# Shattering and growth of cold clouds in galaxy clusters: the role of
radiative cooling, magnetic fields and thermal conduction

Fred Jennings,1,2 Ricarda S. Beckmann,2 Debora Sijacki2 and Yohan Dubois3
1Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh, EH9 3HJ, United Kingdom
2Institute of Astronomy and Kavli Institute for Cosmology, University of
Cambridge, Madingley Rd, Cambridge CB3 0HA, United Kingdom
3Institut d’Astrophysique de Paris, UMR 7095, CNRS, UPMC Univ. Paris VI, 98
bis boulevard Arago, 75014 Paris, France E-mail<EMAIL_ADDRESS>
(Accepted for Publication in MNRAS)

###### Abstract

In galaxy clusters, the hot intracluster medium (ICM) can develop a striking
multi-phase structure around the brightest cluster galaxy. Much work has been
done on understanding the origin of this central nebula, but less work has
studied its eventual fate after the originally filamentary structure is broken
into individual cold clumps. In this paper we perform a suite of 30
(magneto-)hydrodynamical simulations of kpc-scale cold clouds with typical
parameters as found by galaxy cluster simulations, to understand whether
clouds are mixed back into the hot ICM or can persist. We investigate the
effects of radiative cooling, small-scale heating, magnetic fields, and
(anisotropic) thermal conduction on the long-term evolution of clouds. We find
that filament fragments cool on timescales shorter than the crushing
timescale, fall out of pressure equilibrium with the hot medium, and shatter,
forming smaller clumplets. These act as nucleation sites for further
condensation, and mixing via Kelvin-Helmholtz instability, causing cold gas
mass to double within 75 Myr. Cloud growth depends on density, as well as on
local heating processes, which determine whether clouds undergo ablation- or
shattering-driven evolution. Magnetic fields slow down but don’t prevent cloud
growth, with the evolution of both cold and warm phase sensitive to the field
topology. Counter-intuitively, anisotropic thermal conduction increases the
cold gas growth rate compared to non-conductive clouds, leading to larger
amounts of warm phase as well. We conclude that dense clumps on scales of
$500$ pc or more cannot be ignored when studying the long-term cooling flow
evolution of galaxy clusters.

###### keywords:

galaxies: clusters: intracluster medium

††pubyear: 2021††pagerange: Shattering and growth of cold clouds in galaxy
clusters: the role of radiative cooling, magnetic fields and thermal
conduction–B

##  1 Introduction

Galaxy clusters are the largest virialized structures in the Universe. Most of
their baryon content ($\geq 90\%$) can be found in the form of a hot
intracluster medium (ICM) that fills the galaxy cluster and extends out to the
virial radius (McNamara & Nulsen, 2007; Kravtsov & Borgani, 2012). The
remaining baryons are bound in a collection of mostly early-type galaxies with
old stellar populations, with the most massive brightest cluster galaxy (BCG)
residing at their centre. The ICM is observed to possess a complex multi-phase
structure, with hot X-ray emitting gas ($\geq 10^{8}$ K) co-existing with
cold, dense structures ($\lesssim 10^{4}$ K) which likely form via local
thermal instabilities at radii of several tens of kpc from the cluster center
(McDonald et al., 2011; McCourt et al., 2012; Yang & Reynolds, 2016; Beckmann,
R. S. et al., 2019; Das et al., 2021). This cold gas is found in and around
the BCG, where it can be observed as visually spectacular H$\alpha$ emission
nebulae which may fuel cold, clump accretion onto the BCG and their central
supermassive black hole (SMBH) (McDonald et al., 2010; Gaspari et al., 2013;
Tremblay et al., 2016, 2018). The most well-known example of this is the
H$\alpha$ filaments in the Perseus cluster (Fabian, 1994; Peterson & Fabian,
2006; Fabian et al., 2008; McDonald et al., 2018), although filaments in many
other clusters have now been observed (e.g. Russell et al., 2017; Olivares et
al., 2019; Jimenez-Gallardo et al., 2021; North et al., 2021). Cold molecular
gas is also observed with radio observations of CO lines surrounding the BCG
at radii of $50$ kpc or so, as a result of inflows (Edge, 2001; Salomé &
Combes, 2004; North et al., 2021), and most notably up to within 10 kpc of the
central BCG with ALMA (McNamara et al., 2014; Russell et al., 2014; Fogarty et
al., 2019). This inflow is thought to increase the growth efficiency and the
magnitude of the feedback from the SMBH (DeGraf et al., 2016), with important
implications for the morphology of the gas around the central object.

The existence of this cold gas reservoir is closely related to one of the
major open problems in cluster physics today is the cooling flow problem.
Naïve estimates of gas cooling in the ICM predict large scale cooling flows,
but assume an idealized, homogeneous cooling flow (Fabian, 1994), assumptions
that are quite untrue in the observed multi-phase ICM, which has a much more
complex substructure. The cooling flow model also neglects any heating
processes which can inject thermal energy into the cold gas, for example
through feedback processes. When comparing the cooling rate of hot gas via
direct X-ray observations to the star formation rates in clusters, the
discrepancy between the two suggests that there must in fact exist at least
one non-gravitational source of thermal energy which offsets some of the
cooling in these flows (e.g. Fabian, 1994; Peterson et al., 2001; Peterson et
al., 2003; Hudson et al., 2010). This effect must be strong enough to suppress
cooling rates by at least an order of magnitude. Understanding the lack of
cold gas actually seen in clusters compared to estimates based on naïve
cooling flow calculations is a crucial hurdle to be overcome by any complete
model of galaxy cluster evolution. Much work on solving the cooling flow
problem has been directed towards understanding this heating mechanism, and in
particular towards understanding how energy injected by the central active
galactic nucleus (AGN) powered by the SMBH in the BCG offsets radiative
heating losses in the cluster over long periods of time (see for example
Sijacki & Springel, 2006; Vernaleo & Reynolds, 2006; McNamara & Nulsen, 2012;
Gaspari et al., 2013; Gaspari, 2015; Talbot et al., 2021; Bourne et al., 2019;
Bourne & Sijacki, 2021). AGN feedback mainly operates in two modes; kinetic,
which generates jets and bubbles; and a radiative/quasar mode in luminous AGN,
where powerful winds may be driven by magnetic or radiation pressure,
potentially displacing the cold gas. Feedback within galaxy clusters has been
observed directly, as evidenced by jet-inflated bubbles with correspond to the
so-called “X-ray cavities”, by e.g. Boehringer et al. (1993); Churazov et al.
(2000); McNamara et al. (2000); Fabian et al. (2000); Bîrzan et al. (2004);
Randall et al. (2011); Liu et al. (2020); Ubertosi et al. (2021).

One crucial component of the overall cooling cycle of galaxy clusters is the
evolution of cold gas once it has condensed from the hot phase. Large-scale
cluster simulations have concluded that extended cold structures tend to
fragment into smaller clumps over time (Gaspari et al., 2013; Li et al., 2017;
Beckmann, R. S. et al., 2019), but lack the resolution to follow the evolution
of individual small, cold clouds and ultimately to predict their fate. There
are a wide range of potential scenarios for their evolution, with different
consequences for large-scale cluster cooling flows. On one extreme, clouds
could be quickly destroyed and mixed back locally into the hot phase. On the
other extreme, small clumps could act as nucleation sites for cold gas
condensation, which would make clouds grow in mass over time and eventually
deliver large quantities of cold gas to the central BCG. If they survive for
long periods of time, cold clouds could form part of a "fountain" of cold gas
in galaxy clusters; formed when the feedback from a central AGN breaks up
inflowing filaments, they are accelerated in outflows and follow a ballistic
trajectory only to slow down and eventually fall back towards the cluster
centre, along with newly formed cold gas that has condensed via instabilities
at radii of up to $50$ kpc (McNamara & Nulsen, 2007). Clouds that reach the
BCG may be then accreted by its central SMBH and ultimately help to determine
the behaviour of the very feedback that initially created them by increasing
gas accretion rates and reorientation the spin axis of the AGN jet.
Furthermore, cold clumps should be sufficiently destroyed by jets and/or mixed
through further interaction with the ICM if feedback models are to
successfully solve the cooling flow problem and explain the lack of cold gas
in these systems. The simulations of Beckmann, R. S. et al. (2019) show that a
large fraction of cold gas originally in filaments can survive on small scales
after interaction with the jets. Therefore understanding how cold clumps might
survive or perish in the hot gas within an explicit cluster-centric parameter
space is key to the building of consistent models of both cluster cooling and
heating, and of the coupling of feedback to the ICM.

Studying individual clouds embedded in a hot background medium that is moving
at a relative velocity to the cloud is known as the cloud-crushing problem
based on work by Klein et al. (1994) who calculated the analytic timescale on
which small clumps get destroyed in such a scenario. Early simulations
confirmed analytic expectations that clouds get disrupted and mix into the hot
background medium (Klein et al., 1994; Cooper et al., 2009; Zhang et al.,
2017; Scannapieco & Brüggen, 2015). More recently, several authors have shown
that in the presence of radiative cooling cold clouds can grow in some regions
of the circumgalactic medium (CGM) parameter space (Armillotta et al., 2016;
McCourt et al., 2017; Sparre et al., 2019; Gronke & Oh, 2020; Kanjilal et al.,
2021) due to efficient mixing, but get still get destroyed if they are too
small. At the surface between the hot and cold gas, a mixed warm phase forms
due to the Kelvin-Helmholtz instability (Fielding et al., 2020). When the
cooling times of this warm gas are sufficiently short in comparison to the
cloud crushing time, the cloud can grow over time (see Section 2 for details).
Other non-thermal physics, such as magnetic fields and thermal conduction
(Armillotta et al., 2016; Brüggen & Scannapieco, 2016; Li et al., 2020; Sparre
et al., 2020) can also influence the long-term mass evolution of cold clouds.

