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Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed17January2011 (MNLATEXstylefilev2.2) Magnetothermal and magnetorotational instabilities in hot accretion flows 1 De-Fu Bu1⋆, Feng Yuan1† and James M. Stone2 1 0 1Key Laboratory for Research inGalaxies and Cosmology, Shanghai Astronomical Observatory, 2 Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China 2Department of Astrophysical Sciences, Peyton Hall, Ivy Lane, Princeton, NJ 08544, USA n a J 4 1 17January2011 ] E ABSTRACT H In a hot, dilute, magnetized accretion flow, the electron mean-free path can be much . greater than the Larmor radius, thus thermal conduction is anisotropic and along h magnetic field lines. In this case, if the temperature decreases outward, the flow may p be subject to a buoyancy instability (the magnetothermal instability, or MTI). The - o MTIamplifiesthemagneticfield,andalignsfieldlineswiththeradialdirection.Ifthe r accretion flow is differentially rotating, the magnetorotational instability (MRI) may t s alsobe present.Using two-dimensional,time-dependent magnetohydrodynamicsimu- a lations, we investigate the interaction between these two instabilities. We use global [ simulations that span over two orders of magnitude in radius, centered on the region 2 around the Bondi radius where the infall time of gas is longer than the growth time v of both the MTI and MRI. Significant amplification of the magnetic field is produced 1 by both instabilities, although we find that the MTI primarily amplifies the radial 3 component, and the MRI primarily the toroidal component, of the field, respectively. 3 Mostimportantly,wefindthatiftheMTIcanamplifythemagneticenergybyafactor 5 Ft, and the MRI by a factor Fr, then when the MTI and MRI are both present, the 1. magnetic energy can be amplified by a factor of Ft ·Fr. We therefore conclude that 1 amplificationofthemagneticenergybytheMTIandMRIoperatesindependently.We 0 also find that the MTI contributes to the transport of angular momentum, because 1 radialmotionsinducedbythe MTI increasethe Maxwell(by amplifyingthe magnetic : field) and Reynolds stresses. Finally, we find that thermal conduction decreases the v i slope of the radial temperature profile. The increased temperature near the Bondi X radius decreases the mass accretion rate. r a Key words: accretion,accretiondiscs–conduction–magnetohydrodynamics:MHD – ISM: jets and outflow – black hole physics 1 INTRODUCTION conduction can have significant influence on the dynamics (Quataert2004;Johnson&Quataert2007),resultinginthe Hot, diffuse accretion flows are likely present in low- transport of thermal energy from the inner (hotter) to the luminosity active galactic nuclei (AGN) and the quiescent outer (cooler) regions. If the energy flux carried by ther- andhardstatesofblackholeX-raybinaries(Narayan2005; mal conduction is substantial, the temperature of the gas Yuan 2007; Narayan & McClintock 2008). When the mass in the outer regions can be increased above the virial tem- accretion rate is very low, the electron mean-free path will perature. Thus, gas in the outer regions is able to escape beverylarge,sotheaccretingplasmaisnearlycollisionless. fromthegravitationalpotentialofthecentralblackholeand For example, from Chandra observations of the accretion form outflows, significantly decreasing the mass accretion flow at the Galactic Center, Sgr A*, theelectron mean-free rate (Tanaka& Menou 2006; Johnson & Quataert 2007). path is estimated to be 0.02−1.3 times the Bondi radius (∼ 105 −106 Schwarzschild radii). In such cases, thermal Time-dependenthydrodynamicalsimulationsofhotac- cretion flows show that they are convectively unstable, no matter whether the flow is radiatively inefficient (Igumen- ⋆ E-mail:[email protected] shchev & Abramowicz 1999; Stone, Pringle & Begelman † E-mail:[email protected] 1999;Igumenshchev&Abramowicz2000)orefficient(Yuan 2 Bu, Yuan & Stone & Bu 2010). This is consistent with the prediction of one- 2 NUMERICAL SIMULATION dimensional analytical models of advection-dominated ac- We adopt two-dimensional, spherical coordinates (r,θ) and cretion flows (ADAFs; Narayan & Yi 1994; 