The Effects of Anisotropic Viscosity on Turbulence and Heat Transport in the Intracluster Medium

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by  Ian J. Parrish

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed5January2012 (MNLATEXstylefilev2.2)
The Effects of Anisotropic Viscosity on Turbulence and
Heat Transport in the Intracluster Medium
Ian J. Parrish1⋆, Michael McCourt, Eliot Quataert, and Prateek Sharma2
1Department of Astronomy and Theoretical Astrophysics Center, University of California Berkeley, Berkeley,CA 94720
2 2Present Address: Department of Physics, Indian Institute of Science, Bangalore 560012, India
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5January2012
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ABSTRACT
3

] Intheintraclustermedium(ICM)ofgalaxyclusters,heatandmomentumaretrans-
O ported almost entirely along (but not across) magnetic field lines. We perform the first
C fully self-consistent Braginskii-MHD simulations of galaxy clusters including both of
these effects. Specifically, we perform local and global simulations of the magnetother-
.
h mal instability (MTI) and the heat-flux-driven buoyancy instability (HBI) and assess
p the effects of viscosity on their saturation and astrophysical implications. We find that
-
viscosity has only a modest effect on the saturationof the MTI. As in previous calcula-
o
r tions, we find that the MTI can generate nearly sonic turbulent velocities in the outer
t parts of galaxyclusters,althoughviscosity somewhatsuppresses the magnetic field am-
s
a plification.Atsmallerradiiincool-coreclusters,viscositycandecreasethelineargrowth
[ rates of the HBI. However, it has less of an effect on the HBI’s nonlinear saturation, in

part because three-dimensional interchange motions (magnetic flux tubes slipping past
1
each other) are not damped by anisotropic viscosity. In global simulations of cool core
v
4 clusters,weshowthattheHBIrobustlyinhibitsradialthermalconductionandthuspre-
5 cipitatesacoolingcatastrophe.Theeffectsofviscosityare,however,moreimportantfor
7 higher entropy clusters. We argue that viscosity can contribute to the global transition
0 of cluster cores from cool-core to non cool-core states: additional sources of intracluster
. turbulence, such as can be produced by AGN feedback or galactic wakes, suppress the
1
0 HBI, heating the cluster coreby thermalconduction; this makes the ICMmore viscous,
2 which slows the growth of the HBI, allowing further conductive heating of the cluster
1 core and a transition to a non cool-core state.
:
v Key words: convection—galaxies: clusters: intracluster medium—instabilities—
i turbulence—X-rays: galaxies: clusters
X
r
a
1 INTRODUCTION parametersthemeanfreepathofelectronsalongthemagnetic
field line is & 1012 times larger than the gyroradius at all
Clusters of galaxies are thelargest gravitationally boundob-
radiiintheICM.Ionshaveasimilarseparationofscales.The
jects in the universe, and as such, they potentially provide
meanfreepathis,however,alwaysshorterthanthelocalscale
sensitivetestsofcosmologicalparameters.Theyarefilledwith
height;therefore,afluiddescriptionoftheplasma(asopposed
a hot, dilute, magnetized plasma, the intracluster medium
toacollisionless description)isappropriateandtheICMcan
(ICM), that emits copious X-rays. Observations of the ICM
be described by the Braginskii-MHD equations (Braginskii
provideaninterestinganduniquewindowonproblemsrang-
1965).TheseequationsarethestandardidealMHDequations
ing from constraining dark energy to understanding the ac-
supplementedwithanisotropicconductionduetotheelectron
cretion and feedback processes for supermassive black holes.
heat flux and anisotropic momentum transport due to the
The plasma in the ICM has temperatures ranging from
ion viscosity along the magnetic field. In theICM, transport
1–15keVandnumberdensitiesfrom10−4to10−1 cm−3.This
perpendicular tothe local magnetic field is negligible.
dilute plasma has a magnetic field that has been estimated
torangefrom 0.1–10 µG(Carilli & Taylor2002).Withthese As a result of the anisotropic heat transport in the
ICM, the Schwarzschild criterion for convective instability—
that the entropy increase in the direction of gravity—is re-
⋆ E-mail:[email protected] placed by a criterion on temperature. In recent years, two
(cid:13)c 0000RAS
2 Parrish, McCourt, Quataert, and Sharma
buoyant instabilities have been discovered that drive con- theeffectofviscosityontheshapesofrisingAGN-blownbub-
vection with the temperature gradient as a source of free bles. More recently, Dong & Stone (2009) showed that it is
energy. The first instability, the magnetothermal instability criticaltoconsideranisotropicviscosity,ratherthanisotropic
(MTI), was described in Balbus (2000) and has been sim- viscosity,insuchcalculationsbecausetheformerismuchless
ulated in two and three dimensions (Parrish & Stone 2005, effectiveat suppressing the Rayleigh-Taylor instability.
2007; McCourt et al. 2011a). The MTI is unstable when the Inthispaper,wepresentfullyself-consistent 2Dand3D
temperature gradient and gravity are in the same direction Braginskii-MHDsimulationsoftheICM,focusingontheevo-
and grows fastest for a magnetic field perpendicularto grav- lutionoftheMTI andHBI.Weintroduceourcomputational
ity.TheMTI operates intheoutskirtsofgalaxy clusters and methods in §3. In §4 we present local 2D and 3D simula-
has been found to drive vigorous convection that can pro- tions of the HBI and MTI to provide physical insight into
vide over 30% of the pressure support near the virial radius theroleofviscosity.Wethenuseglobalcalculationstostudy
(Parrish et al. 2011). The second instability, the heat-flux- the effect of viscosity on the MTI in the outskirts of galaxy
drivenbuoyancyinstability (HBI)wasdescribed inQuataert clustersandtherole oftheviscousHBIin clustercoresin§5
(2008)andhasbeensimulatedinlocalsimulationsin2Dand & §6, respectively. In the appendix we describe our numeri-
3D(Parrish & Quataert2008).TheHBIisunstablewhenthe cal method for anisotropic viscous transport and discuss the
temperaturegradientandgravitypointinoppositedirections numerical verification of this algorithm.
and has the fastest growth for a magnetic field parallel to
gravity.TheHBIoperatesin thecentersofcool-core clusters
andsaturatesbyreorientingthemagneticfieldtobeperpen-
2 METHOD AND MODELS
dicular to gravity, greatly reducing the effective radial con-
ductivity and hastening a cooling catastrophe (Parrish et al. We solve the usual equations of magnetohydrodynamics
2009; Bogdanovi´c et al. 2009). The interaction of the HBI (MHD) with the addition of anisotropic thermal conduction
with turbulence can help explain the bimodality observed and anisotropic viscous transport. The MHD equations in
between cool core and non-cool core clusters (Parrish et al. conservativeform are
2010; Ruszkowski& Oh 2010).