However, most of the work in the field has been conducted for a parameter
space appropriate for the CGM and associated cold clouds, rather than the
remnants of H$\alpha$ filaments embedded in the ICM of a massive galaxy
cluster. In this paper, we build on work in Beckmann, R. S. et al. (2019), who
demonstrated how cooling filaments in simulations are broken up into small
(100 pc-scale) clumps by interactions with the bimodal jets produced by a
central SMBH. Being limited at a minimum resolution of $120\rm\ pc$, this
cluster-scale work was unable to follow the evolution of clouds on smaller
scales. In the paper presented here, we re-simulate individual clouds with
parameters typical to those seen in Beckmann, R. S. et al. (2019) and the
magnetised version of the same cluster Beckmann et al. (2022a); Beckmann et
al. (2022b) to understand the long-term evolution of such dense gas fragments,
including the impact of radiative cooling, small-scale cloud reheating,
magnetic fields and (anisotropic) thermal conduction.

The paper is structured as follows; in Section 2 we review the theory behind
the cloud crushing problem and present the setup of our simulations in Section
3. The non-magnetised cloud-crushing problem is investigated in Section 4.1,
with the impact of magnetic fields studied in Section 4.2 and that of thermal
conduction in Section 4.3. Discussion and our conclusions are presented in
Section 5.

##  2 The Cloud-Crushing Problem

The classic set-up usually studied in the field is the cloud-crushing problem.
This is an idealised set-up that studies how an individual cloud, embedded in
a (usually) uniform background wind, evolves over time. The characteristic
timescale for the shredding of the cloud of uniform gas density $\rho_{\rm
cl}$ and size $r_{\rm cl}$ within a uniform wind density $\rho_{\rm wind}$ and
relative velocity of $v_{\rm wind}$ is the cloud-crushing timescale (Klein et
al., 1994),

$t_{\text{cc}}\equiv\sqrt{\frac{\rho_{\text{cl}}}{\rho_{\text{wind}}}}\frac{r_{\text{cl}}}{v_{\text{wind}}}=\sqrt{\chi}\
t_{\text{cr}}\,,$ (1)

over which Kelvin-Helmholtz instabilities and shocks will cross the cloud
radius and shred the cloud material. The non-magnetised, non-radiative cloud
crushing problem is scale-free and can be entirely parameterised by the
dimensionless density contrast $\chi$ and the cloud-crossing time
$t_{\text{cr}}$. The actual physical size, wind velocity, and cloud/wind
densities can then be re-scaled once these two parameters are determined.
However, this approach breaks down in environments where radiative cooling in
and around the cloud becomes important. This happens when the cooling time,

$t_{\text{cool}}\equiv\frac{\rho\varepsilon}{\mathcal{L}}=\frac{3nk_{\rm
B}T}{2n_{\rm H}^{2}\Lambda(T,Z)}\,,$ (2)

is comparable to or shorter than the cloud-crushing time. Here
$\Lambda(T,Z)\equiv\mathcal{L}/n_{\rm H}^{2}$ is the cooling function, $k_{\rm
B}$ is the Boltzmann constant, $\varepsilon$ the specific energy, $T$ the
temperature, $n$ the number density, and $n_{\rm H}$ is the hydrogen number
density (Mo et al., 2010, Chapter 8).

The ratio of cooling to cloud-crushing timescales for clouds is

$\theta\equiv\frac{t_{\text{cool}}}{t_{\text{cc}}}=\frac{3n_{\text{cl}}k_{\rm
B}T_{\text{cl}}v_{\text{wind}}}{2n^{2}_{\text{H,cl}}\Lambda(T,Z)r_{\text{cl}}\sqrt{\chi}}\,.$
(3)

The cloud-crushing is sub-dominant when the cloud is at much higher density
than the wind (large $\chi$), and when the cooling is efficient (i.e.
$\Lambda(T,Z)$ is large); for the metallicity of $Z=0.33Z_{\odot}$ that we use
(for both cloud and wind), the cooling function is strongly peaked between
$10^{5}-10^{6}$ K (Sutherland & Dopita, 1993), meaning that the radiative
loses should not be ignored, unlike studies of largely pristine environments
such as the CGM where cooling may not be so important. This metallicity that
we use is a well-known typical value for observations of the ICM, at least
locally (see for example Mushotzky et al., 1978; Allen & Fabian, 1998; Tozzi
et al., 2003; De Grandi et al., 2004; Leccardi & Molendi, 2008; Werner et al.,
2013; Simionescu et al., 2015; Urban et al., 2017).

We can also equate the cooling and cloud-crossing timescales to obtain a
characteristic length at which the cloud should be able to respond dynamically
to cooling-induced perturbations in the temperature and pressure (McCourt et
al., 2017). This length $l_{\text{cloudlet}}\sim c_{\rm s}t_{\text{cool}}$ can
be straightforwardly shown, for a fully ionised plasma with hydrogen fraction
$X$ (assuming an ideal gas speed of sound), to be equal to

$l_{\text{cloudlet}}=\frac{3}{64}\left[\frac{\gamma k_{\rm
B}^{3}T_{\text{fl}}^{3}}{m_{\rm
p}X^{4}n^{2}\Lambda^{2}(T_{\text{fl}},Z)}(3+5X-Z)^{5}\right]^{1/2}\,,$ (4)

where $\gamma$ is the adiabatic index of the gas and $T_{\rm fl}$ is the
temperature floor that the gas settles at, and $m_{p}$ is the proton mass. The
length-scale on which clumps are dynamically responsive is inversely
proportional to the number density and the value of the cooling function, and
proportional to the temperature raised to the $3/2$ power. We plot this length
scale as a function of temperature in Fig. 1 for reference, using the cooling
tables for dust-free, no cosmic rays, and no self-shielding model of
Ploeckinger & Schaye (2020), who have generated cooling rates using the
spectral synthesis code cloudy (Ferland et al., 2017). Fig. 1 shows the
analytical solution for the minimum theoretical lengthscale (given by Equation
(4)) that is expected in cloud shattering, if the cloud shatters continuously
down to a floor temperature (given on the horizontal axis), and thus gives the
rough size of the smallest clumps which should be seen in simulations,
resolution-permitting. Generally $l_{\rm cloudlet}<1\rm\ pc$ but very diffuse
clouds in the temperature range $10^{3}-10^{4}\rm\ K$ can have minimum sizes
of $5\rm\ pc$ or more. The minimum cell length on our fine grid in the
simulations presented in this paper is $8$ pc for the majority of the runs,
and we test a variety of temperature floor temperatures.

From the analytical result we see that we can expect to be able to only
resolve clumps within a narrow range, that are around $0.5$ orders of
magnitude larger than our resolution lengthscale, although we get close to
resolving the length floor over a broad range for diffuse clouds. We would
expect further shattering beyond most of the smallest clumps produced in our
simulation runs, and to gauge what effect this may have we perform a
convergence test in Appendix B using a range of resolutions that are
computationally feasible. We find that with increased resolution less cold gas
forms initially (by up to $40\%$ for t < $t_{\text{cc}}$), but at later times
more cold gas forms due to the further shattering and higher mixing surface
area-to-volume ratio. Therefore, we would expect that the our findings on the
cold phase growth based on lower resolution simulation represent a
conservative lower bound on the magnitude of this effect

Figure 1: The minimum expected cloud size according to Equation (4) as a
function of the cold gas temperature for three different cloud number
densities $n$. This plot assumes a metallicity of $Z=0.33Z_{\odot}$, with
solar abundances of elements, and an adiabatic index of $\gamma=5/3$. In
general, denser clouds fragment to smaller clumps, but the evolution with
temperature is not linear. The black horizontal line denotes our maximum
spatial resolution of $8$ pc adopted for our main production runs, but see
Appendix B for higher resolution simulation results.

##  3 Numerical method

###  3.1 Nomenclature

Throughout this paper, we define a cold phase to include any gas at a
temperature lower than the initial cloud temperature $T<T_{\rm cl}$, a warm
phase to include gas between the initial cloud and wind temperatures ($T_{\rm
cl}<T<T_{\rm wind}$), and a hot phase to include any gas at or above the
initial wind temperature $T\geq T_{\rm wind}$. For all quantitative measures
of the three phases we see in the simulations, we use a straightforward
temperature cut on the gas. For all simulations presented here, $T_{\rm
cl}=10^{6}\rm\ $ and $T_{\rm wind}=10^{8}\rm\ K$. This wind/ICM temperature is
consistent with X-ray observations of clusters (Reiprich & Böhringer, 2002;
Hudson et al., 2010; Frank et al., 2013) which measure X-ray temperatures
mainly in the range $1-10$ keV. The cloud temperature we use is consistent
with the maximum temperature of so-called dense gas used to identify the clump
objects generated within the cluster in Beckmann, R. S. et al. (2019).

We also define as $T_{\rm fl}$ the minimum temperature to which gas is allowed
to cool via radiative loses, the so-called cooling floor temperature.
Furthermore we note another important temperature measure, the mixing
temperature $T_{\rm mix}\simeq\sqrt{T_{\rm cl}T_{\rm wind}}$ given by Begelman
& Fabian (1990), which is convenient for measuring some length scales of the
problem (see Gronke & Oh, 2020).

###  3.2 General setup

The simulations presented in this paper were produced using the ramses code
(Teyssier, 2002), an adaptive-mesh refinement magneto-hydrodynamic (MHD) code.
The simulations are conducted in 3D and include radiative cooling for all of
our production runs, apart from a numerical experiment where we test the non-
radiative setup to highlight the differences. A subset of simulations also
includes magnetic fields with different strengths and orientation, and
(anisotropic) thermal conduction. To solve the MHD equations, ramses employs a
second-order Godunov method (see Toro, 2009), with a MUSCL-Hancock scheme to
solve the Euler equations and compute fluxes across cells. ramses uses an
octree refinement method to adaptively refine the space, and models magnetic
fields as face-centred quantities with constrained transport to satisfy the
solenoidality constraint $\mathbf{\nabla\cdot B=0}$ (Fromang et al., 2006). A
HLLC Riemann solver (Toro, 2009) was used for simulations without magnetic
fields, with a HLLD solver (Miyoshi & Kusano, 2008) used for later MHD runs.
The Courant factor was set to 0.8 for adaptive time-stepping, for which ramses
uses a second-order midpoint scheme. We use the yt Project Python package
(Turk et al., 2010) for all analysis.