1995). Convec- assume axisymmetry ∂/∂φ ≡ 0. We use the Zeus-2D code tive instability occurs because the entropy of the flow in- (Stone&Norman1992a,1992b)tosolvetheMHDequations creasesinthedirectionofgravity.However,inamagnetized with anisotropic thermal conduction: weaklycollisionalaccretionflow,inwhichtheelectronmean- free path is much greater than the Larmor radius, thermal dρ +ρ∇·v=0, (1) conductionisanisotropicandprimarilyalongmagneticfield dt lines. In this case, Balbus (2000; 2001) found that the sta- dv 1 bilitycriterion forconvectionisfundamentallyaltered:con- ρ =−∇p−ρ∇ψ+ (∇×B)×B, (2) dt 4π vection occurs when the temperature – not the entropy – ∂B increases inthedirection ofgravity.Convectiveinstabilities =∇×(v×B), (3) ∂t in this regime are referred to as the magnetothermal instability (MTI). d(e/ρ) ρ =−p∇·v−∇·Q, (4) dt In a weakly magnetized, non-rotating atmosphere, lo- Q=−χb(b·∇)T. (5) b b cal simulations have demonstrated that the MTI can am- Hereρisthemassdensity,visthevelocity,pisthepressure, plifythemagneticfieldandleadtoasubstantialconvective ψ=−GM/risthegravitationalpotential(Gisthegravita- heat flux (Parrish & Stone 2005; 2007). Sharma, Quataert tionalconstantandMisthemassofthecentralblackhole), & Stone (2008; hereafter SQS08) investigated the effects of Bis themagnetic field,e=p/(γ−1) istheinternalenergy the MTI on accretion flows in two-dimensions using global (γ istheadiabaticindex),Qistheheatfluxalongmagnetic simulationsinsphericalpolarcoordinates.Theirsimulations focused on the region around the Bondi radius, where the field lines, χ is the thermal diffusivity, b is the unit vector b MTIgrowthtimeisshorterthangasinfalltime.Theyfound along magnetic field and T is thetemperature of thegas. theMTI canamplifythemagneticfieldandalign fieldlines For convenience, we adopt χ=κp/T and assume with the direction of the temperature gradient (radially), that κ = ac(GMr)1/2 which has the dimension of a consistent with theresultsobtained bypreviouslocal simu- diffusion coefficient (cm2s−1). Anisotropic thermal con- lations. The effects of the MTI on the intracluster medium duction is implemented using themethod based on limiters (ICM)havealsobeeninvestigated(Parrish,Stone&Lemas- (Sharma & Hammett 2007) to guarantee that heat always ter 2008). In that study also, the MTI drives the magnetic flows from hot to cold regions. fieldlinestoberadial, leadingtoaconductivefluxwhichis We initialize our simulations as follows. The radial ve- an appreciable fraction of the classical Spitzer value. Such locity,internal energy,and density of theaccretion flow are a high rates of thermal conduction make the slope of the adopted from the hydrodynamic Bondi solution. For the temperature profile flatter. other two components of the velocity (vθ and vφ), we set vθ =0 and vφ =l0/r with l0(=const.) (a parameter of our simulations)thespecificangularmomentumoftheaccretion The studies of the MTI described above all focus on a flow.Theinitialmagneticfieldisuniformandperpendicular non-rotatingaccretionflow.Itiswell-knownthatinaweakly totheequatorialplane,(i.e.B=B ,wherez isthevertical z magnetized, differentially rotating hot accretion flow, the cylindrical coordinate). We have run four models. The pa- magnetorotationalinstability(MRI;Balbus&Hawley1991; rametersusedinthesemodelsarelistedinTable1.Inmodels 1998) will be present, and MHD turbulence driven by the L1 and L2, we set l0 to be 0.1 times the Keplerian angular MRIisthoughttobethemostpromisingmechanism ofan- momentum at the inner boundary. Thus, in models L1 and gular momentum transport. In principle the MTI and MRI L2,thecentrifugalbarrierisinsideourinnerboundary,and shouldcoexist insuchflows.Therefore, aninterestingques- rotationally supported region is never formed in these two tion for investigation is whethertheMTI andMRI interact models. In models H1 and H2, however, we set l0 to be 10 with one another, i.e., does one instability enhance or sup- timestheKeplerianangularmomentumattheinnerbound- press the other, or do the two instabilities operate entirely ary. Thus, in models H1 and H2, the centrifugal barrier is independently.For example, it is of interest to ask whether