Magnetothermal and magnetorotational instabilities in hot accretion flows

0.26 MB·English

by  De-Fu Bu

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 insta-
bility (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 simi-
larly.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 mag-
netic 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 an-
gular 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 mag-
netic 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 tem-
perature in the outer regions decreases the mass accretion
rate.
Lastly, we would like to mention that in a weakly col-
lisional 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 magnetohydrodynam-
ical (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
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