###  3.3 Initial conditions and refinement

All simulations presented in this paper have a similar setup: a spherical cold
cloud of radius $r_{\rm cloud}=500$ pc and temperature $T_{\rm cl}=10^{6}\rm\
K$ is placed in a uniform hot background medium of temperature $T_{\rm
wind}=10^{8}\rm\ K$. We fix the initial cloud density contrast to
$\chi=\rho_{\rm cl}/\rho_{\rm w}=100$, but vary both the initial cloud density
$\rho_{\rm cl}$ and the initial wind density $\rho_{\rm w}$ (see Table 1 for
details). This cloud temperature was selected in order to initialise the
clouds in pressure equilibrium with the surrounding hot medium, whose
temperature is typical for the ICM of galaxy clusters. This overdensity was
chosen as the typical overdensities in the simulations of Beckmann, R. S. et
al. (2019) range from around 2-4 (see their Fig. 4). We select the lowest
value in this range because this corresponds to the smallest crushing
timescale, meaning we can simulate longer into the dynamical lifetime of the
cloud.

Simulations are conducted in the frame of the cold cloud, which is initialised
with zero bulk velocity. Instead, the background medium is given a bulk
velocity of $200$ km s-1, in line with observations of line-of-sight
velocities in cluster centres (Salomé et al., 2006; McDonald et al., 2012;
Tremblay et al., 2016; Olivares et al., 2019), and corresponding to the peak
of the velocity distribution of the clumps produced in Beckmann, R. S. et al.
(2019) (see Fig. 7 of that work). This is equivalent to a cloud Mach number of
$\mathcal{M}=2$. A random velocity dispersion field of normal distribution
centred around zero and with standard deviation of $50$ km s-1 within the hot
medium and $30$ km s-1 within the cloud was added to the cloud and wind bulk
velocities in order to avoid grid-locking effects and roughly model the
turbulence which may play an important role in galaxy clusters (e.g. Churazov
et al., 2012; Gaspari & Churazov, 2013; Schuecker et al., 2004; Walker et al.,
2015; Hofmann et al., 2016; ZuHone et al., 2018; Hitomi Collaboration et al.,
2018). We calibrated the mean and variance of the velocity dispersion to
values found in and around clumps produced by the simulations presented in
Beckmann, R. S. et al. (2019). No further turbulence driving is added
throughout the simulation.

The simulation is initialised with a passive scalar, initially set to unity
within the cloud and zero outside, which is advected with the flow throughout
the simulation. This scalar is used both during the simulation run and in
post-processing to help distinguish mixed material from pristine background
gas, as cells with high $f_{\rm scalar}=\rho_{\rm scalar}/\rho_{\rm gas}$ are
dominated by gas originally found in the cloud, while those with low $f_{\rm
scalar}$ are dominated by gas initially found in the hot background medium.
Note however that quantitative measures of masses for the three fluid phases
that we will discuss later on, use only a cut on the temperature of the
plasma, as discussed above.

At the start of our simulations the cloud is setup to be in pressure
equilibrium with the ICM in all runs, as this is appropriate in a galaxy
cluster environment. All simulations were initiated with a metallicity of
$Z=0.33Z_{\odot}$, as is typical for clusters (Baumgartner et al., 2005;
Lovisari & Reiprich, 2019), and this metallicity was kept constant throughout.
The metallicity of the cloud is not expected to deviate significantly from the
background medium, since clouds originally likely condense from the hot ICM.
In practice, a metallicity gradient within the cluster and star formation
occurring in and around cold clumps may lead to a range of different
metallicity values for dense clumps, but note that the observed metallicity
gradients of the ICM are typically shallow (Leccardi & Molendi, 2008).

No gravity is included in our simulations. The free-fall time of our clouds is
$t_{\text{ff}}=\sqrt{3\pi/32G\bar{\rho_{\text{cl}}}}$ which, for our fiducial
run (the lowest density we test), means the free-fall time is approximately
$150$ Myr which corresponds to around $6\ t_{\text{cc}}$. As $t_{\rm
cool}<t_{\rm cc}$ in all of our radiative setups, there is a pressure-driven
collapse on sub-crushing timescales during which gravity is sub-dominant and
therefore not included explicitly here.

In MHD simulations, the magnetic field was intialised as a uniform field that
is either initially parallel (“aligned”) or perpendicular (“normal”) to the
flow. Thermal conduction is modelled following Dubois & Commerçon (2016), with
a thermal conduction coefficient equal to $\kappa_{\rm Spitzer}=n_{\rm
e}k_{\rm B}D_{\rm C}$ (Spitzer, 1962), where $n_{\rm e}$ is the electron
number density, $k_{\rm B}$ is the Boltzmann constant and $D_{\rm C}=8\times
10^{30}\big{(}\frac{k_{\rm B}T_{\rm e}}{10^{8}\rm\
K}\big{)}^{\frac{5}{2}}\big{(}\frac{n_{\rm e}}{10^{-2}\rm\
cm^{-3}}\big{)}^{-1}\rm\ cm^{2}s^{-1}$ the thermal diffusivity and $T_{\rm e}$
the electron temperature. In the case of anisotropic thermal conduction, the
perpendicular conductivity is set to $\kappa_{\rm perp}=0.0001\kappa_{\rm
Spitzer}$.

The size of the simulation box is chosen such that the flow crossing time at
the wind velocity is several tens of crushing times. In practice we never
simulate for this long, and all runs are stopped before the mixed tail of the
cloud reaches the edge of the simulation domain, so no material is lost from
the simulation. Most simulations were performed in a box size of $65$ kpc,
with a root grid of $32^{3}$ (refinement level 5) cells which was adaptively
refined by 8 more levels up to a maximum resolution of $\Delta x=8$ pc. The
initial conditions are refined using concentric spheres centered on the cloud,
with the highest refinement level incorporating the entire initial cloud.
Refinement then proceeds adaptively where a cell is refined at the highest
level of refinement if $f_{\rm scalar}>10^{-6}$, and progressively derefined
otherwise.

All simulation parameters are summarised in Table 1 for the hydrodynamical
runs and in Table 2 for MHD simulations with and without conduction.

Table 1: The parameters of the hydrodynamical runs, with the columns showing:
(1) the run labels, (2) whether the run includes radiative cooling, (3) the
initial cloud density $\rho_{\mathrm{cl}}$, (4) the initial wind (background
ICM) density $\rho_{\mathrm{w}}$, (5) the length of the simulation box
$d_{\text{box}}$, (6) the size of the highest resolution element $\Delta x$,
(7) the value of the minimum temperature achieved through cooling $\rm T_{\rm
fl}$, (8) the initial mass of the cold cloud. The crushing timescale
$t_{\text{cc}}$ for all runs is $24.4$ Myr, and the background wind velocity
is $200$ km s-1 in all cases. All simulations have an initial background
temperature of $T_{\rm wind}=10^{8}$ K, and an initial cloud temperature of
$\rm T_{\rm cl}=10^{6}$ K,except for HD0.1_1e4T0, which was initialised with
$\rm T_{\rm cl}=10^{4}$ K

.  Label Cooling $\rho$${}_{\text{cl}}$ [H/cm3] $\rho$${}_{\text{w}}$ [H/cm3]
$d_{\text{box}}$ [kpc] $\Delta x$ [pc] $\rm T_{\rm fl}$ [K] $\rm M_{\rm 0}$
[$\rm M_{\odot}$] HD10_nc ✗ $10$ $0.1$ 150 18 N/A $1.28\times 10^{8}$ HD0.1_nc
✗ $0.1$ $0.001$ 150 18 N/A $1.28\times 10^{6}$ HD10 ✓ $10$ $0.1$ 65 8 10
$1.28\times 10^{8}$ HD1.0 ✓ $1$ $0.01$ 65 8 10 $1.28\times 10^{7}$ HD0.1 ✓
$0.1$ $0.001$ 65 8 10 $1.28\times 10^{6}$ HD0.1_1e4T0 ✓ $0.1$ $0.001$ 65 8 10
$1.28\times 10^{6}$ HD0.1_$F$5.5 ✓ $0.1$ $0.001$ 65 8 $3\times 10^{5}$
$1.28\times 10^{6}$ HD0.1_$F$5.0 ✓ $0.1$ $0.001$ 65 8 $1\times 10^{5}$
$1.28\times 10^{6}$ HD0.1_$F$4.5 ✓ $0.1$ $0.001$ 65 8 $3\times 10^{4}$
$1.28\times 10^{6}$ HD0.1_$F$4.0 ✓ $0.1$ $0.001$ 65 8 $1\times 10^{4}$
$1.28\times 10^{6}$ HD0.1_$F$3.5 ✓ $0.1$ $0.001$ 65 8 $3\times 10^{3}$
$1.28\times 10^{6}$ HD0.1_$F$3.0 ✓ $0.1$ $0.001$ 65 8 $1\times 10^{3}$
$1.28\times 10^{6}$

###  3.4 Cooling

Radiative cooling in ramses is computed using values given in the cooling
tables of Sutherland & Dopita (1993) above $10^{4}\,\rm K$ and those from
Rosen & Bregman (1995) for temperatures below. In our simulations, gas cools
to a minimum fiducial temperature of $T_{\rm fl}=10\rm\ K$ (although we do run
a suite of tests investigating the effect of changing this value in Section
4.1.3).

In galaxy clusters, both cloud dynamical and cloud crushing timescales for
cold clouds are comparable to or longer than the cooling time of the hot ICM.
However, observationally, the ICM temperature does not change significantly
over sufficiently long timescales, as the bulk of cooling of the ICM in galaxy
clusters is offset by various heating sources, from energy injected by the AGN
to possibly thermal conduction. To mimic the impact of such background heating
in the simulations presented here, radiative cooling is only permitted in gas
above a minimum concentration of a passive scalar originally only placed in
the cold cloud (see Section 3.3). This approach prevents the catastrophic
cooling of the hot ICM background gas (as opposed to clump gas and mixed gas,
which we allow to cool freely) which would otherwise occur over timescales
which are short compared to the dynamical timescale, at variance with
observations. We set the required passive scalar value to $f_{\rm
scalar}>0.001$, and still permit hot gas to cool via in-cell mixing with
colder fluid. In MHD simulations, spurious magnetic heating occurred in a very
small number of cells in and around the clouds. To prevent such “hotspots”
from occurring as well as from slowing the simulation, we enforced a minimum
density floor of $0.0005\rm\ cm^{-3}$. Convergence tests showed no
statistically significant impact on the long-term evolution of simulations
from such a density floor.

