Relativistic AGN jets I. The delicate interplay between jet structure, cocoon morphology

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Relativistic AGN jets I. The delicate interplay between jet structure, cocoon
morphology and jet-head propagation
Walg, S.; Achterberg, A.; Markoff, S.; Keppens, R.; Meliani, Z.
DOI
10.1093/mnras/stt823
Publication date
2013
Document Version
Final published version
Published in
Monthly Notices of the Royal Astronomical Society
Link to publication
Citation for published version (APA):
Walg, S., Achterberg, A., Markoff, S., Keppens, R., & Meliani, Z. (2013). Relativistic AGN jets
I. The delicate interplay between jet structure, cocoon morphology and jet-head propagation.
Monthly Notices of the Royal Astronomical Society, 433(2), 1453-1478.
https://doi.org/10.1093/mnras/stt823
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Relativistic AGN jets I. The delicate interplay between jet structure,
cocoon morphology and jet-head propagation
S. Walg,1,2‹ A. Achterberg,1 S. Markoff,2 R. Keppens3 and Z. Meliani4
1AstronomicalInstitute,RadboudUniversityNijmegen,Heyendaalseweg135,6525AJNijmegen,theNetherlands
2AstronomicalInstitute‘AntonPannekoek’,UniversityofAmsterdam,SciencePark904,1098XHAmsterdam,theNetherlands
3CentreformathematicalPlasmaAstrophysics,DepartmentofMathematics,KULeuven,Celestijnenlaan200B,3001Heverlee,Belgium
4LUTH,ObservatoiredeParis,5placeJulesJanssen,92195MeudonCedex,France
Accepted2013May8.Received2013May7;inoriginalform2012December13
D
ABSTRACT ow
n
Astrophysical jets reveal strong signs of radial structure. They suggest that the inner region lo
a
d
ofthejet,thejetspine,consistsofalow-density,fast-movinggas,whiletheouterregionof e
d
the jet consists of a more dense and slower moving gas, called the jet sheath. Moreover, if fro
m
jets carry angular momentum, the resultant centrifugal forces lead to a radial stratification. h
Citsurarcetnutaolbpsreorfivlaet.ioWnseaprerensoenttabthlereteofaucltliyveregsaollavcetitchenuracdleiailjesttrumcotudreel,ssionli2t.t5leDisokfnwowhinchabtowuot ttp://m
n
havebeengivenaradialstructure.Thefirstmodelisahomogeneousjet,theonlymodelthat ras
.o
doesnotcarryangularmomentum;thesecondmodelisaspine–sheathjetwithanisothermal xfo
equation of state; and the third jet model is a (piecewise) isochoric spine–sheath jet, with rdjo
u
constantbutdifferentdensitiesforjetspineandjetsheath.Inthispaper,welookattheeffects rn
a
ofradialstratificationonjetintegrity,mixingbetweenthedifferentjetcomponentsandglobal ls
.o
morphologyofthejet-headandsurroundingcocoon.Weconsidersteadyjetsthathavebeen rg
a/
active for 23Myr. All jets have developed the same number of strong internal shocks along t U
n
theirjetaxisatthefinaltimeofsimulation.Theseshocksarisewhenvorticesarebeingshed iv
e
bythejet-head.Wefindthatallthreejetsmaintaintheirstabilityallthewayuptothejet-head. rs
ite
Theisothermaljetmaintainspartofitsstructuralintegrityatthejet-headwherethedistinction it v
between jetspine and jetsheath materialcan stillbe made. Inthis case, mixing between jet an
A
spine and jet sheath within the jet is fairly inefficient. The isochoric jet, on the other hand, m
s
loses its structural jet integrity fairly quickly after the jet is injected. At its jet-head, little terd
a
structure is maintained and the central part of the jet predominantly consists of jet sheath m
o
material.Inthiscase,jetspineandjetsheathmaterialmixefficientlywithinthejet.Wefind n
J
u
thatthepropagationspeedforallthreemodelsislessthanexpectedfromsimpletheoretical ly
1
predictions.Weproposethisisduetoanenlargedcross-sectionofthejetwhichimpactswith 0
, 2
theambientmedium.Weshowthatinthesemodels,theeffectivesurfaceareais16timesas 0
1
4
largeinthecaseofthehomogeneousjet,30timesaslargeinthecaseoftheisochoricjetand
canbeupto40timesaslargeinthecaseoftheisothermaljet.
Keywords: hydrodynamics–relativisticprocesses–intergalacticmedium–galaxies:jets.
associatedwithactivegalacticnuclei(AGNs),wheregasisaccreted
1 INTRODUCTION
ontoasupermassiveblackhole(SMBH)of106–1010M(cid:2).Inthis
Astrophysicaljetsarehighlycollimatedoutflowsofplasma,gen- paper,wewillonlyfocusonjetsarisingfromSMBHsystems.
erated near a compact object from its accretion disc in accreting Observations show strong signs that astrophysical jets have a
systems.Jetsonparsec(pc)scalesareknowntoarisefromastellar transverse(radial)structure(seeforinstanceSol,Pelletier&Asseo
mass compact object in close binaries, such as a white dwarf, a 1989;Girolettietal.2004;Ghisellini,Tavecchio&Chiaberge2005;
neutronstarorblackhole(BH),whilejetsonkpc–Mpcscalesare Go´mez et al. 2008). It has been suggested that most jets consist
oftwodifferentregions,namelyalow-density,fast-movinginner
region called the jet spine, thought to emerge from a region very
(cid:2)E-mail:[email protected] closetotheBH,andadenserandslowermovingouterregioncalled
(cid:3)C 2013TheAuthors
PublishedbyOxfordUniversityPressonbehalfoftheRoyalAstronomicalSociety
1454 S.Walgetal.
the jet sheath, thought to emerge from the inner accretion disc. method,numericalschemesandtheparameterregime.InSection4,
NumericalsimulationsofaccretionnearBHsalsoshowsucharadial wepresenttheresultsofthedifferentsimulations.Discussionand
structureemerging(e.g.Hardee,Mizuno&Nishikawa2007;Porth conclusionsarefoundinSections5and6.
&Fendt2010).However,theformation,propertiesandevolution
ofjetspineandjetsheatharenotwellunderstood.Infact,whether
theobservedradialstructureisactuallytheresultofanunderlying 2 THEORETICAL BACKGROUND
spine–sheathjetstructurehasnotbeenverifiedbyobservations.
Large-scalejetsareusuallydividedintotwocategories,namely 2.1 Motivationforthisresearch
FRIandFRIIjets(Fanaroff&Riley1974).Thedistinctionisbased
Anumberofnumericalsimulationshavebeenconductedthatstudy
onjet/lobeluminosity(at178MHz)andradiomorphology.FRI
the interaction of (relativistic) jets with their ambient medium.
jets have low luminosity (<1041ergs−1) and diffusive jets/radio
Thesestudiesincludethepurehydrodynamical(HD)case,aswell
lobeswithnoprominenthotspots.FRIIjetshaveahighluminos-
asthemagnetohydrodynamical(MHD)case,withthejetmodelsset
ity (>1041ergs−1), are generally thought to be more stable and upin2D,2.5Dor3D.SeeforexampleMartietal.(1997),Rosen
collimatedanddohaveprominenthotspots.
et al. (1999), Aloy et al. (2000), Meliani et al. (2008), Mignone
Supersonicandunderdense1jetsinflateahotandoverpressured
et al. (2010), Perucho et al. (2011), Bosch-Ramon et al. (2012),
cocoon through which shocked jet and ambient material flows.
