Anisotropic Thermal Conduction and the Cooling Flow Problem in Galaxy Clusters

Abstract

We examine the long-standing cooling flow problem in galaxy clusters with 3D MHD simulations of isolated clusters including radiative cooling and anisotropic thermal conduction along magnetic field lines. The central regions of the intracluster medium (ICM) can have cooling timescales of ~200 Myr or shorter--in order to prevent a cooling catastrophe the ICM must be heated by some mechanism such as AGN feedback or thermal conduction from the thermal reservoir at large radii. The cores of galaxy clusters are linearly unstable to the heat-flux-driven buoyancy instability (HBI), which significantly changes the thermodynamics of the cluster core. The HBI is a convective, buoyancy-driven instability that rearranges the magnetic field to be preferentially perpendicular to the temperature gradient. For a wide range of parameters, our simulations demonstrate that in the presence of the HBI, the effective radial thermal conductivity is reduced to less than 10% of the full Spitzer conductivity. With this suppression of conductive heating, the cooling catastrophe occurs on a timescale comparable to the central cooling time of the cluster. Thermal conduction alone is thus unlikely to stabilize clusters with low central entropies and short central cooling timescales. High central entropy clusters have sufficiently long cooling times that conduction can help stave off the cooling catastrophe for cosmologically interesting timescales. Comment: Submitted to ApJ, 14 pages, 14 figures

Full text

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arXiv:0905.4500v1 [astro-ph.CO] 27 May 2009
Draft version May 27, 2009
Preprint typeset using LATEX style emulateapj v. 12/01/06
ANISOTROPIC THERMAL CONDUCTION AND THE COOLING FLOW PROBLEM IN GALAXY CLUSTERS
Ian J. Parrish1,2, Eliot Quataert1, & Prateek Sharma1,2
Draft version May 27, 2009
ABSTRACT
We examine the long-standing cooling flow problem in galaxy clusters with 3D MHD simulations of
isolated clusters including radiative cooling and anisotropic thermal conduction along magnetic field
lines. The central regions of the intracluster medium (ICM) can have cooling timescales of ∼ 200 Myr
or shorter—in order to prevent a cooling catastrophe the ICM must be heated by some mechanism such
as AGN feedback or thermal conduction from the thermal reservoir at large radii. The cores of galaxy
clusters are linearly unstable to the heat-flux-driven buoyancy instability (HBI), which significantly
changes the thermodynamics of the cluster core. The HBI is a convective, buoyancy-driven instability
that rearranges the magnetic field to be preferentially perpendicular to the temperature gradient. For
a wide range of parameters, our simulations demonstrate that in the presence of the HBI, the effective
radial thermal conductivity is reduced to ? 10% of the full Spitzer conductivity. With this suppression
of conductive heating, the cooling catastrophe occurs on a timescale comparable to the central cooling
time of the cluster. Thermal conduction alone is thus unlikely to stabilize clusters with low central
entropies and short central cooling timescales. High central entropy clusters have sufficiently long
cooling times that conduction can help stave off the cooling catastrophe for cosmologically interesting
timescales.
Subject headings: convection—galaxies: clusters:
galaxies: clusters
general—instabilities—MHD—plasmas—X-rays:
1. INTRODUCTION
X-ray observations of the intracluster medium (ICM)
of relaxed galaxy clusters show a centrally peaked sur-
face brightness distribution. The observed temperatures
and densities are high enough that the plasma can have
a cooling time much less than 500 Myr (Sarazin 1986).
The standard isobaric cooling flow model predicts mass
dropping out of the X-ray emitting ICM at rates in ex-
cess of 100–500M⊙yr−1. However, X-ray spectroscopic
observations with Chandra and XMM-Newton have ruled
out classical cooling flows of material below 1 keV as it
would copiously emit lines such as Fe XVII that are not
observed (Peterson & Fabian 2006). Therefore, a mech-
anism is required to heat the ICM to avert this cooling
catastrophe.