###  3.5 Clump finding

To gather statistics on the distribution of individual clumps in our
simulations, we use the water-shed segmentation-based structure finding
algorithm PHEW (Bleuler et al., 2015). Clumps are identified in all gas with
$T<3\times 10^{7}\rm\ K$, and densities above $n>0.02$ cm-3. All clumps with a
relevance (peak-to-saddle ratio) of less than 30 are merged through the saddle
into the neighbouring clump.

##  4 Results

###  4.1 Radiative Cooling and Cloud Growth

In this section we study the impact of varying cloud and background gas
parameters to understand how local gas properties impact the evolution of cold
clumps in galaxy clusters. All simulations presented here have no magnetic
fields, while the simulations with magnetic fields are discussed in Section
4.2. Specific simulation parameters are outlined in Table 1.

####  4.1.1 The impact of radiative cooling

Two runs were performed with cooling turned off in order to quantify the
impact of cooling and compare our work against the large existing body of work
that does not consider cooling.

Figure 2: Number density slices showing the difference in evolution after 3
cloud-crushing times between the run without cooling (HD0.1_nc, left) and an
identical cloud run with cooling (HD0.1, right). Radiative cooling promotes
shattering of the cloud during the early evolution, with cloud morhology being
distinctly different. Figure 3: The fractional change in cold gas mass shown
for cooling and non-cooling simulations at different cloud densities. The
cloud is destroyed within a few $t_{\text{cc}}$ if cooling is not present
consistent with a large body of previous literature. However, in radiative
simulations significant cloud mass growth is seen for all densities explored.
Figure 4: Time evolution of the warm gas fraction normalised to the original
cloud mass for different initial densities. In the presence of radiative
cooling, the amount of warm gas is reduced in comparison to the non-cooling
runs as the warm gas is depleted by gas cooling into the cold, dense phase.
However, warm gas gradually builds up over time, with initially more diffuse
clouds producing a larger fraction of warm gas.  Figure 5: Density-weighted
projections of the number density $n$ shown at different times for simulation
HD0.1. The cloud collapses and fragments within one crushing timescale, and
the resulting clumplets then interact dynamically with the wind, forming an
extended tail. Note that each projection is centred on the most upstream clump
material. Figure 6: Distribution of clump masses (left) and clump radii
assuming spherical clumps (right) for simulation HD0.1 at four different
epochs. Insets in the left panel show the slopes of the power law fit for
clump masses above $10^{-5}\rm\ M_{\odot}$.

The cloud crushing timescale for the clouds we use is $24.4$ Myr, and the
cooling timescale is initially a factor $10^{4.6}$ shorter than this for the
densest cloud HD_10, and $10^{3.6}$ and $10^{2.6}$ times shorter than the
crushing time for HD_1.0 and HD_0.1, respectively. As shown in Fig. 2, cooling
completely changes the morphology of cold clouds. Without cooling (left
panel), cloud gas is mixed rapidly into the hot phase, while with cooling
(right panel) the cold cloud is shattered into many small fragments that
persist. More quantitatively, Fig. 3 shows that the clump in the HD0.1_nc run
loses 70% of its initial mass by $t=2\,t_{\rm cc}$, which rises to 99% by
$t=3\,t_{\rm cc}$. The cloud being destroyed on a timescale of the order of
$t_{\text{cc}}$ is in agreement with standard results of the cloud-crushing
problem. This evolution is independent of the initial clump density that we
vary by two orders of magnitude (compare the low density non-cooling cloud
HD0.1_nc to the high density non-cooling cloud HD10_nc in Fig. 3).

When radiative cooling is included, the cloud follows a completely different
evolution, as shown in Figs. 3 and 4, which quantify the time evolution of the
cloud and wind mass, respectively, and visualized in Fig. 5 which illustrates
the time evolution of the projected number density. Most notably, rather than
being destroyed, the total mass of the cloud grows significantly over time.
This long-term cloud growth is expected from previous work by Li et al. (2020)
and Gronke & Oh (2020), who both predict a minimum cloud survival radius (of
$6.5$ and $3.3$ pc, respectively, computed using Equations (3) and (5) from
Kanjilal et al., 2021) well below the initial $500$ pc radius of the cloud
studied here.

Initially, for $t\lesssim$ $t_{\text{cc}}$ the cold mass in the clouds grows
slowly, but then it accelerates, doubling within 5 $t_{\text{cc}}$ even for
the lowest initial cloud density explored (see Fig. 3). This cold mass growth
is more pronounced for the runs with higher initial cloud densities (see
Section 4.1.2 for a more detailed discussion) and for simulations performed at
higher resolution (see Appendix B).

In Fig. 6, we investigate whether this increased growth is due to a larger
number of individual clumps, or an increased mass per clump. To do so, we
identify individual cloudlets in the wake as described in Section 3.5 and then
study the statistics of this ensemble of cloudlets for each simulation over
time. As can be seen in Fig. 6, the total growth in cloud mass is both due to
a higher number of individual clumps, and due to a higher mass per clump.

The process commences with an initial collapse phase, during which the cloud
cools rapidly and compacts on sub-crushing timescales. Density inhomogeneities
are seeded in the collapse, and material starts to ablate from the main cloud
in the form of individual fragments with radii in the range $5-100$ pc, as
shown in Fig. 6. Over time, the original cloud is entirely broken up and the
resulting clumps, as well as the warm, diffuse material between them, forms an
extended clumpy wake, as shown in Fig. 2 and the late-time panels of Fig. 5.
The typical slope of $dN_{\rm clumps}/dM$ is of the order of -0.7 (see fitted
slopes in Fig. 6), which is somewhat shallower than the -1 measured in Gronke
et al. (2022), which suggests that the higher levels of turbulence in Gronke
et al. (2022) accelerate shattering. We find, by analysing the typical range
of clump sizes and densities seen post-shattering, that the characteristic
crushing timescale of the clumps is roughly the same as that of the initial
parent cloud, allowing us to continue use of $t_{\text{cc}}$ as a robust
characteristic time even after the shattering process. The overdensity $\chi$
does however increase, to approximately $100\times T_{cl,0}/T_{cl,ps}$ for our
initial overdensity of $100$ and a post-shattering temperature $T_{cl,ps}$
when the cloud attains approximate hydrostatic equilibrium again.
We run one cloud in initial hydrostatic non-equilibrium, HD0.1_1e4T0, by
setting the initial cloud temperature to $T_{cl}=10^{4}$ K, to examine how
sensitive the shattering and further evolution is to the initial temperature
and pressure of the clump. As can be seen in the mass evolution plots of Fig 3
and 4, shattering occurs quickly enough that evolution is insensitive to the
initial cloud temperature, and the only cloud property we find to be important
is the density.

Figure 7: Left: Binned temperature-density phase space occupied by the clouds
for simulations HD0.1, and HD10, with the colourmap showing the ratio of
cooling time to crushing time $\theta$ at three different times (left to
right). We also show the regions we define throughout the paper as ’warm’ gas
($10^{6}<T<10^{8}$ K) and ’cold’ gas ($T<10^{6}$ K). The change in initial
cloud density (as well as wind density) leads to a different phase space
region of the cooling function being sampled by the cloud, particularly in the
warm phase; at higher densities the cooling times are much shorter, and
cooling can take place as much as nearly six orders of magnitude faster than
cloud crushing for the densest cloud we test. The densest cloud HD10 cools
isochorically in the warm phase by approximately one and a half orders of
magnitude more than the least dense cloud, and settles later onto the isotherm
that will take the temperature down to $T_{\rm fl}$. Right: We instead show
the mass probability density of the two clouds each after 10 $t_{\text{cc}}$.
Though the initial overdensities were identical, the more dense cloud HD_10
holds more of its mass in dense, cold fluid, and has a more bimodal
distribution between the cold and hot phase, whereas HD_0.1 has a mass
distribution more evenly distributed through the warm phase as well.

The late-time evolution after several crushing times is driven by the
evolution of the wake in the cooling case. The mixing of the cold fragments
with the hot ICM generates a warm, mixed phase, which makes up the wake of the
clump, and trails any high-density remnants of the original clump that remain
at the ’head’ of the cloud. The amount of gas contained in this wake grows
over time but remains small in comparison to the cold gas mass, reaching about
about $25\%$ of the original cloud mass by $t=3t_{\rm cc}$ for HD0.1 (see Fig.
4). This warm gas has a very short cooling time, as can be seen in Fig. 7, due
to a peak in the cooling function $\Lambda(T)$ between $10^{5}\,\rm K$ and
$10^{6}\,\rm K$. In the most extreme cases, the cooling time can be $5$-$6$
orders of magnitude shorter than the crushing timescale. For such a short
cooling time, cooling dominates the evolution of the gas, and gas transfers
quickly through the temperature-density phase space from high temperature,
diffuse to cold, newly condensed cloud111We note that the bimodal feature in
the temperature range $10^{4}-10^{6}\rm\ K$ in Fig. 7 is due to the structure
of the flow, and results from a combination of cooling and mixing at the
surface of individual cold clumps. The left-hand feature is formed by gas up-
stream of individual clumps, and consists predominantly of gas that originates
in the hot wind. The right-hand feature is composed of gas down-stream of
individual clumps, which contains a higher percentage of gas ablated from the
clumps and is therefore at higher density.. The mixed phase condenses onto the
clump as new cold gas, which means that the cold fragments act as nucleation
sites for cold condensate within the hot ICM. We find that this is dominant
over mixing losses, resulting in the growth of the total cold gas mass over
time (see Fig. 3). The no-cooling cases, display no such growth, even though a
significant amount of warm gas is produced, with the fraction of warm gas more
than an order of magnitude larger in the no-cooling runs in comparison to the
initial cloud mass (see Fig. 4). This markedly different evolution with
cooling shows the importance of including radiative cooling in driving the
evolution of cold clouds in hot atmospheres of galaxy clusters.

