisbeing tensionforcesareseentocompresstheprojectile(withmean resolved;weseehereclearevidencethatwiththeresolutionusedinthissim- ulationthedrapedlayerisbeingadequatelyresolved. Inthelow-resolution density 150 times that of the magnetizedmedium) before ≈ R=1simulation, whichalsowasruninasomewhatsmallerbox, towards repellingit. theendoftherunthedrapinglayerbeginstoleavethetopofthesimulation This outcome is hinted at in Fig. 4 of Asai et al. (2005), domain,leadingtothesuddenrapiddropinmagneticfieldstrength. wherein2dthemagneticenergyincreaseslinearlyandwith- outbound, while the 3d modelsreach a maximummagnetic The magnetic layer in Run G is under-resolved; while the energy. valueofR/∆xisthesameasotherruns,thevelocityishigher, sothatbyEq.2 thelayeristhinner. We includethisrunbe- 3.2.6. ThreeDimensionalSimulations cause it demonstrates certain robustness of results; although A listing of the eight main runs done for this work are thelayerstructureisnotadequatelyresolvedatthestagnation shownin Table1, andthebasic setupfollowsthediscussion point,otherglobalpropertiesofthemagneticlayer(geometry earlier.Theparametersvariedarethesizeoftheprojectile,its anddynamicaleffects)otherwiseremainrobust. velocity,andthestrengthoftheambientmagneticfield.Other Indeed,oneshouldbecarefulaboutwhatonemeansby‘re- runs(the equivalentof runB butwith halfthe resolution, or solved’. Thisdiscussionshouldnotbetakentomeanthatthe with the same resolution but differing boundary conditions) othersimulationsareinallrespectsresolved.Inparticular,as were runto confirmthat the resultsdid notchange; they are we will see in §4 and§5, and assuggestedby Dursi(2007), notlistedhere. theflowaroundthebubbleinthexzplane(e.g.,transverseto Toshowthattheserunswereproducingresultsindependent theinitialmagneticfield)isunstabletoKelvin-Helmholtzand ofresolution,themaximummagneticfieldstrengthalongthe Rayleigh-Taylor instabilities, not stabilized by the presence stagnationlineforalltherunswiththesameP andρu2 are of magnetic field. Since we have not prescribed any small- B0 0 plottedversustimeinFig.7. Themaximumfieldstrengthis scale dissipative physicsin these simulations, these instabil- sensitivetotheresolutionofthemagneticfieldlayer,butwe ities will never properly converge with increased resolution seeherethatvaryingtheresolutionbyafactoroftwodoesnot (e.g., Calder et al. 2002) as new unstable scales are added. effecttheresults,asthefieldlayerisadequatelyresolved(but Thiscanonlybecorrectedbyaddingsmall-scalephysics,e.g. only marginallyin the case of R/∆x=32). We also see, as thermaldiffusion;thisis leftforfuturework, as the relevant we’dexpectfromthediscussionsin§2,thatthefieldstrength microphysics is itself a current research problem (Lyutikov inthelayerdoesnotdependonthesizeofthecore. 2007;Schekochihinet al. 2007). While the rate of develop- MagneticDrapinginClusters 7 mentoftheinstabilitiesandtheirpropertiesisveryimportant icantly larger than the initial radius of the projectile; in this forlong-termmixingofthematerialofthemovingobjectwith example,theexpansionisarelativelymodest15%. thatofthesurroundingmedium,werestrictourselveshereto studyingthedevelopmentofthemagneticlayeranditsglobal properties. A notehereonourAMRisinorder. InTable1, we show thefinestresolutionachievedinthesimulation. FLASHplaces resolution elements based on a second-derivativerefinement criterion, in an attempt (appropriate for a second-order ac- curate code) to reduce error; see Fryxell et al. (2000) for more details. Here we refined based on pressure, density, FIG. 8.— Shown is, left, the magnetic field along the stagnation line and composition (e.g., fraction of projectile material); early in the simulation (’+’) and a fitted theory prediction, with the two fitting parameters being theposition ofthe peakandaradius giving acharacter- experimentswithalsorefiningonmagneticfieldcomponents istic fall off of the field strength. On the right are cut-planes along and showedlittleornodifference. Theresultisthattheentirere- acrosstheinitial magnetic fieldofthedensityoftheprojectile, withacir- gion immediately surroundingthe projectile is fully refined, cle of radius and position given by the fit to the magnetic field structure, andthewakeregionisnearlyfullyrefined.However,because left. Theradiusgivenbythefitcorrespondswiththeradiusofcurvatureat theworkingsurfaceoftheprojectile. ResultsaretakenfromrunBattime no interesting flow features are occurringin most of the do- t=38.75;resultsfromothersimulationsandothertimesgivesimilarlygood mainatanygiventime, thesavingsfromusingAMRcanbe fits. This figure has been somewhat degraded. ForaPDFversion ofthis substantial; for runG, forinstance, the verysimple, laminar manuscriptwithhigherresolutionfiguresandinteractive3dgraphics,please initial conditions require only 9.7 105 zones; by time 40, seehttp://www.cita.utoronto.ca/∼ljdursi/draping/. this has increased to 8.1 106 z×ones, and by time 75, an approximately-steady≈9.1 ×106 zonesarebeingused; thisis to be compared to the 46×9.8 106 zones, or a factor of 51 4.2. Comparisonofthevelocityfield × greater,thatwouldberequiredtoresolvetheentiredomainat To compare the potential flow calculations, done in the thissamefinestresolution. frameofthesphericalbody,withthoseofthenumericalsimu- A final feature worth noting in our runs is that for run E, lations,wetransformthenumericalsimulationsintotheframe M2A=1/2(ρu2/PB0)≈0.78<1;thatis,thisrunhasthepro- oftheprojectile.Becausetheprojectileslowsdownovertime, jectilemovingsub-Alfvénically,ifonlymarginally.However, we do not use the initial velocity u for this transformation, 0 becauseofthefieldgeometry,wewillseethatthismakeses- butmeasuretheinstantaneousmass-weightedvelocityofthe sentiallynodifferenceforthedraping.Thisissimplybecause projectile,bymakinguseofthefactthatwearetrackingthe theAlfvénspeedinthedirectionofmotionoftheprojectileis fluid that initially resided in the projectile by use of an ad- zero–nocomponentofthemagneticfieldpointsinthatdirec- vectedpassivescalar, a. Thuswe measuretheinstantaneous tion.Whiletheexactimpositionofthisconditioninourinitial conditionsissomewhatartificialinthiscase,itisalwaystrue that it is only the component of the magnetic field that lies transversetothedirectionofmotionthatwillbedraped. 4. COMPARISONOFTHEORYANDSIMULATIONS 4.1. Magneticfieldalongthestagnationline The first comparison we make is to the one-dimensional predictions made along the stagnation line, for instance in Lyutikov(2006),whereaveryspecificpredictionismadefor therampup,withaparticularfunctionalform,ofthemagnetic fieldstrengthgiveninEq.1,anditisnotnecessarilyclearthat suchapredictionwillholdwhentheprojectilebeginstodevi- atesignificantlyfromspherical. To make this comparison, we extract the magnetic field strength along the stagnation line for an output, and fit it to theequation B B 1 | | = 0 (8) ρ ρ0 1- R 3 r (z- z0) FIG.9.—Comparisonofthevelocityfieldoftheanalyticalsolutioninthe (cid:16) (cid:17) kinematic approximation (top panels) with our numerical simulation (bot- where B0, ρ0 are known, and the fit is for the parameters R, tom panels) in the plane of the initial magnetic field. While the velocity which would correspondto the radius of the sphere, and z , fieldsresembleeachotherverywellintheupperhalf-space,therearedistinct 0 whichwouldbethepositionofthecentreofthesphere. Re- differences inthe lowerhalf-space. Theseareduetothenon-linear back- reaction of the dynamically important magnetic field in the draping layer sultsforatypicaloutputareshowninFig.8. Notonlydothe ontheMHDflowthatgenerates vorticity inthewakeoftheprojectile (cf. fits well representthe behaviourmagnetic field strength, but §5.4). Shownhereandinthenext figures isazoom-in onasmallregion they also suggest a physical interpretationfor the functional ofourcomputationaldomainthatextendsupto112lengthunitsandisfour formsinterpretationevenwhentheprojectilebecomessignif- timeslargerinxandydirection. Notethatwesymmetrizedthecolormap icantly non-spherical; R becomes the radius of curvature of of the υx-component in order not to be dominated by one slightly larger eddy. Thisfigurehasbeensomewhatdegraded. ForaPDFversionofthis theworkingsurfaceoftheprojectileatthestagnationline. As manuscriptwithhigherresolutionfiguresandinteractive3dgraphics,please the projectile becomesmoredistorted, R can becomesignif- seehttp://www.cita.utoronto.ca/∼ljdursi/draping/. 8 Dursi&Pfrommer FIG. 10.— Comparison of the ram pressure in our numerical simula- tion (left panel) with the analytical solution in the kinematic approxima- tion (right panel). Aheadofthe projectile, the rampressure resembles an exact potential flow behavior up to the draping layer which can be seen FIG. 11.—Comparisonofthemagneticfieldinournumericalsimulation as a black layer around the projectile with a deficit of hydrodynamical (bottompanels)withtheTaylorexpansionoftheanalytical solutioninthe pressure. Non-linear magnetic back-reaction of the draping layer causes kinematicapproximationthatstrictlyappliesonlynearthesphere(toppan- the flow to depart from the potential flow solution and to develop vortic- els).WeshowtheCartesiancomponents(lefttoright,x,y,z)ofthemagnetic ity. This figure has been somewhat degraded. For a PDF version of this fieldintheplanethatisparalleltotheinitialmagneticfield. Thereisanice manuscriptwithhigherresolutionfiguresandinteractive3dgraphics,please agreement between both solutions intheupper half-space, whilethere are seehttp://www.cita.utoronto.ca/∼ljdursi/draping/. againdistinctdifferencesinthelowerhalf-space.Themagneticshouldersbe- hindtheprojectilecanbeidentifiedthatpreventsthedrapinglayerfromcon- velocityoftheprojectileas tractingtowardsthesymmetryaxis.Inaddition,MHDturbulencestartstode- velopinthewakeoftheprojectile.Thisfigurehasbeensomewhatdegraded. u= hρaυzi. (9) FacotrivaeP3DdFgvraeprshiiocns,opflethaissemseaenuhsttcpr:i/p/twwwiwth.chiitga.huetrorroesnotolu.ctiao/n∼filjgduurressi/danradpiinntge/r.