The plasma in the ICM has temperatures of 1–15 keV
and number densities of 10−3–10−1cm−3.
netic field in the ICM is estimated to be in the range
of 0.1–10 µG depending on where the measurement is
made (Carilli & Taylor 2002). Under these conditions,
the Coulomb mean free path is many orders of magni-
tude larger than the gyroradius; e.g., at T = 3 keV,
ne= 10−2cm−3, and B = 1µG, the mean free path is
λmfp≈ 0.3 kpc, while the electron gyroradius is ρe≈ 108
cm. The mean free path is, however, smaller than the
temperature gradient scale length. As a result, a fluid de-
scription of the ICM plasma, e.g. MHD, is appropriate,
but the effects of anisotropic heat and momentum trans-
port must also be included. The Braginskii-MHD equa-
tions (Braginskii 1965) are a modification of the standard
MHD equations to include anisotropic transport due to
The mag-
1Astronomy Department & Theoretical Astrophysics Center,
601 Campbell Hall, University of California, Berkeley, CA 94720;
iparrish@astro.berkeley.edu
2Chandra/Einstein Fellow
the magnetic field.
AGN feedback and conduction from the thermal bath
at large radii are two of the most often discussed mech-
anisms for heating cool cluster cores. This work shall
only briefly consider the former.
mal conduction, has been studied by many authors
(e.g., Binney & Cowie 1981; Narayan & Medvedev 2001;
Zakamska & Narayan 2003; Guo et al. 2008). Because of
uncertainties associated with the magnetic geometry, the
heat flux is often parameterized as an effective thermal
conductivity given as a fraction, fSp, of the ideal (field-
free) Spitzer heat flux. Previous work, however, has not
considered the dynamic consequences of anisotropic con-
duction. The plasma in galaxy clusters is unstable to two
different convective instabilities driven by anisotropic
thermal conduction along magnetic field lines. The first,
the magnetothermal instability, or MTI (Balbus 2000;
Parrish et al. 2008), occurs when the temperature gra-
dient and gravity are in the same direction, as is true
at large radii in galaxy clusters. Well inside the cooling
core, the heat-flux-driven-buoyancy instability, or HBI,
occurs where the temperature gradient is in the opposite
direction of gravity (Quataert 2008; Parrish & Quataert
2008). The HBI occurs in the cooling core of the ICM
where the temperature increases outward.
simulations of the MTI and HBI have shown that they
significantly modify the thermal conductivity, because
they saturate by rearrangingthe magnetic field geometry.
Thus, determining the effective conductivity in the ICM
requires considering the back-reaction of the anisotropic
heat flux on the magnetic field geometry.
In this work, we examine the stability of the cooling
cores of galaxy clusters using three-dimensional MHD
simulations including anisotropic thermal conduction
and cooling. In particular we assess the interplay be-
tween cooling and the HBI. This paper is organized as
The latter, ther-
Nonlinear

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2 Parrish, et al
follows. In §2 we summarize the physics of the HBI and
the thermal instability and their nonlinear saturation in
local calculations. In §3 we then describe the equations
of Braginskii-MHD and the numerical tools we utilize to
solve them. §4 presents our fiducial cluster model (based
on Abell 2199) in detail and examines its evolution over
cosmic time. In §5, we examine a variety of variations
of our fiducial model including cluster halo parameters,
magnetic field strength and geometry, and the central
cluster entropy. We also show a few experiments with
a simple AGN heating model (§5.7). Finally, in §6 we
discuss the implications of this work for the cooling flow
problem and highlight some of our plans for future work.
In parallelto our
Bogdanovic et al.(2009)
study using similar numerical methods. Their simula-
tions start with different initial conditions and cover
a different part of parameter space but reach broadly
similar conclusions.
work
has
describedhere,
similarconducteda
2. PHYSICS OF THE HBI AND COOLING
The linear physics of the HBI and its nonlin-
ear evolution are outlined in Quataert (2008) and
Parrish & Quataert (2008), respectively. We briefly re-
view them here for clarity. The heat-flux-driven buoy-
ancy instability is a convective instability driven by a
background heat flux with the temperature gradient as
the source of free energy. In contrast, the entropy gra-
dient drives the more familiar Schwarzschild convection.
