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Mon.Not.R.Astron.Soc.000,000–000(2017) Printed17January2017 (MNLATEXstylefilev2.2) Generation of toroidal magnetic fields in accretion disks Mohammadtaher Safarzadeh1(cid:63), Smadar Naoz2, Alexander Sadowski3, Lorenzo Sironi4, Ramesh Narayan5 1SchoolofEarthandSpaceExploration,ArizonaStateUniversity,Tempe,AZ85287-1404,USA; 2DepartmentofPhysicsandAstronomy,UniversityofCalifornia,LosAngeles,CA90095,USA 3MITKavliInstituteforAstrophysicsandSpaceResearch,77MassachusettsAve,Cambridge,MA02139,USA 4DepartmentofAstronomy,ColumbiaUniversity,550W120thSt,NewYork,NY10027,USA 5Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA02138,USA 7 1 0 17January2017 2 n ABSTRACT a Magnetic fields play an important role in the dynamics of accretion disks, however, the ori- J ginofthefieldsisoftenobscured.Hereweshowthatmagneticfieldscanbegeneratedinan 3 initiallynon-magnetizedaccretiondisksthroughtheBiermannbatterymechanism,wherethe 1 radialtemperatureprofileandtheverticaldensityprofileofthesesystemsprovidetheneces- sarilyconditionsforthisprocesstooperatenaturally.Weconsiderthegenerationoffieldsina ] E protoplanetarydisksanddisksaroundBlackHoles(BHs).Forprotoplanetaryaccretiondisks H we find that the generated magnetic fields can be as strong as few Gauss over the lifetime of the disk (5 Myr), for a solar mass star, and weakly dependent on the accretion rate. The . h generatedseedshavetoroidalstructurewithoppositesignintheupperandlowerhalfofthe p disk. The same pattern exists in a thin accretion disk around a rotating BH, where the field - o generationrateincreasesforlargerBH’sspinparametersinaco-rotatingdiskandspinconfig- r uration.Atafixedr/risco,whereristheradialdistancefromtheBH,thebatteryscalesasM−2 t forboththinaccretiondisksandAdvectionDominatedAccretionFlows(ADAF).ForADAFs s a afteronecharacteristicaccretiontimethebatterycangenerateafieldinthegalacticcenterat [ theorderof5×10−6G.Inaddition,wetestthismechanisminGRMHDsimulationsandfind ittobeinagoodagreementwithouranalyticalestimates.Thefactthatthebatteryworksin 1 v thelimitofzeroaccretionrate,itmakesthismechanismaviablecandidatetoprovidetheseed 0 fieldsfromaninitiallynon-magnetizedaccretiondisksuchthatlateronMagnetoRotational 0 Instability(MRI)takesover. 8 3 Keywords: accretion,accretiondiscs—dynamo—instabilities—magneticfields—MHD 0 . 1 0 7 1 INTRODUCTION turbulentdynamoduringthegravitationalcollapseforthefirstgen- 1 eration of protostars (e.g., Schleicher et al. 2010; Schober et al. v: Accretiondisksareassociatedwithawiderangeofastrophysical 2012).Althoughthesemechanismscanplayaroleinprovidingthe phenomena from disks around protostars to Black Holes (BHs). i seed magnetic fields, here we propose that magnetic fields in ac- X These disks are assumed to be magnetized and the accretion to cretiondiskscanbenaturallyformedthroughtheBiermannbattery happenthroughtheMagnetoRotationalInstability(MRI)process r process(Biermann1950). a (Balbus&Hawley1991).However,theoriginofthemagneticfield inthediskisrarelydiscussed.Thereareafewapproachesthatare Biermann(1950)showedthatifaplasmahasarotationalmo- adoptedintheliteraturetoaddressthesourceofmagneticfieldin tion, current must exists, which leads to the generation of mag- accretiondisks.ForthestellarmassBHs,themagneticfieldistypi- netic