- ρa h i ThetoppanelsofFig.9showtheanalyticalsolutionofthe velocityfield aroundthe sphericalbodywith radiusR in the kinematic approximation. For convenience and to simplify the comparisonto the magnetic field visualization, we show theCartesiancomponentsofthevelocityfield. Atinfinity,the fluid is characterized by a uniform velocity υ =- e u. The z quadrupolar flow structure results from the fluid decelerat- ing towards the stagnation line, the successive acceleration around the sphere until θ =π/2 and mirroring this behavior in the lower half-plane by symmetry. The bottom panels of Fig.9showthenumericalsolutionofthevelocityfieldaround an initially spherical projectile that deformed in response to thenon-linearevolutionofthemagnetizedplasma.Thewhite line reflects the 0.9 contour of the ‘projectile fluid’ and cor- responds to an iso-density contour.2 The quadrupolar flow structureintheupperhalf-planeresemblesnicelytheanalytic potential flow solution. As the flow approaches the projec- tile and surroundsit, there are importantdifferencesvisible. In the analytical solution, the flow accelerates for θ π/2 ≤ and decelerates for larger angles θ. In the numerical solu- tion, the magneticdrapinglayer is stationarywith respectto FIG.12.—Sameaspreviousfigure,butinthetheplanethatisperpendicular totheinitialmagnetic field. Shownis,lefttoright, thex,y,zcomponents. the projectile. This causes the flow almost comes to rest in Asexpected fromouranalytic solutions, thedrapinglayerformsbypiling the magnetic draping layer. The back-reaction of the mag- upmagneticfieldlinesaheadoftheprojectile. Theirregularmagneticfield neticdrapinglayerontheflowcastsa‘shadow’onthewake inthewakeisgeneratedbythevorticitythatisabsentbydefinitioninour oftheprojectile. Itpreventstheflowtoconvergetowardsthe potentialflowsolution.Thisfigurehasbeensomewhatdegraded. ForaPDF versionofthismanuscriptwithhigherresolutionfiguresandinteractive 3d symmetryaxisandsuppressesthedecelerationoftheflow.In- graphics,pleaseseehttp://www.cita.utoronto.ca/∼ljdursi/draping/. stead,vorticityisgeneratedatthedrapinglayerwhichwillbe studiedindetailin§5.4. Thecomparisonoftherampressure 4.3. Comparisonofthemagneticfield inFig.10underpinsthisargument. It is instructive to compare the analytic solution of the 2Notethattheapparentgridstructureseenintheupperpartofoursimu- frozen-inmagneticfieldinthekinematicapproximationtothe latedυx-componentisanartifactofourplottingroutineaswellassmallgrid numericalsolutionintheplanesthatareparallelandperpen- noise. Theinterpolationschemeoftheplottingroutinefalselyinterpolatesa diculartotheinitialmagneticfield. Wecomparetheindivid- smoothvelocitygradientwithanentireAMRblockwhileitactuallydrops quicklytothevelocityvalueatinfinity. ualCartesiancomponentsofthefield(Fig.11and12)aswell MagneticDrapinginClusters 9 FIG. 14.— A plot for the 3d runs presented here showing the mag- neticpressureonthestagnationlineonceasteadyvaluehadbeenachieved for this quantity as a function of the mean ram pressure (ρhui2) as seen by the projectile. Omitted is run G, for which the magnetic layer was under-resolved andthus themaximum magnetic field strength inthelayer falls much lower; however, as we will see, even this under-resolution does not strongly effect other global properties of the magnetic drape. FIG. 13.—Comparisonofthemagneticenergydensityinournumerical This figure has been somewhat degraded. For a PDF version of this simulation(leftpanel)withtheanalyticalsolutioninthekinematicapproxi- manuscriptwithhigherresolutionfiguresandinteractive3dgraphics,please mation(rightpanel). Thetop(bottom)panelsshowtheplanethatisparallel seehttp://www.cita.utoronto.ca/∼ljdursi/draping/. (perpendicular)totheinitialmagneticfield.Intheanalyticalsolutionthereis ing cone causes the stationary flow not to convergetowards anarrowmagneticlayerdrapedaroundthesphericalbody,whileinoursim- ulationsthedrapinglayerpeelsoffbehindtheprojectileduetovorticitygen- thesymmetryaxisandprotectstheregioninthewakeagainst eration. Thegeometryofthemagneticdrapinglayerintheupperhalf-plane theincreaseofthemagneticenergywithoutbounds. Thenu- is very similar inboth planes suggesting there an approximately spherical merical solution can qualitatively be obtained by remapping symmetry. Inthewakeoftheprojectile, thedrapinglayerformsacharac- the analytic solution for θ >π/2 onto the coordinate along teristicopeninganglewhilethefieldlinescanswipearoundtheprojectilein theperpendicularplaneandthedrapinglayerclosestowardsthesymmetry themagneticdrapingcone. Intheperpendicularplanetothe axis. Thisfigurehasbeensomewhatdegraded. ForaPDFversionofthis initialmagneticfield,thereisevenbetteragreementbetween manuscriptwithhigherresolutionfiguresandinteractive3dgraphics,please the analytic and the numerical solution. The magnetic field seehttp://www.cita.utoronto.ca/∼ljdursi/draping/. in thatplane lies primarilyin its initialy-direction. This be- haviour can easily be understood in terms of the field lines asthemagneticenergydensityinFig.13. Notethatweonly sweeping aroundthe sphere in a laminar flow. Numerically, showaTaylorexpansionofthehighlycomplexexactsolution wesimulatetheresponseofthegeometryoftheprojectileto asderivedinAppendixA. Strictly,thissolutionappliesonly near the sphere with an accuracy to ((r- R)3/2) as well as thehydrodynamics. Vorticesinthe wake deformthe projec- O tile leadingto a cap-geometryand a mushroomshape ofthe for flow lines that have small impact parameters initially at y-componentofthemagneticfield. Thisimpliesthattheflow infinity.Usingadifferentexpansion,weverifiedthatthegen- lines detach from the dense material of the projectile gener- eralsolutionhastheappropriatebehaviorofthehomogeneous atingfurthermorevorticityandMHDturbulenceinthewake. magneticfieldatinfinityintheupperhalf-spacepointingto- TheturbulentfieldmixestheCartesiancomponentswhichcan wardsthepositivey-coordinateaxis,i.e.rightwardsinFig.11. benicelyseenintheFig.12. Themagneticpressuresumma- Asexpected,they-componentofthemagneticfieldincreases rizesourresultsnicelyshowingthedrapingconeintheparal- as we approach the sphere since the field lines are moving lelplaneandthe mushroomshapedmagneticenergydensity closertoeachother. Intheimmediatevicinityofthesphere, intheplaneperpendicularto that(cf.Fig.10). Note thatwe theBfieldattainsadipolarz-componentasthefieldlinesare choosethe same color scale as derivedfromthe simulations carried around the sphere with the fluid and causes them to whichleadstoasaturationofthemagneticenergydensityin bendinreactiontotherampressureofthesphere.Aspointed the kinematic approximation at the contact of the spherical out by Bernikov and Semenov (1980) the magnetic lines of bodyandontheaxisinthewake. forcethatendatthestagnationpointarestronglyelongatedas theswipearoundthesphereparalleltothelineofflowreach- 5. CHARACTERISTICSOFMAGNETICDRAPING ingfromthe stagnationpointinto the rear. Thisleads to the 5.1. Fieldstrengthindrapinglayer unphysicalincreaseofthemagneticfieldasitapproachesthe lineofsymmetryinthewakeandeventuallytoalogarithmic The kinematic solution predicts the magnetic pressure di- divergenceofthemagneticenergydensitythere. verges at the stagnation point, which is clearly unphysical. In the upper half-plane, the analytic solution matches the Fromour discussionsin §2 and §4, we expectthat the mag- numericaloneclosely. Interestingly,intheregionbehindthe neticpressureinthedrapinglayershouldbeonorderρu2,at deformed projectile, a magnetic draping cone develops that whichpointthemagneticback-reactionbeginstostronglyef- stemsfromthedynamicallyimportantdrapinglayerthathas fect the flow; to first order there is no dependence on other swipedaroundthesphereandadvecteddownstreamthepro- parameters, such as backgroundmagnetic field. One would jectile. Inaddition,themagneticpressureinthewakeofthe expect, too, from looking at figures such as Fig. 4 that the projectile is also amplified by a moderate factor of roughly maximummagneticpressureshouldexceedtherampressure five(cf.Fig.13).Wewillshowfurtherdown,thatthisfieldis by some factor, as the magnetic pressure distribution at the generatedtogetherwith vorticityin the drapinglayer. In the headofthedrapeisresponsibleforredirectingtheflowinthe parallelplanetotheinitialmagneticfield,themagneticdrap- planeofthedraping. 10 Dursi&Pfrommer FIG.17.—Plotofmagneticenergydensityinthey,zplaneforsimulations withR=0.5and,lefttoright,u=0.125,0.25,0.5;shownwithblacklinesare thefittedopeninganglesofthemagneticdrapinglayer,omittingtheregion includingthematerialfromtheprojectile.Thefitslopes(e.g.,tanθ)are1.24, 0.515, 0.261, and those predicted byυA0/hui are 1.13, 0.566, 0.283; this agreementiswithin10%. Thisfigurehasbeensomewhatdegraded. Fora PDFversionofthismanuscriptwithhigherresolutionfiguresandinteractive FIG. 15.— As in Fig. 1, but for Run E; that is, with the projec- 3dgraphics,pleaseseehttp://www.cita.utoronto.ca/∼ljdursi/draping/. tile’s velocity reduced by a factor of one-half (so that u=0.125 in code units). A PDF version of this manuscript with an interactive 3d ver- sion of this figure, following Barnes and Fluke (2007), is available at http://www.cita.utoronto.ca/∼ljdursi/draping/. 