Qualitatively, our results on cold cloud growth agree with recent results by
Jennings & Li (2020), Gronke & Oh (2020) and Li et al. (2020), as well as with
results by Gronke & Oh (2018) even though the latter do not see the same
amount of shattering as they impose a higher temperature floor of $4\times
10^{4}\rm\ K$ (see Section 4.1.3 for a discussion on the impact of cooling
floors). The cold clumps simulated here exceed the pressure condition required
for shattering put forth by Gronke & Oh (2020), namely
$P_{\text{cl,ps}}/P_{\text{cl,0}}\sim T_{\text{cl,ps}}/T_{\text{cl,0}}\lesssim
0.33$, where $T_{\text{cl,ps}}$ is the post-shattering cloud temperature,
equal to the cooling floor temperature. We do not find any significant post-
shattering coagulation of fragment clumps, probably due to the extreme
pressures generated by the large temperature gradients, with the fragments
being well-separated after the post-shattering expansion. This can be clearly
seen at late times in Fig. 5.

The final size of clumplets needs to be considered in this context too.
McCourt et al. (2017) put forth an argument that clouds will shatter until
their clumps are small enough that the sound-crossing time is of order the
cooling time, and thus cooling can occur isobarically. This occurs when clumps
reach a physical lengthscale of $l\sim c_{\rm s}t_{\rm cool}$, which occurs
approximately on the timescale of $t_{\rm cool}$. McCourt et al. (2017) show
that clouds should shatter down to a size of $l_{\text{cloudlet}}\sim c_{\rm
s}t_{\rm cool}\sim(0.1\text{ pc})/n$, and this has been further confirmed by
other simulation works (see e.g. Gronke & Oh, 2020). The largest
characteristic length of these cloudlets in our simulations is expected to be
around a parsec, an order of magnitude smaller than our minimum resolved
scale, even with the high resolution we have adopted. Ideally, a resolution
two order of magnitude better than our simulation runs should be used, to
investigate whether shattering stops at this characteristic scale, or if
clumps will fragment even more, but this is computationally unfeasible with
our current set-up since $r_{\rm cl}/l_{\text{cloudlet}}\gg 1$.

In summary, we reproduce the destruction or growth of the cloud on crushing
timescales found by similar studies. For the parameter space of cold clouds in
a hot cluster environment, we find that clouds grow in mass on timescales
relevant for their trajectories when radiative cooling is taken into account.
This growth is significant, and leads to cold clouds acting as nucleation
sites within the hot ICM, with condensation beginning after only a few
crushing timescales, which in cluster environments translates to a few tens of
Myr.

####  4.1.2 Density Dependence

We now compare the effects of altering the initial cloud density on the
evolution of clouds themselves, by comparing runs with initial cloud densities
of $\rho_{\rm cl}=10\ {\rm cm}^{-3}$ (HD10) and $\rho_{\rm cl}=1\ {\rm
cm}^{-3}$ (HD1.0) to the fiducial run HD0.1 which has cloud density $\rho_{\rm
cl}=0.1\ {\rm cm}^{-3}$. All three simulations have the same overdensity
$\chi=\rho_{\rm cl}/\rho_{\rm w}=100$, and are therefore identical within the
context of scale-free considerations.

As expected from analytic work, the absolute density of the cloud does not
change the mass evolution in the absence of cooling as long as $\chi$ is
constant. This can be clearly seen by the identical time evolution of both the
warm and cold gas for simulations HD10_nc and HD0.1_nc in Fig. 3 and Fig. 4.

It is only with the addition of cooling that the absolute density of the cloud
makes a difference to its evolution. An initially lower density cloud forms
lower density fragments post-shattering as would be expected, and the lower
density of the clumplets in the wind results in greater mixing of cold gas
into a warm phase. This can be clearly seen in Fig. 4 by the larger amount of
warm gas generated by the lower density cloud, in comparison to its initial
mass. This confirms that the presence of cooling breaks the scale-free nature
of the problem, by introducing an additional characteristic timescale which
depends on both the cloud density and temperature (as well as metallicity, but
this is kept constant in our runs). As the cooling time is shorter for denser
material, initially denser clouds show faster mass growth due to the more
rapid condensation of gas from the warm to the cold phase. The impact of the
higher density mixed gas on the cooling time for all three simulations can be
seen in Fig. 7, with the fastest cooling regime in the warm phase only
occupied by the densest clouds. Interestingly, all three simulations reach a
rough equilibrium between mixing new gas into the warm phase, and cold gas
condensing into the cold phase, but the equilibrium amount of warm gas is
lower for initially higher density clouds (see Fig. 4).

For our densest cloud modelled here, HD10_nc, the initial cloud conditions are
such that Li et al. (2020) predict that clouds with an initial radius of
$r_{\rm cl,init}<651\rm\ pc$ (computed using Equation (5) in Kanjilal et al.,
2021) should not survive, which is larger than our initial cloud radius of
$500$ pc. By contrast, Gronke & Oh (2020) predict a minimum cloud survival
radius of $334$ pc (computed using Equation (3) in Kanjilal et al., 2021), and
therefore predict cloud HD10_nc should survive. With only a single cloud in
this borderline region between the two regimes, we refrain from strong
statements about either criterion. Instead, we simply note that HD10_nc
exhibits robust growth in our simulations and that this range of radii would
be an interesting range to probe in future work.

We conclude that cooling is the dominant mechanism determining the evolution
of the cloud for all clouds within the two orders of magnitude in density
probed here. We also confirm that in cluster environments, where cooling
dominates the evolution of cold clouds, scale-free studies are insufficient
and the parameter space needs to be explored widely. The smaller the ratio of
cooling time to cloud crushing time $\theta$ (see Equation (3)), the faster
cold clumps in the ICM can condense new gas and grow, with all cases tested
here more than doubling the clouds initial mass on timescales of around
$150\rm\ Myr$, and some increasing it by almost a factor $10$.

####  4.1.3 Cloud crushing and small scale heating

The sub-kpc scale physics which takes place in cold clouds within galaxy
clusters is poorly understood. However, observations show the presence of warm
gas, despite short cooling times, which suggests that clouds need to be
persistently powered. Over the years, many processes have been suggested as
the source of this heating, from photoionization by the central AGN (e.g.
Heckman et al., 1989; McNamara & Nulsen, 2012) or massive stars (Canning et
al., 2014), via thermal conduction (Voit et al., 2008) and magnetic
reconnection (Hanasz & Lesch, 1998; Tanuma et al., 2003; Churazov et al.,
2013) to cosmic ray streaming heating (Ruszkowski et al., 2018), to name a
few. Stars can form in “knots” along filaments (Vantyghem et al., 2018), and
these sites may also provide some heating and feedback to the cold gas once
fragmented. So far, no consensus has been reached on the origin of cloud
heating, but such energy injections at the cloud radius would off-set some of
the cooling and prevent clouds cooling below an effective temperature floor.
This floor would of course be expected to be time-varying, with the
characteristic timescale of the most dominant heating mechanism setting the
timescale of variation. However, without knowing where the dominant energy
input comes from, we choose to keep the floor constant throughout our
simulations.

In the simulations presented here, we test the impact of such small-scale
heating and feedback processes, as well as the heating effects of a radiation
background within the cluster, by varying the cooling floor, i.e. the minimum
temperature that gas can reach through radiative cooling. The implicit
assumption is that at this temperature, the energy losses from radiative
cooling match the combined heating effects of the processes we do not
explicitly model. To test the robustness of our general results against this
choice of cooling floor, we select one simulation (HD0.1) to be repeated with
a variety of cooling floor temperatures $T_{\text{fl}}$ . These runs are
detailed in Table 1.

Figure 8: The normalised cold gas mass fraction for variations of HD0.1 run
with different cooling floor temperatures $T_{\rm fl}$. They are compared to
our flagship run HD0.1, which has an effective temperature floor of $T_{\rm
fl}=10\rm\ K$. Trends are non-linear, with cold gas mass growth the fastest
for both high ($T_{\rm fl}\geq 10^{5}\rm\ K$) and low ($T_{\rm fl}\leq
10^{3}\rm K$) cooling floors, but suppressed for $10^{3}<T_{\rm fl}<10^{5}\rm\
K$. Figure 9: The normalised warm gas mass fraction for each variant of HD0.1
run with different cooling floor temperatures $T_{\rm fl}$. Lower cooling
floors produce more warm mass in the temperature range $10^{3}\ {\rm K}\leq
T_{\rm fl}<10^{5.5}\ {\rm K}$, but significantly lower cooling floors again
reduce the warm gas mass.

We find that clouds grow for all choices of cooling floor tested here. During
the initial cloud disruption phase ($t<0.5\ t_{\rm cc}$), a higher cooling
floor produces less cold gas (Fig. 8) and more warm gas (Fig. 9). The cloud
with a cooling floor at $3\times 10^{5}$ K hardly shrinks in radius during
$t<$ $t_{\text{cc}}$, as the initial cooling and resulting loss of pressure is
strongly reduced, while clouds with lower cooling floors contract
significantly as can be seen in the images in Fig. 5 for our fiducial HD0.1
run.

Beyond 1 $t_{\text{cc}}$, the evolution becomes more complex. There is a
general trend for higher cooling floors to produce more cold gas for $0.5$
$t_{\text{cc}}$$<t<4$ $t_{\text{cc}}$, while for lower cooling floors the cold
gas mass increase is lower initially but then accelerates after several
crushing times. The lower the cooling floor, the faster the turn-up occurs,
but results converge for minimum temperatures of $T_{\rm fl}\leq 10^{3}\rm\
K$. As a result of this speed-up in condensation for low values of $T_{\rm
fl}$, by 5 $t_{\text{cc}}$, the highest and lowest $T_{\rm fl}$ have a similar
amount of total cold gas mass, while the intermediate-floor runs have a
smaller amount.