fields. This process has a wide range of applications from callyassociatedwiththemagneticfieldoftheprogenitorstar(e.g., the generation of magnetic fields in stars (e.g., Biermann 1950; Bisnovatyi-Kogan & Ruzmaikin 1974, 1976), while for a super- Doi&Susa2011)togalacticscalemagneticfields(e.g.,Mestel& massiveBHsthefieldisbelievedtobeaccretedfromthesurround- Roxburgh1962;Widrow2002;Subramanianetal.1994;Subrama- inggas(Narayanetal.2003;Eatoughetal.2013).Themagnetic nian2010;Naoz&Narayan2013).Forexample,Naoz&Narayan fieldsaroundthefirstgenerationofprotostarscanbetheresultof (2013)showedthatmagneticfieldsarenaturallygeneratedinthe largescalecascades,whereweakmagneticfieldsareamplifiedvia earlyuniverse,atintergalacticscales,whichisconsistentwiththe observedfields’strength.TheBiermannbatterywasalsoshownto beabletogeneratemagneticfieldsinsidesupernovabubbles(e.g., (cid:63) E-mail:[email protected] Hanayamaetal.2005). (cid:13)c 2017RAS 2 Safarzadeh,Naozetal. Forthegenerationofseedmagneticfieldsinthismechanism, asthetemperaturesofthedisksareestimatedfromfewthousands anonalignedgradientofelectronsdensityandgradientofelectrons KelvinintheinnerregionsofthediskdowntotensofKelvininthe pressureisneeded.Thissituationnaturallyarisesinthestructureof outerparts.However,asmentionedabove,theBiermannbatteryis accretiondiskswherethepressuregradientismostlyinthevertical independentontheionizationfraction.Weadoptasimpledensity directionandtemperaturehasaradialgradientinthedisk.These andtemperatureprofileforageometricallythin,Keplerianproto- seedmagneticfieldscangrowinlineartimescalesandinadiffer- planetary disk, (for review of the derivation see Armitage 2013). entially rotating disk can lead to a strong magnetohydrodynamic Theaxisymmetricdiskisdescribedbyitsverticalandradialstruc- (MHD) instability through MRI that makes the field to grow ex- ture. Assuming a hydrostatic equilibrium the vertical structure is ponentially with time to reach its equipartition value in the disk determinedbythepressuregradientequationinthezˆdirection (e.g.,Bret2009).Herewefocusongeneratingseedmagneticfield dP in an initially non-magnetized accretion disk. We do not specify =−ρΩ2z, (5) dz K thenaturethatcausestheaccretionandweassumethataccretion whereΩ istheKeplerianfrequencydefinedas: takesplacewhilethediskisnotinitiallymagnetized(i.e.,MRIdoes K notoperate).Turbulentviscositycouldbethoughtofaasplausible (cid:114) GM mechanismforourpurposes(e.g.,Shakura&Sunyaev1973).As ΩK = r3 (6) wewillshowbelow,generationoftheseedmagneticfieldsthrough Assuminganidealgasequationofstate,thedensityprofilein Biermannbatterydependsonlyveryweaklyontheaccretionrate. theverticaldirectionis Therefore,aprioriwedonotrequireMRItobetakingplacetopro- videaccretionforthebatterytowork. ρ= √Σ e2−Hz22 , (7) Shiromoto et al. (2014) studied the generation of magnetic 2πH fields in a first generation proto-circumstellar disk based on 2D WiththescaleheightdefinedasH=c /Ω andthespeedofsound s K radiation hydrodynamics simulations. Unlike their work, here we hasonlyaradialdependency(e.g.,Armitage2013) study a range of applications from generation of seed magnetic (cid:115) fieldsaroundaprotostar,tosupermassiverotatingblackholeatthe k T(r) c = B . (8) centerofthegalaxy. s µm p Thepaperisorganizedasfollow:In§2weintroducetheBier- mannbatteryequation.In§3weapplytheBiermannbatteryequa- Thesurfacedensity,Σ,isgivenby tion in protoplanetary disk setting. In §4 the battery is applied in  (cid:114)  thinaccretiondiskaroundKerrblackholes.In§5wecomparethe Σ(r)= 3Mπ˙ν1− Rr∗ , (9) ratespredictedforADAFswithonesobtainedforstate-of-the-art generalrelativistic(GR)MHDsimulationsofradiativelyinefficient whereν = Hc α(Shakura&Sunyaev1973).