1.2 1 tan θ0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1 1.2 FIG.18.—Aplotforthe3drunspresentedhereshowingthetangentofthe FIG. 16.— As in Fig. 1, but for run G; that is, with the pro- fitopeninganglesofthedrapeintheyzplaneversusvA/hui. Datafromall jectile’s velocity increased by a factor of two (so that u = 0.5 in code runsareshown. units). A PDF version of this manuscript with an interactive 3d ver- sion of this figure, following Barnes and Fluke (2007), is available at rectionofthesimulations,and–omittingtheregionsaboveor http://www.cita.utoronto.ca/∼ljdursi/draping/. neartheprojectileitself,andtheregionbelowwhichthedrape We can test this by plotting, for all our runs, the steady becomesweakerthantransientfeaturesinthewake–linesare maximummagnetic pressure at the stagnation line (the field fit, and the slope givesthe (half-)openingangle. The results quantitythatiseasiesttoconsistentlycharacterize)versusthe ofthefittingprocedureareshownforthesamethreesimula- meanrampressureseenbytheprojectile,ρ u 2,where u is tions in Fig. 17, and a scatter plot for all are runs are given themeanoftheprojectilevelocity(calculatehdiasinbyEqhni.9) in Fig. 18. The scatter forthis quantity,and agreementwith duringtherun,andρistheambientdensity.Theplotisshown theprediction,issomewhatworsethanfortheotherquantities inFig.14andverifiesourexpectation. weconsider,possiblybecausethelarge-scalegeometryofthe drapingismoresensitivetotheboundariesandthefinitesize 5.2. OpeningAngle ofthedomainthanother,morelocal,quantities. The magnetic bow wave behind the projectile is expected 5.3. Decelerationbymagnetictension topropagatetransverselyawayfromtheprojectileatυ along A the field lines, and of course to fall behind the projectile at In the scatter plots presented above, we use the mean ve- velocityu. Thissuggestsanaturalopeningangleintheplane locity u oftheprojectileovertime,becausethereisamea- h i along the magneticfield, tanθ=υ /u. Thatthe direction of surable deceleration of the projectile. An example, for run A thescalingiscorrectcanbedeterminedbyqualitativeinspec- F, is shown in Fig. 19. Before the projectile encounters the tion of a sequence of 3d renderingsof simulationoutputsas magneticfieldatz=10,hydrodynamicdrag–inprincipleei- thevelocitychanges;e.g.,Figs.15,1,16foru=1/8,1/4,1/2 ther(numerical)viscousdragorthethedragforcecausedby andυ fixedat0.1414. the creation of a turbulentwake – is all thatcan play a role, A Althoughthefieldlinesarestretchedduringthedraping,it andforthesimulationspresentedhere,itisthesecondwhich is the initialυ that is relevant, as the stretching of the field dominates. Thewell-knownformforthedragonasphereis A linesinthez-directiondonoteffectthepropagationspeedin F =1/2ρu2AC ,orintermsofadeceleration, D D the y-direction. For instance, consider a zˆ-velocity shear in gy,ivυes=u(s0B,˙0,=y/τ),(wυithBB)==((00,,B00,,B0)/.τT)h,esointdhuatcttihoenmeqauganteiotinc u˙D=- 83 ρρpu2RCD (10) 0 h i υfielBˆd iyˆs=o(nlBy/∇c√h×a4nπgρe)d×(Binyˆt)h/eBzˆ-d=irBec/ti√on4;πtρh=usυυAy =υA·yˆ= wbehtewreeeCnD0i.s07th-e0d.r5a,gwciotheffi0c.5ienfot,reaxptuerrbimuleennttalwlyakken,oρwnitsothbee A · | | · | | y A,0 p One can quantifythe agreementwith thisscaling by mea- density and A is the cross-sectional area of the projectile in suring the opening angle for the drapes in our simulations. thedirectionofmotion. Themaximaofmagneticfieldoneithersideofthestagnation However,oncethemagnetizedregionisreachedandamag- lineinthe y- zplanearefoundandtabulatedalongthezdi- netic layer built up, then another force acts on the projectile

ebook img

Draping of Cluster Magnetic Fields over Bullets and Bubbles -- Morphology and Dynamic Effects PDF

0.73 MB·English
by  L. J. Dursi
#additional_collections #journals #arxiv
Save to my drive
Quick download
Download
Upgrade Premium
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Draping of Cluster Magnetic Fields over Bullets and Bubbles -- Morphology and Dynamic Effects

DRAFTVERSIONFEBRUARY2,2008 PreprinttypesetusingLATEXstyleemulateapjv.10/09/06 DRAPINGOFCLUSTERMAGNETICFIELDSOVERBULLETSANDBUBBLES–MORPHOLOGYANDDYNAMIC EFFECTS L.J.DURSIANDC.PFROMMER CanadianInstituteforTheoreticalAstrophysics,UniversityofToronto,Toronto,ON,M5S3H8,Canada DraftversionFebruary2,2008 ABSTRACT High-resolution X-ray observations have revealed cavities and ‘cold fronts’ with sharp edges in tempera- ture and density within galaxy clusters. Their presence poses a puzzle since these featuresare not expected to be hydrodynamicallystable, or to remain sharp in the presence of diffusion. However, a moving core or 8 0 bubblein evena veryweakly magnetizedplasma necessarilysweeps upenoughmagneticfield to buildup a 0 dynamicallyimportantsheath;thelayer’sstrengthissetbyacompetitionbetween‘plowingup’andslipping 2 aroundoffieldlines,anddependsprimarilyontherampressureseenbythemovingobject. Inthisinherently three dimensionalproblem, our analytic argumentsand numericalexperimentsshow that this layer modifies n thedynamicsofa plungingcore, greatlymodifyingthehydrodynamicinstabilitiesandmixing,changingthe a geometryofstrippedmaterial,andslowingthecorethroughmagnetictension.Wederiveanexpressionforthe J maximummagneticfieldstrengthandthicknessofthelayer,aswellasfortheopeningangleofthemagnetic 8 wake.Themorphologyofthemagneticdrapinglayerimpliesthesuppressionofthermalconductionacrossthe layer,thusconservingstrongtemperaturegradients.Theintermittentamplificationofthemagneticfieldaswell ] h astheinjectionofMHDturbulenceinthewakeofthecoreisidentifiedtobeduetovorticitygenerationwithin p themagneticdrapinglayer. Theseresultshaveimportantconsequencesforunderstandingthecomplexgasdy- - namicalprocessesoftheintra-clustermedium,andapplyquitegenerallytomotionsthroughothermagnetized o environments,e.g.theISM. r t Subjectheadings:hydrodynamics—magneticfields—MHD—turbulence—galaxies: clusters: general— s a diffusion [ 2 1. INTRODUCTION through the intercluster medium, as seen at the centers of v manycool-coreclusters(e.g.,McNamaraetal.2005;Bîrzan Recent observations of very sharp ‘cold fronts’ in galaxy 3 et al. 2004). In this case, the bubble would be quickly dis- clustersraiseunansweredquestionsinthehydrodynamicsof 1 rupted on rising absent some sort of support (e.g., Robin- galaxyclusters(seeforinstancethereviewofMarkevitchand 2 son et al. 2004). However, the draping of a pre-existing 0 Vikhlinin2007),forsuchabrupttransitionsarenotexpected magneticfield maystronglyalterthedynamicsandsuppress . to be stable against either hydrodynamicalmotionsor diffu- 1 hydrodynamic instabilities, as seen recently in simulations sionforextendedperiodsoftime. 1 (Ruszkowski et al. 2007b). The morphology of the draped It has been known for some decades in the space science 7 magnetic field may be able to suppress transport processes community that an object moving super-Alfvénically in a 0 acrossthe bubble interfacesuch as cosmic ray diffusionand magnetizedmedium can very rapidly sweep up a significant : v magneticlayerwhichisthen‘draped’overtheprojectile(e.g., heatconduction.Thishasimportantconsequencesforcosmic i BernikovandSemenov1980).Forconcretenessindiscussing ray confinement in these buoyantly rising bubbles and may X explainre-energizedradio‘relic’sources,broadcentralabun- the process, we show in Fig. 1 a picture of this mechanism r danceprofilesofclusters,andtheexcitationoftheHαlinein a takenfromoneofoursimulations,whichwillbedescribedin filaments trailing behind bubbles(Ruszkowski et al. 2007a). moredetailinlatersections. Although the analytics and simulations we discuss here fo- There has been significant interest in applying this idea cusonthecaseofanoverdense‘core’movinginanexternal of magnetic draping in galaxy clusters (e.g., Vikhlinin et al. field,weexpectthebasicmagneticdynamicstoalsoextendto 2001; Lyutikov 2006; Asai et al. 2004, 2005, 2007) as such thecaseofanunderdensebubbleprobablydependingonthe a magnetic field could naturally inhibit thermal conduction magneticenergydensityoftheplasma. across a front (e.g., Ettori and Fabian 2000) allowing it to Becauseweareinterestedhereinthefundamentalsofaba- remain sharp over dynamically long times. Although such sic process—thatofthedrapingofa fieldaroundanobject draping has been explored in the past, in the space sciences andtheresultinghydrodynamicaleffectontheobjectandits the resulting dynamicsis relatively simpler, as generally the interactionwiththeexternalmedium—weconsiderforthis objectbeingdrapedisasolidbody,withlittleinteriordynam- paper,inbothouranalyticandcomputationalwork,thesim- icsofitsown. However,inthecaseofforinstanceamerger plestpossiblecase; anoverdense,non-self-gravitating‘blob’ ofgas-richclusters, thehydrodynamicsofthe drapedplung- moving through a quiescent medium with a magnetic field ingcorecanalsobemodified,withthestrongmagneticlayer uniformonthe scalesconsidered. (Thetermwe willuse for providing some stabilization against instabilities that would thisblobwilldependonthesituation;whendiscussingastro- otherwiseoccur(Dursi2007). physical implications, we will speak of ‘cores’ or ‘bullets’, Theeffectofastrongdrapedmagneticlayercouldbeeven dependingonthecircumstances,or‘bubbles’forunderdense greater for underdense objects, such as for bubbles moving regions;forthecaseofournumericalsimulations,wewillre- Electronicaddress:[email protected],[email protected] fer to ‘projectiles’, as the overdense fluid in the simulations 2 Dursi&Pfrommer differsinstructurefromcoresorbulletsinlackingselfgrav- Because in this case the flow falls quickly to zero at the ity;inouranalyticworkwheretheblobisarigidsphere,we surface of the