∂ρ
None of these previous numerical studies have self- +∇·(ρv)=0, (3)
∂t
consistently included viscosity because the ratio of viscous
to thermal diffusion, the Prandtl number,1 for a hydrogenic
∂(ρv) B2 BB
plasma is +∇· ρvv+ p+ I− +Π +ρg=0, (4)
∂t 8π 4π
ν (cid:20) (cid:18) (cid:19) (cid:21)
Pr= ≈0.01. (1)
χ ∂E B2 B(B·v)
+ ∇· v E+p+ − +Π·v (5)
where ν and χ are the diffusion coefficients for momentum ∂t 8π 4π
(cid:20) (cid:18) (cid:19) (cid:21)
(by ions only) and thermal energy (by electrons only), re- + ∇·Q+ρ∇Φ·v=−L,
spectively. Pr ∼0.01 is relatively small, and thus the effects
of viscosity were expected to be small in comparison with ∂B
thoseofconduction.2 However,theReynoldsnumberisfairly −∇×(v×B)=0, (6)
∂t
small in theICM:
whicharetheequationsofconservationofmass,momentum,
U L n k T −5/2 andenergyandtheinductionequation,respectively.Thetotal
Re=4 e B ,
100kms−1 100kpc 0.05cm−3 3keV energy E is given by
(cid:18) (cid:19)(cid:18) (cid:19)(cid:16) (cid:17)(cid:18) (cid:19)(2)
v·v B·B
and thus it is not so clear that the effects of viscosity can E =ǫ+ρ + , (7)
2 8π
beneglected.Kunz(2011)(hereafter K11) recentlyextended
thelineardispersionrelationfortheMTIandHBItoinclude where ǫ=p/(γ−1). Throughout this paper, we assume γ =
anisotropicviscosityandprovidesanintuitive,physicalexpli- 5/3. The anisotropic electron heat fluxis given by
cation of its effects. The most important is that the growth Q=−κ ˆbˆb·∇T, (8)
Sp
ratesoftheHBIcanbesuppressedbyviscosityinthelimitof
ˆ
veryrapidconduction(andthusveryrapidviscousdamping). where κ is the Spitzer conductivity (Spitzer 1962) and b
Sp
Isotropicviscosityhasbeenutilizedinasmallnumberofpre- is a unit vector in the direction of the magnetic field. The
vious numerical studies; e.g., Reynoldset al. (2005) studied Spitzer conductivity can be written as κ = n k χ, where
Sp e B
χ is the actual diffusion coefficient (in units of L2T−1). The
viscous stress tensor is given by
1maTghniestiqcuPanrtaintydtslhnouumldbenrotinbewhcoicnhfutsheedtwhietrhmtahledoiffftuesniv-ditiyscuisssreed- Π≡−3ρν ˆbˆb:∇v− ∇·v ˆbˆb− I , (9)
3 3
placed by the electrical resistivity. The magnetic Prandtl number (cid:20) (cid:21)(cid:20) (cid:21)
isverylargeintheICM. whereν isthemicrophysicalmomentumdiffusioncoefficient,
2 The effective value of the Prandtl number that enters the per-
often termed the kinematic viscosity. Both transport coef-
turbedtotal energy equation inthe linearanalysiscorresponds to
ficients are functions of temperature proportional to T5/2,
Preff=0.02.Thisisbecauseµ=0.5:onlyelectronsparticipatein
where we presume T = T . The microphysics fixes the ra-
conduction, while both electrons and ions contribute to the total e i
thermal energy. The net effect is that χ is smaller for the MHD tio of the transport coefficients to be Pr = 0.01 as given by
fluid by a factor of 2. In our calculations we take Pr = 0.01 and Equation (1).
µ=0.5. Theenergyequationalsoincludesacoolingterm,L.The
(cid:13)c 0000RAS,MNRAS000,000–000
Anisotropic Viscosity in the Intracluster Medium 3
cooling function we adopt is from Tozzi & Norman (2001) dispersion relation herein a slightly different coordinate sys-
with thefunctional form tem. We define gravity to be in the z-direction, the initial
magnetic field is in the x-y plane and k2 = k2 +k2 is the
⊥ x y
L=n n Λ(T), (10)
e p component of the wavevector perpendicular to gravity. The
withunitsofergcm−3 s−1.Thetemperaturedependenceisa dispersion relation for thegrowth rate, σ,is given by
fittocooling dominatedbyBremsstrahlungabove1keVand ˆ ˆ 2
−ω 1− b·k (12)
metal lines below 1 keV with visc
(cid:20) (cid:16) (cid:17) (cid:21)
Λ(T)= C1(kBT)−1.7+C2(kBT)0.5+C3 10−22, (11) σ˜2 σ˜2(σ+ωcond)+σN2kk⊥22 +ωcondωb2uoyK
= ,
w10h−e2r,efCor1 a=(cid:2)m8e.6ta×llic1i0t−y3o,fCZ2 == 50..83Z×1,0w−i2t,ha(cid:3)unnditCs3of=[C6.3] =× σ σ˜2h(σ+ωcond)+ σN2+ωcondωb2uoy 1−b(2xˆbk·kyˆ2i)2
⊙ i h (cid:16) (cid:17) i
ergcm3s−1. We use a mean molecular weight of µ ∼ 0.62 whereσ˜2 ≡σ2−(k·vA)2,andvA=B/(4πρ)1/2istheAlfv´en
whichcorrespondstoametallicityofapproximately1/3solar. velocity.The Brunt-V¨ais¨al¨a oscillation frequency is given by
For our simulations we use the Athena MHD code
1 ∂P ∂lnS
(Gardiner & Stone 2008; Stone et al. 2008) combined with N2 =− , (13)
γρ ∂z ∂z
the anisotropic conduction methods of Parrish & Stone
(2005) and Sharma & Hammett (2007). The anisotropic vis- where S ≡Pρ−γ, and corresponds to buoyant oscillations in
cosityisimplementedinaverysimilarmannertoconduction an unmagnetized plasma (g-modes). The characteristic fre-
(seetheappendix).Theheating,cooling,andanisotropiccon- quencyat which conduction and viscosity act are given by
duction and viscosity are operator split and sub-cycled with
2 ˆ 2
respect to the MHD timestep. The cooling simulations are ωcond = 5χ b·k , (14)
ilmowplwemhiecnhteUdVwliitnhesabteecmompeeriamtuproertflaonotr,aonfdTth=e0c.o0o5liknegVc,ubrvee- ω = 3vt2(cid:16)h ˆb·(cid:17)k 2, (15)
visc 2 ν
fitis nolonger accurate. Thistemperaturefloor preventsthe ii
(cid:16) (cid:17)
cooling catastrophe from going to completion. where χ is the thermal diffusivity and νii is the ion collision
This paper will cover a variety of initial conditions from frequency.The buoyancyfrequency is given by
localCartesianboxestoglobalclustermodels.Ourinitialcon-
∂lnT
ditions will thus be described briefly in subsequent sections ωb2uoy = g ∂z , (16)
withappropriatereferencesformoredetails.Ineachcase,we (cid:12) (cid:12)
(cid:12) (cid:12)
havecarriedoutaleastoneresolutionstudytoassurenumer- which is roughly the fastest g(cid:12)rowth ra(cid:12)te of the MTI or HBI.