αisadimensionless s accretionflowsandIn§6wesummarizeandgiveconclusions. parameterandourresultsdoesnotdependonitsspecificvalue. Thetemperatureradialprofileisdeterminedbytheaccretion rateandandthebalanceofheatingandcoolinginthediskwhich 2 BIERMANNBATTERY leadstoradialeffectivetemperatureprofileof  (cid:114)  Tnohne-vgaenniesrhaitniogncroofssmpargondeuticctofifeledlsecitnroBnienrummabnenrbdaetntesirtyyigsradduieentto, Te4ff(r)= 38GπMσrM3˙ 1− Rr∗ . (10) ∇n andelectronpressuregradient,∇P e e whereσistheBoltzmannconstant.Wecomputethebatteryatz= Hforallthecalculations.Tokeeptheanalysissimple,wesetT ∼ ∂B ∇n ×∇P e =∇×(u×B)−c e e . (1) Teff ineq(4)andT ∼Teff ineq(8).1Since∇ρand∇Tebothhave ∂t en2 e rândzˆcomponents,thefieldisgeneratedinthetoroidaldirection, Assumingidealgasequationofstate, P = n k T andassuming i.e., e e B e that the temperature of the electrons is equal to that of the gas, (cid:32) (cid:33) ∂B ck ∂T ∂ρ ∂ρ∂T T =T wecanre-write∇n ×∇P as =− B e − e φˆ (11) e disk e e ∂t eρ ∂r ∂z ∂r ∂z ∇n ×∇P =∇n ×(n ∇T +T ∇n )k , (2) e e e e e e e B The Biermann battery will work so long as the density and where k is Boltzman constant. Given n = χ ρ, where χ is the temperaturegradientsarenotparalleltoeachother.Thiscondition B e e e ionizationfraction,wefind willalmostalwaysbesatisfiedinadisk.Toavoidgettingintode- ∇n ×∇P =n k ∇n ×∇T =n k χ ∇ρ×∇T , (3) tails,weestimatetheorderofmagnitudeoftheBiermannbattery e e e B e e e B e e bycrossingtheverticalgradientofdensitywiththeradialgradient Wherewithouttheinductiontermwehave oftemperature.ThereforeEquation(11)canbewrittenas: ∂B =−ckB∇ρ×∇Te (4) ∂B ck (cid:32)∂T ∂ρ(cid:33) ∂t e ρ =− B e φˆ, (12) ∂t eρ ∂r ∂z Notethattheionizationfraction,χ ,cancelsoutandthustheequa- e which,canbewrittenas: tiondependsonlyonthedensityandtemperatureofthesystem. √ ∂B cGMm µ7R +r( R /r−6) = p ∗ ∗ zφˆ (13) ∂t 8e r4(r−R ) ∗ 3 PROTOPLANETARYDISK Planetsareformedfromtheprotoplanetarydisksmadeofgasand 1 Thediskmid-planetemperatureisoftenjustafactorofafewlargerthan dust surrounding young stars. These disks are not fully ionized, Teff.Wefeelthereforethatthisapproximationisreasonable. (cid:13)c 2017RAS,MNRAS000,000–000 Generationoftoroidalmagneticfieldsinaccretiondisks 3 ��×��-�� =� �=�(��) ��-�� =�� ×��-�� =-�(��) �=�� (ϕ)(/)�� |/∂(/)�� -�� ∂| ��-�� -��×��-�� -�� -��×��-�� /�* �� Figure1.Magnitudeoftherateofmagneticfieldgenerationasafunction ofr/R∗fora1M(cid:12)staratz=Honthedisk.Generatedfieldschangetheir �� signatrturn =49/36R∗.Theseedshaveanoppositesignontheupperand Figure 2. Shows the magnitude of magnetic field generation rate as a lowerhalfofthedisk.TherateofseedgenerationthroughBiermannbattery function of r/risco for a 4 × 106M(cid:12) black hole with accretion rate of fwoeraaklpyrodteoppelnandeetnatryondtihskeiasccprreotpioonrtiroanteal(∝toMt˙h1e/8a)c.cWreetinadgomptaµss=M2a.3ndinotnhliys