outsideourinnerboundary,sothatarotationallysupported convectivemotionsdrivenbytheMTIcantransportangular disk can form, which is subject to the MRI. We set β (the momentum,andsimilarlywhetherturbulentmotionsdriven ratio of gas pressure to magnetic pressure) to be β = 1012 by theMRI can transport heat. at the outer boundary in our initial conditions. We assume that the adiabatic index γ =1.5. We use the Bondi radius, Motivatedbythequestionabove,wepresenttheresults rB = GM/c2∞ (c∞ is the sound speed at infinity and we of two-dimensional magnetohydrodynamical (MHD) simu- set c2 =10−6c2 where c is the speed of light) to normalize ∞ lation of hot accretion flows with anisotropic thermal con- all of thelength scales in our paper. Times reported in this ductionand rotation.Weinvestigateflowswhich areslowly paperareallintheunitoftheinverserotationfrequencyat rotating (not rotationally supported) at the Bondi radius, theouter radius of thedomain, Ω−1 =[r3 /GM]1/2. out out wherethegrowth timeofboththeMTI andMRIisshorter All of our calculations span a domain bounded by an than the infall time. This paper is organized as follows: we inner boundary at r = 0.05r and an outer boundary in B introduce our numerical methods and initial conditions in at 20r . As in SQS08, we adopt a numerical resolution of B Section 2, most of our results are presented in Section 3, 60×44.Theradialgridislogarithmicallyspaced.Auniform while Section 4 is devotedto a summary and discussion. polar grid extends from θ = 0 to π. At the poles, we use MTI and MRI in 2-dimensional accretion flow 3 Table 1.Parametersforallofourmodels Models l0† a‡c description L1 0.1lin 0 noMTI,noMRI L2 0.1lin 0.2 onlyMTI H1 10lin 0 onlyMRI H2 10lin 0.2 withMRI,withMTI † l0 istheinitialangularmomentumoftheflow.lin isthe Keplerianangularmomentum attheinnerboundary. ‡ χ=κp/T isthethermaldiffusivityandκ=ac(GMr)1/2. axisymmetric boundary conditions. In the radial direction, weuseinflow/outflowboundaryconditionsatboththeinner andouterboundaries.Forthetemperatureattheinnerand outer boundaries, we use outflow boundaryconditions. Figure 1.Evolutionofthevolumeaveraged magneticenergyin both the r and φ components of the magnetic field normalized 3 RESULTS to the initial magnetic energy in a shell from rout/40 to rout in modelL1.Inthis model,magnetic fieldisamplifiedonlybyflux 3.1 Magnetic field amplification freezing. Since vθ ∼ 0, there is almost no amplification of the θ component of the field, so it is not shown. The growth of each Aweaklymagnetized,conductive,differentiallyrotatingac- cretion flow is subject to magnetothermal and magnetoro- componentisdefinedasδBr2,θ,φ=(hB2r,θ,φiV −Br2,θ,φ,0),where hiV denotes volumeaverageandB0 istheinitialfieldstrength. tational instabilities when thecondition, 1dpdlnT dΩ2 − + <0, (6) ρdr dr dlnr freezing. First, we investigate the magnetic field amplifica- is satisfied, where Ω is the angular velocity of theaccretion tion by flux freezing by carrying out model L1. We note flow.Providedthatthemagneticfieldisweak,thecondition that in models L1 and L2, the centrifugal barrier is inside for the existence of MTI alone is ourinnerboundary,thusthetimescalefortheMRItogrow 1dpdlnT is much longer than the gas infall timescale. Furthermore, − <0. (7) ρdr dr in model L1 we set thecoefficient of thethermal diffusivity a = 0, i.e., there is no MTI in model L1. Therefore, the This condition is always satisfied in the initial conditions c magneticfieldamplificationinmodelL1canonlybedueto of our calculations, thus as long as the MTI growth time flux freezing. The results below show that the effect of flux is smaller than the gas infall time, the MTI will grow. The freezingonmagneticfieldamplificationisverysmall,allow- condition for the existence of theMRI is ing us to ignore this effect in theanalysis of models L2, H1 dΩ2 and H2. <0. (8) dlnr In model L2, we set the coefficient of the thermal dif- This condition is again always satisfied in the initial condi- fusivity ac = 0.2 so that it is unstable to the MTI. In this tions of ourcalculations. Defining model, the magnetic field is amplified only by the MTI. In modelsH1andH2,thecentrifugal barrierisoutsideourin- γ2 =(cid:12)(cid:12)−1dpdlnT(cid:12)(cid:12), (9) ner boundary. The timescale for the growth of the MRI is MTI (cid:12) ρdr dr (cid:12) (cid:12) (cid:12) muchshorter than thegas infall time, so that the MRI will 2 (cid:12)dΩ2(cid:12) affect the flow in both of these two models. In model H1, γMRI =(cid:12)(cid:12)dlnr(cid:12)(cid:12), (10) we set the coefficient of the thermal diffusivity ac = 0, so (cid:12) (cid:12) thattheMTI issuppressedandthemagneticfieldisampli- theMTIandMRIgrowthtimescalesarethentMTI =γM−1TI fied only by the MRI. In model H2, we set the coefficient andtMRI =γM−1RI,respectively.Thewavelengthofthemax- of the thermal diffusivity ac = 0.2, so that both the MTI imum growth rate for MTI and MRI are andMRIcoexist, andthemagneticfieldisamplifiedbythe nonlinearevolutionofboth.Thepropertiesofthefourmod- λ =2πv /γ (11) MTI a MTI elsaresummarizedinTable1.Themodelsdemonstratethe and interplay between the MTI and MRI. We have also tried various other values of a , and find that changing a by a λ =2πv /γ , (12) c c MRI a MRI factor of a few does not affect our results, consistent with respectively. Hereva(≡pB2/4πρ) is theAlfven speed. theconclusion in SQS08. WeinvestigatetheinterplaybetweentheMTIandMRI Figures 1-4 show the change of magnetic energy as a byexaminingthemagneticfieldamplificationbythetwoin- functionoftimeinthesefourmodels.Theenergyisaveraged stabilities. Note that in addition to the MTI and MRI, the fromr=r /40tor .Thesolid,dotted,anddashedlines out out magnetic field can also be amplified by the geometric con- correspond to the energy in the r, θ, and φ components of vergenceoftheaccretionflowinsphericalgeometryandflux themagnetic field,respectively. 4 Bu, Yuan & Stone Figure1showsthemagneticenergyevolutioninmodel L1. The field is smoothly amplified by flux freezing. At the endof thesimulation, thefieldis onlyamplified byafactor of2.Becausev ∼0,thereisalmost noamplification ofthe θ θ component of the field. From this model, we can see that the effect of flux freezing on magnetic field amplification is very small. We nowinvestigate themagnetic field amplification by theMTI(modelL2).TheonlydifferencebetweenmodelsL1 and L2 is that in the latter we include thermal conduction sothattheMTIispresent.Fig.2showstheevolutionofthe magnetic energy in model L2, it shows significantly more amplification of the magnetic field in comparison to model L1. This amplification is mainly due to motions driven by theMTI.Initially,thefieldisrapidlyamplifiedbytheMTI. After time t = 6, the magnetic field ceases to grow expo- nentially,andthegrowthratebecomessmall,forthefollow- ing reason. The growth rate of the fastest growing mode of Figure2.Changeofvolumeaveragedmagneticenergyinther, the MTI is given by γ ≃|[1/ρ][dp/dr][dlnT/dr]|1/2b where θandφcomponentsofthemagneticfieldnormalizedtotheinitial θ b = B /B. The growth rate decreases with time because magnetic energy in a shell from rout/40 to rout as a function of θ θ of the decrease of b with time. In this model, we use pa- timeinmodelL2.Inthismodel,themagneticfieldisamplifiedby θ theMTI.Thegrowthofeachcomponent ofthemagneticfieldis rameters F , F and F to represent the amplification factor of thre,tr, θθ,tand φ cφo,mt ponents of the magnetic field, defined as δBr2,θ,φ =(hB2r,θ,φiV −Br2,θ,φ,0), where hiV denotes volumeaverageandB0 istheinitialfieldstrength. respectively. Wefind 2 2 Fr,t =Br,f/B0 ∼100, (13) 2 2 Fθ,t =Bθ,f/B0 ∼10, (14) 2 2 Fφ,t =Bφ,f/B0 ∼100, (15) whereB ,B andB correspondtother,θandφcom- r,f θ,f φ,f ponents of the magnetic field at the end of the simulation, respectively. The radial component of the magnetic field is amplified by the MTI significantly. The MTI saturates by aligning field lines with the temperature gradient (radial direction), which is consistent with the results of previous work (Parrish & Stone 2005; 2007; SQS08). SQS08 found thatthemagneticfieldamplificationbyMTIisquasi-linear. If the magnetic field is weak initially, MTI can amplify the fieldbyafactorof∼10−30,independentoftheinitialfield strength. Ourcalculation confirms their result. Weinvestigatefield amplification bytheMRIin model H1. In this model, we turn off thermal conduction so the MTI does not