(cid:12) (cid:12)
ical convergence. Anydeviations will be noted. Finally, thedimensionless geometric factor is given by
k2 2b b k k
3 PHYSICS OF THE MTI AND HBI WITH K= 1−2b2z k⊥2 + xkz2x z, (17)
VISCOSITY
where b = B /(cid:0)B is th(cid:1)e dimensionless magnetic field
x x
We begin with a qualitative description of the physics of the strength.
HBI and MTI. Despite the mathematical similarities of the Wecan greatly simplify thisexpression andimproveour
instabilities, it isusefultodiscusseach instabilityseparately. physicalintuitionbyassumingthatthemagneticfieldisweak
The MTI is most unstable for horizontal (B ⊥ g) magnetic (or, equivalently, that the perturbation wavelength in the
fieldswiththetemperaturegradientdT/dz<0.Anupwardly clustercoreλ≫vA/τ ∼2kpc,whereτ isatypicaltimescale
displacedfluidelementisconnectedbymagneticfieldlinesto for the instabilities to grow). We also assume the perturba-
ahotterregion deeperintheatmosphere.Heatflowingalong tion lies in theplane containing B and g, i.e. ky =0. In this
the field expands the fluid element relative to its surround- limit, thedispersion relation simplifies to
ings, lowers its density and buoyantly destabilizes the per-
ˆ ˆ 2
turbation. The upward motion causes the magnetic field to σ(σ+ωcond) σ+ωvisc 1− b·k (18)
be more aligned with the background temperature gradient, (cid:20) (cid:18) (cid:16) (cid:17) (cid:19)(cid:21)
leading toan instability. + σN2k⊥2 +ω ω2 K=0.
The HBI, on the other hand, is most unstable for ver- k2 cond buoy
tical (B k g) magnetic fields with the temperature gradient Without viscosity, there is a fast conduction limit in
dT/dz > 0. Imagine a small displacement of fluid elements which the growth rate asymptotes to a constant σ =
with a wavevector that has a component both parallel and ω k2/k2; with viscosity, however, no such limit exists.
buoy ⊥
perpendicular to gravity (and B). This configuration, illus- In Figure 1 we plot the theoretical curves for the growth
tratedinFigure1ofQuataert(2008),hasregionsinwhichthe rates of both the HBI and MTI as a function of the ratio
magneticfieldlinesbunchtogetherandspreadapart,leading ω /ω for several Prandtl numbers. These curves as-
cond buoy
to a converging and diverging heat flux. Rapid heat conduc- sume a wave vector k at 45 degrees from the magnetic field
tion along the perturbed magnetic field lines causes an up- B; these represent typical, not maximum, growth rates. The
wardly displaced fluidelement to beheated (by tappinginto ratio ω /ω ∝k2 so that larger values of thisratio cor-
cond buoy
thebackground heat flux),leading to a buoyantrunaway. respond physically to smaller scales at fixed conductivity.
K11 carried out a full linear perturbation analysis on Figure 1 shows that the inviscid solution reaches an
equations3–8and presentsadispersion relation for theMTI asymptotic growth rate in the fast conduction limit; how-
andHBIincludingtheeffectsofanisotropicviscosity.Forclar- ever, with viscosity the growth rate slowly decreases as one
ity when comparing with our results, we reproduce the K11 moves to the infinite conduction limit. For the HBI, the
(cid:13)c 0000RAS,MNRAS000,000–000
4 Parrish, McCourt, Quataert, and Sharma
ˆ ˆ
HBI, bz = 1 MTI, bz = 0
11
✽ ✽ ✽ ✽ ✽
✽✽ ✽✽ ✽✽ ✽
✽✽ ✽ ✽ ✽ ✽
✽
✽✽ ✽ ✽
✽
oy ✽
bu−1−1
ω 00
/ 11 ✽
σ
Pr=0.00
✽
Pr=0.01
Pr=0.06
1100−−11 11 1100 110022 11 1100 110000
ω /ω ω /ω
cond buoy cond buoy
Figure 1.Thelinesshowthetheoretical lineargrowthratesoftheMTI(right)andHBI(left)foravarietyofconductionfrequenciesand
severalPrandtlnumbersforkat45◦ relativetothemagneticfield.Thisgeometryisjustanexampleanddoesnotshowthefastestgrowing
mode for either the MTI or HBI. In particular, for the MTI there are modes withgrowth rates comparable to the inviscid(Pr = 0) case
evenforωcond ≫ωbuoy (K11); thisisnottruefortheHBI.RealastrophysicalplasmashavePr=0.01.Pr=0.06isshownonlyasatest
of our numerical methods. Low ωcond corresponds to larger scales at fixed conductivity. Measured growth rates from 2D simulations are
plottedasstarsymbolsandagreewellwiththeanalyticresults.