Md˙er=so1f0m−1a0gMni(cid:12)t/uydeeasrtrfoonrgdeiffreinrean=t0sp.9in9pcaarsaemceotmerpsa.rTehdewbiathttearyScishawbaoruzstc2hoilrd- calculation. blackholeatallradii.Theradiusatwhichthegeneratedseedfieldschange signinazimuthaldirection(rturn)becomessmallerasthespinparameteris Notethatthemagneticfieldchangesitsazimuthaldirectiongoing increased.rturn =2×riscoforaSchwarzschildblackhole.Wesetz=Hin thiscalculation.Thefieldchangesitssignacrosstheequatorialplaneinφˆ fromtheuppertothelowerhalfofthedisk.Thereisnodependence direction. onaccretionrateinthisequation,however,aswecomputethebat- teryatz=H,itwouldmeanaweakdependenceonaccretiondate (∝M˙1/8). which assuming a hydrostatic equilibrium leads to the following The magnetic field generated over an accretion timescale densityprofile (Trh/eurb)aitsterayt cthaenoorpdeerratoefov5e0rµtGhefolirfeatim1eMo(cid:12)fsthtaerdwisikthanRd∗c=an1reRac(cid:12)h. ρ= √Σ e2−Hz22BC (cid:114)C , (15) aboutfewtotensofGaussinlargepartsofthedisk.InFigure1we 2πH B showtheseedsgenerationrateinaproptoplanetarydisk.Theverti- where ci.ael.,dzist=ancHe.adTohpetegdenineroauterdcasleceudlasticohnasnsgcealtehseiarsstihgenscfraolemhueipgphetr, Σ= M˙ (cid:32)1− (cid:114)risco(cid:33) , (16) half to lower half of the disk and also at the characteristic radius 3πν r rturn=49/36R∗. and H,ν,Σ are a function of r and a where a is specific angular We note that the change of azimuthal sign of ∂B/∂t at rturn momentumoftheblackhole. arisesbecauseTe hasamaximumatthisradius.Thisispredicted Thetemperatureprofilefromthebalanceofheatingandcool- bythethindiskmodel,butwhetherthisexactshapeisfollowedby inginthediskis realdisksisdisputed. Theequipartitionfieldcanbeestimatedbyequatingthemag- T4(r)= 3GMM˙ (cid:32)1− (cid:114)risco(cid:33)D (17) netic pressure ∼ B2/8π and the pressure of the ionized gas. For e 8πσr3 r B protoplanetarydisksitisintheorderof103GatthevicinityofR , ∗ A,B,CandDarerelativisticcorrection(Novikov&Thorne1973; whichisordersofmagnitudelargerthantheBiermannseedfield. Page&Thorne1974;Doerreretal.1996)definedas: Furthermore,wealsoestimatewhenMRIstartstodominatedthe accretionprocess(cid:112).Takingthemostunstablewavelength,∼2πvA/Ω, A = 1− 2GM + a2 where v = B/ 4πρ2 is the Alfven wave velocity, and equating c2r c2r2 A √ it to H = cs/Ω, for z ∼ H, we find magnetic field peaking at B = 1− 3GM + 2a GM r/R(cid:12) ∼ 1.5, with strength of ∼ 200 G, which is order of magni- c2r c2r3/2 √ tudelargerthantheBiermannfieldatthisradiusafter5Myr,and 4a GM 3a2 staysinsimilarorderofmagnitudedifferenceatallradii. C = 1− + c2r3/2 c2r2 √ 1 (cid:90) r x2c2−6xGM+8a xGM−3a2 D = √ √ (cid:16) √ (cid:17) dx . (18) 4 SEEDSINDISKAROUNDKERRBLACKHOLES 2 r risco x x2c2−3xGM+2a xGM Following Shakura & Sunyaev (1973) and Novikov & Thorne whereriscoistherootofthefollowingequationforthecaseof (1973),theverticaldensityandradialtemperatureprofilesofthese aco-rotatingdisk: disksisdescribedbysimilarequationsaspresentedabove,withthe (cid:114) MGc a MGc a followingrelativisticcorrections: 1−6 +8 −3 =0 (19) r c r3 cr dP GM C =−ρ z , (14) Here we explore a simple example of a thin accretion disk dz r3 B withaconstantverticaltemperatureprofile(thus,Equation(12)is applicable)inthegalacticcenter.Wenotethatthetheaccretionsys- 2 Forsimplicityweassumeacompletelyionizedgas. temaroundthesupermassiveblackholeatthecenterofourgalaxy (cid:13)c 2017RAS,MNRAS000,000–000 4 Safarzadeh,Naozetal. expressthediskprofileusingthefollowingscalingrelations:  ��-�� =�×��=��-��  2 γ−1 �=�×��=��-� c2 = v2 (20)  s 3γ−5/9 k �=�×��=��-�� -�� (cid:32) γ−1 (cid:33) /)�� vr = − γ−5/9 αvk (21) ( |/∂�� -�� M˙ ∂| ρ = − , (22) 4πRHv r ��-�� wherethescaleheightHisalongthezcoordinate,v istheradial √ r componentofthevelocityofthegas,v = GM/ristheKeplerian k velocity,andγisthespecificheatofthegasanditsvalueisbetween ��-�� 4/3to5/3,foraradiation-pressuredominatedandanon-relativistic � gas-pressuredominatedaccretionflow,respectively.Weassumea �� gaspressuredominateddisk,i.e., P = c2ρ,andthedensityofthe s Figure3.Showstheabsolutemagnitudeofmagneticfieldgenerationrateas gasrelatestotheelectronnumberdensityvia:ne = χeρ/µ,where afunctionofradiusfromthelaststablecircularorbit(risco)aSchwarzschild χeistheionizationfractionandµisthemeanmolecularmass.We black hole with different values of M in M(cid:12) units and M˙ in (M(cid:12)/year) adoptµ=1.18mpandχe =1wherempisthemassoftheproton. units.ThesolidlinecorrespondstoaSchwarzschildblackholewithM = UsingEquation(1)wefindthat 4×106M(cid:12) andM˙ = 10−10M(cid:12)/year.Thedashedlinecorrespondstothe samemassbutwith10timeshigheraccretionrate.Thedottedlineisfora ∂B ∼ c ∂ρ ∂c2sφˆ ∼ µ6cGM(γ−1)φˆ, (23) blackholewith10timessmallermassbutthesametheaccretionrate.At ∂t en ∂H ∂r er3(9γ−5) e afixedr/riscothebatteryisproportionaltoM−2andisweaklydependent whereinourlasttransitionwehavepluggedintherelationin onaccretionrate(∝M˙1/8).Theradialturningpoint(rturn)isindependentof bothmassandaccretionrate.Weassumez=Hinthiscalculation. Equations(20)-(22).NotethatinourADAFmodelthescaleheight H describes the vertical component, and due to the “puffed up" natureofthediskwesetH∼r.Thishassimplescalingof: ∂B (cid:32) r (cid:33)−3(cid:32) M (cid:33)−2 γ−1 ∼3.6×106 G/sec, (24) ∂t r M (9γ−5) sc (cid:12) isprobablybestmodeledasanADAF(e.g.,Narayan&Yi1994; Narayanetal.1995;Abramowiczetal.1988),whichareexpected wherer = 2GM/c2 istheSchwarzschildradius.Foracharacter- sc toresultinanevenlargerfieldstrength.WediscussADAFsinsec- isticaccretiontimet ∼r/v wefind: acc r tion5. (cid:32)M (cid:33)(cid:18)r (cid:19)3/2 γ−1 Figure2showsthemagneticfieldgrowthratefora4×106M B∼17.9 (cid:12) sc G, (25) (cid:12) M r (9γ−5) black hole with different spin parameters as a function of r/r . isco Thebatteryisstrongerforblackholeswithlargerspinparameterat whichisordersofmagnitudesmallerthantheequipartitionfieldin afixedr/r . anADAFwhichcanbeestimatedas: isco We find that the Biermann battery in the thin accretion disk M (cid:18)r (cid:19)5/4 modelisweaklydependentonaccretionrate(∝M˙1/8atz=H)and Beq ∼ 7.8×107G M(cid:12) rsc × proportionalto M−2 atafixedr/risco.ThisisillustratedinFigure (cid:32) M˙ (cid:33)1/2(cid:32) 1−γ (cid:33)1/2 3.Asinthecaseofprotoplanetarydisk,thegeneratedseedshave . (26) 10−10M /yr α(5/9−γ) oppositesignintheupperandlowerhalfofthedisk.Thechangeof (cid:12) azimuthalsignofdB/dtatrturnarisesbecauseTehasamaximumat Fromtheserelationsweseethatforextremelylowaccretionrates thisradiuswhichispredictedbythethindiskmodel(seeShafeeet (∼ 2.3×10−17(r/r )1/4 M yr−1)thetwomagneticfieldstrengths sc (cid:12) al.(2008);Nobleetal.(2009);Kulkarnietal.