moving sphere, magnetic field rapidly ‘builds willrefertotheblobasasphereorsphericalbody.) up’ around the projectile, and in the kinematic limit eventu- We furtherconsiderthecaseoftheobjectmovingsubson- allybecomesinfinite.Thehighdegreeofsymmetryalongthe ically;whilethecaseofsupersonicmotionisinterestingand stagnation line (the axis of symmetry of the object pointing highly relevant, we anticipate that in the usual case where inthedirectionofmotion)greatlysimplifiesthemathematics, the bow shock is well separated from the magnetic layer – and as shown in for instance Lyutikov (2006), the magnetic that is where the standoff distance ∆ ρ0R R, (where R fieldstrengthdirectlyalongthestagnationlineisgivenby ≈ ρs ∼ tihsethsehoracdkieudsdoefntshitey)coisrem, ρu0chisgtrheeataemrtbhiaenntthdeenmsaitgyn,eatnicdlρasyeisr |B| = B0 1 (1) thicknessl - 2R R, (wherel is the approximatemag- ρ ρ0 1- R 3 ≈MA ≪ R+s netized layer thickness and the alfvénic Mach number, asdiscussedinmoredetailinMthAenextsection)thattheargu- whereB0istheambientmagnqeticfi(cid:0)eld,(cid:1)ρ0istheambientden- sity, R is the radius of the solid sphere projectile, and s is ments here will also hold, so we save the more complicated thedistancealongthestagnationlinefromthesurfaceofthe geometryandlargerparameterspaceofthecompressiblecase sphere. forfuturework. The analytic works cited, and presented here, considered purelyincompressibleflow;foroursimulations,weconsider only very modest compressibility, with projectile motions throughtheambientfluidquitesubsonic,soitsufficesforthe momenttoconsiderintheexternalmediumρ=ρ . Thefluid 0 here is further considered to be infinitely conducting; how- ever, the buildup of magnetic field without a corresponding buildupofmassdoesnotviolatethe‘flux-freezing’condition, asshowninthecartoonFig.2asincomingfluidelementsare ‘squished’alongthesidesoftheincomingsphere,sothatthe magnetic flux coming out the sides of the fluid element re- mainsconstant,evenastheconcentrationoffieldlinesbuilds upalongthestagnationline. Furtherincreaseinmagneticen- ergycomesfromthe stretchingoffield linesin the direction FIG.1.—Arenderingofoneofthethree-dimensionalsimulations(referred toasRunFlaterinthiswork)performedforthiswork,discussedinmorede- ofmotionofthecore. tailinlatersectionsbutincludedheretoillustratethephysicalpicture. An overdenseprojectileissentthroughauniformlymagnetizedmedium,sweep- ingupmagneticfieldaheadofit.Plottedisadensityisosurface,correspond- ingtothemeandensityofthebullet,andsomefiducialmagneticfieldlines. Thecut-planeiscolouredbymagneticenergydensity,asarethefieldlines. Theminimumofthecolourmapforthemagneticenergydensityissettobe themagneticenergydensityintheambientmedium;energieslessthanthat (whichoccurinthewake,duetoentrainmentofunmagnetizedmaterialasthe bubblerises)issettothecoloroftheambientenergydensity. Themagnetic fieldis‘draped’intoathinlayerformingabowwave,leavingturbulenceina FIG.2.—Acartoonshowingthedistortionofincomingfluidelementsand wakebehindthebullet.Magneticfieldlinespileupalongthestagnationline stretchingoffieldlinesasaredsphericalprojectilemovesupwardsthrough ofthisinitiallyaxisymmetricbullet,whileintheplaneperpendiculartothe theambientmedium. initialfield,thefieldlinescansliparoundthebullet. APDFversionofthis manuscriptwithaninteractive3dversionofthisfigure,followingBarnesand Inreality,ofcourse,themagneticfielddoesback-reactonto Fluke(2007),isavailableathttp://www.cita.utoronto.ca/∼ljdursi/draping/. the flow, and the kinematic potential flow solution fails for tworeasons–buildupofastrongmagneticfieldlayer(which In§2wegiveanoverviewofthephysicsofdraping,putting violates the kinematic assumption) and creation of vorticity ourworkinthecontextofpreviousresults;in§3wedescribe (inconflictwiththepotentialflowassumptions). ouranalyticandcomputationalapproaches;in§4wecompare Themagneticfieldshouldexertasignificantback-reaction theresultsofourtwoapproaches,andfromtheunderstanding whentheresultingmagneticpressureiscomparabletotheram gainedtherein§5wedescribecharacteristicsofdraping;we pressureoftheincomingmaterial: B2/8π ρ u2,whereuis discusstheeffectofinstabilitiesin§6,considerthelimitations ∼ 0 thespeedofthecorethroughthequiescentambientfluid.The ofourresultsandconsiderapplyingthemtolatertimesin§7, firstplacethiswillhappenisalongthestagnationline,which andfinallyconcludein§8. bysymmetrywillbethelocationofthelargestmagneticen- ergydensity. Thelayerofmagneticfieldwiththismagnitude 2. GENERALPHYSICSOFMAGNETICDRAPING isexpected(fromEq.1 andassumingl R)to beofthick- Previouswork(e.g.,BernikovandSemenov1980;Lyutikov ness ≪ 2006) has looked at the basic picture of magnetic draping 1 l= R (2) in a simplified way in some detail; we summarize some of 6α 2 theirkeyresultsaswellasournewinsightintothisproblem MA here. Intheseworks,theknownpotentialflowaroundasolid where A=u/υA istheAlfvénicMachnumberofthecore, M sphereistakenasaninput,andapurelykinematicmagnetic υA2 =B20/4πρ0 istheambientAlfvénspeed,andαisthecon- field, uniform and perpendicular to the direction of motion, stant of proportionality describing the maximum magnetic is added. The derivation of Bernikov and Semenov (1980) pressure in units of the incoming ram pressure, B2 /8π = max is clarified, and a novelset of useful approximationsfor the αρ u2. Wewillseethatα 2andfiducialvaluesforthesitu- 0 resultingfieldnearthesolidspherearegiven,inAppendixA. ationsconsideredherewil≈linvolve 2 3,sothatatypical MA≈ MagneticDrapinginClusters 3 FIG. 3.—Acartoonshowingtheexpectedgeometryofthedrapedmag- neticfield(blue)overtheobject(red). Seenintheplaneofthedirectionof theambientfield,panel(a),withthedirectionoftheambientfieldshown,a cleanbowwaveispresentedwithawell-definedopeningangle.Intheplane perpendiculartotheambientfield,panel(b),thefieldlinescansliparound theprojectile, andtheflowwouldclosebackinonastagnationlineonthe FIG. 4.—Wecomparethedrapedmagneticpressureandrampressurein othersideoftheobjectexceptforlargely-2dvorticalmotionsinducedbyin- theplanethatisparallel (perpendicular) totheinitial magnetic fieldinthe stabilitiesatthemagneticinterface. Thegeometryoftheflowintheregion top(bottom)panelswithalogarithmiccolorscale. Intheparallelplane,the indicatedbydashedboxdependsheavilyonthefinalshape,andthusinternal overpressure of the magnetic draping layer is only partly compensated by structure,ofthemovingobject. adeficit inrampressure andeventually isresponsible fordecelerating the coreduetomagnetictension.Intheperpendicularplane,therampressurein valueforlwillbeapproximatelyR/36.Evensuchaverythin thewakeofthecoreattainsmuchhighervaluesandthedrapinglayercloses towards thesymmetry axis unlike theparallel plane where anice opening layer can have important effects, both in terms of suppress- coneforms.Shownhereisazoom-inonasmallregionofourcomputational ing thermalconduction(Ettoriand Fabian 2000)and hydro- domain.Thisfigurehasbeensomewhatdegraded. ForaPDFversionofthis dynamicinstabilities(Dursi2007). manuscriptwithhigherresolutionfiguresandinteractive3dgraphics,please seehttp://www.cita.utoronto.ca/∼ljdursi/draping/. ItshouldbenotedherethatwhenweusetheAlfvénicMach number throughthisworkitshouldreallybeconsidered A in Fig. 3 and for our simulation in Fig. 4. This shows that M a dimensionless ratio of ram pressure to magnetic pressure thedrapinglayerbecomesdynamicallyimportantandfillsin (uMsed2Aw=hρe0nui2n/t(e2rPpBre0)t)in,goritaatsthaeralteiaosto,fsvoemloecictaieusti(oun/υshAo)ualsdthbee tphreesdseufirecisthoofwrasmanproevsesru-rper.esTshueresuamheaodftohfetmheagcnoereticthaantdleraadms velocitiesareorientedindifferentdirections;intheworkpre- toadecelerationoftheprojectile. sented here, the velocityof the drapedobjectwill always be completelyorthogonaltotheambientdirectionofpropagation 3. METHODOLOGY of Alfvén waves. Thusthere is an importantsense in which In order to understandthe full non-linearphysicsof mag- our projectiles are always (infinitely) super-Alfvénic, which neticdrapingaroundadynamicallyevolvingdenseprojectile isnotcapturedintheratio . A moving in a magnetized plasma we perform our analysis in M Sweepingupsuchamagneticfieldwilloccuronatimescale two steps. First, we analytically study the properties of the t/t √α(l/R) (√α )- 1,wheret =2R/uisthepro- c A A c flow of an ideally conducting plasma with a frozen-in mag- ∼ M ∼ M jectile’s own crossing time. This result means that, because neticfieldaroundaspheretoexplorethecharacteristicsofthe the magnetic layer is very thin, a strong field can be built magneticfield near the surface of the body. To this end, we up extremely quickly. Crucially, particularly for the prop- disregardanypossiblechangeintheflowpatternbymeansof agation of bubbles, the sweep-up time can be significantly theback-reactionofthemagneticfield. Whilethederivation smaller than a single crossing time; this is relevant because ofthisproblemcanbefoundinAppendixA, we summarize apurelyhydrodynamicbubblewillgenerallyself-disruptinto thekeyresultsinthissection.Inthesecondstep,wecompare a torus, or smaller fragments in a turbulent medium, in on thisanalyticalsolutiontoanMHDadaptivemeshrefinement ordera crossing time (Robinsonet al. 2004;Pavlovskiet al. simulationandexploreitquantitatively. 