The evolution in warm gas mass, shown in Fig. 9 is similarly non-linear. Again
the early evolution is directly dependent on the cooling floor temperature
$T_{\rm fl}$, with lower $T_{\rm fl}$ producing less warm gas for $t<$
$t_{\text{cc}}$. For later times, the long-term evolution of the total warm
gas mass $M_{\rm w}$ is generally a function of $T_{\rm fl}$, with higher
$T_{\rm fl}$ producing less $M_{\rm w}$. The outlier to this trend is
HD0.1_F5.0, which despite having the second highest $T_{\rm fl}$ produces a
similar amount of $M_{\rm w}$ as runs with $T_{\rm fl}\leq 10^{3.5}\rm\ K$.

Figure 10: Time evolution of the number of clumps (top panel) and the mass of
the most massive clump as a function of initial cloud mass (bottom panel) for
variations of the HD0.1 run with different $T_{\rm fl}$. In general, higher
minimum temperatures lead to less fragmentation and a more stable main clump
but the evolution is not linear with $T_{\rm fl}$.

This somewhat complex behaviour in the warm phase is due to how cloud
shattering proceeds for each simulation. As can be seen in Fig. 10, the cloud
evolution changes fundamentally with increasing $T_{\rm fl}$. For low values
of $T_{\rm fl}$ we see an evolution very similar to that shown for HD0.1 in
Fig. 5: the original cloud shrinks rapidly very early on as it loses pressure
support, and breaks up into a large number of individual clumps. During this
time, the total cold gas mass increases, but the mass of the most massive
clump $\rm M_{\rm mm}$ actually reduces over time as shattering progresses.
This occurs within half a crushing timescale and produces small clumplet
fragments, which interact with the wind over the next few crushing times to
produce individual, but often connected, wakes of warm gas. This process has
been found recently in other studies too, notably by McCourt et al. (2017),
and Gronke & Oh (2020), though Jennings & Li (2020) notes that it is still
uncertain as to whether fragmentation is shock-induced due to the Richtmyer-
Meshkov instability (Meshkov, 1972), or if some other process is at play, such
as pressure interactions between two neighbouring clouds.

For higher $T_{\rm fl}$, the shattering is less strong and the number of
individual clumps remains much lower. This produces less mixed gas before
$0.5$ $t_{\text{cc}}$ as a result of hot gas not being pulled violently into a
mixing layer by the pressure gradients during the collapse. In general, higher
$T_{\rm fl}$ means a more intact main clump with a higher mass (see evolution
of the mass of the most massive clump, $M_{\rm mm}$, in Fig. 10). For the
highest values of $T_{\rm fl}$ tested here, the main clump remains intact and
actually grows in mass over time. The cooling floor temperature has such an
important impact because the initial cooling of the cloud, once placed in the
ICM, occurs rapidly enough that it is essentially isochoric, which means that
the pressure gradient produced at the boundary is linearly proportional to the
value of the temperature floor where the cloud settles, and so there is a
range of orders of magnitude between the forces these clumps experience across
runs. We also note that the number of clumps after the initial ablation event
is closely linked to $\rm l_{\rm cloudlet}$ shown in Fig. 1: runs with $T_{\rm
fl}$ corresponding to maxima of $\rm l_{\rm cloudlet}$ (for example HD0.1_F5.5
or HD0.1_F4.0) have less clumps early on than those corresponding to local
minima (such as HD0.1_F4.5 or HD0.1_F5.0).

After reaching an initial equilibrium, the clouds at the higher temperature
floor tend initially to ablate more than mix, with clumplets breaking or
shredding off from the main clump, increasing the effective mixing surface,
and generating mixed gas at the boundary layer. This is perhaps a more
traditional cloud-crushing scenario, and one that approaches the behaviour in
the no-cooling limit case. Mixed gas does condense to form new cold gas, but
the physical extent of the wake at very high temperature floors appears to be
suppressed, and there is a larger consolidated “head” of the cloud that
remains in the wind.

The amount of ablation and shattering is directly related to the amount of
warm gas produced, as can be seen by studying the outlier simulation
HD0.1_F5.0, which produces many more individual clumps around 1.5
$t_{\text{cc}}$ than either HD0.1_F5.5 or HD0.1_F4.5. This early ablation
event is also the exact point in time when the warm gas mass $\rm M_{\rm w}$
for this simulation increased significantly in Fig. 9.

We conclude that internal heating mechanisms could play an important role in
the long-term evolution of cold clouds by suppressing ablation and shattering,
and allowing the main cloud to survive for longer. However, in a cluster
environment, mixing at the cloud surface remains sufficiently efficient that
even with reduced shattering clouds grow in mass over time, albeit slower than
those that originally were originally mixed more efficiently into the warm
phase.

####  4.1.4 Kelvin-Helmholtz vs Pressure Gradients

The generation of a mixed phase at the boundary layer of the cloud is the key
process that drives the evolution and long-term growth of the clouds via
condensation in our simulations. Gronke & Oh (2019), who find robust cloud
growth in a galactic setting, argue that rapid cooling can generate pressure
gradients large enough that mixing in the boundary layer is dominated by
pressure-driven mixing rather than by Kelvin-Helmholtz processes in the shear
layer. If pressure mixing dominates, the cooling rate directly determines the
growth rate of the cloud, and growth continues even when the cloud is almost
co-moving.

Figure 11: The root mean square velocity of the cold gas against the growth
rate of the cold cloud, normalised to the total cold mass in the box at the
present time for each data point, in units of $t_{cc}^{-1}$, for our three
hydro runs. The initial collapse and shattering phase is not included. Note
that the mean square velocity of the cloud relative to its initial zero
velocity is plotted, and that the wind velocity sits at $200$ km s-1. The
mass-normalised growth rate reduces as the cloud becomes co-moving in the
wind, indicating that the mixing is largely Kelvin-Helmholtz instability-
driven and that the rate of condensation onto the cloud reduces as the cloud
decelerates and the Kelvin-Helmholtz mixing becomes less effective.

To understand how mixing proceeds in our simulations, we plot the growth rate
of the cold gas fraction normalised to the current cold mass
${\dot{\mathrm{M}}\mathrm{{}_{c}/M_{c}}}$ against the velocity of the cloud
(which is initially at rest in the simulation frame) in Fig. 11. Note that
‘cloud’ refers to the entire wake of mixed gas, not the most massive remnant
of the original cloud. As can be seen in Fig. 10, the most massive clump
looses mass over time, and therefore has a _negative_ growth rate, while the
cloud has _positive_ growth rate as the total gas mass increases over time.
Random motions not along the axis of flow are included in this root-mean-
square velocity, since this velocity dispersion can still contribute to
instability mixing, although in practice the velocity magnitudes are in
general small compared to the bulk flow. (This can be seen by only a small
excess velocity of the gas in the approximately co-moving state reached by
HD1.0 and HD10.)

We find that though the total cloud growth rate continues to increase with
time as the cloud becomes approximately co-moving with the wind, the
normalised rate – which takes into account the fact that at later times the
cloud is larger and has a greater mixing area – reduces. For example, when the
overall cloud reaches approximately co-moving status, the normalised growth
rate is around $5-10\%$ the value at a relative velocity of 170 km s-1. This
supports the argument that the condensation and cloud growth is driven by
Kelvin-Helmholtz mixing, which acts to a lesser extent when the cloud slows
down relative to the wind, rather than the pressure mixing. In the growth rate
of the warm fraction (not shown), we see a similar decline as the cloud
becomes co-moving, but a peak in growth rate between $75$ and $100$ km s-1;
most likely this corresponds to the point where the cold gas area becomes
completely covered in a warm mixing layer.

In clusters this slowing down occurs due to both the ram pressure experienced
by the cloud gas as it punches through the ICM, and also due to the
gravitational effect of the cloud climbing out of the central potential in the
cluster. The cloud can also slow down/become entrained due to cooling gas
imparting momentum to the cloud as it condenses. Treating this case as an
inelastic collision and conserving moementum, the approximate velocity
relative to the wind at a given time will be $\frac{\Delta
v(t)}{v_{wind}}\approx\frac{m_{cl,0}}{m_{cl}(t)}$ (Gronke & Oh, 2018).

###  4.2 Magnetic fields and cloud growth

Observational studies of galaxy clusters using Fast Radio Burst (FRB) probes
(e.g. Prochaska et al., 2019) and rotation measures (e.g. Farnes et al., 2017;
Malik et al., 2020) have shown that the ICM is magnetised with a field
strength of about $2\,\rm{\mu}$G. However, field strengths in the centres of
cluster cooling flows may be an order of magnitude higher (see Carilli &
Taylor, 2002), and recent observations of the Smith cloud (a high velocity
cloud around $3$ kpc below the galactic plane of the Milky Way with a size and
temperature comparable to our simulated clouds) by Hill et al. (2013) and
Betti et al. (2019) give lower bounds on the peak line-of-sight field strength
of $8\,\mu$G and $5\,\mu$G, respectively. Magnetic fields have long been
assumed to be dynamically unimportant in the cloud crushing problem, with the
magnetic pressure often orders of magnitude lower than the thermal pressure.
However, with mixing being recognised as the dominant mechanism for cloud
evolution, more and more simulations of the CGM and ICM have begun to consider
their effect in recent years. Such simulations have shown that magnetic fields
can help prevent shredding by stabilising perturbation modes aligned with the
field (via draping, see e.g. Mac Low et al., 1994; McCourt et al., 2015;
Sparre et al., 2020), and by establishing anisotropic thermal conduction (e.g.
McCourt et al., 2012; Jennings & Li, 2020), with draping then able to suppress
conduction across the boundary layer (Dursi & Pfrommer, 2008a).

In this section, we investigate how magnetic fields of different strengths and
orientations influence the evolution of cold clouds as they travel through the
hot ICM. The additional impact of thermal conduction will be studied in
Section 4.3. Magnetic fields of initial strength $B$ were initialised in one
of two configurations: either aligned (“parallel”) with the direction of flow,
or normal (“perpendicular”) to the direction of flow. We test three different
field strengths, with detailed information on parameter choices for magnetised
simulations given in Table 2. All simulations presented in this Section and
the next are magnetised variations of the HD10 simulation.