(2011)foradiscus- equate. sionofthezero-torqueconditionattheISCOandtheeffectithas Wenowcomparetheestimatesofthegrowthrateoftheaz- on the viscous heating profile in a thin accretion disk). However, imuthalcomponentofthemagneticfieldduetothebatteryobtained Bφ changesitssignacrossthemidplaneandthisresultisarobust aboveundersimplifyingassumptions,withratescalculatedusinga predictionofthetheory. moredetailedaccretionflowstructureproducedbyGRMHDsim- ulations. For this purpose, we took the model of radiatively inef- ficient,two-temperatureaccretionflowpresentedanddiscussedin Sadowskietal.(2016)(modelRad8).Thisparticularsolutionde- scribesanaccretionflowonanon-rotating,10M BHwithaverage (cid:12) accretionrateoftheorderof4×10−9M˙ .Tocalculatethebattery- Edd relatedgrowthofthemagneticfieldwetookthetime-averagedout- 5 ADVECTION-DOMINATEDACCRETIONFLOWS putofthesimulationandappliedEq.(4).Resultsarepresentedin DISKS Fig4. Inmanyaccretiondisksaroundblackholestheviscouslydissipated The top panel presents the growth rate multiplied by radius accretionenergycangointoheatingtheaccretionflowratherthan cubed,i.e.,dB/dt(r/r )3,onthepoloidalplaneofthesimulation. sc beingradiatedaway.Thisisthemostimportantcharacterizationof The values are given in G/s. The most profound feature is the advection-dominatedaccretionflows(ADAFs)disks(seeforrecent asymmetryagainsttheequatorialplane.Thisreflectsthefactthat review: Yuan & Narayan 2014). Following Narayan & Yi (1994) thegradientsofdensityandtemperaturearesymmetricwithrespect self-similarsolution,forα << 1andanaxis-symmetricflow,we to the equatorial plane, and therefore their cross product changes (cid:13)c 2017RAS,MNRAS000,000–000 Generationoftoroidalmagneticfieldsinaccretiondisks 5 • Protoplanetary disks: Assuming a simple disk profile, we foundthatatoroidalmagneticfieldscannaturallygrowfromini- tiallynotmagnetizedaccretiondisksinprotoplanetarydisksover thelifetimeofthedisk(attheorderoffewtotensofGauss,seeFig- ure1).Thesefieldsarelinearlyproportionalwiththecentralmass and weakly dependent of the accretion rate. The generated seeds haveoppositesignsintheupperhalf(+φˆ)andlowerhalf(−φˆ)of thedisk.Wealsofindthatthesefieldschangesignatacharacteris- ticradiusofr =49/36R .Notethattheradialchangeofsignat turn ∗ r arisesbecauseT ismaximumatthisradius,aspredictedbythe turn e thindiskmodel.However,thispictureisdisputedintheliterature. • Thin accretion disk around a rotating BH: We adopted a disk temperature and density profile, which includes relativistic corrections (following Novikov & Thorne 1973; Page & Thorne 1974;Doerreretal.1996).Theseedshaveoppositesingintheup- per and lower half of the disk and also change their sign in the azimuthal direction at r = 2×r for a Schwarzschild black turn isco hole.LargerBHspinsyieldsmallervaluesofr andlargerrateof turn seedgeneration(seeFigure3).WenotethatthevanishingofT at e r=r ,(aspredictedbyShakura&Sunyaev1973)iscontrover- ISCO sialbecauseitcomesfromthezero-torqueboundaryconditionat theISCOandfromignoringadvection.Thus,thebehaviorofsign Fgiivgeunrein4.GT/hs,eignroawGthRrMatHeDofstihmeualzaitmiounthoaflamraagdniaettiivcefilyelidn,edffiBc/ident(tr/arcsccr)e3-, flipmaynotberealistic.Atafixedr/riscothebatteryscalesasM−2 andthebatteryisstrongerforBHswithlargerspecificangularmo- tionflow(modelRad8fromSadowskietal.2016)fora10 