exist, and the magnetic field amplification Figure3.Changeofvolumeaveragedmagneticenergyinther, must be due to motions driven by the MRI. Figure 3 θandφcomponentsofthemagneticfieldnormalizedtotheinitial shows the time evolution of the magnetic energy. Initially magnetic energy in a shell from rout/40 to rout as a function of (t<16.5), the field is very weak, and at our resolution the time in Model H1. In this model, magnetic field is amplified by fastestgrowingmodeoftheMRIisnotresolved,sothatfield theMRI.Thegrowthofeachcomponentofthemagneticfieldis amplification occurs primarily by flux freezing. The initial defined as δBr2,θ,φ =(hB2r,θ,φiV −Br2,θ,φ,0), where hiV denotes (t < 1) rapid amplification of the azimuthal component of volumeaverageandB0istheinitialfieldstrength. themagneticfieldisduetoshear.Att=16.5,themagnetic field strength is amplified by flux freezing to a high enough valuethattheMRIisresolved,andthefieldundergoessub- t = 18, we find k2v2 +dΩ2/dlnr ∼ 0 in the region where a stantial and rapid amplification by the MRI. To illustrate theMRIgrowthwavelengthcanberesolved,sothegrowth this point, we define a parameter Re = λ /∆x, where rate of MRI decreases significantly. Using parameters F , MRI r,r ∆x is the grid size, which measure how well the MRI is re- F , and F to represent the magnetic field amplification θ,r φ,r solvedonournumericalgrid.Figure5showscontoursofRe factor of the r, θ and φ components of the magnetic field, at t=16.5. From this figure,it is clear that unstable wave- respectively,we find lengthsoftheMRIarewellresolvedatthattime.TheMRI 2 2 amplifies the field rapidly, and shortly after t = 16.5 the Fr,r =Br,f/B0 ∼10, (16) MRI saturates and the field begins to grow algebraically. Fθ,r =Bθ2,f/B02 ∼10, (17) Saturation of the MRI can be understood as follows. The condition for MRI tooperate is k2va2+dΩ2/dlnr<0.After Fφ,r =Bφ2,f/B02 ∼104, (18) MTI and MRI in 2-dimensional accretion flow 5 becomes algebraic. Using parameters F , F and F to rep- r θ φ resenttheamplificationfactorofther,θandφcomponents of themagnetic energy,respectively, we find 2 2 3 Fr =Br,f/B0 ∼10 , (19) 2 2 Fθ =Bθ,f/B0 ∼100, (20) 2 2 6 Fφ =Bφ,f/B0 ∼10 , (21) whereB ,B andB correspondtother,θandφcom- r,f θ,f φ,f ponents of the magnetic field at the end of the simulation, respectively. Comparing models L2, H1 and H2, we find, F ∼F ·F , (22) r r,t r,r F ∼F ·F , (23) θ θ,t θ,r F ∼F ·F . (24) φ φ,t φ,r Figure 4.Changeofvolumeaveragedmagneticenergyinther, θandφcomponentsofthemagneticfieldnormalizedtotheinitial Let us first consider the amplification of the radial compo- magnetic energy in a shell from rout/40 to rout as a function of nent of the magnetic field. If the MTI alone amplifies the time in Model H2. In this model, magnetic field is amplified by radialcomponentof themagneticenergybyafactor ofFr,t both the MTI and MRI. The growth of each component of the (model L2), and the MRI alone by a factor of F (model r,r magnetic field is defined as δBr2,θ,φ = (hB2r,θ,φiV −Br2,θ,φ,0), H1), then when the MTI and MRI coexist the radial com- where hiV denotes volume average and B0 is the initial field ponent of the magnetic field can by amplified by a factor strength. of F ·F (model H2). The amplification factors for the r,t r,r other two components of the magnetic energy behave similarly.Wethereforeconcludethatinthenonlinearregimethe MTI and MRI amplify the field independently, most likely because the MTI is buoyancy instability driven by radial gradients in the temperature, while the MRI is a dynam- ical instability driven by the radial gradient of the orbital frequency.Providedthatthemagneticfieldisweak andthe plasma is dilute, thermal conduction from hotter to cooler regionswillmaketheplasmabuoyantlyunstable,regardless of whether the flow also contains turbulent eddies driven by the MRI. On the other hand, provided that the magnetic field is weak and differential rotation exists, the MRI can operate regardless of the presence of convective eddies driven bythe MTI. 3.2 The role of MTI in angular momentum transfer Figure 5. Contour of Re = λMRI/∆x at time t = 16.5, where The MRI is now believed to be the most promising mecha- ∆x is the