Table Of ContentDRAFTVERSIONFEBRUARY2,2008
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