Table 2: The parameters of MHD runs, with columns showing: (1) the names, (2) the magnetic field strength $B_{0}$, (3) the initial orientation of the magnetic field relative to the initial bulk flow velocity (either normal to the flow or aligned with the flow), and (4) the type of thermal conduction type (if at all). All runs are variations of HD10 and share its parameters where not explicitly stated otherwise. Name |  $B_{0}$ [$\mu$G] | Orientation | Conduction
---|---|---|---
MHD_1N | 1.0 | normal | ✗
MHD_1A | 1.0 | aligned | ✗
MHD_3N | 3.0 | normal | ✗
MHD_3A | 3.0 | aligned | ✗
MHD_5N | 5.0 | normal | ✗
MHD_5A | 5.0 | aligned | ✗
MHD_1N_i | 1.0 | normal | isotropic
MHD_1A_i | 1.0 | aligned | isotropic
MHD_3N_i | 3.0 | normal | isotropic
MHD_3A_i | 3.0 | aligned | isotropic
MHD_5N_i | 5.0 | normal | isotropic
MHD_5A_i | 5.0 | aligned | isotropic
MHD_1N_a | 1.0 | normal | anisotropic
MHD_1A_a | 1.0 | aligned | anisotropic
MHD_3N_a | 3.0 | normal | anisotropic
MHD_3A_a | 3.0 | aligned | anisotropic
MHD_5N_a | 5.0 | normal | anisotropic
MHD_5A_a | 5.0 | aligned | anisotropic
Figure 12: The density-weighted projected number-density (top panel), and the
magnetic field magnitude in a slice of the simulation box for the six MHD only
runs. On the lower panels the line integral convolution is plotted to give an
idea of the orientation of the magnetic field at each region. One can clearly
see that if the field is normal to the flow then the field lines drape more
around the head of the cloud, and become more non-uniform in the wake. Also,
compared to the no MHD simulations at a similar time (see e.g. Fig. 5 for
comparison, though recall that the initial densities are $100$ times smaller
in the pure hydro plot), the amount of dense, cold gas present is highly
suppressed, with higher $B$ field leading to more suppression. Figure 13: Time
evolution of the normalised cold gas mass fraction for six magnetised
variations of the HD10 simulation, as well as the corresponding hydrodynamical
run. Even a weak field suppresses cloud growth, with suppression being
stronger for higher field strengths. Magnetic fields initially perpendicular
to the flow seem to be more effective at stunting the cloud growth than
initially parallel fields, by a larger factor in the $1\mu$G case and to a
lesser extent for stronger field strengths. Note for that for this plot, and
all following magnetic field run mass/curl plots, only every fifth data point
is shown for clarity. Figure 14: Time evolution of the normalised warm gas
mass fraction for six magnetised variations of the HD10 simulation. Magnetic
fields reduce the amount of warm gas by up to two orders of magnitude over a
few crushing timescales for the range of field strengths tested here.
Reduction in the warm gas mass is more pronounced for the simulations where
magnetic field lines are initially normal to the flow.

As can be seen in Fig. 12, magnetic fields have a significant impact on the
morphology of cold clouds and their wake over time, but not enough impact to
prevent the growth of the cold cloud mass (see Fig. 13). In comparison to the
reference simulation HD10, mass growth is slower for all magnetic field
strengths tested here, and this suppression is greater for stronger magnetic
fields irrespective of the initial field configuration. The warm gas mass
fraction is similarly suppressed, as shown in Fig. 14, with initially stronger
magnetic fields leading to less warm gas mass. Interestingly, the trend in
mass growth is not reversed between the warm and cold gas phases like it is
for the simulations of clouds with different density (see Section 4.1.2),
where the clouds that grow fastest in cold gas mass tended to generate the
least warm gas.

Figure 15: Time evolution of the number of clumps (top) for six simulations
with different magnetic field orientations and magnetic field strengths of the
HD10 simulation. Higher magnetic field strengths, and fields initially
orientated normal to the direction of flow, lead to less fragmentation.

We also see a distinct difference in cloud evolution between the two magnetic
field configurations tested here, with the perpendicular field suppressing
mixing and allowing for more efficient cooling at all field strengths tested
than the equivalent runs with initially aligned fields to the flow. For our
highest field strength ($5\rm\ \mu$G), there is a difference of approximately
$8\%$ in the cold gas mass and a difference in the warm fraction of
approximately $53\%$ at $3$ $t_{\text{cc}}$ between the normal and aligned
configurations. For a $1\ \mu$G field these differences are larger at 3
$t_{\text{cc}}$, at $110\%$ and $238\%$ in the cold and warm fractions,
respectively. Under the assumption that mixing is driven by the Kelvin-
Helmholtz instability, such as in the analytical solution by Chandrasekhar
(1961), this trend is surprising. If mixing is instability driven, a magnetic
field with fields lines parallel to the fluid flow should stabilise the
Kelvin-Helmholtz instabilities and slow the growth rate of unstable modes,
with larger field strengths providing greater stabilising action. This is due
to the magnetic tension force resisting the bending of fields lines.

To explain the trends in the warm gas reported here, it is necessary to
instead consider the morphology of the cloud and the topology of the field
lines that result from the two initial field set-ups. The impact of magnetic
fields on the warm gas in particular can be seen visually in Fig. 12, which
shows that at fixed field strength, the aligned configuration produces a
significantly more extended wake than the normal configuration. This trend is
confirmed in Fig. 15 which shows that at fixed field strength, there are fewer
individual fragments downstream in the wake than for magnetic fields
perpendicular to the flow. As discussed in Section 4.1.3, a more extended wake
and greater fragmentation leads to a greater surface area of interface between
the cold and hot gas, which increases mixing and production of the warm phase.
This, in turn, drives increased condensation.

There are a several mechanisms suppressing the wake in the perpendicular field
configuration. Firstly, there is a strong magnetic draping effect taking place
near the head/s of the clump/s, such as that described by Dursi & Pfrommer
(2008b) and more recently by Sparre et al. (2020). As the clump moves through
the hot ICM, it ‘sweeps up’ magnetic field lines, which become pinched
together near the head of the cloud. Dursi & Pfrommer (2008b) find that the
magnetic tension associated with this draped layer can dominate over
hydrodynamic drag forces in slowing the clump down and reducing fragmentation.
Draping of the field does not happen as significantly in the case where the
magnetic field is parallel to the flow velocity, since the cloud material can
slip along with the field lines. In our case, magnetic draping is present at
the head of the cloud, as can be seen by the increased field magnitude at the
head of the clumps in the perpendicular field configuration in Fig. 12.
Perhaps the most important draping effect however is the suppression of
Kelvin-Helmholtz instabilities when the field is tangled, which is not as
pronounced if the field is uniform. This is due to a draping layer which
suppresses Kelvin-Helmholtz instabilities (Sparre et al., 2020), which is more
effective in a tangled-field scenario because in a uniformly magnetised wind
the instabilities are suppressed only along the axis which is parallel to the
field orientation, rather than along all axes.

Figure 16: The average value of the curl of the magnetic field (mass-weighted)
for all ‘non-hot’ gas (i.e. all gas below the wind temperature). The
perpendicular field configuration tends to have a larger value of the curl
after the early cloud collapse than the parallel field equivalent.

The second process acts on the wake behind the clump. As can be seen in the
line-integral convolutions which trace the magnetic field direction in Fig.
12, the field morphology in the wake depends strongly on the initial magnetic
field topology. For an initially parallel field, the magnetic field wraps
around the cloud and wake, and a field configuration that is mostly parallel
to the flow is maintained. In the perpendicular case, however, the field
becomes very turbulent downstream of the wake. This tangling of the field
lines creates magnetic tension, which prevents the wake from elongating as
efficiently in the perpendicular field case, due to the Lorentz force on the
plasma in this region. This effect is probably enhanced by the suppression of
mixing instabilities by draping, with less clumpy material then being swept
downstream in the wake.

To quantify this non-uniformity of the field in the wake and compare between
the aligned and perpendicular field configurations, we plot the average curl
of the magnetic field in non-wind ($<10^{8}$ K) fluid in Fig. 16. We find that
the perpendicular field always has a higher mass-weighted average of the
magnitude of the $B$-field curl than the parallel field case in the mixed gas,
by up to a factor of a few. This trend persists at later times as the value of
the curl converges.

Figure 17: Distribution of plasma $\beta$ versus temperature for all
magnetised simulations at $t=2.6$ $t_{\text{cc}}$. Solid (dashed) lines mark
the upper $90$th percentile of the distribution at a given temperature for all
runs with initially parallel (perpendicular) magnetic fields while dotted
lines mark the lower $10$th percentile. The shaded regions highlight the range
of values for MHD_3N and MHD_3A, respectively. Except in the range
$10^{5}\,\mathrm{K}<T<10^{6}\,\mathrm{K}$, lower limits for MHD_3N and MHD_3A
are very similar. For runs with an initially perpendicular magnetic field,
magnetic pressure support dominates for cold gas, while for initially aligned
field the distribution is much wider.

As discussed in Section 4.1.4, mixing in our simulations is primarily driven
by Kelvin-Helmholtz instability, but there is still some growth in the cold
phase even when the cloud is approximately co-moving, which is likely due to
the pressure-driven effects. With an initial plasma $\beta=10^{3}-10^{4.5}$
($\beta=2P_{\rm th}/B^{2}$, where $P_{\rm th}$ is the thermal pressure and
$B^{2}/2$ is the magnetic pressure), magnetic pressure effects are negligible
in the hot wind but $\beta$ naturally decreases as the gas cools until
$\beta<1$ in the cold clumps (see Fig. 17). The relative strength of magnetic
pressure at a given temperature depends on the field configuration. Originally
perpendicular fields produce a much wider range of $\beta$ values at a given
temperature in comparison to aligned fields. However, for all simulations
tested here, much of the warm gas remains on average dominated by thermal
pressure ($<\log(\beta)>\gtrsim 0$). The presence of magnetic fields,
therefore, does not prevent the pressure-gradient-driven mixing of warm gas
into cold gas, and at best merely slows the process down. Interestingly, the
minimum $\beta$ as a function of temperature is the same for both simulations,
but MHD_3N has a much higher maximum $\beta$ than gas of the same temperature
in MHD_3A. As a result, all cold clumps are magnetically dominated for MHD_3A,
while some cold clumps remain thermal pressure dominated for MHD_3N.