Gilkis & Soker (2012), Prokhorov et al. (2012), Refaelovich &
These jets deposit a large amount of energy into the surround- D
Soker (2012), Soker et al. (2013) and Wagner et al. (2012). The o
ingmediumandwillaltertheirdirectenvironmentdrastically.This w
dependenceoftheenergyfeedbackfromahomogeneousjettothe n
phenomenontiesincloselytothestudyofAGNfeedback,theques- lo
ambientmediumonthefiniteopeningangleofajethasbeenstudied a
d
tionofhowpartoftheenergyproducedbyAGNsisputbackinto e
indetailbyMonceau-Baroux,Keppens&Meliani(2012).More- d
theintergalacticmedium(IGM)andhowthisinfluencesgalaxyevo- over, Aloy et al. (2000) have studied jets with a spine–sheath jet fro
lution(e.g.Ciotti&Ostriker2007;Schawinskietal.2007;Sijacki m
structure;however,thesejetmodelsdonotincludeangularmomen- h
etal.2007;Rafferty,McNamara&Nulsen2008;Fabian2012;Gitti, tum.Theydo,however,includemagneticfields. ttp
Brighenti&McNamara2012). Aglobalpictureoftheflowpatternswithinajetanditssurround- ://m
Eventhoughthereisstrongevidenceofaradialstructurewithin ingcocoonhasemerged,butamoredetaileddescriptionoftheflow nra
AGNjets,theconnectionbetweenthisstructureanditsimpacton dynamics,andtheroleofaspine–sheathjetstructureinparticular s.ox
tehxeacItGfMormatolfaragetrasncsavleesrssetislltraretimficaaintisonlarpgreolfiyleunmkingohwtnh.avSeinacelarthgee iwsisltliilmlmprisosviengo.uHrvavieiwngoanbAetGteNrufeneddebrsatcakndininggeonferthale.sReeflloewvapnattqtuerenss- fordjo
influenceontheevolutionofthejetatlargescales,astudyabout tions are: How does the jet impact the ambient medium exactly? urn
thisaspectisclearlycalledfor. Whatpartoftheambientmediumundergoesstronginteractionwith als.o
thejetandwhatpartismerelydeflected?Howmuchmixingisthere rg
1.1 Mainfocusofthisresearch betweenshockedambientmediumandshockedjetmaterial?What at U/
effectwilladifferentradialstratificationhaveonthejetintegrity n
AGNjetsgenerallyremaincollimatedoverhugedistances,reaching and possibly the formation and development of internal shocks? ivers
lengths up to hundreds of kpc or even several Mpc. This implies Andinthecaseofstructuredjets,howdoesspineandsheathma- ite
thatthesejetseitherremainverystableinternallyandarenoteasily terialmixinternallywithinthejet,aswellasintheirsurrounding it v
a
disruptedbyinstabilitiessuchastheKelvin–Helmholtzinstability, cocoon? n A
orareconfinedbyexternalpressureforces. Havingabetterunderstandingoftheinterplaybetweenjet,co- m
s
Inthispaper,weexplorethreedifferentjetmodels,oneradially coonandambientmedium,aswellastheeffectofradialstratifica- terd
uniformjet(fromthispointonreferredtoasthehomogeneousjet) tiononjetintegrityandmixingeffectscouldhelpustosearchfor am
andtwojetswithadifferenttypeofspine–sheathjetstructure.We andcomparewithobservationalfeatures. on
study the effect of radial stratification on transverse jet integrity J
u
and quantify the mixing between jet components in detail. Also, ly
1
we closely look at the flow patterns that emerge within the jet- 2.2 Jetmodels 0, 2
head.Moreover,westudyhowthesejets(initiatedastypicalFRII 0
1
Whendealingwithjets,itisconvenienttoexpresstheirlength-scales 4
jets) and their surrounding cocoons have evolved after they have
beenactiveforaperiodof∼107 yr.Itisknownthatajetandits intermsofthegravitationalradiusoftheBHinthe‘centralengine’
that feeds jet activity, R = GM /c2, with G the gravitational
surroundingcocoonquicklyachieveapproximatepressurebalance g BH
constantandcthespeedoflight.2
asthejetpenetratesintotheambientmedium.Asaresult,thejet
Theoreticalconsiderationstogetherwithsomeobservationalev-
adaptstopressurevariationsthattraveldownthecocoon.Wewill
idence (e.g. Hada et al. 2011) point at a situation where jets in
lookinmoredetailatthesepressurewavesandhowtheyrelatetothe
generalhavedistinctregions,characterizedbyprocessesthattake
formationofstronginternalshockswithinthejet.Finally,wewill
placeatdifferentdistancesfromthecentralengine.Ifthejetlaunch-
comparetheactualpropagationofthejet-headtothepropagation
ingmechanismsforBHbinaries(BHBs)andAGNsareintrinsically
predictedbysimpletheory.
similar,thenweexpecttheprocessesthattakeplacealongthejet
axistobeapproximatelyscaleinvariant.Inthatcase,thesecharac-
1.2 Outlineofthispaper teristicregionsarelocatedatapproximatelythesamedistance,when
measuredinunitsoftheBHgravitationalradiusR .Lobanov(2011)
This paper is outlined as follows. In Section 2, we present back- g
ground theory for our models. Then in Section 3, we discuss the
2Togiveasenseforthedimensions,thegravitationalradiusforaBHwith
1ComparedtothelocalIGM. MBH=108M(cid:2)isRg∼1.48×108km≈1au.