fastest growth occurs on scales satisfying ω ∼ ω , i.e., methodology described in McCourt et al. (2011b) with the
visc buoy
ω ∼6ω ,andthemaximumgrowthrateforPr=0.01 clusters listed in Table 2 of that work. These examples span
cond buoy
is about 70% of the asymptotic inviscid value. This is in- relatively massive NCC clusters all the way down to groups.
dependent of k unless k ≪ k . For higher viscosities, Cool-core (CC) clustershaveω /ω ∼5–30 inthebulk
⊥ ⊥ k cond buoy
the maximum growth rate scales as ∼ ω2 /ω . The in- of the core between a few–200 kpc. By comparing the real
buoy visc
fluence of viscosity on the MTI is much more modest. In CC clusters to Figure 1 we see that viscosity only modestly
particular, the fastest growing mode is achievable even for reduces the growth rates of the HBI. Non-cool-core (NCC)
ω ,ω ≫ ω ; this is not seen in Figure 1 because of clustersaremoreconducting/viscousasaresultofthehotter
visc cond buoy
theparticularperturbationchosen.Onecan,however,always and lower density cores. The effects of viscosity on the HBI
find short wavelength MTI modes that grow at ∼ω . aretheoreticallymorelikelytobesignificantinNCCclusters.
buoy
The microphysical plasma physics fixes the ratio of the The star symbols in in Figure 1 show good agreement
diffusion coefficients, the Prandtl number (eqn. [1]), to be betweenthemeasuredgrowthratesinoursimulationsandthe
Pr = 0.01. This value corresponds to ω /ω = 1/6. analyticresults;thisrepresentsastrongtestofouralgorithms
visc cond
Because there are MTI modes that grow rapidly even for for anisotropic conduction and viscosity. We discuss this in
ω ≫ω ,wewouldapriorinotexpectviscositytohave more detail in theappendix.
cond buoy
asignificant effectontheevolutionoftheMTI.FortheHBI,
the viscous and inviscid growth rates are similar only when
ω /ω . 10; these larger scales in a physical system
cond buoy 4 LOCAL SIMULATIONS
are only minimally affected by viscosity. We thus expect the
HBI to be modified by viscosity only if ω ≫10ω on 4.1 Initial Conditions
cond buoy
thelargest scales of thesystem.
It is illustrative to start with the simplest possible experi-
InFigure2weshowtheratioofω /ω forobserved
cond buoy
ments, namely local two- and three-dimensional boxes with
clusters using k = 2π/r as the conduction length scale and
ˆ ˆ systemsizesL.thescale-heightH.Forthissectionwework
not including any geometric factors, i.e. k·b = 1. The left
in units with k =m =1, g =−1, and with a hydrogenic
panel focuses on the core of the cluster where the HBI op- B p 0
plasma that has µ=1/2. Ourinitial conditions are fully de-
erates while the right panel is at larger radii where the MTI
tailed in McCourt et al. (2011a). For the HBI, we start with
is present. The values of ω /ω in Figure 2 are charac-
cond buoy
a simple hydrostatic equilibrium that is linearly unstable:
teristic of the largest scales in the system. Smaller scales are
more viscous/conducting, but are also where magnetic ten- T(z) = T (1+z/H), (19)
0
sion is most likely to suppress the MTI/HBI. Figure 2 also ρ(z) = ρ (1+z/H)−3, (20)
showstheratioω /ω (definedinthesameway)forour 0
cond buoy
model clusters used in the global simulations in §5 & 6. The wherewesetT =ρ =1andchooseascaleheightofH =2.
0 0
models are generally representative of real clusters. OurboxesareofsizeL=0.2ineachdimensionwitharesolu-
The data in Figure 2 comes from the ACCEPT sample tionof (96)2 or(96)3.Resultsat thisresolution areverywell
(Cavagnolo et al. 2009); the analysis of the data follows the converged by all metrics. The vertical boundary conditions
(cid:13)c 0000RAS,MNRAS000,000–000
Anisotropic Viscosity in the Intracluster Medium 5
CC(ACCEPT) CC(ACCEPT)
30 NCC(ACCEPT) NCC(ACCEPT)
1
CC(HBI) MTI
NCC(HBI)
)
r
=
λ20
( 1
y
o
u
b
ω
/
d
n
o
c
ω
0
1
50 100 150 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
r (kpc) r (Mpc)
Figure 2. Left: The measured ratio of ωcond/ωbuoy for the cores of clusters in the ACCEPT database, taking an average value of
|dlnT/dlnr|=1/3 in ωbuoy and k =2π/r in ωcond. The latter choice corresponds to the conduction time across the local radius r and
represents the smallest value of ωcond/ωbuoy at a given radius. Cool-core clusters have ωcond/ωbuoy ∼ 10 in the bulk of the cool core.
Non-cool-core clusters have somewhat more rapid conduction and are thus more viscous as well. We have also overplotted the values of
ωcond/ωbuoy inourglobalclustermodelsusedforHBI(theCCandNCCHBImodels).Right: Themeasuredvaluesofωcond/ωbuoy inthe
outskirts of clusters in the ACCEPT sample. Our model cluster for the global MTI simulation is overplotted as well. Our fiducial model
clustersarereasonablysimilartoobservedclusters.