2007). This buildup of magnetic field will greatly effect the flow 3.1. Analyticalsolution in the direction of the ambient field lines, and the projectile will leave a magnetic bow wave behind it; by analogy with The potential flow solution for an incompressible flow aroundasphericalbodyreadsas othersimilarbowwaves,weexpectittohaveanopeningan- gle of tanθ υA/u. In the plane perpendicular to the am- R3 R3 bientmagne≈ticfield, however,the magneticfieldwillhavea υ=e - 1 ucosθ+e +1 usinθ, (3) r r3 θ 2r3 much less direct effect as field lines can simply slip around (cid:18) (cid:19) (cid:18) (cid:19) theprojectileandinstabilitiescanoccur. Inthepotentialflow whereRdenotestheradiusofthesphereanduisthespeedof simulation,theflowsmoothlyreattachesattherearofthepro- thecorethroughthequiescentambientfluid. Usingthissolu- jectile;however,inthiscase,vorticalmotionsgeneratedatthe tion,wesolvefortheresultingfrozen-inmagneticfieldwhile magneticcontact(wherethemagneticpressureandmagnetic neglectingits back-reactiononto the flow. For convenience, tensionforceismisalignedwiththedensitygradient)andby weshowheretheapproximatesolutionwhichisvalidnearthe instabilitiesatthemagneticinterface(whicharenotstabilized sphere, inthisplane)detachthewakefromtheobject,leavinglargely two-dimensionalvorticalmotionsalongthefieldlinesinthis 2 3s sinθ B = B sinφ, (4) plane.Theresultingexpectedgeometryisshowninacartoon r 3 0 R 1+cosθ r 4 Dursi&Pfrommer theprojectileischosensothatthematerialisinitiallyinpres- sureequilibrium. Theprojectileinitialvelocityistypicallychosentobe1/4, for a Mach numberinto the the ambientmedium of 0.32. Thesimulationinthetransversedirectionsrangefrom≈[- 4,4], and in the direction of motion of the projectile ranges from [0,28] for an aspect ratio of 2:7; in most of the simulations with projectiles larger than the fiducial R = 1, the domain size is increased proportionately. The initial magnetic field strength can be defined in terms of α =ρu2/P ; a typical 0 B0 valueusedin these simulationsis25/4, orP =1/100. Pe- B0 riodicboundaryconditionsareusedinthedirectionsperpen- diculartothedirectionofmotion,andzero-gradient‘outflow’ conditionsareusedin thezdirection. Experimentswithdif- ferenthorizontalboundaryconditionsproducednomajordif- ferencesinresults. Theprojectilefluidisinitiallytaggedwithapassivescalar, so that the material corresponding to the projectile can be tracedthroughoutthesimulation. FIG.5.—Diagramshowingthegeometryofthesimulationspresentedhere. Asphericalprojectilewithasmoothdensityprofileρmax(1+cos(πr/R))/2is 3.2.2. CodeChoice sentinthe+zdirectionwithaninitialvelocityυthroughanambientmedium with density ρ0 and a uniform magnetic field pointed in the +y direction. As can be seen from analytic arguments, and is shown Themagneticfieldstrength‘turnson’throughthedomainwithatanh-profile in some detail in §2, two features characterize the problem inthedirection ofmotionoftheprojectile, asindicated bytheshading of of magnetic draping: the formation of a narrow strongly- thebox;thisallowsustostartthecoreinanessentiallyfield-freeregionand smoothlyenterthemagnetizedregion.Periodicboundaryconditionsareused magnetized layer, and the relative simplicity of the dynam- inthedirectionsperpendiculartothedirectionofmotion,andzero-gradient ics, inthata potentialflowsolutionwithonlymagneticfield ‘outflow’conditionsareusedinthezdirection. kinematics captures much of the problem, lacking only the R magneticfieldback-reaction. B =B sinφ , (5) θ 0 3s Becauseoftheseparationofscales(arelativelylargeobject r moving through an ambient medium and a relatively small R B =B cosφ , (6) layerformingaroundit), thehighestresolutionrequirements φ 0 3s r wouldimposealargecostonthesimulationsiftheresolution where we introduceda radialcoordinatefrom the surface of hadtobeeverywhereuniform;indeed,itisonlyasmallpor- thesphere,namelys=r- R.Theseapproximatesolutionsuni- tion of the simulation domainwhich needsto be resolvedat formly describe the field near the sphere with respect to the thehighestlevel. Thisisespeciallytruesincethesimulations angle θ. As describedin AppendixA, the energydensity of we will need to perform are three-dimensional (as we will themagneticfieldforminginthewakebehindthebodyispre- seeinthenextsub-section,itisimpossibletodomeaningful dictedtodivergelogarithmically.Wepointoutthatthevalid- simulationsofmagneticdrapingintwodimensions). Thus,a ityofthepotentialflowsolutionheavilyreliesonthesmooth simulationcodewhichallowssomeadaptivityofmeshingis irrotationalfluidsolutionwherethemagneticback-reactionis extremelyhelpfulforapproachingthisproblem. negligible. We will see that these assumptions are naturally The relatively straightforward magnetic field dynamics violatedinthewake. meansthat,unlikeinproblemsof(forinstance)studyingthe details of MHD turbulence, we do not require high-orderfi- 3.2. Numericalsolution nitedifferencemethods;thisisparticularlytruebecauseofthe 3.2.1. Setup sharpnessofthethinlayersandthelargedensitygradientsin this problem. Instead, an MHD solverwhich can accurately Thesimulationspresentedinthispaperaresetupasshown dealwithsharpgradientsisvalued. in Fig. 5. For clarity of understanding the physical pic- AsaresultoftheimportanceofAMRforthesesimulations, ture, we consider only the magnetohydrodynamics(MHD); thecodewe’vechosentoperformthesesimulationswithisthe noexternal-orself-gravityisconsidered,andwedeferother FLASHcode(Fryxelletal.2000;Calderetal.2002). FLASH physics such as self-consistent inclusion of thermal conduc- is an adaptive-mesh general purpose astrophysical hydrody- tivitytofuturework. Inthisreport,wealsoconsideronlythe namics code which is publicly available1. The MHD solver magneticfield of the externalmedium, and assume that it is we use here is a dimensionally-split second-order accurate uniformoverthescalesofinteresthere. 8-waveGodunov-typesolverwhichisdescribedinmorede- In the code units we consider here, the ambient material tail in Powell et al. (1999). The smallness of spuriousmag- has a density of ρ = 1, and a gas pressure P = 1. The 0 neticmonopolesisensuredbyadiffusion-type‘div-B’clean (unmagnetized) fiducial projectile has a radius that we vary operation. This diffusive cleaning approach can be prob- in our runs between R = 0.5 and 2, and a maximum den- lematic near strong shocks, where diffusion cannot operate sity of ρ = 750. With the density profile chosen ρ(r) = max quicklyenough;however,nosuchshocksoccurinthesesimu- ρ (1+cos(πr/R))/2, the mean density of the projectile is max lations. FLASHhasbeenoftenusedforrelatedproblemssuch (1- 6/π2)ρ /2 0.2ρ . Both the ambient and projec- max max as hot and magnetized bubbles in the intercluster medium ≈ tile material are treated as ideal, perfectly conducting fluids (e.g.,Robinsonetal.2004;Pavlovskietal.2007;Heinzetal. withratioofspecificheatsγ=5/3,andsotheadiabaticsound speed in the ambientmedium is 5/3. The pressure inside 1http://flash.uchicago.edu p MagneticDrapinginClusters 5 2006;Roediger et al. 2006;Gardini 2006; Pope et al. 2005; field,sothedoubletransientofbeingshockedandexposedto Dalla Vecchia et al. 2004;Heinz et al. 2003;Brüggen2003; magneticfieldatthesametimewouldnotbeappropriatefor BrüggenandKaiser2002). thecasewewishtoconsider. Othermorerecentwork,whichinlargepartmotivatedthe 3.2.3. Parameters research presented here, considers squarely the case we are interestedin–magneticdrapinginclusterenvironmentsAsai Performingsimulationsofdrapingoveraprojectilewithan et al. (2004, 2005, 2007). In these works, the authors simi- explicit hydrodynamicscode (so that compressibility effects larly consider a spherical projectile moving through an am- will be included, for ease of comparison with later, super- bient magnetized medium. These simulations include more sonic, work)with a finite resolutionplacessomerestrictions physics (e.g. anisotropic thermal conduction, tangled mag- ontherangeofparameterspacewhichcanbeexplored. neticfields)thanweconsiderhere,asthegoaloftheseworks For simulating these cases with no leading shock, we re- wastofindoutifthedrapingofthemagneticfieldcouldsig- quire that the velocity of the projectile, u, be less than the nificantlysuppressthermalconductionintoacoldcore.These sound speed in the ambient fluid – but to take a reasonable simulationsshowthatmagneticdrapingispotentiallyimpor- number of timesteps (avoiding computational expense and tantfor suppressingthermalconductionacrossthe interface, spuriousdiffusion)requiresthattheprojectilevelocityremain of the order of the sound speed; thus u.c , or ρ u2 .γP, despitethefactthattheirdrapinglayersareunder-resolvedby s 0 atleastafactorof100(cf.Eqns.2and7). where ρ and P are the unperturbeddensity and pressure of 0 Inaslightlydifferentclusterenvironment,MHDeffectscan the ambient medium. For the hydrodynamicsof draping to alsostabilizethemorphologiesofbuoyantlyrisingradiobub- berealistic,themagneticpressureinthefluidmustbesignifi- blesfromAGNinclustercenters. Ruszkowskietal.