RelativisticAGNjetsI.Steady2.5Djets 1455
discussesfivesuchdistinctregions.Verylongbaselineinterferome- momentumandenergyareconserved.Thatmeansthatthefunda-
try(VLBI)observationsofAGNsusuallyprobethecollimationand mentalequationscanbecastinconservativeform,whichina3+1
acceleration region, which occurs at a distance of about ∼103R , formulationread:
g
wheremagneticfieldsarestillthoughttoplayasignificantrole.In ∂U
somecases,VLBIobservationsofAGNjetsareabletoresolveup ∂ti +∇·Fi =0. (1)
to even much smaller distances from the central engine [in Hada
Here, the U (with i =1−5) are the conservative variables, and
etal.(2011),M87isobservedonlyafewtensofgravitationalradii i
fromthecentralengineandrecentlyDoelemanetal.(2012)have the Fi their corresponding fluxes. These relations can be derived
fromthecovariantformulation,andinparticularfromthevanishing
beenabletoresolvethejetbaseandestimatethisregiontolieata
distanceof∼5.5R fromtheSMBH].However,inoursimulations divergenceoftheenergy–momentumtensor,seeforinstanceWein-
g berg(1972,chapter2.10).Employingunitswithc=1fromhere
wewillfocusonthekineticenergyflux-dominated(KFD)region
ofthejets,whichtypicallyoccursat∼106–1011R .There,themag- on,theconservativevariablesemployedinMPI-AMRVACaredefined
g as
neticfieldisweak,soitdoesnotsignificantlyaffectthedynamics ⎛ ⎞
ofthejetflow.3Therefore,wewillnotbeprimarilyconcernedwith ⎛D⎞ γρ
⎜ ⎟
thedynamicaleffectofmagneticfields. U =⎝ S⎠ ≡⎜⎝ γ2ρhv ⎟⎠ . (2)
It is often assumed that a hot and tenuous plasma is present τ
in the innermost regions of accretion, close to the BH horizon γ2ρh−P −γρ D
o
and the innermost stable circular orbit, with magnetic field lines Here,ρ isthemas(cid:8)sdensityinthejetrestframe,v isthevelocity wn
threadingtheBHhorizon.IftheBHisspinning,gasandmagnetic vectorandγ =1/ 1−|v|2theassociatedLorentzfactor(proper loa
d
field lines are carried along by a general relativistic effect called speed:γv).ThevectorSisthemomentumdensity,Pisthepressure ed
‘framedragging’,extractingangularmomentumfromthespinning andτ isthekineticenergydensitythatincludesthekineticenergy fro
BH(Blandford&Znajek1977).Itisthereforeexpectedthatifjets ofthebulkandthermalmotion.4The(relativistic)specificenthalpy hm
indeedconsistofaspine–sheathjetstructure,thejetspineemerges his5 ttp
from this region, and consists of a hot, tenuous and fast-rotating e+P ://m
gasF.urtherout,butstillwithintheinneraccretiondisc,materialis h= ρ , (3) nras.o
tthhoanugmhattteoribaelilnesthsehdoitreacntdvmicionrietydoenfsthee,rBoHta.tiTnhgeajtetloswheearthveilsolcikiteielys twhietrhmeal=eneethrg+y ρde,ntshietytoettahlainntdertnhaelceonnetrrgiybudtieonnsitoyf,tihnecluredsitn-gmtahses xfordjo
toemergefromthisregion(Blandford&Payne1982).Therefore,it energyρ.Moreover,Pisthegaspressureand(cid:3) isthepolytropic urn
isexpectedthatthejetsheathconsistsofadenserandcolderflow, indexofthegas.Thecorrespondingfluxesare: als
withlowerazimuthalvelocitiesthanthejetspine.Atlargedistances ⎛ Dv ⎞ .org
fbruolmkvtheelocceitnyt,rabluetnagilnoew,ethreLjoertesnhtezaftahcmtoartethriaanlsthtiallthoafsthaerejleattisvpiisntiec. F=⎜⎜⎝Sv+P I⎟⎟⎠ , (4) at Un/
Fjeotrcwonofirkgurrealtaitoinng,sreaedifaotrivinesfteaantcuereGshoifseAllGinNiejteatsl.t(o2a00s5p)i.ne–sheath (τ +P)v iversite
Inthispaper,weconsiderbothahomogeneousjetwithconstant withIthe3×3identitymatrix. it v
density and pressure over its cross-section, as well as jets with a Inordertoobtainacompletedescriptionoftherelativisticfluid, an
A
spine–sheathjetconfiguration.Itshouldbenotedthatnotmuchis the system is closed with an EOS, relating gas pressure to mass m
knownabouttheactualradialstructureofaspine–sheathjet,sowe density. Instead of simply putting (cid:3) ≡ dlnP/dlnρ equal to 5/3 ste
willassumethatalljetsstartoutinpressureequilibriumwiththeir (for a classical ideal gas), or 4/3 (for a relativistically hot ideal rda
m
ambientmedium.Weconsidertwodifferenttypesofstructuredjets: gas),weemploythesameinterpolationfunctionthatwasusedin o
Thefirstmodelusesapolytropicindex(cid:3) =5/3andispiecewise Meliani et al. (2008), describing a realistic transition between a n J
u
isochoric:aconstantbutdifferentdensityforjetspineandjetsheath, relativisticallyhotgasanda‘cold’non-relativisticgas.Thisinter- ly
1
whichwewillrefertoastheisochoricjetfromnowon.Theother polationfunctioniscalledtheMathewsapproximation(Blumenthal 0
, 2
model is set up with an isothermal equation of state (EOS) and &Mathews1976)andisbasedontheSyngeEOS(Synge1957).The 0
1
assumes a constant temperature across the jet cross-section. We Mathewsapproximationusesaneffectivepolytropicindexequalto 4
willrefertothismodelastheisothermaljetfromnowon.These 5 1 (cid:9) (cid:10)ρ(cid:11)2(cid:12)
twocasesresultinadifferentradialstructure,aswillbediscussed (cid:3) = − 1− . (5)
eff 3 3 e
inSection2.4.Jetspineandjetsheatharegivendifferentvaluesfor
density,pressureandvelocity,andweallowforrotationaroundthe Withthisdefinitionfortheeffectivepolytropicindex,therelativistic
jetaxis,sothatthejetcarriesangularmomentum. specificenthalpycanbewrittenas
(cid:9) (cid:12)
1 e ρ
h= 4 − , (6)
3 ρ e
2.3 Hydrodynamics:basicequationsandmethods
andthecorrespondingclosurerelationfollowingfrom(3)becomes:
Wehavesimulatedthedifferentjetmodelsmakinguseofthespe- (cid:13) (cid:14)
cialrelativistic,grid-adaptiveMHDcodeMPI-AMRVAC(Keppensetal. P = 1 e− ρ2 . (7)
2012).Inclassicalandrelativisticidealhydrodynamics,totalmass, 3 e
4Orequivalently,τ isthetotalenergydensityinthelab-frame,withthe
3ThemagneticfieldintheKFDregiondoesofcourseinducetheobserved lab-framerest-massenergyγρsubtracted.
synchrotronemission. 5Thequantityρhisreferredtoasrelativisticenthalpy.
1456 S.Walgetal.
Thetotalinternalenergyofthegasperparticleis(cid:7)=e/n,withn assumesaself-similarrotationprofileoftheform:
tahseisnuthmebcearsdeefnosrityahoafdthroengicasj.eFt,otrhaeprelasst-mmaascsonesniesrtginygpoefrppraorttoicnlse, ⎧⎪⎪⎪⎨Vφ2,sp (cid:13)RR (cid:14)asp jetspine:0≤R≤Rsp,
ismp andthethermalenergyofthegasperparticleis(cid:7)th =eth/n. v2(R)= (cid:13) sp(cid:14) (9)
Not much is known about the composition of an AGN jet at kpc φ ⎪⎪⎪⎩V2 R ash jetsheath:R <R≤R .