For an arbitrarily oriented magnetic field, the dispersion
relation of the HBI is (Quataert 2008):
0 = ω˜ ω2+ iωcond˜ ω2− N2ωk2
⊥
k2
z)k2
−iωcondg
?dlnT
dz
??
(1 − 2b2
⊥
k2+2bxbzkxkz
k2
?
, (1)
where N is the Brunt-V¨ ais¨ al¨ a frequency,
N2= −1
γρ
∂P
∂z
∂ lnS
∂z
, (2)
and
˜ ω2= ω2− (k · vA)2,(3)
where vA= B/(4πρ)1/2is the Alfv´ en speed, and
ωcond=2
5χ
?ˆb · k
?2
(4)
is the frequency for conduction to act on a given scale,
withˆb the unit vector directed along the magnetic field
and χ the thermal diffusivity3in units of cm2s−1. This
dispersion relation is written without loss of generality
for a geometry in which gravity and the initial atmo-
spheric gradients are in the ˆ z-direction and the initial
magnetic field lies in the ˆ x–ˆ z plane.
For a weak magnetic field, equation (1) has unstable
solutions for either sign of the temperature gradient. The
case of dT/dz > 0 corresponds to the HBI. For a weak,
initially vertical magnetic field (bz = 1, bx = 0), the
growth rate of the HBI is given by
?dlnT
ω2≈ −g
dz
?k2
⊥
k2,(5)
3The literature is not consistent regarding the use of χ and κ.
We will use χ to represent a true diffusion coefficient and κ to
represent a conductivity in units erg cm−1s−1K−1.
where k⊥ is with respect to gravity. Qualitatively, one
can picture the HBI as being driven by regions of con-
verging and diverging perturbed magnetic field lines.
Regions of converging magnetic field are conductively
heated and become buoyant. In local simulations, the
HBI generates MHD turbulence and a modest magnetic
dynamo that amplifies the field. The most prominent
method by which the HBI saturates is via a significant
reorientation of the magnetic field geometry. The HBI
takes a largely vertical field and reorients it to become
largely horizontal. This fact is crucial for cluster cores
since this reorientation of the magnetic field can signifi-
cantly reduce the heat transport across an HBI unstable
region.
In addition to driving the HBI, thermal conduction
also has a significant impact on thermal instability as
originally shown in Field & Saslaw (1965).
criterion states that wavelengths longer than
The Field
λF =
?
Tκ
nenpΛ(T)
?1/2
,(6)
are thermally unstable, where Λ(T) is the cooling func-
tion discussed later. For modes with wavelengths shorter
than λF, the conduction time is shorter than the cool-
ing time, and local perturbations are stabilized. In the
ICM, the plasma is often locally stable to thermal in-
stability, but unstable to global modes even in the pres-
ence of conduction (Kim & Narayan 2003). Equation (6)
was derived under the assumption of isotropic conduc-
tion. The results are similar for anisotropic conduction,
except that conduction can only stabilize perturbations
with short wavelengths along the magnetic field.
An interesting way to examine the physics of cooling in
clusters comes from the recent work of Voit et al. (2008),
who examined the role of the central entropy of the ICM
as an indicator of feedback and star formation in galaxy
clusters. The entropy is defined as K = kBTn−2/3
Voit et al. (2008), the entropy profile for a cluster is fit
using
e
. In
K(r) = K0+ K100
?
r
100kpc
?α
,(7)
where K0is approximately the central entropy, K100is
the power law normalization and α > 0 is the power
law exponent. They find that low entropy clusters, those
with K0≤ 30 keV cm2, have stronger Hα emission, star
formation indicators, and AGN activity than higher en-
tropy clusters, K0≥ 30 keV cm2(Cavagnolo et al. 2008).
As a matter of terminology, clusters with an inwardly
decreasing temperature are referred to as cool-core or re-
laxed clusters. Clusters with an isothermal or inwardly
increasing temperature profile are referred to as non cool-
core clusters.