(2007b) cantlylessthanthegaspressure,P P. Finally,resolution B0 ≪ carefully set up magnetically isolated bubbles and tangled requirementsforresolvingthethicknessofthemagneticlayer clustermagneticfieldsintheambientmediumasrequiredfor willputsomeconstraintonthethicknessofthemagneticlayer causally or temporally disconnectedgenerating processes of l>R/NfromEq.2,withNbeingthenumberofpointswhich these fields. They showed that magnetic draping of cluster resolvetheradiusoftheprojectile;typicallythesizeofthedo- magnetic fields possibly supported by helicity of magnetic main(ifatfullresolution)willbe8N 8N 28N. Thiscon- × × fields internal to the bubble can strongly alter the dynamics straint,expressedintermsoftherelevantpressures(rampres- sureandinitialmagneticpressure)P &3α(ρ u2)/N. Com- andsuppresshydrodynamicinstabilities;thusresemblingthe B0 0 observedmorphologicallyintactX-raycavities. bined,theseconstraintsgive Knowing that under some circumstances thermal conduc- 3αρ u2 tion can be suppressed by magnetic draping and hydrody- γP&ρ0u2≫PB0 & N0 (7) namicstabilitycanbeenhanced,itbecomesworthwhiletoun- derstandtheprocessofdrapingitselfinsomedetail,inwhich For a given u – which is more or less arbitrary, fixed to be casetheadditionalmicro-physics(atthisstage)makesunder- near the (arbitrary) sound speed – there is thus a relatively standingthisbasicprocessmore,notless, difficult. Thuswe narrowrangeofinitialmagneticpressuresintermsoftheram considerhereacasewithnoexplicitmicro-physicalconduc- pressureoftheambientmaterialontotheprojectilewhichcan tionandsimplifiedmagneticfieldgeometry.Withthischoice, be efficiently simulated. As we will see, α 2, and for the our simulationsremain scale free and can be applied to var- simulationspresentedhere,N 32- 64,mea≈ningwearecon- ious astrophysical environments. Additionally, our simula- ∼ strainedto studyroughlythatpartofparameterspace where tions can be directly compared to analytics and we are able ρ0u2/PB0 ≈1- 10. to carefully examine the process of the draping itself. Fur- ther,themoresimplifiedphysicsalongwiththeuseofAMR 3.2.4. ComparisontoPreviousNumericalWork allowsustorunsimulationsforsignificantlylongertimesas Havingdescribedthe setup of the simulationswe perform themorecomplicatedearliersimulations. forthiswork,itisworthcomparingthisto previousnumeri- 3.2.5. TwoDimensionalResults cal work. One body of work (e.g., Miniati et al. 1999;Gre- goriet al. 1999, 2000)considereda very similar problemin In two dimensions, the imposition of a symmetry greatly thecontextoftheISM,whereadensecloudmaybemoving limitspossiblemagneticfieldgeometries.Inanaxisymmetric through(for instance) the magneticfield associated with the geometry,theonlymeaningfuluniformfieldgeometryispar- Galactic spiral arms. In this context, a cloud of cold-phase allel to the axis of symmetry, which in this case would also ISM gas may very well not be globally self-gravitating (or bethedirectionofmotionoftheprojectile;inthissomewhat onlyweaklyso)andsoaroughly‘top-hat’densityfunctionis artificialcasemagneticfieldcouldsomewhatconstrainapro- usedtodescribethecloud. Inaddition,asappropriateto the jectile(orabubble;Robinsonetal.(2004))butdrapingcould ISM, supersonic motion was imposed on the ambient mate- notoccur. rial,withthecloudexperiencingahigh-Machnumbershock In planar symmetry, the field can have componentsout of and an ambient magnetic field at the same time; this might the plane, in the plane parallelto the directionof motion, or beappropriateforinstanceasa cold-phasecloudencounters intheplaneperpendiculartothedirectionofmotion. Acom- a hot-phase medium for the first time. In our work, we are ponent out of the plane will only have the dynamical effect consideringsubsonic motion initially, so that one significant of adding to an effective gas pressure ((e.g., Chandrasekhar differenceis the flow speed of the drapingmaterial, in addi- 1981)); the component along the direction of motion of the tion to the much flatter density profile used in the previous projectilecannotbedraped. work. Whilewedeferthesupersoniccasetofuturework,we Previouswork(e.g.,Asaietal.2005)hasexaminedthecase note that in our context, for example in the case of (for in- withtwodimensionalplanarsymmetrywithamagneticfield stance)acorefallingintoacluster,wewouldexpectthecore intheplaneofthesimulationandperpendiculartothedirec- toslowlyacceleratebeforeencounteringsignificantmagnetic tion of motion of the projectile. However, in this case, field 6 Dursi&Pfrommer Run R u Pb,0 ρu2/Pb,0 R/∆x A 1 1 1 6.25 64 2 4 100 B 1 1 1 6.25 64 4 100 C 2 1 1 6.25 32 4 100 D 1 1 1 3.125 64 4 50 E 1 1 1 1.5625 32 2 8 100 F 1 1 1 6.25 32 2 4 100 G 1 1 1 25 32 2 2 100 H 2 1 1 15.625 128 4 250 TABLE1 DETAILSOF3-DIMENSIONALSIMULATIONSRUNFORTHISWORK. SIMULATIONSWERERUNWITHANAMBIENTDENSITYANDPRESSURE OF1INCODEUNITS,ANDγ=5/3.SIMULATIONSWERERUNUNTIL 2 MAXIMUMMAGNETICFIELDONSTAGNATIONLINEWAS 16 1 3 APPROXIMATELYCONSTANT,TYPICALLY40-80TIMEUNITS. noitisoP telluB111024 8 6 0 20 40 60 80 100 120 140 Time FIG. 6.—Theaboveisaplotoftheprojectile position(calculated here bythemaximumheightatwhichthereissignificantprojectilematerialatany giventime)overtimeinatwodimensionaldrapingsimulation.Atabouttime 75,theprojectileisactuallybouncedbackundertheextremelystrongmag- netictensionwhichintwodimensionsmustgrowaheadoftheprojectile.The toppanelshowsB2inacloseupofthesimulationdomainatthreerepresen- tativetimesduringthebounce,withawhitecontourindicatingtheposition oftheoriginalprojectilematerial. Thisfigurehasbeensomewhatdegraded. ForaPDFversionofthismanuscriptwithhigherresolutionfiguresandinter- active3dgraphics,pleaseseehttp://www.cita.utoronto.ca/∼ljdursi/draping/. linescannotcannotsliparoundtheprojectile,andsoasmore andmoremagneticfieldgetssweptupbytheprojectile,mag- FIG.7.—Aplotofthemaximummagneticfieldstrengthalongthestagna- netic tension grows monotonicallyand linearly ahead of the tionlineforrunswiththreedifferentprojectilesizesandtwodifferentresolu- projectileuntiltheforcesbecomescomparablenotonlytothe tionsacrosstheprojectile,allwiththesamevelocityintotheambientmedium ram pressure seen by the projectile of the ambient medium, andthesameambientmagneticfield. Plottedisthemaximummagnitudeof themagneticfieldalongthestagnationlinevsthetime(scaledtothecrossing buttotherampressureoftheprojectileasseenbytheambient timeoftheprojectile).Theprojectileinitiallysitsinanunmagnetizedregion. medium. Atthispoint,theprojectiletrajectoryisreversed.A Themaximummagnetic fieldstrengthalongthestagnation lineisasensi- figuredescribingthisisshowninFig.6,wherethemagnetic tivemeasureofwhetherthestructureofthedrapedmagneticfieldisbeing tensionforcesareseentocompresstheprojectile(withmean resolved;weseehereclearevidencethatwiththeresolutionusedinthissim- ulationthedrapedlayerisbeingadequatelyresolved. Inthelow-resolution density 150 times that of the magnetizedmedium) before ≈ R=1simulation, whichalsowasruninasomewhatsmallerbox, towards repellingit. theendoftherunthedrapinglayerbeginstoleavethetopofthesimulation This outcome is hinted at in Fig. 4 of Asai et al. (2005), domain,leadingtothesuddenrapiddropinmagneticfieldstrength. wherein2dthemagneticenergyincreaseslinearlyandwith- outbound, while the 3d modelsreach a maximummagnetic The magnetic layer in Run G is under-resolved; while the energy. valueofR/∆xisthesameasotherruns,thevelocityishigher, sothatbyEq.2 thelayeristhinner. We includethisrunbe- 3.2.6. ThreeDimensionalSimulations cause it demonstrates certain robustness of results; although A listing of the eight main runs done for this work are thelayerstructureisnotadequatelyresolvedatthestagnation shownin Table1, andthebasic setupfollowsthediscussion point,otherglobalpropertiesofthemagneticlayer(geometry earlier.Theparametersvariedarethesizeoftheprojectile,its anddynamicaleffects)otherwiseremainrobust. velocity,andthestrengthoftheambientmagneticfield.Other Indeed,oneshouldbecarefulaboutwhatonemeansby‘re- runs(the equivalentof runB butwith halfthe resolution, or solved’. Thisdiscussionshouldnotbetakentomeanthatthe with the same resolution but differing boundary conditions) othersimulationsareinallrespectsresolved.Inparticular,as were runto confirmthat the resultsdid notchange; they are we will see in §4 and§5, and assuggestedby Dursi(2007), notlistedhere. theflowaroundthebubbleinthexzplane(e.g.,transverseto Toshowthattheserunswereproducingresultsindependent theinitialmagneticfield)isunstabletoKelvin-Helmholtzand ofresolution,themaximummagneticfieldstrengthalongthe Rayleigh-Taylor instabilities, not stabilized by the presence stagnationlineforalltherunswiththesameP andρu2 are of magnetic field. Since we have not prescribed any small- B0 0 plottedversustimeinFig.7. Themaximumfieldstrengthis scale dissipative physicsin these simulations, these instabil- sensitivetotheresolutionofthemagneticfieldlayer,butwe ities will never properly converge with increased resolution seeherethatvaryingtheresolutionbyafactoroftwodoesnot (e.g., Calder et al. 2002) as new unstable scales are added. effecttheresults,asthefieldlayerisadequatelyresolved(but Thiscanonlybecorrectedbyaddingsmall-scalephysics,e.g. only marginallyin the case of R/∆x=32). We also see, as thermaldiffusion;thisis leftforfuturework, as the relevant we’dexpectfromthediscussionsin§2,thatthefieldstrength microphysics is itself a current research problem (Lyutikov inthelayerdoesnotdependonthesizeofthecore. 2007;Schekochihinet al. 2007). While the rate of develop- MagneticDrapinginClusters 7 mentoftheinstabilitiesandtheirpropertiesisveryimportant icantly larger than the initial radius of the projectile; in this forlong-termmixingofthematerialofthemovingobjectwith example,theexpansionisarelativelymodest15%. thatofthesurroundingmedium,werestrictourselveshereto studyingthedevelopmentofthemagneticlayeranditsglobal properties. A notehereonourAMRisinorder. InTable1, we show thefinestresolutionachievedinthesimulation. FLASHplaces resolution elements based on a second-derivativerefinement criterion, in an attempt (appropriate for a second-order ac- curate code) to reduce error; see Fryxell et al. (2000) for more details. Here we refined based on pressure, density, FIG. 8.