scales. The plasma might consist of electrons and positrons (see φ,sh R sp jt
sp
for example Reynolds et al. 1996 or Wardle et al. 1998), but it
ThesameprofileisusedinMeliani&Keppens(2009).Here,R is
mightalsobeanelectron–protonplasma,oramixtureofboth.We sp
theradiusofthejetspineandR istheouterradiusofthejetsheath,
willhoweverassumethatatthelength-scalesweareconsidering,a jt
significantamountofmixingwiththeambientmediumhastaken whichcoincideswiththejetradius.Vφ,spisaconstantthatgivesthe
maximumrotationwithinthejetspineforR−→R ,andsimilarly
place, so that the jet can effectively be described by a hadronic sp
plasma. fortheconstantVφ,sh.Theconstantsasp andash areself-similarity
constants.Itcanbeseenimmediatelythattheconstanta needsto
It can easily be seen that the effective polytropic index for a sp
non-relativistically‘cold’gaswith(cid:7) (cid:8)m reducesto(cid:3) =5/3, bepositiveinordertoavoidsingularitiesatR−→0.Furthermore,
th p eff
whileforarelativisticallyhotgaswith(cid:7) (cid:9)m ,itreducesto(cid:3) = theRayleighcriterionforstabilityofflowsrotatingonacylinder
th p eff
4/3.6SeeMelianietal.(2008)foramorecompletedescriptionof againstaxisymmetricperturbationsis
theMathewsapproximationtotheSyngeEOS. d (cid:15) (cid:16) D
dR γhRvφ >0, (10) ow
n
lo
see for instance Pringle & King (2007, chapter 12). From this, a
d
e
2.4 Radialpressureprofileforspine–sheathjets it follows that both self-similarity constants need to satisfy the d
Since AGN jets remain collimated over huge distances, they are conditionasp >−2andash >−2.Wesetash =−2,correspond- from
ingtoajetsheathflowwithconstantspecificangularmomentum: h
eroxupnedctiendgst.oIbnefiancta,pipfraoxjiemtadtoeepsrensostusretaertquoiulitbirniupmrewssiuthretheeqiurisluibr-- λRa≡yleγighhR’vsφcr=itecroionns.taAnst,thmeaskeinlfg-siitmmilaarrigtiynaclolnysstatanbtloefatchceojredtisnpgintoe ttp://m
rium,unbalancedpressureforcesatitsjet-ambientmediuminter- needs to be positive, we adopt the same value that was used in nra
mfparceeedssiucuamreusieesqdutheilefiibnjreeitdutmeoxiaescittrhleyearcisheexledps.asnTcdhl,eeaoqrr.uIcenostntihtorenac‘osttfauhnnodtiwalrdathpmipsrooadmxeilbm’iefaontert MkineWdliseanosiof&ljveeKtste.hpTephreeandsfiia(r2lst0fo0jer9ct)eia-sbngadlivasenetnceacsepoqn=usat1atin/o2tn,.b(u8t)dfoifrfetwreontdidfefenrseintyt s.oxfordjo
doubleradiogalaxies(e.g.Blandford&Rees1974;Scheuer1974; for jet spine and jet sheath. There, we set up the radial pressure urn
B19e9g0e)lm,uannd&erdCeinosfefij1e9ts8a9t;lLaergahery,dMistuaxnlcoewsf&roSmtetphheecnesn1tr9a8l9e;nDgianlye p5r/o3fi.lTehoef stheecojnedt mkiankdinogfujseetoisf athpeoilsyotrthoeprimcainldjeext weqhuearletowe(cid:3)fi=x als.org
createastrongbow-shock.Thisbow-shockenclosesahotandover- a/
pressuredcocoon(comparedtotheundisturbedambientmedium). tthheetveamrypienrgatpurreessoufrteh,eanjedtibnyitiaadljiuzisntigngthtehejedteancsciotyrdainccgotrodi(cid:3)ngl=y t1o. t Un
Sincewedonotknowtheexactconditionsforsuchcocoonswhen iv
westartoursimulations,wesetthejetsupindirectpressureequilib- Fveiglo.c1itsyh,opwresstshuerei,nnituiamlbtrearndsvenerssitey(raanddiatle)mpproerfialteusreofththaetwazeirmeuutsheadl ersite
riumwiththeundisturbed‘ambient’IGM,whichwewillindicate in these simulations. In the following Sections (2.4.1 and 2.4.2), it v
withasubindex‘am’fromnowon. the actual radial pressure profiles will be derived. The choice for an A
Inthecaseoftheradiallyuniform,orhomogeneousjet(which theparametersofjetandambientmediumthathavebeenusedto m
wtheecjeatllpcraessesuHre),towetheseptrtehsesuprreesosfurteheeqaumilbibierniutmmeudpiubmy.eFqouratjientgs generatetheexactjetprofilesarediscussedinSections2.5and3. sterda
m
with a spine–sheath jet structure on the other hand (which we o
n
callcaseAfortheisochoricjetandcaseIfortheisothermaljet), 2.4.1 Pressureprofilefortheisothermaljet(I) Ju
thepressureprofileisnottrivial.Itcanbeobtainedbysolvingthe ly
specialrelativistichydrodynamic(SRHD)radialforceequationthat Tosolvetheradialforcebalanceequationfortheisothermaljet,we 10
balancestheradialpressureforcewiththecentrifugalforcedueto firstusetheidealgaslawtowrite(8)as , 20
1
twhiethrothtaetijoentaoxfisthaeloflnugidth.eWze-auxsiesacnydlinndergilceacltcthoeorldaitnearatelsex(Rpa,nφs,iozn) s2dP = vφ2dR , (11) 4
ofthejet(assumedtobeslowsothatvR(cid:8)vz)sothatthevelocity P (1−vz2−vφ2)R
isv=(0, vφ, vz).Onehas wheretheisothermalsoundspeedinarelativisticgassisgivenby
ddPR = ρhRγ2vφ2 = (cid:15)1−vρ2h−vφ2v2(cid:16)R . (8) s2= RμhT, (12)
z φ
withRthegasconstantandμtheparticlemassinunitsofhydro-
In this paper, the index ‘sp’ refers to variables and constants
gen mass. The temperature is taken constant. To solve the radial
belongingtothejetspine,whereastheindex‘sh’referstovariables
force balance equation, the temperature for the jet spine and the
and constants belonging to the jet sheath. An analytical solution
jetsheathdoesnotnecessarilyhavetobethesame.However,we
for the SRHD radial force balance equation can be found if one
adoptaconstantTacrossjetspineandjetsheathhere,wherewe
assumethatanydifferencesintemperaturehavebeenwashedoutat
largedistancesfromthecentralengine.Wewillassumethevertical
componentofthevelocitytobeconstantvz =Vz (alsonotneces-
6For an electron–positron plasma, the energies at which the gas would sarilythesameforjetspineandjetsheath)andusetheself-similar
becomerelativisticarelowerbyafactorme/mp∼5×10−4. azimuthal velocity profile (9). In that case, the pressure profile is
RelativisticAGNjetsI.Steady2.5Djets 1457
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mFcrioogsduser-lec.uI1nts.rseIhndoi,twtihaelintlrobagnl1as0cvkoerftshteheejpeprtorepfisrlsoeufiroleefsPazfioinmruutnhtihetasilsoorfotththaeetrimochnaalvrjaφec(ttRe()rsio(spltiiadcnpleirlneAesss))u.raTenhPdicsthhr=oetap1ti.ie5oc0new×priso1efi0l−ies6ohecarhsgobrciemce−nje3tu(s(pedadansefholerBdb)l;oiintnhesbt)hl,ueienisthocetahsleeorgmV10φalo=afnth1de.0thn×eumis1bo0ec−rh3docerin.csTijtehyte niversite
ninunitsofthecharacteristicnumberdensitynch=10−3cm−3(panelC);andingreenthelog10ofthethermaltemperatureTinunitsofthecharacteristic it v
ttwemopveerrattiucraeldTacshh=ed1l.i0n9es×.T1h0e13prKes(spuarneeolfDth)eoafmthbeiejentt.mInedadiudmitiiosnd,ethneotiemdabgyesthsehodwashtheedjheotrriazdoinutsalaltinRejti=np1aknpeclBan.dthejetspineradiusatRsp=Rjt/3asthe an Am
s
easilyintegratedto ThesetworelationsdetermineAspandAsh.Moreover,thepressure terd
atthecentreofthejetP(R=0)=P alsodeterminestheconstant am
P =A(cid:21)1−α(cid:13) R (cid:14)a(cid:22)−σ. (13) Aspby 0 on J
Rsp Asp=P0. (16) uly 1
0
Here,A,a,α andσ areallconstantswiththelatterthreegivenin Therequirementforthepressuretoremainpositivethroughoutthe , 2
0
thejetspineby jet’s cross-section is satisfied when αsp < 1 and αsh < 1, which 14
leads to the physically obvious condition V2+V2<1, the total
z φ
a=a , α = Vφ2,sp , σ = 1 . (14) speedattheinterfacesmustbelessthanthespeedoflight.