These observational results can be qualitatively under-
stood in light of the Field criterion (eq. [6]). When cool-
ing is pure Bremsstrahlung, the Field length becomes a
function of entropy, scaling as λF ∝ K3/2
cooling can take place on small length scales when the
entropy is low. The HBI decreases fSp, making smaller
wavelengths unstable to cooling. Fully non-linear simu-
lations, such as those presented here, are needed to un-
derstand the combined dynamics of cooling and the HBI.
0
f1/2
Sp. Thus,

Page 3

Thermal Conduction and the Cooling Flow Problem3
3. METHOD
3.1. Equations of MHD with Anisotropic Heat
Conduction
We solve the usual equations of ideal MHD with the
addition of a vector heat flux, Q, and a gravitational
acceleration g = −∇Φ:
∂ρ
∂t+ ∇ · (ρv) = 0, (8)
∂(ρv)
∂t
+ ∇ ·
?
ρvv +
?
p +B2
8π
?
I −BB
4π
?
+ ρ∇Φ = 0,
(9)
∂E
∂t+∇ ·
+∇ · Q + ρ∇Φ · v = H − L,
?
v
?
E + p +B2
8π
?
−B (B · v)
4π
?
(10)
∂B
∂t
− ∇ × (v × B) = 0, (11)
where the symbols have their usual meaning. The total
energy E is given by
E = ǫ + ρv · v
2
+B · B
8π
,(12)
and the internal energy, ǫ = p/(γ − 1). We assume γ =
5/3 throughout.
The anisotropic heat flux is given by
Q = −nkBχC(T,n)ˆbˆb · ∇T,(13)
where the thermal diffusivity is given by the Spitzer value
(Spitzer 1962) andˆb is a unit vector in the direction of
the magnetic field.The Spitzer thermal diffusivity is
given by
χC(T,n) = 8×1031
?
T
10keV
?5/2?
n
5 × 10−3
?−1
cm2s−1.
(14)
Note that χ is a thermal diffusivity, and it depends
inversely on the density.The Spitzer conductivity is
κSp = nkBχC which has only the well-known T5/2de-
pendence and no density dependence.
The energy equation also includes heating (H) and
cooling (L) terms. The cooling function we adopt is from
Tozzi & Norman (2001) with the functional form
L = nenpΛ(T), (15)
with units of erg cm−3s−1. The temperature dependence
is a fit to cooling dominated by Bremsstrahlung above 1
keV and metal lines below 1 keV with
Λ(T) =?C1(kBT)−1.7+ C2(kBT)0.5+ C3
where C1 = 8.6 × 10−3, C2 = 5.8 × 10−2, and C3 =
6.3 × 10−2, for a metallicity of Z = 0.3Z⊙, with units
of [Ci] = ergcm3s−1. The majority of our runs do not
include additional heating terms. For these runs H = 0
in equation (10).
In runs with heating, we adopt a heating profile of the
form
H = H0e−(r/rH)2,
?10−22, (16)
(17)
TABLE 1
Timescales in Model T1
TimescaleSymbolTimea(Myr)
Sound Crossing
Alfv´ en Crossing (1µG)
Conduction
HBI Growth
Cooling
τS
τA
45
1.1 × 103
20
120
1.4 × 103
τcond
τHBI
τcool
aEvaluated for L = 50 kpc as volume-averaged
quantities.
where rH is the scale radius of heating and the normal-
ization is chosen as
H0=Ltherm
r3
Hπ3/2,(18)
where Ltherm is the total thermal heating input. This
model is motivated by a simple description of AGN feed-
back. We discuss simulations with heating in more detail
in §5.7.
3.2. Timescales
We now discuss several of the key timescales in galaxy
clusters in order to provide some intuition for the impor-
tant physical processes (See Table 1). We examine these
timescales for volume-averaged quantities in our fiducial
atmosphere of A2199.The generation of the fiducial
atmosphere is discussed in §4.1. The volume-averaged
sound speed is approximately 108cm s−1, corresponding
to a sound crossing time over 50 kpc of τS ≈ 45 Myr.