fix the temperature to its initial value, and the horizontal time t = 30 t . This is well into the non-linear regime, in
HBI
boundary conditions are periodic. The initial magnetic field which the statistical properties of the plasma do not evolve
is weak and vertical, B =10−6zˆ. stronglywithtime.Weareabletosimulateavarietyofphys-
0
Forthelocal MTI simulations weutilize theset-upfrom icalscalesintheselocalsimulationsbychangingtheconstant
§3.3 of McCourt et al. (2011a) given by conductivity. We then label the simulation with a ratio of
ω /ω calculated using the box size, λ = L, for the
Sω2 cond buoy
T(z) = T exp buoy 1−ez/S , (21) conduction length in ωcond. For ωcond/ωbuoy = 2, the left-
0 " g0 # most column shows that viscosity does not strongly change
(cid:16) (cid:17)
the saturated state. The inviscid cases shown in the top row
g(z) = g e−z/S, (22)
0 all show qualitatively no difference in the nonlinear satu-
with T0 = g0 = 1 and S = 3. We select ωb2uoy = 1/2 and rated state as a function of ωcond/ωbuoy; however, the ef-
solve numerically for ρ(z) to ensure hydrostatic equilibrium. fectofanisotropicviscositybecomesclearfortheviscouscase
TheseboxesareofsizeH/2withaninitially weakhorizontal withωcond/ωbuoy =60(bottomright).Thiscaseshowspromi-
magneticfield,B =10−6zˆ.Convergenceisabitmoresubtle nent, highly-bunched vertical magnetic field structures that
0
in these simulations, but 1283 is reasonably well-converged are never seen in the inviscid case. This bunching occurs be-
(McCourt et al. 2011a). causethefieldlinescannotslippasteachotherin2D,apoint
For the HBI, the atmosphere satisfies the Schwarzschild we will return to shortly.
criterion (dS/dz > 0) and would be buoyantly stable in the In order tobetterunderstand thelimitations of 2D sim-
absenceofanisotropicconduction.Inthesecalculationswefix ulations, it is useful tocompare the2D and 3D evolution for
the diffusivities so that the conduction frequency across the identical parameters. Figure 4 shows the exact same simula-
box is a chosen constant relative to the buoyancy frequency, tions performed in 2D and 3D for three different conductivi-
and show results for different values of ω /ω for both ties.FortheHBI,theresultsforsimulationswithdifferentbox
cond buoy
theinviscid case and thephysical case with Pr=0.01. sizesareessentiallythesameprovidedthatthesamevalueof
ω (k = 2π/L)/ω is used (McCourt et al. 2011a). We
cond buoy
4.2 Nonlinear Saturation
demonstratethisresultexplicitlyinFigure5whichcompares
the magnetic geometry evolution for the viscous HBI in two
4.2.1 HBI
different box sizes. To keep the ratio of ω /ω = 50
cond buoy
We begin by examining the nonlinear saturation of the HBI fixed,thesimulation with L/H =0.3 has aconductivityand
with 2D single mode simulations. The magnetic field lines viscosity that are 9 times larger than the simulation with
from thesesimulations arevisualized inFigure3.Eachverti- L/H =0.1.Theevolution,especiallyatlinearandearlynon-
cal column compares inviscid and viscous simulations at the linear times, is nearly indistinguishable. At very late times,
(cid:13)c 0000RAS,MNRAS000,000–000
6 Parrish, McCourt, Quataert, and Sharma
ω /ω = 2 ω /ω = 6 ω /ω = 60
cond buoy cond buoy cond buoy
0.20
0.15
Pr = 0
z 0.10
0.05
0.00
0.20
0.15
Pr = 0.01
z 0.10
0.05
0.00
0.00 0.05 0.10 0.15 0.00 0.05 0.10 0.15 0.00 0.05 0.10 0.15 0.20
x x x
Figure3.Snapshotsofthemagneticfieldlinesfrom2DHBIsimulations.Thetoprowdisplaysinviscidsimulations,whilethebottomrow
includes anisotropic viscosity. The conductivity increases from left to right; the snapshots at a given conductivity are shown at the same
absolutetimeforboththeinviscidandviscoussimulation(30HBIgrowthtimesfortheinviscidmodel).Atlowωcond/ωbuoy,viscosityhas
littleeffectonthenonlinearevolutionoftheHBI;however,forωcond/ωbuoy∼60,anisotropicviscositycausesthemagneticfieldtobunch
intopronouncedverticalstructures.Thiseffect ismuchweakerin3Dsimulationsrelativetothe2Dresultsshownhere(seeFig.4).
the larger box has experienced slightly more magnetic field What is the cause of the striking difference in the 2D
evolution as a result of the different modes in the domain and 3D evolution of the HBI with viscosity? Recall in Fig-
relative to the scale height. We conclude from Figure 5 that ure 3 that the magnetic field lines strongly bunched up in
local simulations of the HBI with L . H are independent 2D and could not slip past each other. This type of motion
of domain size; the results depend instead on the value of is known as the interchange mode and can be thought of as
ω /ω , the fundamental dimensionless parameter that twomagneticfluxtubesslippingpasteachother.Interchange
cond buoy
quantifiesthe influence of conduction and viscosity. Thus al- motions have v ⊥B and therefore are not damped at all by
though the boxes in Figure 4 are only ∼ 0.1H in size, the anisotropic viscosity, although they would be damped by an
local simulations with ω /ω = 18 & 50 are quite in- isotropic viscosity. In 3D the magnetic field lines are able to
cond buoy
dicative of how the HBI would evolve on large scales in the slip past each other, and the HBI is able to continue to re-
ICM(seeFig.2).Thelargescalesarethemostimportantbe- orient the magnetic field. This effect is seen in a different
cause both tension and viscosity suppress the growth of the guise in studies of the magnetic Rayleigh-Taylor (RT) insta-
HBI on smaller scales. We justify this interpretation of the bility(Stone& Gardiner2007).In2D,magnetictensionsup-
local HBI simulations using global cluster models in §6. presses the RT instability below a critical wavelength; how-
ever, in 3D magnetic tension does nothing to suppress the
ˆ
Figure4showsthemeanmagneticfielddirection,h|b |i,
z interchangemotionsoffluxtubes.Thus,theinstabilitygrows
whichisrelated totheeffectiveverticalthermalconductivity
despite the apparent wavelength cut-off in the dispersion re-
ˆ2
as fSp ∼ hbzi. In 2D the ability of the HBI to reorient that lation.Thisuniquely3D effectofanisotropic viscosity makes
magnetic field is retarded by viscosity relative to the invis- 2D studies of nonlinear saturation irrelevant, and greatly re-
cid case, especially for ωcond/ωbuoy =50 (right). However, in ducestheeffectofviscosityrelativetowhatonemightna¨ıvely
3D the effect of the viscosity is much less pronounced. The expectfromthedispersionrelation.Itisparticularly striking
biggest effect of viscosity is that for ωcond/ωbuoy = 50, the for ωcond/ωbuoy = 50 (right panel in Fig. 4) the non-linear
initial growth of the HBI is slower; however, the instability evolution in 3D is quite similar with and without viscosity,
stillproducesasignificantrearrangementofthemagneticfield althoughtheviscousevolutionisdelayedbytheconsiderably
by t ∼ 50tbuoy. We note that our initial velocity perturba- longer linear growth time.
tionshavewhitenoise amplitudeswith amean magnitudeof
5×10−4c . For a larger and more realistic perturbation, the
s
linear phasewould complete more quickly.