sp sp 1−V2 sp a s2
z,sp sp sp
2.4.2 Pressureprofileforthepiecewiseisochoricjet(A)
Expressions in the jet sheath are analogous and can be found by
changingthesubscriptsp −→ sh.TheconstantA inthejetspine Instead of assuming a constant temperature T ∝ P/ρ, we now
sp
assumeapiecewiseisochoric(orconstantdensity)jetwithinthejet
and the corresponding constant A in the jet sheath follow from
requiring (1) pressure balance at tshhe jet spine–sheath interface at spineadensityρsp,polytropicindex(cid:3)spandspeedvz=Vz,sp,and
R=R and(2)requiringpressurebalancewiththepressureP of similarlyρsh,(cid:3)sh andVz,sh inthejetsheath.Then,theradialforce
sp am
thesurroundingmediumatthejetouterradiusR=R . balanceequation(8)canberewrittenas
jt
Thisleadstotwoconditions: R dP˜ − (cid:3) (cid:15) vφ2 (cid:16) P˜ =0. (17)
A (cid:23)1−α (cid:24)−σsp =A {1−α }−σsh, dR (cid:3)−1 1−vz2−vφ2
sp sp sh sh
Here,
(cid:21) (cid:13) (cid:14) (cid:22)
Ash 1−αsh RRsjpt ash −σsh =Pam. (15) P˜(R)≡ (cid:3)(cid:3)−1 ρh=P(R)+ (cid:3)(cid:3)−1 ρ . (18)
1458 S.Walgetal.
Usingrotationprofile(9),onecansolvethisequation: thejetmaterial,seeequation(3).Therefore,thekineticluminosity
(cid:21) (cid:13) R (cid:14)a(cid:22)−τ ofaradiallyuniformjetcanbewrittenas
P˜(R)=A˜ 1−α R . (19) L =A n m γ v (h γ −1). (26)
sp jt jt jt jt jt jt jt jt
Here, a and α have the same meaning as in the isothermal case. In the case of a structured spine–sheath jet, we approximate its
TheconstantA˜ isdeterminedfromrequiringpressureequilibrium kinetic luminosity by adding the contributions from the jet spine
attheinterfacesR andR ,aswasrequiredintheisothermalcase. andthejetsheathtothekineticluminosityseparately:
sp jt
Moreover,τ isaconstant,whichforthejetspineisgivenby
L =L +L , (27)
jt sp sh
(cid:3)
τsp= a ((cid:3)sp−1) . (20) where Lsp and Lsh are defined in the same way as (26), but with
sp sp theirindicesreferringtothecorrespondingcomponents.Intherest
Asbefore,expressionsinthejetsheathareanalogoustotheexpres- of this derivation, we will just focus on the case of the radially
sionsinthejetspineandcanbefoundbychangingthesubscript uniform,homogeneousjet.
sp −→ sh. The constant A˜ in jet spine (and the corresponding Now suppose that we know the following jet parameters from
sp
constantA˜ inthejetsheath)inthiscasearedeterminedbysolving observationsforaparticularAGNjet:kineticluminosity,jetradius
sh
(cid:3) −1 and jet velocity (or equivalently a Lorentz factor γjt). Suppose
A˜sp{1−αsp}−τsp− s(cid:3)p ρsp thatwecanalsodetermineanumberdensitynam andtemperature Dow
sp Tamoftheambientmedium(fromwhichwecanderivetheambient nlo
=A˜sh{1−αsh}−τsh− (cid:3)s(cid:3)h−1ρsh, imteisdipuomsspibrleesstuorceaPlcamulwatieththtehediednesailtygarastliaowb)e.tWweitehnthjeetsempaaterarimaleatenrds aded
A˜sh(cid:21)1−αsh(cid:13)RRjt(cid:14)ash(cid:22)−sτhsh = Pam+ (cid:3)s(cid:3)h−1ρsh. (21) ajemtFbaiinresdnt,tawmmeebdiaeisunsmtumm. eedpiuremss.uTrheeenq,uuisliibnrgiuemquaattitohnesi(n3te)rafancde(2b6et)w,oenene http from
sp sh canshowthattheratioofnumberdensitycanbewrittenas ://m
ThesetworelationsdetermineA˜spandA˜sh,withtheconstraintthat (cid:13) (cid:14) (cid:25) (cid:10) (cid:11) nra
pP˜h0ys=icaA˜llsypa≥llo(cid:3)ws(cid:3)peds−pso1luρtsipon.smusthaveP0≥0,orequivalently(22) nnajmt = ((cid:3)−1) π((cid:3)Rj2t−Lnjatm1m)j(tγjt−−(cid:3)1)γ(cid:25)jt γj2tγj−2t −11 kbmTjatm , s.oxfordjourn
(28) als
.o
withk theBoltzmannconstant. rg
b a/
2.5 Jetproperties:densityratioandkineticluminosity Aswementionedbefore,weassumethejettobehadronic,sothat t U
thenumberdensityratiocanbewrittenasa(proper)massdensity n
iv
Observations of AGN jets yield a few basic parameters, such as ratio,givenby7 ers
the jet length and diameter, the luminosity of jets, lobes and (in ρ n ite
FRIIsources)hotspotsandpossiblythesynchrotronagebasedon ηR= ρjt = njt . (29) it va
the observed spectrum of the non-thermal radiation. In addition, am am n A
itispossibletoderivecocoonparametersfromtheX-raycavities JetswithηR<1arecalledunderdenseandjetswithηR>1are m
s
observed around some of the stronger sources. In principle, one called overdense. Underdense jets are less stable than overdense te
rd
can estimate the advance speed of the jet from these data and, jetsanddevelopinternal(diamond-shaped)shocksmoreeasily.For am
usingamodel,getcluesonjetcomposition,e.g.thequestionofan underdensejets,thepropagationspeedofthejet-headismuchlower o
n
electron–positronjetplasmaversusahydrogenplasma. thanthevelocityofthebulkmaterialofthejet(seeSection2.6). Ju
Inthissection,wewillexplainhowobserveddatacanbeusedto Forthesejets,atthejet-headthejetflowisterminatedbyastrong ly 1
calculatethemassdensityratiobetweenjetmaterialandmaterial shockcalledtheMachdisc. 0, 2
oftheambientmedium.Then,inSection2.6,wewillusethisto TheIGMinthevicinityofgalaxiesandinsideclustersofgalaxies 01
4
estimatethejet-headadvancespeedforaradiallyuniformjet. (theso-calledintraclustermedium,orICM)isusuallydenotedasa
Inordertodoso,wefirstdefinethekineticluminosityofajetL warm–hotintergalacticmedium(WHIM).Intheseregions,number
jt
asthetotalpowerL thatisproducedbythejet,withitsrest-mass densitiesrangefrom∼5×10−6 to∼10−3cm−3 andtemperatures
tot
energydischargethroughthejetsubtracted.Stillworkinginunits areoftheorderof105–107K(seee.g.Dave´etal.2001,2010;Kunz
wherec=1: etal.2011).
Since many powerful AGN jets are formed inside clusters of
Ljt=Ltot−M˙. (23) galaxies(e.g.Begelman,Blandford&Rees1984;Smithetal.2002),
wechoosetofocusontheICMastheambientmediumforourjets
ThetotalpowerL foraradiallyuniformjetisgivenby
tot andtakeforthenumberdensityn =1×10−3cm−3 andfixthe
am
L =A n m h γ2v , (24) temperatureoftheambientmediumtoT =107K.
tot jt jt jt jt jt jt am
Forthejet,wewilltakeapowerfulradiosource,withaluminosity
andtherest-massenergydischargethroughthejetM˙ by ofL =afew×1046ergs−1(atypicalluminosityforFRIIandBL
jt
M˙ =A n m γ v . (25) Lacsources,seeforinstanceItoetal.2008orMaetal.2008).Also,
jt jt jt jt jt
Here,A =πR2isthecylindricalradialcross-sectionofthejet,n
jt jt jt
isthenumberdensityofthejetmaterial,mjtistheaveragedmassof 7Notethattheinertiaofthematerialinthelab-framescalesasγ2nmfora
theparticlesinthejetandhjtisthespecificrelativisticenthalpyof givenparticlemassm.