M(cid:12) BH.The toppanelshowsthegrowthrateinthepoloidalplane.dB/dtchangessigns mentum.Therelativisticdiskprofileleadstoaweakdependency over the equatorial plane because of symmetry in gradients of density ontheblackholeaccretionrate∝M˙1/8(seeFigure3). and temperature. The bottom panel shows the average absolute value of • ADAF: We also analyzed the generation of magnetic field dB/dt(r/rsc)3averagedverticallyover−1<z/r<1. in an ADAF, where our analytic estimates, based on Narayan & Yi (1994) self-similar solution, are in good agreement with with GRMHDsimulations(seeFigure4).UsingEquation(25)wecan sign.Forthesamereasonofdiskstructuresymmetry,thegrowth estimate the magnetic field generated after one characteristic ac- rate at the equatorial plane is zero. Far from the plane, the field cretiontimeforthediskaroundthecentralBHinourgalacticcen- does not change significantly and reaches values of the order of ter. We find that the Biermann battery can naturally give rise to dB/dt(r/rsc)3≈1000G/s. ∼ 5×10−6 GforthesupermassiveBHatthecenterofourgalaxy The bottom panel of Fig 4 shows the absolute value of the within one accretion timescale. After one year of operation this growth rate averaged vertically over the region −1 < z/r < 1. It mechanismcannaturallyreachamagnitudeof7Gverynearthe confirmsthetypicalvalueandshowsthatthegrowthratescalewell accretionradius(r ∼ r ),whichisattheorderofthefewGesti- sc withthethirdpowerofradius.Theaveragevalueshouldbecom- mationsofthemagneticfieldstrengthinthevicinityofthegalactic pared to the analytical prediction based on Eq. (24) which gives, center(e.g.,Mosćibrodzkaetal.2009). for10M ,γ=5/3,andχ =1, (cid:12) e Wenotethatwhilethereisamplediscussionintheliterature ∂B(cid:32) r (cid:33)3=2400G/s. (27) abouttheroleofpoloidalfield(e.g.,Narayanetal.2003)insub- ∂t r sequentevolutionofaccretiondisks,thereislittlediscussionabout sc Bˆ .Ourstudyshowsthatthisfieldnaturallyevolvesindisksand Havinginmindthatthisestimatewasderivedundersimplifiedas- φ thereforeitwouldbeinterestingtosimulatetherolethatBiermann- sumptions about the vertical and radial structure of ADAFs, one generated Bˆ might play in the evolution of the accretion disks. shouldconsidertheagreementverygood. φ Thismayalsohelptobetterunderstandtheroleofazimuthalsign reversalintheupperandlowerhalfofthediskandpotentiallyat r=r . turn 6 DISCUSSION The generation of magnetic fields through the radiation pressure 7 ACKNOWLEDGEMENTS on the electrons in the inner edge of the accretion disks has beendiscussedinpreviousworks(Contopoulos&Kazanas1998; MSissupportedbytheNationalScienceFoundationundergrant Bisnovatyi-Koganetal.2002;Contopoulosetal.2015).Inthislet- AST14-07835 and by NASA under theory grant NNX15AK82G. terweshowedthatlargetoroidalmagneticfieldscouldnaturallybe SNacknowledgespartialsupportfromaSloanFoundationFellow- generatedthroughBiermannbatteryprocessstartinginitiallyfrom ship. zeromagneticfield.Theverticaldensityprofileofanaccretiondisk andtheradialtemperatureprofilewhicharisesduetothebalance between heating and cooling, lead to generation of toroidal seed REFERENCES magneticfieldthroughtheBiermannbattery.Thesetoroidalfields changetheirsignattheequatorialplaneofthedisk. 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