(cid:13)c 0000RAS,MNRAS000,000–000
Anisotropic Viscosity in the Intracluster Medium 7
Figure 4. The nonlinear evolution of the magnetic field due to the HBI for ωcond/ωbuoy = 2, 18, and 50 (left to right), in 2D and 3D
localsimulations(L=0.1H).We quantifythefieldevolutionusingthevertical component of themagnetic field.Thedifferences between
the viscous and inviscid simulations are much smaller in 3D (blue lines) than in 2D (black lines). We attribute this to the presence of
interchange modes in 3D. Even for ωcond/ωbuoy ∼50 the nonlinear evolution of the HBI in3D is similarto that in inviscid simulations,
withtheprimarydifferencebeingthatthenonlinearsatuationisdelayedbecauseofthereducedlineargrowthrate.
4.2.2 MTI slightly more vertical than isotropic (about a 10% change
in h|b |i). We can gain insight into this behavior by consid-
z
Next,we also examine thenonlinear evolution of theMTI in
ering the evolution of simulations with an initially vertical
3DCartesiansimulations.Thedispersionrelationshowsthat
field, a case that is linearly stable, but non-linearly unstable
thefastestgrowingmodesoftheMTIarelessaffectedbyvis-
(see Figure 6 of McCourt et al. 2011a). An initial horizontal
cosity than they are for the HBI. Figure 6 shows the results
perturbation is purely Alfv´enic in nature, and thus is unaf-
of our Cartesian simulations of the MTI with and without
fected by Braginskii viscosity. A simulation with an initially
viscosity andreveals several interestingproperties. First, the
vertical field and ω /ω = 1 (blue dotted line) evolves
cond buoy
saturatedkineticenergiesarelargelyindependentofboththe
towards an isotropic magnetic field configuration. However,
ratio of ω /ω and whether anisotropic viscosity is in-
cond buoy in doing so, the field develops a component parallel to the
cluded. Note that the simulation with ω /ω = 50 is
cond buoy velocity, which is susceptible to viscous damping. Thus, for
consistent with the range of clusters in the ACCEPT sam-
an initially vertical magnetic field, the higher viscosity sim-
ple that we have examined (see the right panel of Figure 2).
ulation (ω /ω = 50, blue dashed line) evolves much
cond buoy
The fact that the kinetic energies are so similar even in the
more slowly than its less viscous counterpart. It is precisely
high-viscosity simulation, is likely the result of interchange-
thispartialstabilization ofmodesthatinitially havethefield
like turbulence.
alignedwithgravitythatresultsintheradialbiasseeninthe
Second, we find the very interesting result that the high-conductivity,horizontal simulation.
growth of the magnetic energy (middle panel of Figure 6)
is suppressed as theviscosity is increased. This result can be
understood in relatively easy terms:theincrease in themag-
5 GLOBAL MODELS OF THE MTI IN
netic field strength in this typeof turbulenceis proportional
CLUSTER OUTSKIRTS
to the increase in the length of the magnetic field line, as it
is stretched and tangled. These parallel stretching motions In this section we discuss the effect of Braginskii viscosity
are precisely those motions that are damped by anisotropic on the evolution of the MTI in the outskirts of galaxy clus-
viscosity, thussuppressing thegrowth of themagnetic field. ters using global cluster models. Beyond approximately the
Finally,weconsiderthereorientationofthemagneticge- scale radius of a cluster, the temperature profile of the ICM
ometry by theMTI (bottom panel of Figure 6). In theinvis- almost always declines with radius, thus making it unstable
cid case, an initially horizontal magnetic field is re-oriented to the MTI. Recent work has shown that the MTI is able
to be largely isotropic, h|b |i = 1/2 in 3D. For modest vis- todrivelargeturbulentvelocitieswhich providenon-thermal
z
cosities ω /ω = 1, there is little change in the sat- pressuresupportinhot,massiveclusters(Parrish et al.2011).
cond buoy
urated geometry; however, for higher conductivity and vis- Itiscriticaltounderstandthisnon-thermalpressuresupport
cosities (e.g., ω /ω = 50) the magnetic field becomes and its implications for measuring cluster masses for use in
cond buoy
(cid:13)c 0000RAS,MNRAS000,000–000
8 Parrish, McCourt, Quataert, and Sharma
Figure 5. The evolution of the magnetic field orientation for the Figure6.ThenonlinearevolutionoftheMTIforωcond/ωbuoy=1
viscous HBI(Pr=0.01) for two different domain sizes relative to and50in3D (L=H/2) withandwithout viscosity. Weomitthe
thetemperaturescaleheight(L/H).Tokeeptheratioofωcond(k= ωcond/ωbuoy = 50 inviscid simulation, as its evolution is almost
2π/L)/ωbuoy = 50 fixed, the simulation with L/H = 0.3 (dotted exactly thesameas theωcond/ωbuoy =1inviscidsimulation.The
line)hasaconductivity andviscositythatare9timeslargerthan kinetic energy (top panel) islargelyindependent of viscosity. The
the simulation with L/H =0.1 (solid line). For local simulations magneticenergygeneration(middlepanel)issuppressedforhigher
(L .H)thelinearandnonlinearevolutionarenearlyindependent viscosities.Themagneticgeometry(bottompanel)forinitiallyhor-
ofboxsizeforfixedωcond/ωbuoy. izontalmagneticfields(blacklines)becomesrelativelyisotropicfor
allthesimulations,althoughslightlymoreverticalwithhighervis-
cosity. Statistical isotropy corresponds to h|bz|i = 0.5 (red, long-
cosmological parameter estimation through either X-ray or dashed line). Simulations with initially vertical fields (blue lines)
arelinearlystablebutnonlinearlyunstable(McCourtetal.2011a);
Sunyaev-Zelovich(SZ) methods.
viscosityslowsthereorientationofmagneticfieldinthiscase.