RelativisticAGNjetsI.Steady2.5Djets 1459
Table1. Freeparametersthatwereusedforthejetinflowpropertiesandtheinitializationoftheambientmediumforthethreejet
modelsH,IandA.Kineticluminosity(Ljt),numberdensity(n),Lorentzfactor(γ),azimuthalvelocity(Vφ),polytropicindex((cid:3)),gas
pressure(P).InthecaseofmodelH,thejetishomogeneousintheradialdirectionandisdescribedbysingle-valuedquantities.The
pressureintheambientmediumfollowsfromthenumberdensitynamandassumingatemperatureoftheambientmediumofTam=
107K.TheparametersformodelsIandAareinitializedseparatelyforjetspine(denotedas‘sp’)andjetsheath(denotedas‘sh’).In
thecaseofthemodelsIandA,thepressurevariesradially,asindicated.InthecaseoftheImodel,thedensityvariesradiallyinorder
tokeepthetemperatureconstant.
Models Ljt(1046ergs−1) n(10−6cm−3) γ Vφ(10−3c) (cid:3) P (10−12ergcm−3)
sp | sh sp | sh sp | sh sp | sh sp | sh
H(homogeneous) 3.82 4.55 3.11 0.0 1 1.38
I(isothermal) 1.82 3.35 P/ρ=constant 6.0 3.0 1.0 1.0 5/3 5/3 Accordingtoequation(13)
A(isochoric) 0.44 3.39 1.0 5.0 6.0 3.0 1.0 1.0 5/3 5/3 Accordingtoequation(19)
Externalmedium – 1.0×103 – – 5/3 1.38
thebulkmaterialofthejetsinoursimulationsiscold(bywhichwe ofη ∼10−3 andaLorentzfactorofγ ∼3withcorresponding
R jt
meanthatthegassatisfiesaclassicalEOS,(cid:3)=5/3).Wewilltake β =0.943),wefindthatthejet-headpropagationspeedisapprox- D
jt o
fortheradiusofthejetRjt =1kpc,correspondingtoajetwitha imatelyβhd∼8×10−2.Thejet-headadvancespeed,togetherwith wn
typicalhalf-openingangleof1◦(Pushkarevetal.2009)atadistance thelengthofthejet,yieldsanestimateforthetimethatthecentral loa
d
of57kpcfromthecentralengine.8Andfinally,wewilltakethisjet enginehasbeenactive.Wewillusethismethodforanalyticallypre- ed
tobetrans-relativisticwithamoderateLorentzfactorofγjt=3. dictingthejet-headadvancespeedtocomparewithoursimulations fro
m
Substitutingthesevaluesinto(28),wefindamassdensityratio inSection4. h
Tofabthlee1ordsheorwofsηthRe∼ex1a0c−t3,jectoprraersapmoentdeirnsgthtoatvaerreyuunsedderdfoernstehejejtest. ttp://m
modelsinthispaper.9 Somepropertiesofunderdensejetswillbe 2.7 Jetproperties:rotation nra
s
treatedintheSection2.6. 2.7.1 Jetangularmomentum .o
x
fo
In steady, axisymmetric hydrodynamic flows the specific angular rd
2.6 Jet-headadvancespeed momentumλ≡γhRVφ(neglectinggeneral-relativisticcorrections) journ
Thevelocitywithwhichajetpenetratesintotheambientmediumis is conserved. Its value is set by the rotation of the wind source. als
lessthanthebulkvelocityofjetmaterial.Thisisespeciallytruefor Then,theazimuthalfour-velocitydecaysas .org
underdensejets.Nearthepointwherethejetimpactswiththeam- λ a/
bientintergalactic/interstellarmedium,astructureformsincluding γhVφ = R . (31) t Un
asreefvpoearrrwsaetairsndhgobscohkwo(cMskheaodcchkamdtihbsacite)npthtreagctaedsdeefcsreoltmehreastjheesot,ctahkeecdojenjtetafltcomtwda.itTsecrhioiansltwiannhudoitlyea (cid:14)IBnpoa+fxipsBoyφlmoeˆimdφa,elttrhfiiecelasdin/tflduoaiwtdioelaninlieMss,dHfiofDfremflreaonlwlty.sTdwheifiethrnee,adtmhbeyagannegtuiclafirevledloBcit=y iversiteit v
systemcomprisesthejet-head. an
V B A
Thejet-headadvancespeedcanactuallybeestimatedfromram (cid:14)= φ −κ φ , (32) m
R R s
pressure arguments in the rest frame of the head, where the flow te
ismore-or-lesssteady(Martietal.1997;Rosenetal.1999).The is constant along flow lines. Its value is set by conditions at the rda
m
jet-headγad√vηancβespeedfoundinthiswayequals vsoeulorccietyoafntdhemwaginnde.tiHcfiereel,d,κag≡aiVnpa/cBopnisstatnhtearloatnigofloofwthleinpeosl.oidal on Ju
β = jt R√ jt , (30) SuchaxisymmetricMHDwindsbehaveroughlyasfollows:close ly
hd 1+γ η 1
whereagainjηt =Rρ /ρ istheratioofmassdensityofjetmaterial γtoVth(cid:8)e s(cid:15)oBurc/e√, 4wπhρereh(cid:16)th(cid:15)e1−wi(cid:14)nd2Ris2/scu2b(cid:16)-Awliftvhe´ρnic=inρt/hγe tsheensperotpheart 0, 20
R jt am p p 0 0 1
4
andmassdensityofambientmediummaterial.Inthecasewhere density,thewindrotatesalmostrigidlywith
the gas is relativistically hot, so that h > 1, the same expression
Vφ ∼(cid:14)R. (33)
holds,butthentheratioofmassdensitiesissubstitutedbytheratio
ofrelativisticenthalpiesη −→ρ h /ρ h . Thissolidrotationisenforcedbystrongmagnetic torquesonthe
R jt jt am am
Fromequation(30),itisimmediatelyclearthatunderdensejets wind material. Although one can define a conserved specific an-
with η (cid:8)1 have propagation speeds much less than their bulk gular momentum λ that has a mechanical, as well as a magnetic
R
velocities,unlesstheyareveryrelativisticwithγ (cid:9)1.Usingthe contribution,themechanicalangularmomentumisobviouslynot
jt
sameparametersaswedidinSection2.5(resultinginadensityratio conserved!
Well beyond the(cid:15)so-c√alled Al(cid:16)fv(cid:15)e´n point, the(cid:16)point on a flow
line where γV = B / 4πρ h 1−(cid:14)2R2/c2 , the flow speed
8Atthisdistance,thejetisdominatedbykineticenergyflux. p p 0
issuper-Alfve´nicandmagnetictorquesbecomedynamicallyunim-
9Itisworthwhiletonotethatthechoiceinparameterspaceisfairlylarge portant.There,thewindsatisfies(31),butwiththevalueofλnow
andthatdifferentchoicesforLjt,Tamorγjtcouldinprincipleresulteasily setby(cid:14)andtheradiusR oftheAlfve´npoint:
indifferentdensityratios.However,itturnsoutthatformostsetsofrealistic A
parameters,thedensityratiowillingenerallieintherangeofηR∼10−3 λ=μ(cid:14)R2 , (34)
−1,mostofwhichcorrespondtounderdensejets.Ourchoiceistherefore A
reasonableandcorrespondstoanunderdensejetatthelowerendofthe where μ≡E/c2≥1 with E the conserved total energy per unit
spectrum. mass in the wind. This means that the value of λ can be much
1460 S.Walgetal.
higherthaninthehydrodynamiccase,leadingtoalargerrotation Table2. Listofcharacteristicquantitiesshownincgs
speedfarfromthesource. units.Thesecharacteristicquantitiesapplythroughout
thepaper.