A weak magnetic field of 1 nG corresponds to an Alfv´ en
crossing time over 50 kpc of τA≈ 1012years.
For a more typical magnetic field of 1 µG, the Alfv´ en
speed is 4.4×106cm s−1, leading to a crossing time across
50 kpc of 1.1 Gyr. We will see shortly that the Alfv´ en
timescale is the longest timescale in the problem. For this
cluster the central magnetic beta, the ratio of thermal
pressure to magnetic pressure is β0= 8πp/B2≈ 6,600
for B = 1µG. Even for B = 10µG, the central magnetic
beta is significantly greater than unity. Further out in
the core, where the thermal pressure has dropped, the
beta parameter is typically of order several hundred.
Also of interest is the heat conduction timescale. Our
model cluster has a volume-averaged thermal diffusiv-
ity of ?χ? ≈ 3.8 × 1031cm2s−1, which yields the scale-
dependent conduction time
τχ=L2
χ
0.80Myr (L = 10kpc).
=
?
20Myr(L = 50kpc)
(19)
The HBI growth time in the fast conduction limit is
τHBI=
?dlnT
dr
dφ
dr
?−1/2
≈ 126Myr.(20)
As we will see later, the HBI has an opportunity to grow
significantly compared to the average time between ma-
jor mergers, roughly 5 Gyr, for a typical cluster. The
final timescale of interest is the cooling time at the cen-
ter of the cluster
γ
γ − 1
τcool=
e
nenpΛ(T)≈ 1.4Gyr.(21)
This cooling time estimate is in general too long (by al-
most a factor of 2), as it does not account for the increase

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4 Parrish, et al
in the cooling rate as temperature decreases and the den-
sity and line emission increase. Nonetheless, the cooling
time is much longer than the HBI growth rate for the
fastest growing, short-wavelength modes. Thus, the HBI
growth is likely to play a significant role in the thermal
evolution of the cluster core.
3.3. Numerical Tools
For our simulations we use the
code (Gardiner & Stone 2008; Stone et al. 2008) com-
bined with the anisotropic conduction methods of
Parrish & Stone (2005) and Sharma & Hammett (2007).
In particular, we use harmonic averaging of the conduc-
tivity and the monotonized central difference limiter on
transverse heat fluxes to ensure stability. The heating,
cooling, and thermal conduction are operator split and
sub-cycled with respect to the MHD timestep. The cool-
ing simulations are implemented with a temperature floor
of T = 0.05 keV, below which UV lines become impor-
tant, and the cooling curve fit is no longer accurate. This
temperature floor prevents the cooling catastrophe from
going to completion.
Most of the simulations in this work are done on uni-
form Cartesian grids of (128)3. One high-resolution run
is done at (256)3. We use modified reflecting boundary
conditions for all MHD variables, in which the pressure
and density are extrapolated in the ghost zones, but the
other variables are reflected. This prevents the gravita-
tional source term from introducing a spurious acceler-
ation at the boundary. For the temperature boundary
condition, we introduce a thermal bath at r ≥ rmaxwith
a fixed temperature Touter. For almost all of our runs
rmax = 200 kpc. This thermal bath physically repre-
sents the massive thermal energy available in the ICM
outside the cluster core. Since we are simulating the en-
tire cluster in a Cartesian geometry, there is no inner
boundary condition at the cluster center.
To seed multiple modes of the HBI and break symme-
try, we add Gaussian white noise perturbations to the
initial velocity field such that the applied perturbation is
everywhere a fixed fraction of the sound speed, typically
≈ 1%. In the absence of the HBI, cooling, or an ap-
plied perturbation, we find that we can hold hydrostatic
equilibrium to better than one part in 104.
All of our runs, unless otherwise noted, are run to a
time of 9.5 Gyr, a large fraction of the age of the universe.
In a small number of runs, the simulations do not go
to completion as a result of very severe cooling flows
concentrating large quantities of mass into the central
few zones of the grid. These exceptions are noted.