— Shown is, left, the magnetic field along the stagnation line and composition (e.g., fraction of projectile material); early in the simulation (’+’) and a fitted theory prediction, with the two fitting parameters being theposition ofthe peakandaradius giving acharacter- experimentswithalsorefiningonmagneticfieldcomponents istic fall off of the field strength. On the right are cut-planes along and showedlittleornodifference. Theresultisthattheentirere- acrosstheinitial magnetic fieldofthedensityoftheprojectile, withacir- gion immediately surroundingthe projectile is fully refined, cle of radius and position given by the fit to the magnetic field structure, andthewakeregionisnearlyfullyrefined.However,because left. Theradiusgivenbythefitcorrespondswiththeradiusofcurvatureat theworkingsurfaceoftheprojectile. ResultsaretakenfromrunBattime no interesting flow features are occurringin most of the do- t=38.75;resultsfromothersimulationsandothertimesgivesimilarlygood mainatanygiventime, thesavingsfromusingAMRcanbe fits. This figure has been somewhat degraded. ForaPDFversion ofthis substantial; for runG, forinstance, the verysimple, laminar manuscriptwithhigherresolutionfiguresandinteractive3dgraphics,please initial conditions require only 9.7 105 zones; by time 40, seehttp://www.cita.utoronto.ca/∼ljdursi/draping/. this has increased to 8.1 106 z×ones, and by time 75, an approximately-steady≈9.1 ×106 zonesarebeingused; thisis to be compared to the 46×9.8 106 zones, or a factor of 51 4.2. Comparisonofthevelocityfield × greater,thatwouldberequiredtoresolvetheentiredomainat To compare the potential flow calculations, done in the thissamefinestresolution. frameofthesphericalbody,withthoseofthenumericalsimu- A final feature worth noting in our runs is that for run E, lations,wetransformthenumericalsimulationsintotheframe M2A=1/2(ρu2/PB0)≈0.78<1;thatis,thisrunhasthepro- oftheprojectile.Becausetheprojectileslowsdownovertime, jectilemovingsub-Alfvénically,ifonlymarginally.However, we do not use the initial velocity u for this transformation, 0 becauseofthefieldgeometry,wewillseethatthismakeses- butmeasuretheinstantaneousmass-weightedvelocityofthe sentiallynodifferenceforthedraping.Thisissimplybecause projectile,bymakinguseofthefactthatwearetrackingthe theAlfvénspeedinthedirectionofmotionoftheprojectileis fluid that initially resided in the projectile by use of an ad- zero–nocomponentofthemagneticfieldpointsinthatdirec- vectedpassivescalar, a. Thuswe measuretheinstantaneous tion.Whiletheexactimpositionofthisconditioninourinitial conditionsissomewhatartificialinthiscase,itisalwaystrue that it is only the component of the magnetic field that lies transversetothedirectionofmotionthatwillbedraped. 4. COMPARISONOFTHEORYANDSIMULATIONS 4.1. Magneticfieldalongthestagnationline The first comparison we make is to the one-dimensional predictions made along the stagnation line, for instance in Lyutikov(2006),whereaveryspecificpredictionismadefor therampup,withaparticularfunctionalform,ofthemagnetic fieldstrengthgiveninEq.1,anditisnotnecessarilyclearthat suchapredictionwillholdwhentheprojectilebeginstodevi- atesignificantlyfromspherical. To make this comparison, we extract the magnetic field strength along the stagnation line for an output, and fit it to theequation B B 1 | | = 0 (8) ρ ρ0 1- R 3 r (z- z0) FIG.9.—Comparisonofthevelocityfieldoftheanalyticalsolutioninthe (cid:16) (cid:17) kinematic approximation (top panels) with our numerical simulation (bot- where B0, ρ0 are known, and the fit is for the parameters R, tom panels) in the plane of the initial magnetic field. While the velocity which would correspondto the radius of the sphere, and z , fieldsresembleeachotherverywellintheupperhalf-space,therearedistinct 0 whichwouldbethepositionofthecentreofthesphere. Re- differences inthe lowerhalf-space. Theseareduetothenon-linear back- reaction of the dynamically important magnetic field in the draping layer sultsforatypicaloutputareshowninFig.8. Notonlydothe ontheMHDflowthatgenerates vorticity inthewakeoftheprojectile (cf. fits well representthe behaviourmagnetic field strength, but §5.4). Shownhereandinthenext figures isazoom-in onasmallregion they also suggest a physical interpretationfor the functional ofourcomputationaldomainthatextendsupto112lengthunitsandisfour formsinterpretationevenwhentheprojectilebecomessignif- timeslargerinxandydirection. Notethatwesymmetrizedthecolormap icantly non-spherical; R becomes the radius of curvature of of the υx-component in order not to be dominated by one slightly larger eddy. Thisfigurehasbeensomewhatdegraded. ForaPDFversionofthis theworkingsurfaceoftheprojectileatthestagnationline. As manuscriptwithhigherresolutionfiguresandinteractive3dgraphics,please the projectile becomesmoredistorted, R can becomesignif- seehttp://www.cita.utoronto.ca/∼ljdursi/draping/. 8 Dursi&Pfrommer FIG. 10.— Comparison of the ram pressure in our numerical simula- tion (left panel) with the analytical solution in the kinematic approxima- tion (right panel). Aheadofthe projectile, the rampressure resembles an exact potential flow behavior up to the draping layer which can be seen FIG. 11.—Comparisonofthemagneticfieldinournumericalsimulation as a black layer around the projectile with a deficit of hydrodynamical (bottompanels)withtheTaylorexpansionoftheanalytical solutioninthe pressure. Non-linear magnetic back-reaction of the draping layer causes kinematicapproximationthatstrictlyappliesonlynearthesphere(toppan- the flow to depart from the potential flow solution and to develop vortic- els).WeshowtheCartesiancomponents(lefttoright,x,y,z)ofthemagnetic ity. This figure has been somewhat degraded. For a PDF version of this fieldintheplanethatisparalleltotheinitialmagneticfield. Thereisanice manuscriptwithhigherresolutionfiguresandinteractive3dgraphics,please agreement between both solutions intheupper half-space, whilethere are seehttp://www.cita.utoronto.ca/∼ljdursi/draping/. againdistinctdifferencesinthelowerhalf-space.Themagneticshouldersbe- hindtheprojectilecanbeidentifiedthatpreventsthedrapinglayerfromcon- velocityoftheprojectileas tractingtowardsthesymmetryaxis.Inaddition,MHDturbulencestartstode- velopinthewakeoftheprojectile.Thisfigurehasbeensomewhatdegraded. u= hρaυzi. (9) FacotrivaeP3DdFgvraeprshiiocns,opflethaissemseaenuhsttcpr:i/p/twwwiwth.chiitga.huetrorroesnotolu.ctiao/n∼filjgduurressi/danradpiinntge/r.- ρa h i ThetoppanelsofFig.9showtheanalyticalsolutionofthe velocityfield aroundthe sphericalbodywith radiusR in the kinematic approximation. For convenience and to simplify the comparisonto the magnetic field visualization, we show theCartesiancomponentsofthevelocityfield. Atinfinity,the fluid is characterized by a uniform velocity υ =- e u. The z quadrupolar flow structure results from the fluid decelerat- ing towards the stagnation line, the successive acceleration around the sphere until θ =π/2 and mirroring this behavior in the lower half-plane by symmetry. The bottom panels of Fig.9showthenumericalsolutionofthevelocityfieldaround an initially spherical projectile that deformed in response to thenon-linearevolutionofthemagnetizedplasma.Thewhite line reflects the 0.9 contour of the ‘projectile fluid’ and cor- responds to an iso-density contour.2 The quadrupolar flow structureintheupperhalf-planeresemblesnicelytheanalytic potential flow solution. As the flow approaches the projec- tile and surroundsit, there are importantdifferencesvisible. In the analytical solution, the flow accelerates for θ π/2 ≤ and decelerates for larger angles θ. In the numerical solu- tion, the magneticdrapinglayer is stationarywith respectto FIG.12.—Sameaspreviousfigure,butinthetheplanethatisperpendicular totheinitialmagnetic field. Shownis,lefttoright, thex,y,zcomponents. the projectile. This causes the flow almost comes to rest in Asexpected fromouranalytic solutions, thedrapinglayerformsbypiling the magnetic draping layer. The back-reaction of the mag- upmagneticfieldlinesaheadoftheprojectile. Theirregularmagneticfield neticdrapinglayerontheflowcastsa‘shadow’onthewake inthewakeisgeneratedbythevorticitythatisabsentbydefinitioninour oftheprojectile. Itpreventstheflowtoconvergetowardsthe potentialflowsolution.Thisfigurehasbeensomewhatdegraded. ForaPDF versionofthismanuscriptwithhigherresolutionfiguresandinteractive 3d symmetryaxisandsuppressesthedecelerationoftheflow.In- graphics,pleaseseehttp://www.cita.utoronto.ca/∼ljdursi/draping/. stead,vorticityisgeneratedatthedrapinglayerwhichwillbe studiedindetailin§5.4. Thecomparisonoftherampressure 4.3. Comparisonofthemagneticfield inFig.10underpinsthisargument. It is instructive to compare the analytic solution of the 2Notethattheapparentgridstructureseenintheupperpartofoursimu- frozen-inmagneticfieldinthekinematicapproximationtothe latedυx-componentisanartifactofourplottingroutineaswellassmallgrid numericalsolutionintheplanesthatareparallelandperpen- noise. Theinterpolationschemeoftheplottingroutinefalselyinterpolatesa diculartotheinitialmagneticfield. Wecomparetheindivid- smoothvelocitygradientwithanentireAMRblockwhileitactuallydrops quicklytothevelocityvalueatinfinity. ualCartesiancomponentsofthefield(Fig.11and12)aswell MagneticDrapinginClusters 9 FIG. 14.— A plot for the 3d runs presented here showing the mag- neticpressureonthestagnationlineonceasteadyvaluehadbeenachieved for this quantity as a function of the mean ram pressure (ρhui2) as seen by the projectile. Omitted is run G, for which the magnetic layer was under-resolved andthus themaximum magnetic field strength inthelayer falls much lower; however, as we will see, even this under-resolution does not strongly effect other global properties of the magnetic drape. FIG. 13.