Our initial condition for this section is a spherically-
symmetric, hot, massive cluster that resembles Abell 1576
with a mass of 1.6×1015 M . We use a softened NFW pro-
⊙
file with a scale radius of r = 600 kpc and a softening ra- gradient, we fix the temperature at the peak of the cluster
s
dius of 70 kpc. We initialize an atmosphere in hydrostatic temperature profile (approximately 10 keV at 200 kpc) and
equilibrium using the entropy power law in the ACCEPT atthemaximumradius(approximately6.5keVat1225 kpc)
databaseforAbell1576:acentralentropyK =186keVcm2, of our model cluster to the initially-computed temperature
0
K = 98 keVcm2, and power-law exponent, α = 1.38 values.Thus, we are imposing Dirichlet boundaryconditions
1
(Cavagnolo et al.2009).Ifweassumethatourfiducialcluster on the temperature profile. This fixed temperature gradient
is located at z =0.1, then for the appropriate WMAP5 cos- then drivesthecontinued evolution of theMTI.
mological parameters r = 1.09 Mpc, and the virial radius Figure7showsthenon-linearevolutionoftheMTIinthe
500
is r = 1.6 Mpc, where r corresponds to an overdensity global galaxy cluster context. The evolution of the kinetic
200 ∆
of ∆ timesthecritical density.Wedonot includecooling, as energy with time is quite similar for both the viscous and
wefocusontheportionoftheclusterthatiswelloutsidethe inviscid case just asin thelocal simulations (Figure6);how-
cooling radius. For full simulation details, see Parrish et al. ever,theamplificationofthemagneticfieldisagainsomewhat
(2011). By examining Figure 2, we can see that the ratio of suppressed in theviscous case. Themagnetic field stretching
ω /ω in our model is consistent with the range ob- necessary to amplify the field is exactly the motion that is
cond buoy
served in real clusters in the ACCEPT database at the radii damped by anisotropic viscosity. This damping of magnetic
of interest (r&200 kpc). field amplification makes it more difficult to use a turbulent
The simulations are carried out on a (196)3 Cartesian dynamo,regardlessofthesourceoftheturbulence,toamplify
gridinacomputationaldomainthatextendsfromthecenter the magnetic field from primordial values to those observed
of the cluster out to ±1300 kpc. Within this Cartesian do- todayingalaxyclusters.Themagneticgeometryevolutionis
main, we define a spherical subvolume with a radius of 1225 verysimilar with andwithout viscosity,asweinitially began
kpcfrom which we extract cluster properties. In thisvolume with a geometrically isotropic magnetic field. A very slight
we initialize tangled magnetic fields with h|B|i = 10−8 G radial bias exists in the steady state in both cases.
(plasmaβ ∼104–106)andaKolmogorovpowerspectrum.In Figure 8 shows the saturated and azimuthally-averaged
ordertosimulateclusterswithanegativeradialtemperature rms Mach number profile for our model cluster at a time of
(cid:13)c 0000RAS,MNRAS000,000–000
Anisotropic Viscosity in the Intracluster Medium 9
Figure7.Thenon-linearevolutionofvolume-averagedproperties Figure 8. The saturated and azimuthally-averaged rms Mach
oftheMTIinclustersinourglobalmodelwith(dotted lines)and numberprofileduetotheMTIinourglobalmodelcluster(model
without(solidline)viscosity.Theevolutionofthevolume-averaged MTIinFig.2). Theinviscid(solid line)and viscous (dotted line)
kinetic energy (top panel) and the the magnetic field geometry simulationsdifferonlybyasmallamount.
(bottom panel) are largely unchanged by the viscosity. The mag-
netic geometry is measured with respect to the radial direction:
θ =0◦ corresponds to radial,and θ =60◦ (red, long-dashed line)
corresponds to a random field. The amplification of the volume- further? There has been speculation that thermal conduc-
averaged magnetic energy (middle panel) is modestly suppressed tionfrom largeradiicouldoffsetthecoolingluminosity (e.g.,
byanisotropicviscosity,similartowhatwas observedinourlocal Narayan & Medvedev2001);however,theHBIpresentsase-
simulations. rious problem to this scenario. Alternatively, thereare many
observations of AGN feedback that present a more plausible
heating mechanism.
8.3 Gyr. The turbulence profiles with and without viscosity
Thefirst clustermodelweconsider ismodeled on obser-
are almost identical beyond 400 kpc. More quantitatively,at
vationsof Abell 2199 (Johnstone et al. 2002). Weinitialize a
r500 therms Mach numbersdifferbyonly 5%. Nearthevery cluster in a static NFW halo with a mass of 3.8×1014 M
⊙
centerofthecluster,thermsvelocityisactuallylargerforthe
and a scale radius of 390 kpc. In this potential we calcu-
viscous simulation. Although we have no definitive explana-
late a spherically-symmetric atmosphere in hydrostatic equi-
tion,thisisperhapsduetotheturbulenteddiesbecomingless
librium and thermal equilibrium with conduction (at 1/3 of
aligned with the magnetic geometry. Since our cluster is on
Spitzer) exactly balancing cooling. The model cluster has a
themassiveandhotendoftheclustermassfunction,thisclus-
central temperature and electron density of ≃ 2.0 keV and
terhasaparticularlylargevalueofωcond/ωbuoy,whichyields ≃0.021cm−3,respectively,andatemperatureanddensityof
the greatest effect of viscosity on the linear MTI (see Fig- 5keVand1.67×10−3 cm−3 at 200kpc.This correspondsto
ure 1). Since viscosity has little effect on the MTI-generated
a central cooling time of 1.7 Gyr. In this region we initialize
turbulence in this hot cluster, we expect viscosity to make
a tangled magnetic field with a Kolmogorov power spectrum
almost no difference for the MTI in lower mass (and cooler) andan amplitudeof h|B|i∼10−8.Themagnetic fieldistan-
clusters.