gridsize. It is clear that the wavelength of the fastest nismofangularmomentumtransportinfullyionizedaccre- growingmodeoftheMRIiswellresolvedattimet=16.5. tion flows. It is interesting to investigate whether the MTI cancontributetoangularmomentumtransferinhot,diffuse accretion flows. whereB ,B andB correspondtother,θandφcom- We first compare the angular momentum profiles in r,f θ,f φ,f ponents of the magnetic field at the end of the simulation, modelsL1andL2at theend ofbothcalculations. Theflow respectively. The toroidal component is most significantly in model L1 is laminar, while there is (only) MTI in model amplified as expected. L2. Angular momentum profiles at the end of the simula- In model H2, we turn on thermal conduction so both tions are shown in Figure 6. It is clear that the angular the MTI and MRI coexist. Figure 4 shows the time evolu- momentum distribution of model L1 is almost the same as tion of the magnetic energy. Initially (t < 3), there is only the initial distribution. This is because the flow in model the MTI. This is because initially the field is too weak for L1islaminar,andthemagneticfieldisveryweak:therefore unstable wavelengths of the MRI to be resolved. At t = 3 boththeReynoldsandmagneticstressesareverysmalland the magnetic field has been amplified by the MTI enough there is almost no angular momentum transfer. It is also that unstable modes of theMRI are resolved, and the MRI cleat that in model L2, angular momentum is transferred begins to drive motions. After t = 3, the field begins to be from the inner to the outer regions. In order to investigate amplifiedexponentiallybyboththeMTI andMRI,untilat themechanismfortheangularmomentumtransportinthis t∼20bothinstabilitiessaturateandthegrowthofthefield case,weplottheMaxwellandReynoldsstressesinFigure7. 6 Bu, Yuan & Stone Figure6.Radialprofilesofspecificangularmomentumformod- Figure 7. Radial profiles of stress. The dotted and solid lines elsL1(dottedline)andL2(solidline).Thedashedlineshowsthe correspond to the Maxwell stress in models L1 and L2, respec- initialangularmomentumdistribution. tively.ThetriangleandsquareareReynolds stressinmodelsL1 andL2,respectively. ThedottedandsolidlinescorrespondtotheMaxwellstress (−B B /4π) in models L1 and L2, respectively. Because of r φ magneticfieldamplificationbytheMTI,theMaxwellstress in model L2 is much larger than that in model L1. The tri- anglesandsquaresdenotetheReynoldsstressinmodelsL1 and L2 respectively. Due to the presence of motions driven bytheMTIinmodelL2,theReynoldsstressisthreeorders of magnitude higher than in model L1. Because the mag- neticfieldisstillveryweakinmodelL2,theMaxwellstress is much smaller than the Reynolds stress. We conclude the angular momentum transport in model L2 is dominated by theReynoldsstress.WenotethatalthoughtheMTIinduced Reynoldsstressdoestransferangularmomentum,theeffect is quite low, so that there is only a small change of the angular momentum profilecompared totheinitial conditions. Next we compare the angular momentum distribution inmodelsH1andH2atlatetimesinFigure8.Bothprofiles arequitesimilar.Recallthatbothmodelsareunstabletothe MRI, while only H2 is unstable to the MTI. The similarity Figure8.Radialprofilesofspecificangularmomentumformod- in theangularmomentum profilesindicates that therole of elsH1(solidline)andH2(dashedline). the MTI in angular momentum transport is less important compared to theMRI. Quataert (2007). The decrease is caused by the increased heating of the outerregions by thermal conduction. 3.3 Effects of MTI on the structure of the accretion flow 4 SUMMARY AND DISCUSSION Figure9showsthetime-averageddensity(upper-leftpanel), temperature (upper-right) and mass accretion rate (lower Inextremely low accretion rate systems, theplasma is very panel)formodelsH1andH2.Intheupperpanels,thesolid dilute, the collisional mean-free path of electrons is much and dashed lines correspond to models H1 and H2, respec- greaterthantheirLarmorradius;thusthermalconductionis tively.Inthelowerpanel,thesolid,dottedanddashedlines anisotropicandalongmagneticfieldlines.Anisotropicther- correspond to the net accretion rate, inflow rate and out- mal conduction can fundamentally