2.7.2 Continuousrotationprofile Char.quantities Symbol cgsunits
Validsolutionsoftheradialforcebalanceequation(8)allowfordif- Numberdensity nch 10−3cm−3
ferentvaluesoftheconstantsVφ,spandVφ,sh.Givingtheseconstants Pressure Pch 1.50×10−6ergcm−3
adifferentvaluewillresultinadiscontinuousrotationprofile,where Temperature Tch 1.09×1013K
themostrealisticscenarioistheonewhereVφ,sp >Vφ,sh.Closeto
thecentralengine,wherethedifferentjetregions(spineandsheath) boundary cells at the Z = 0 axis, between R = 0 andR = R .
jt
arethoughttobedrivenbydifferentmechanisms,sucharotation Exceptforthecellsinvolvedininjectingthejetmaterial,allother
profile seems a reasonable one. However, as the jet propagates cellsinthelowerboundaryarefreeoutflowboundaries.Inaddition,
throughtheambientmedium,mixingeffectsbetweenjetspineand theinflowvelocityofthesecellsisreducedto20percentoftheir
jetsheatharelikelytowashoutthediscontinuityoccurringatthe originalvalue,inordertoavoidspuriousnumericaleffectsnextto
jetspine–sheathinterface.Therefore,therotationprofileatlarger thejetinlet.
distances from the central engine is likely to be continuous. This Thesizeofourcomputationaldomainis(250×500)kpc2.We
leads us to choose the rotation constants equal to one maximum chooseabasicresolutionof(120×240)gridcellsandallowforfour Do
value:Vφ,sp=Vφ,sh≡Vφ. aodfd(1it9io2n0a×lr3efi84n0em)gernitdlceevlelsls..TThheirsefroerseu,ltwseincaannreefsfoelcvteivdeerteasilosluutpiotno wnloa
d
(65×65)pc2. ed
3 METHOD Table1givesanoverviewofthefreeparametersthatwereusedfor fro
m
thesesimulations.Moreover,Table2showsalistofcharacteristic h
3.1 Themodels,setupandinitialconditions variablesthatareusedthroughoutthepaper,andwhichapplytothe ttp
Inthispaper,wesimulateAGNjetswithmoderateLorentzfactors plots. ://m
n
ofγ ∼afew,puttingthemintothetrans-relativisticregime.Wesim- ra
s
suplaintee–aschoenattihnujoetussl[yandriisvoetnhehrommaolg(eIn)ejoeutsa(nHd)ajeptiaencdewtwisoesitsruoccthuorreidc 3.2 MPI-AMRVACandnumericalschemes .oxford
(A)jet]ofwhichtheradialprofilesaretreatedinSection2.4. OursimulationsemploythecodeMPI-AMRVAC(Keppensetal.2012). jou
Alljetshaveconstantandsimilarluminosityduringtheirentire Itisaversatilecodethatallowsforvariousdiscretizationschemes, rna
evolution.Inafollow-uppaper,wewillbeconcernedwiththecase involvingtheuseofdifferentlimitersinthereconstructionsfrom ls.o
oftwodistinctepisodesofjetactivityforthesamejetmodelsH,I cellcentretocelledge.Itallowsforadaptivemeshrefinementand arg/
andA.Inordertomakeacleardistinctionbetweenthetwocases, canberunparallelonmultipleprocessors. t U
weintroduceanindex‘1’forthesteadycaseandintroduceanindex The simulations are performed with a special relativistic HD niv
‘2’forthecaseofepisodicactivity.Therefore,thispaperwilltreat module.Wechooseafour-step‘Runge–Kutta’time-discretization ers
thesimulationsH1,I1andA1. scheme,incombinationwithasecond-orderspatialtotalvariation ite
Thesimulationshavebeenperformedonthesamespatialdomain diminishingLax–FriedrichsschemewithaKorenlimiter.Thiscom- it v
a
for a duration of ∼23Myr (22.8Myr) with a kinetic luminosity bination captures shocks well without exhausting computational n A
of L ∼4−5×1046ergs−1. The jets are injected into a WHIM resources. m
withjctonstantdensity(nam=10−3cm−3)andconstanttemperature MPI-AMRVACcanbeinitializedusingconservativevariables,which sterd
(T =107K),whichisareasonableapproximationforthecondi- areadvectedasaccordingtotheirfluxescalculatedthroughequa- am
am
tionsinsideaclusterofgalaxies,atlargedistancesfromthecentral tion(1).However,thevariablescanalsobeinitializedasprimitive on
engine.Thetimestepsaredynamicallydeterminedbythecode,but variables,whichMPI-AMRVACthenconvertsbacktoconservativevari- July
areoftheorderof270yr. ables.Wechoosetodothelatter.Inthatcase,thefreeparameters 1
0
Ourjetsarecylindricallysymmetricwiththeirjetaxisalongthe ofthemodelsarethemass-densityρ,thevelocityv andthepres- , 2
0
Z-axis.Atthestartofthesimulationthejetprotrudesalongitsaxis sureP.Finally,MPI-AMRVACneedstobeinitiatedwithamaximum 14
into the computational domain over a distance equal to its initial valueforthepolytropicindex(cid:3).Weinitializedthepolytropicin-
radius,whichwechooseR =1kpcforallthreemodels.Inthecase dexas(cid:3)=5/3.Thischoicefor(cid:3)isconsistentwiththeMathews
jt
ofstructuredjets,thisisequivalenttotheouterjetsheathradius. approximation(equations5–7)fortheseparameters.
Forthesejets,wechoose(inabsenceofobservationalconstraints,
andinaccordancewithMeliani&Keppens2009)theradiusofthe
3.3 Tracersofjetmaterial
jetspineequaltoR =R /3.
sp jt
WechoosethemaximumrotationofthestructuredjetstobeVφ∼ Injetswithradialstructure,orincaseswherejetactivityisepisodic,
1×10−3.10 itisimportanttokeeptrackofthevariousconstituents(forexample,
Thejetsstartoutinpressureequilibriumwiththeirsurroundings, jet,jetspine,jetsheathorambientmedium).Tothatend,weemploy
asdescribedinSection2.4.Afterinitialization,thejetflowiscreated tracers,θA(t,r),11thatarepassivelyadvectedbytheflowfromcell
bylettingmaterialflowintothecomputationaldomainthroughthe to cell. Appendix A treats the definition of the tracers employed
here. The number of tracers that were used for each simulation
variesfromcasetocase.Basically,everyconstituentwewouldlike
10Thisisafairlyconservativechoicecomparedtothevalueofthecritical to trace is initialized to θ = θ = +1 in the region where this
max
rotationfortheisochoricjet,seeSection5.4.2.Moreover,notethateven
thoughwesimulatepurelyHDjetsatkpcscales,weassumetheyhaveall
startedoutasfullyMHDjets. 11TheindexAreferstoacertainconstituentAinthesimulation.
