UvA-DARE (Digital Academic Repository) 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 General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:24 Feb 2023 MNRAS433,1453–1478(2013) doi:10.1093/mnras/stt823 AdvanceAccesspublication2013June11 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 D o w n lo a d e d fro m h ttp ://m n ra s .o x fo rd jo u rn a ls .o rg a/ t U 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.