alter the Schwarzschild flow rate, respectively. The black and red lines correspond convective instability criterion. Convection sets in when to models H1 and H2, respectively. Asshown in the upper- temperature (as opposed to entropy) increases in the di- right panel, the radial profile of temperature in model H2 rectionofgravity.Convectiveinstabilityinthisregimeisre- (with MTI) is flatter than that in model H1. This is likely ferredtoasthemagnetothermalinstability(MTI).TheMTI because thermal conduction transports energy from the in- canamplify themagneticfieldandalign fieldlineswith the ner to the outer regions of the flow. From the lower panel, temperature gradient (i.e., the radial direction). If the flow we can see that the inflow rate decreases due to the effects has angular momentum, the MRI may also exist. We have of thermal conduction. This is consistent with Johnson & investigated the possible interaction between the MTI and MTI and MRI in 2-dimensional accretion flow 7 mentumtransport.WefindthattheMTI enhancesangular momentum transport simply because it can amplify magnetic field (thus enhancing the Maxwell stress) and induce radial motions (thus enhancing the Reynolds stress). But we note that the effects of the MTI on angular momentum transfer aresmall. Finally, wefindthat thermalconduction does make the slope of the temperature smaller due to the outward transport of energy, and the increase of the temperature in the outer regions decreases the mass accretion rate. Lastly, we would like to mention that in a weakly collisional accretion flow, the ion mean-free path can be much greater than its Larmor radius; thus the pressure tensor is anisotropic.Inthiscase,thegrowthrateoftheMRIcanin- crease dramatically at small wavenumberscompared to the MRI in ideal MHD (Quataert, Dorland & Hammett 2002; Sharma, Hammett & Quataert 2003). In this regime, the Figure 9. Radial profiles of time-averaged variables in models viscous stress tensor is anisotropic, and Balbus (2004; see H1andH2.Upper-leftandupper-rightpanelscorrespondtothe alsoIslam&Balbus2005)hasshownthatwhenanisotropic radial profiles of density and temperature, respectively. In the viscosity is included, the flow is subject to the magnetovis- upper panels, solid and dashed lines correspond to models H1 cousinstability(MVI).TheMVImayalsoamplifymagnetic and H2, respectively. In the lower panel, the solid, dotted and fieldsignificantly. Aninterestingproject for thefutureis to dashedlinescorrespondtothenetaccretionrate,inflowrateand investigate theeffects of theMVI on hot accretion flows. outflow rate, respectively. Theblackand redlinescorrespond to modelsH1andH2,respectively. 5 ACKNOWLEDGMENTS MRI by performing two-dimensional magnetohydrodynamical (MHD) simulations of nonradiative rotating accretion We thank P. Sharma for sending us his code. This work flows with anisotropic thermal conduction. was supported in part by the Natural Science Founda- We have presented the results from four models, with tion of China (grants 10773024, 10833002, 10821302, and focus on the amplification of magnetic energy by the com- 10825314), the National Basic Research Program of China bination of MTI and MRI. As a control case, we computed (973Program2009CB824800), andtheCAS/SAFEAInter- a model (L1) in which neither the MTI nor the MRI was nationalPartnershipProgramforCreativeResearchTeams. present, so that themagnetic field is amplified only by flux freezing and geometrical compression in the flow. We find that field amplification via flux freezing in this case is very small. Next,weinvestigated theeffectsoftheMTI onmag- REFERENCES neticfieldamplification in modelL2.WefindthattheMTI BalbusS.A.,2000, ApJ,534,420 can amplify magnetic field significantly, which is consistent BalbusS.A.,2001, ApJ,562,909 withtheresultobtainedinpreviouswork.Finallyweconsid- BalbusS.A.,2004, ApJ,616,857 eredtheevolutionofrotating flowsthatareunstabletothe BalbusS.A.,HawleyJ.F.,1991, ApJ,376,214 MRI. In model H1 (no thermal conduction), only the MRI BalbusS.A.,HawleyJ.F.,1998, Rev.Mod.Phys.,70,1 is present. In model H2, both the MTI and MRI coexist. 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