gled on scales from 50 kpc down to 30 kpc. The simulations
arecomputedonaCartesiangridinadomainextendingfrom
theclustercenterto240kpcwith(128)3 gridpoints.Aresolu-
6 GLOBAL MODELS OF THE HBI IN tionstudyat(256)3showedthatthebehaviorofthemagnetic
CLUSTER CORES field geometry was almost indistinguishable from the lower
resolution results we focus on here (Fig. 9). The magnetic
We now move inwards to consider the effects of viscosity in field geometry and temperature profiles as a function of ra-
globalmodelsofthecoresofgalaxyclusters.Within200kpc, dius(notplotted)arenearlyidentical;asaresult,weconsider
thecoolingtimesareasshortas100Myrat∼10kpc.These these simulations very well-converged. In these simulations,
shortcoolingtimes,alongwithevidencethatlargemassfluxes the temperature is fixed to the initial temperature at a ra-
of gas are not cooling from a hot to cold phase constitute diusof200kpceverywhere,butthereisnocentralboundary
the modern cooling flow problem (Peterson & Fabian 2006). condition. For more details of the set-up see Parrish et al.
Namely, what is keeping cool core cluster cools from cooling (2009). Our CC atmosphere model has ω /ω ∼ 10,
cond buoy
(cid:13)c 0000RAS,MNRAS000,000–000
10 Parrish, McCourt, Quataert, and Sharma
Figure9.Timeevolutionofthevolume-averagedangleofthemag- Figure 10. Azimuthally-averaged temperature profiles for iden-
neticfieldwithrespecttotheradialdirectioninourglobalcool-core tical cluster cores with different imposed rms turbulent velocities
cluster model(model CCinFig.2). θ=0◦ isradial.θ=60◦ cor- (seelegend)withafixeddrivingscaleofL=40kpc.Averystrong
respondstoarandom,isotropicmagneticfield.TheHBIreorients bimodalityisseeninthestabilitypropertiesofthethermalprofiles.
the magnetic field, reducing the effective conductivity, which pre- Braginskiiviscosityisincluded.Drivingat71.7kms−1producesa
cipitates the cooling catastrophe at 2.4 Gyr. Verylittle difference stabletemperatureprofile;whereasatinychangeinthedrivingto
is seen between the inviscid (solid line) and viscous (dotted line) 70.5 km s−1 leads to the cooling catastrophe. The two thermally
evolution.Wedemonstratethattheseresultsarewell-convergedby stablerunsreportthemeasuredtimeinparentheses;whilethetwo
plottingthenearlyidenticalevolutionofaviscoussimulationwith lowervelocity unstable runs (marked witha*)reportthe timeof
doubletheresolution(red dashed line). coolingcatastrophe inparentheses.
which is reasonably consistent with the cool-core ACCEPT byanyviscosityatall.AstheHBIdrivesrathersmallturbu-
clusterplottedinFigure2.Relativetoobservedclusters,our lent velocities (12 km s−1 at 2 Gyr), the viscous dissipation
modelissomewhat highinω /ω atsmallradiiandon doesnexttonothingtoimpedethecoolingcatastrophewhich
cond buoy
the low-end at larger radii. This discrepancy is due to the occursatalmosttheexactsametimeasintheinviscidsimula-
fact that by enforcing thermal equilibrium, we demand that tion.IntheBraginskii(collisional, anisotropic) limit,heating
conductionmatchcoolingeverywherewithoutanyAGNfeed- by viscous dissipation of turbulence is thermally unstable as
back.Thisis,perhaps,ahintthatotherheatingmechanisms, ∇·(Π·v) ∝ ν ∝ T5/2, so even more vigorous turbulence
such as AGNfeedback, are necessary for real CC clusters. cannot stably balance cooling.
Wefirstevolveourfiducialclustermodelwithanisotropic Turbulence driven by galaxy wakes, AGN feedback,
conduction and cooling only. We diagnose the effect of the structure formation, or any other source can counteract
HBIbycalculatingthemeanmagneticfieldanglefromradial the ability of the HBI to reorient magnetic field lines
hθi = cos−1h|ˆb·rˆ|i. This quantity is relevant to the thermal (McCourt et al.2011a).Quantitatively,turbulenceonascale
evolution as the effective radial thermal conductivity, often L with velocity δv is able to suppress theHBI when
called theSpitzerfraction, is given by
L dlnT −1/2
fSp ≡Qr/Q˜r ≈cos2h|ˆb·rˆ|i, (23) teddy(L)≃ δv .ξtHBI≃ξ g dr , (24)
(cid:18) (cid:19)
where Q˜ ≡ −κdT/dr, which is the radial heat flux if the where t is the HBI growth time, and ξ is a dimension-
r HBI
conduction werepurely isotropic at theSpitzervalue. Figure lessconstantthatisdeterminedbysimulations(Sharmaet al.
9showsthattheHBIactstoreorientthemagneticfieldtobe 2009; Parrish et al. 2010). Since t ∝L2/3, this inequality
eddy
moreazimuthal,thusreducingtheeffectiveradialconductiv- is most difficult to satisfy at the outer scale, which domi-
ity.Inthiscalculationthecoolingcatastrophe(definedasthe natestheturbulentenergy.Possiblesourcesofturbulenceare
central temperature reaching our imposed floor of 0.05 keV) galaxy wakes (Kim et al. 2005) or AGN feedback itself.
occurs at ∼2.4 Gyr. Wesimulate theinteraction of turbulence, theHBI,and
When the effects of Braginskii viscosity are added, the coolinginthefiducialclustercoremodeldescribedpreviously
evolution changes very little as shown by the dotted line in by adding a random velocity forcing. We drive the velocity
Figure 9. As we saw previously in the local calculations, in fields in Fourier space with a flat spectrum on a scale L =
3D interchange-typemotions are able to proceed unimpeded 40±10kpcsuch thatthephasesarerandom andtheforcing
(cid:13)c 0000RAS,MNRAS000,000–000