—Comparisonofthemagneticenergydensityinournumerical This figure has been somewhat degraded. For a PDF version of this simulation(leftpanel)withtheanalyticalsolutioninthekinematicapproxi- manuscriptwithhigherresolutionfiguresandinteractive3dgraphics,please mation(rightpanel). Thetop(bottom)panelsshowtheplanethatisparallel seehttp://www.cita.utoronto.ca/∼ljdursi/draping/. (perpendicular)totheinitialmagneticfield.Intheanalyticalsolutionthereis ing cone causes the stationary flow not to convergetowards anarrowmagneticlayerdrapedaroundthesphericalbody,whileinoursim- ulationsthedrapinglayerpeelsoffbehindtheprojectileduetovorticitygen- thesymmetryaxisandprotectstheregioninthewakeagainst eration. Thegeometryofthemagneticdrapinglayerintheupperhalf-plane theincreaseofthemagneticenergywithoutbounds. Thenu- is very similar inboth planes suggesting there an approximately spherical merical solution can qualitatively be obtained by remapping symmetry. Inthewakeoftheprojectile, thedrapinglayerformsacharac- the analytic solution for θ >π/2 onto the coordinate along teristicopeninganglewhilethefieldlinescanswipearoundtheprojectilein theperpendicularplaneandthedrapinglayerclosestowardsthesymmetry themagneticdrapingcone. Intheperpendicularplanetothe axis. Thisfigurehasbeensomewhatdegraded. ForaPDFversionofthis initialmagneticfield,thereisevenbetteragreementbetween manuscriptwithhigherresolutionfiguresandinteractive3dgraphics,please the analytic and the numerical solution. The magnetic field seehttp://www.cita.utoronto.ca/∼ljdursi/draping/. in thatplane lies primarilyin its initialy-direction. This be- haviour can easily be understood in terms of the field lines asthemagneticenergydensityinFig.13. Notethatweonly sweeping aroundthe sphere in a laminar flow. Numerically, showaTaylorexpansionofthehighlycomplexexactsolution wesimulatetheresponseofthegeometryoftheprojectileto asderivedinAppendixA. Strictly,thissolutionappliesonly near the sphere with an accuracy to ((r- R)3/2) as well as thehydrodynamics. Vorticesinthe wake deformthe projec- O tile leadingto a cap-geometryand a mushroomshape ofthe for flow lines that have small impact parameters initially at y-componentofthemagneticfield. Thisimpliesthattheflow infinity.Usingadifferentexpansion,weverifiedthatthegen- lines detach from the dense material of the projectile gener- eralsolutionhastheappropriatebehaviorofthehomogeneous atingfurthermorevorticityandMHDturbulenceinthewake. magneticfieldatinfinityintheupperhalf-spacepointingto- TheturbulentfieldmixestheCartesiancomponentswhichcan wardsthepositivey-coordinateaxis,i.e.rightwardsinFig.11. benicelyseenintheFig.12. Themagneticpressuresumma- Asexpected,they-componentofthemagneticfieldincreases rizesourresultsnicelyshowingthedrapingconeintheparal- as we approach the sphere since the field lines are moving lelplaneandthe mushroomshapedmagneticenergydensity closertoeachother. Intheimmediatevicinityofthesphere, intheplaneperpendicularto that(cf.Fig.10). Note thatwe theBfieldattainsadipolarz-componentasthefieldlinesare choosethe same color scale as derivedfromthe simulations carried around the sphere with the fluid and causes them to whichleadstoasaturationofthemagneticenergydensityin bendinreactiontotherampressureofthesphere.Aspointed the kinematic approximation at the contact of the spherical out by Bernikov and Semenov (1980) the magnetic lines of bodyandontheaxisinthewake. forcethatendatthestagnationpointarestronglyelongatedas theswipearoundthesphereparalleltothelineofflowreach- 5. CHARACTERISTICSOFMAGNETICDRAPING ingfromthe stagnationpointinto the rear. Thisleads to the 5.1. Fieldstrengthindrapinglayer unphysicalincreaseofthemagneticfieldasitapproachesthe lineofsymmetryinthewakeandeventuallytoalogarithmic The kinematic solution predicts the magnetic pressure di- divergenceofthemagneticenergydensitythere. verges at the stagnation point, which is clearly unphysical. In the upper half-plane, the analytic solution matches the Fromour discussionsin §2 and §4, we expectthat the mag- numericaloneclosely. Interestingly,intheregionbehindthe neticpressureinthedrapinglayershouldbeonorderρu2,at deformed projectile, a magnetic draping cone develops that whichpointthemagneticback-reactionbeginstostronglyef- stemsfromthedynamicallyimportantdrapinglayerthathas fect the flow; to first order there is no dependence on other swipedaroundthesphereandadvecteddownstreamthepro- parameters, such as backgroundmagnetic field. One would jectile. Inaddition,themagneticpressureinthewakeofthe expect, too, from looking at figures such as Fig. 4 that the projectile is also amplified by a moderate factor of roughly maximummagneticpressureshouldexceedtherampressure five(cf.Fig.13).Wewillshowfurtherdown,thatthisfieldis by some factor, as the magnetic pressure distribution at the generatedtogetherwith vorticityin the drapinglayer. In the headofthedrapeisresponsibleforredirectingtheflowinthe parallelplanetotheinitialmagneticfield,themagneticdrap- planeofthedraping. 10 Dursi&Pfrommer FIG.17.—Plotofmagneticenergydensityinthey,zplaneforsimulations withR=0.5and,lefttoright,u=0.125,0.25,0.5;shownwithblacklinesare thefittedopeninganglesofthemagneticdrapinglayer,omittingtheregion includingthematerialfromtheprojectile.Thefitslopes(e.g.,tanθ)are1.24, 0.515, 0.261, and those predicted byυA0/hui are 1.13, 0.566, 0.283; this agreementiswithin10%. Thisfigurehasbeensomewhatdegraded. Fora PDFversionofthismanuscriptwithhigherresolutionfiguresandinteractive FIG. 15.— As in Fig. 1, but for Run E; that is, with the projec- 3dgraphics,pleaseseehttp://www.cita.utoronto.ca/∼ljdursi/draping/. tile’s velocity reduced by a factor of one-half (so that u=0.125 in code units). A PDF version of this manuscript with an interactive 3d ver- sion of this figure, following Barnes and Fluke (2007), is available at http://www.cita.utoronto.ca/∼ljdursi/draping/. 1.2 1 tan θ0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1 1.2 FIG.18.—Aplotforthe3drunspresentedhereshowingthetangentofthe FIG. 16.— As in Fig. 1, but for run G; that is, with the pro- fitopeninganglesofthedrapeintheyzplaneversusvA/hui. Datafromall jectile’s velocity increased by a factor of two (so that u = 0.5 in code runsareshown. units). A PDF version of this manuscript with an interactive 3d ver- sion of this figure, following Barnes and Fluke (2007), is available at rectionofthesimulations,and–omittingtheregionsaboveor http://www.cita.utoronto.ca/∼ljdursi/draping/. neartheprojectileitself,andtheregionbelowwhichthedrape We can test this by plotting, for all our runs, the steady becomesweakerthantransientfeaturesinthewake–linesare maximummagnetic pressure at the stagnation line (the field fit, and the slope givesthe (half-)openingangle. The results quantitythatiseasiesttoconsistentlycharacterize)versusthe ofthefittingprocedureareshownforthesamethreesimula- meanrampressureseenbytheprojectile,ρ u 2,where u is tions in Fig. 17, and a scatter plot for all are runs are given themeanoftheprojectilevelocity(calculatehdiasinbyEqhni.9) in Fig. 18. The scatter forthis quantity,and agreementwith duringtherun,andρistheambientdensity.Theplotisshown theprediction,issomewhatworsethanfortheotherquantities inFig.14andverifiesourexpectation. weconsider,possiblybecausethelarge-scalegeometryofthe drapingismoresensitivetotheboundariesandthefinitesize 5.2. OpeningAngle ofthedomainthanother,morelocal,quantities. The magnetic bow wave behind the projectile is expected 5.3. Decelerationbymagnetictension topropagatetransverselyawayfromtheprojectileatυ along A the field lines, and of course to fall behind the projectile at In the scatter plots presented above, we use the mean ve- velocityu. Thissuggestsanaturalopeningangleintheplane locity u oftheprojectileovertime,becausethereisamea- h i along the magneticfield, tanθ=υ /u. Thatthe direction of surable deceleration of the projectile. An example, for run A thescalingiscorrectcanbedeterminedbyqualitativeinspec- F, is shown in Fig. 19. Before the projectile encounters the tion of a sequence of 3d renderingsof simulationoutputsas magneticfieldatz=10,hydrodynamicdrag–inprincipleei- thevelocitychanges;e.g.,Figs.15,1,16foru=1/8,1/4,1/2 ther(numerical)viscousdragorthethedragforcecausedby andυ fixedat0.1414. the creation of a turbulentwake – is all thatcan play a role, A Althoughthefieldlinesarestretchedduringthedraping,it andforthesimulationspresentedhere,itisthesecondwhich is the initialυ that is relevant, as the stretching of the field dominates. Thewell-knownformforthedragonasphereis A linesinthez-directiondonoteffectthepropagationspeedin F =1/2ρu2AC ,orintermsofadeceleration, D D the y-direction. For instance, consider a zˆ-velocity shear in gy,ivυes=u(s0B,˙0,=y/τ),(wυithBB)==((00,,B00,,B0)/.τT)h,esointdhuatcttihoenmeqauganteiotinc u˙D=- 83 ρρpu2RCD (10) 0 h i υfielBˆd iyˆs=o(nlBy/∇c√h×a4nπgρe)d×(Binyˆt)h/eBzˆ-d=irBec/ti√on4;πtρh=usυυAy =υA·yˆ= wbehtewreeeCnD0i.s07th-e0d.r5a,gwciotheffi0c.5ienfot,reaxptuerrbimuleennttalwlyakken,oρwnitsothbee A · | | · | | y A,0 p One can quantifythe agreementwith thisscaling by mea- density and A is the cross-sectional area of the projectile in suring the opening angle for the drapes in our simulations. thedirectionofmotion. Themaximaofmagneticfieldoneithersideofthestagnation However,oncethemagnetizedregionisreachedandamag- lineinthe y- zplanearefoundandtabulatedalongthezdi- netic layer built up, then another force acts on the projectile

See more

The list of books you might like

Upgrade Premium
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.
DMCA & Copyright: Dear all, most of the website is community built, users are uploading hundred of books everyday, which makes really hard for us to identify copyrighted material, please contact us if you want any material removed.
Copyright @ PDFdrive.to, 2022 | We love our users