RelativisticAGNjetsI.Steady2.5Djets 1461
constituentisinjectedintothesystem.Weputitsvalueequaltoθ= orjetmaterialandambientmediummaterial).Inthatcase,wecan
θ =−1elsewhere.Forcompletenesssake,wewilllisttheexact writethesumofthemassfractionsoftheconstituentsas
min
valuesforeachsimulationbelow. (cid:26)N
H1: for the homogeneous steady jet we use one tracer, θ. We δk(t,r)=δA(t,r)+δB(t,r)+δ(cid:20)(t,r)=1, (37)
initializethistracertoθ =+1forjetmaterial,andθ =−1forthe k=1
ambientmedium. whereδ(cid:20)(t,r)isthesumofallothercomponentswithinδV(t,r).12
A1andI1:forsteadyjetswithstructure,weemploytwotracers; Inthesimplecasewherethesystemonlyconsistsoftwoconstituents
θsp for material from the jet spine and θsh for material from the (e.g. jet and ambient medium), one has δ(cid:20)(t,r)=0. Since this
jetsheath.Thetracerθsp isinitializedasθsp =+1formaterialin derivationappliestoallindividualgridcells,wewilldroptheindex
thejetspineandθsp =−1elsewhere.Equivalently,tracerθsh was (t,r)fromnowon.
initializedasθsh=+1formaterialinthejetsheathandθsh=−1
elsewhere.
3.4.2 Quantifyingtheamountofabsolutemixing(cid:18)
Despitethefactthatthetracersareinitiatedwithvaluesθ (t,r)=
A
±1,assoonastheyareadvected,actualmixingaswellaseffects To study the amount of mixing between the two constituents A
fromnumericaldiscretizationwillyieldtracervalueswithinavol- andB,itisusefultodefineanabsolutemixingfactor(cid:18)AB,which
umeelementδV(t,r)intherange−1≤θ (t,r)≤+1.Wewillin- considers the absolute amount of the mass fractions within that D
terpretthetracervalueθA(t,r)todirectlycAorrespondtotheamount cell. We choose (cid:18)AB = 0 in the case of no mixing by which we own
ofconstituentAinthatvolumeelement. meanthatonlyoneofthetwocomponentsAorBispresentwithin lo
a
δV and therefore δA = 0 or δB = 0. We choose (cid:18)AB = 1 in the ded
3.4 Mixingeffectsforvariousconstituents case of maximum absolute mixing by which we mean the same fro
amount of constituents A and B are present within δV , soδ = m
BelaesmeednotnδVth(et,arm),owunetaoref avbalreiotuossctuodnystitthueenatmsoinunat ogfivmenixvinoglubmee- δfrBa.ctFiounrtshwerimthoinret,hewceelilmapnodsetheaalminoeuarntsocfalainbgsolbuettewmeeixnintgh.eTmhAeasses http://m
tweendifferentconstituents.Thefollowingsectionsgiveadetailed assumptions completely determine the definition of the absolute n
desTchriepfitirosnttoyfpheoowfmmiixxiinnggcisancableleqduaabnstiofiluedte.mixing((cid:18))anddeals mixingfacto(cid:27)rfortwo(cid:27)differentconstituentsinacell:13 ras.ox
(cid:27)δ −δ (cid:27) fo
weleitmhetnhte.Ienxathcattmcaasses,f(cid:18)rac=tio0nsmoefantshothseatctohnesctiotnusetnittuseinntsahvavoelunmoet (cid:18)AB≡1−(cid:27)(cid:27)δAA+δBB(cid:27)(cid:27) . (38) rdjou
mixed at all, while (cid:18) = 1 means that the mass fractions of the Intheory,onecanalsoconsiderthemoregeneralcaseofmixing rnals
cmoanssstiftruaecntitosninsaarveo.lumeelementareequal,regardlessofwhatthose banetdwaeesnectwonodsewtsithofacotontsatlitmueanstss;forancetiwonithδ(cid:20)at.oItnaltmhaatsscafrsaecftioornmδu(cid:20)la1 a.org/
Thesecondtypeofmixingiscalledmass-weightedmixing((cid:19)). (38)stillapplies;however,theindicesAand2Bwillthenbereplaced t U
n
For this type of mixing, the mass fraction of a constituent in a bytheindices(cid:20) and(cid:20) .Inthispaper,wewillonlybeconcerned iv
1 2 e
vtihtoyel:uc(cid:19)mome=peule0tmamtieonenatanilssddnoiovmimdaienidx.ibIntygi,sthwtehhetiroleetfao(cid:19)lrme=aasms1eomafsetuharnaetsocafonhcsootmmituopeglenettneleiyn- withthemixingofindividualconstituents. rsiteit va
n
homogeneousmixture. 3.4.3 Quantifyingtheamountofmass-weightedmixing(cid:19) A
m
s
Absolutemixingisausefulconceptforsituationswhereoneisin- te
3.4.1 Massfractionsofmultipleconstituentsinavolume rd
terestedintheexactamountsoftheconstituentswithinthatvolume. am
When considering fluid volume elements, all material within one Itwill,however,notalwaysgiveanintuitivesensefortheamount o
n
elementδV(t,r)(onegridcell)isthesumofallitsconstituents. ofhomogeneityofthemixture. Ju
Whilesomemodelstreatthecontentsofavolumeelementwitha Toillustratethis,considerafixedvolumeVwhichismadeupof ly 1
0
multiplefluidapproach(i.e.differentconstituentshavingadifferent twoconstituentsAandB,withtotalmassesMA andMB,andtheir , 2
temperature,density,velocity,etc.),ournumericalmethodaverages sumM=MA+MB.Atfirst,thesetwoconstituentsareunmixedand 014
thesequantitiesout,sothateachgridcellcanbecharacterizedby separatedbyawall,dividingVintoequaltwoparts 1V.Wethen
2
one mass density, one pressure, one velocity vector, etc., known removethewallandstiruptheconstituents.Whentheconstituents
astheone-fluidapproximation.Inthatcase,thetotalmassdensity havehadthetimetosettledownandmaximallymixwitheachother,
ρ(t,r)withinδV(t,r)isthesumofmassdensitiesofthedifferent
constituentsρk(t,r).ForasystemwithNconstituents,thiscanbe
writtenas 12Notethatthiscaneithersimplybetheambientmedium,butitcouldin
theoryalsobeawholecollectionofotherconstituents.
(cid:26)N (cid:26)N
ρ(t,r)= ρk(t,r)=ρ(t,r) δk(t,r), (35) 1m3iOxinnegsfuabcttoler.pIofiantvomluumstebeelemmaednetrδeVgawrdoiunlgdtchoisntdaeinfinnietiiothnerofofthceonasbtsiotuluentet
k=1 k=1 AorB(andthereforeδA=δB=0),theabsolutemixingfactorisnotclearly
whereδk(t,r)isthemassfractionofconstituentkwithinδV(t,r), definedsincethesecondtermwouldyieldavalue0/0.Inthisparticular
sothat case,wedefinetheabsolutemixingfactor(cid:18)ABasfirsttakingoneofthetwo
(cid:26)N massfractionsequalto0(sosayδA=0)andthenformallytakingtheother
δk(t,r)=1. (36) mfraacstsiofnraecqtiuoanletoqu0a,lotnoe0is(lseoftswayitδhB1=−0|)±.AδBft|e,rwhhaivcihngaltwakaeynsythieeldfisrsatvmalauses
k=1 of0,regardlessofthevalueofthemassfracδtBionδB.Therefore,theabsolute
We are interested in the effect of mixing of two certain con- mixinginthecaseofabsenceofbothconstituentsisperdefinitionequalto
stituents A and B (e.g. jet spine material and jet sheath material, (cid:18)AB=0.