Athena MHD
4. FIDUCIAL SIMULATION
4.1. Initial Conditions
To introduce the phenomenology of the HBI in cluster
cores, we begin by discussing a simple fiducial calcula-
tion based on observations of the galaxy cluster Abell
2199 as discussed in Zakamska & Narayan (2003) and
Johnstone et al. (2002). This fiducial run is identified
as T1 in the table of Runs (Table 2). Our initial con-
ditions for the cluster core are obtained by constructing
a spherically symmetric atmosphere in both hydrostatic
and thermal equilibrium. We have found that it is quite
advantageous to start from thermal equilibrium. Runs
without thermal equilibrium experience thermal fronts
propagating from the boundaries. Starting in thermal
equilibrium produces a much more physical initial con-
dition. The equations for this equilibrium are
dP
dr=µmHP
kBT
1
r2
drdr
dφ
dr,
(22)
d
?
r2fSpκdT
?
= nenpΛ(T) − H,(23)
where fSpis the initial effective thermal conductivity and
the heating function is neglected for our fiducial case.
Our potential is chosen to be a softened NFW potential
with a dark matter density distribution given by
ρDM=
M0/2π
(r + rc)(r + rs)2,(24)
where M0is the scale mass, rsis the scale radius, and rc
is the softening radius (Navarro et al. 1997). The poten-
tial softening is important for numerically maintaining a
very accurate hydrostatic equilibrium. The potential is
given by
φ=−2GM0
?
rc
(rs− rc)2
ln(1 + r/rs)
r/rc
?
ln
?1 + r/rc
1 + r/rs
?
?
+ln(1 + r/rc)
r/rc
?
−rs(rs− 2rc)
rc(rs− rc)2
The plasma is modeled as a fully ionized ideal gas with
µ = 0.617 and µe= 1.176.
We solve equations (22) and (23) as a two point bound-
ary value problem with the constraints of matching T at
both the inner (Ti) and outer (To) boundary. As these
are a third-order system of equations, we choose a further
symmetry constraint, namely, that the heat flux vanishes
at the center. This system of equations is slightly stiff,
but generally is soluble with a shooting method with a
good initial guess. We only solve these ODEs to estab-
lish our initial condition. The subsequent evolution of the
equilibrium is calculated using the full system of partial
differential equations (PDEs) for MHD, equations (8)–
(11).
For our fiducial model, we use a physical initial mag-
netic field geometry, that of tangled magnetic fields.
First, in constructing the two-point boundary value for
our initial conditions, we set fSp = 1/3. Second, the
initial fields are tangled with a Kolmogorov spectrum
using the method outlined in detail §4.2 of Parrish et al.
(2008). We initialize the fields in Fourier space as
.(25)
˜A(k) =˜A0
?
k
kpeak
?−α
,(26)
where kpeakis chosen as the wavenumber corresponding
to 2–4 times the grid scale. We also choose a low-k cut-
off corresponding to wavelengths of 1/2 to 1/4 of the
domain size. We randomize the phase and use the Fast
Fourier Transform to calculate the vector potential in
real space. We choose α = −17/6, the appropriate k-
space scaling for the 1D power spectrum of Kolmogorov
turbulence. Our last step in initializing the magnetic
field is to difference the vector potential to calculate a
manifestly divergence-free initial field. The normaliza-
tion of the magnetic field for our fiducial run is such that
?|B|? = 10−9G.

Page 5

Thermal Conduction and the Cooling Flow Problem5
TABLE 2
Initial Properties of Nonlinear Runs
Run
M0 (M⊙)
Rs (kpc)
Rca(kpc)
Ti(keV)
To (keV)
B0 (G)
K0 (keV cm2)
R1
T1b
T1-HB
T1-256
T2
T3
E1
E2
E2-NCc
E2-HB
E3
E3-NCc
H1d
I1e
Iso1
Iso2
aSoftening radius of NFW halo (eq. [24])
bFiducial Run
cNo conduction
dSimulation includes additional heating (see §5.7).
eIsotropic conduction only
3.8 × 1014
3.8 × 1014
3.8 × 1014
3.8 × 1014
1.1 × 1015
3.8 × 1014
3.8 × 1014
3.8 × 1014
3.8 × 1014
3.8 × 1014
3.8 × 1014
3.8 × 1014
3.8 × 1014
3.8 × 1014
3.8 × 1014
3.8 × 1014
390
390
390
390
520
390
390
390
390
390
390
390
390
390
390
390
20
20
20
20
26
20
20
20
20
20
20
20
20
20
20
20
2
2
2
2
4
1
3
4
4
4
5
5
5
5
10−9, radial
10−9, tangled
10−6, tangled
10−9, tangled
10−9, tangled
10−9, tangled
10−9, tangled
10−9, tangled
10−9, tangled
10−6, tangled
10−9, tangled
10−9, tangled
3.5 × 10−6, tangled
10−9, radial
10−9, tangled
10−9, tangled
15.5
22.4
22.4
22.4
31.1
5.46
43.6
83.1
83.1
83.1
122.
122.
14.0
1.5
52.5
154.
9.5
6
5
5
5
5
5
5
5
5
4.5
4.5
4.5
4.5
2
2
4.5
4.5
Fig. 1.— The temperature, density, and pressure of our fidu-
cial initial condition are chosen to roughly correspond to those
observed in Abell 2199. We initialize tangled magnetic fields for
this simulation.
Our model cluster, Abell 2199 has an inner tempera-
ture of roughly 2 keV and a temperature of 5 keV near
200 kpc (Johnstone et al. 2002). The gravitational po-
tential is fit to an NFW profile with a scale radius of
rs= 390 kpc, a softening radius of rc= 20 kpc, and a
mass of M0= 3.8×1014M⊙. Figure 1 shows the fiducial
atmosphere that results from these parameters. The cen-
tral density in our thermal equilibrium model is slightly
lower than the observed value.
4.2. Evolution of the Fiducial Case
The evolution of our fiducial cluster model prominently
illustrates the physics of the HBI and cooling in the
galaxy cluster core. This evolution is best understood
through a variety of diagnostics. First, in Figure 2 we
show the time evolution of the magnetic and kinetic en-
ergies in the cluster. There is a very weak magnetic dy-
namo in the first 2 Gyr or so due to the HBI. The initial
drop in magnetic energy is due to reconnection. After
? 3 Gyr, the core loses central pressure support, inflow
begins, and the kinetic energy increases. Correspond-
ingly, the magnetic energy is amplified through flux-
freezing leading to a maximum increase of ∆?B2? ≈ 2.5.
We define the magnetic energy amplification at the final
simulation time, tf, as
∆?B2? ≡
?B2?(tf)
?B2?(t = 0),(27)
where the angle brackets denote a volume average. The
kinetic energy dominates the magnetic energy at all
times. Figure 2 shows that the HBI is not a strong source
of magnetic field amplification in cluster cores.
The real hallmark of the HBI is the reorientation of
the magnetic field geometry. Figure 3 shows the evolu-
tion of the volume-averaged angle of the magnetic field
with respect to the radial direction from its initial tan-
gled state (θ = 60◦) to a final saturated state of 74.6◦.
The angles given here and in Table 3 are volume av-
erages over the entire cluster. The reorientation of the
magnetic field is quite dramatic and takes place on just
a few Gyr. Concomitantly, Figure 4 shows the evolution
of the azimuthally-averaged radial-temperature profile.
We typically bin the temperature into 5 kpc radial bins.
As the magnetic geometry evolves to be more azimuthal,
the thermal conduction from outside the core begins to
shut off and the temperature starts to fall. At ∼ 2.7 Gyr,
the central temperature has hit the cooling floor of 0.05
keV—an effective proxy for the cooling catastrophe. We
do not remove gas from the grid after hitting the tem-
perature floor, and thus, we never see a true ”cooling
flow.” We define the time of the cooling catastrophe, tcc,