###  4.3 Thermal Conduction

In the presence of strong temperature gradients, such as those seen in and
around cold clumps in the intracluster environment, free electrons transport
heat parallel to the magnetic field lines (Cowie & McKee, 1977). This
anisotropic thermal conduction is expected to smooth temperature, pressure and
density gradients in multiphase systems (Vieser, W. & Hensler, G., 2007). For
sufficiently high field strengths, it will also suppress orthogonal
instabilities (e.g. Parrish et al., 2012) as momentum and conductive heat
transfer can be restricted almost exclusively to paths along field lines. As
well as stabilising instabilities and thus reducing the proportion of the gas
in the catastrophically-cooling warm phase, conduction has the potential to
directly slow down the cloud growth by offsetting radiative cooling losses
with energy transferred from the surrounding hot medium.

Anisotropic conduction is computationally expensive to model in simulations
and for this reason has frequently been omitted in previous work on the cloud
crushing problem, or approximated using analytic assumptions based on the
morphology of the magnetic field, which translates to a numerical “fudge
factor” in the classical Spitzer formula (Spitzer, 1962). In this paper, we
explicitly model the conduction within ramses as part of our full MHD set-up.
All simulations with conduction are detailed in Table 2, and include runs with
either isotropic (post-fixed with $X\\_{\rm i}$) and anisotropic conduction
($X\\_{\rm a}$). For each type of conduction we test a strong, an
intermediate, and a weak field in both the perpendicular and aligned
configuration. We shall continue referring to the magnetised runs without
conduction as ‘MHD’ runs.

Figure 18: Time evolution of the normalised cold fraction for cold clouds
simulated with magnetic fields and thermal conduction. Isotropic thermal
conduction causes clouds to evaporate in the hot wind, while anisotropic
thermal conduction is insufficient to prevent their growth. Note that the
growth rate for MHD runs is slow within the first $t_{\text{cc}}$, hence the
linear scale on the vertical axis. Figure 19: The normalised warm mass
fraction for all simulations with magnetic fields and thermal conduction. Both
isotropic and anisotropic thermal conduction increase the amount of warm gas
in comparison to simulations without conduction. This effect is generally
stronger for anisotropic than isotropic thermal conduction. Figure 20: The
density-weighted magnetic curl projection (top panel) and magnetic field
magnitude slice (bottom panel) for the eight MHD runs with thermal conduction,
as well as four corresponding non-conduction MHD runs. The orientation of the
magnetic field lines is shown via the line integral convolution.

The addition of thermal conduction influences both the amount of cold and warm
gas, with isotropic conduction producing significantly wider wakes than
anisotropic conduction (see Fig. 20). While the remaining cloud cloud stays
very compact in the isotropic conduction phase, any material ablated is mixed
into a wide, turbulent wake which is quickly heated to hot gas temperatures.
By contrast, in the anisotropic case, wakes remain much narrower and the
ablation of diffuse material from the back of cloud produces the warm gas seen
here.

The clearest impact of conduction is seen in the cold gas, where cloud
evolution separates into two very distinct modes depending on whether the
conduction is isotropic or not (see Fig. 18): isotropic conduction causes the
cold gas mass to decrease over time, similarly to but more slowly than what is
observed in the non-cooling runs shown in Fig. 3. For all runs with isotropic
conduction, the hot ICM can supply sufficient thermal energy to offset cooling
and prevent the formation of new cold gas from the mixed phase. The peak in
the condensed mass just after $0.15$ $t_{\text{cc}}$ (where the cold mass
losses are briefly almost nullified) is likely due to the high fraction of
warm gas formed within $0.1$ $t_{\text{cc}}$, which may manage to condense
some cold gas before the warm fraction drops after $0.2$ $t_{\text{cc}}$. From
there on, the cold mass is reduced by around $1\%$ by $0.4$ $t_{\text{cc}}$
and by $2-3\%$ by $0.8$ $t_{\text{cc}}$. This mass loss rate is around 10
times slower than the non-cooling run of the same density, HD10_nc whose cold
mass is reduced by $30\%$ by $0.9$ $t_{\text{cc}}$. We conclude that isotropic
thermal conduction is able to offset radiative cooling in and around cold
clouds in galaxy clusters and to prevent their growth on long timescales.

As expected, conduction acts to generally increase the warm gas fraction
compared to the MHD runs, due to its ability to transfer heat from the
background hot wind to the mixed phase. This is shown in Fig. 19. This warm
gas can no longer be classified as a pure "mixed" phase, as heat is now
allowed to flow along temperature gradients, i.e. warm gas is now also
generated through the heat flux from the hot ICM into the cold cloud fluid. As
a result, more warm gas is created early on, up to $\sim 100$ times the amount
seen in simulations without conduction at $0.1$ $t_{\text{cc}}$, since
conduction timescales are shorter than the dynamical mixing timescales. The
conduction runs produce more warm gas than any of the purely hydrodynamical or
MHD runs. For example MHD_1N_a and MHD_1N_i produce $13$ and $37$ times as
much warm gas, respectively, as MHD_1N at 0.6 $t_{\text{cc}}$.

For all anisotropic cases we see evolution more in line with the other
simulations presented in this paper: clouds grow in mass over time, as can be
seen in more detail in Fig. 21. Apart from an early boost in cold gas mass for
the cases with an aligned magnetic field, the growth rate of cold gas in the
presence of anisotropic conduction is reduced in comparison to the equivalent
non-conducting runs. The impact of anisotropic thermal conduction on both the
cold and the warm phase can be understood in the context of the length-scales
of conduction in and around cold clouds. Following Section 3.1 of Li et al.
(2020), the heat precursor in the hot phase is $\lambda_{e}\simeq 100\,\rm
pc$. The skin depth of heat conduction in the cloud is $\lambda_{\rm
skin}=\lambda_{e}/\chi\simeq 1\,\rm pc$. As a result, the thermal conduction
has no direct effect on the cold cloud (such as suppressing KH instabilities
or reheating cold gas to the warm phase), but it warms up the mixing layer on
a $100$ pc scale. This leads to the increased amount of warm gas observed
here, but also explains why this warm gas cools more slowly into cold gas.
Note that the effect of thermal conduction is directly proportional to density
of the cloud. For clouds with $\chi<100$ the skin depth will quickly increase,
and thermal conduction should have a very significant effect on the cloud
structure and the number of cloudlets.

Figure 21: Time evolution of the normalised cold fraction for all cold clouds
that grow in Fig. 18. Anisotropic conduction makes little difference to the
evolution of the cloud for a magnetic field that is normal to the direction of
flow, but boosts the growth of cold gas for an initially aligned magnetic
field. Figure 22: Slice plot of the temperature in MHD_1N_a (left) and
MHD_1A_a (right) at $t=0.18\,t_{\rm cc}$. Limits were chosen to highlight the
structure of the plume pre-cooled by thermal conduction.

To understand the early boost in cold gas mass for aligned magnetic fields in
the presence of anisotropic thermal conduction (compare MHD_1A_a to MHD_1A and
MHD_1N_a in Fig. 21), one needs to look in detail at where anisotropic
conduction extracts the energy from the hot wind. As can be seen visually in
Fig. 22, in the aligned case (MHD_1A_a), thermal energy is extracted upstream
of the cloud as it flows down the magnetic field lines. This creates a cooler
plume of gas ahead of the cloud, which, due to the loss of thermal pressure,
also becomes somewhat denser. Due to the flow direction, this gas then impact
the mixing layer of the cloud, where the early cooling at $t<0.2\,t_{\rm cc}$
occurs. With a perpendicular magnetic field (MHD_1N_a in Fig. 22), a similar
plume forms next to the cloud but due to its location, the plume in this case
flows downstream without interacting further with the cloud. For this reason,
the cloud in the aligned case is fed by pre-cooled, denser gas early on which
reduces the cooling times in the mixing layer. By contrast, the cloud in a
perpendicular field configuration is fed by pristine wind gas, like in the
non-conducting case. As a result, clouds in an aligned field configuration
grow faster. We expect this early boost to cold gas mass to be almost entirely
restricted to magnetic fields parallel to the flow. In most other
configurations, magnetic field lines will quickly drape over the cloud (see
Sparre et al., 2020, for a careful study of draping), suppressing conduction
at the surface and maintaining large temperature gradients with reduced
‘mixing’. Heat transport would also be restricted to chaotic paths in the
tangled field, which may further reduce the efficiency of heat transfer and
the mass fraction of the warm phase.

From our simulations with thermal conduction, it is clear that conduction adds
significant complexity to the problem of cloud growth; on one hand, heat
conduction from the wind to the boundary layer increases the warm fraction,
which, in turn, can increase the growth depending on where the pre-cooled gas
is located. Conversely, the same conduction channel can supply heat from the
wind to this warm fraction (whether generated by conduction or Kelvin-
Helmholtz mixing), and help to offset radiative losses, increasing the cooling
timescale and consequently decreasing the condensation rate. However, we find
that with anisotropic thermal conduction, and a range of different magnetic
field morphologies, cold clouds in the hot ICM continue to grow, albeit more
slowly than in the non-magnetic case.

We do note that the magnetic field and conduction runs are very
computationally expensive, and as such we are only able to push them to a few
crushing times at most. This may mean that we miss important late-stage
evolution while the cloud is entrained, which could be different to the early
stage evolution where the cloud is mainly being mixed and accelerated and
generates a ’tail’ (see e.g. Grønnow et al., 2018; Gronke & Oh, 2019; Kanjilal
et al., 2021). This entrainment phase usually starts later for clouds in
magnetic fields, and begins after multiple $t_{\text{cc}}$, possibly beyond
the time we are able to simulate for for our clouds. We therefore advise that
while our results apply to the initial stages of cloud growth, without
simulations able to follow the late-time evolution of magnetic field and
conduction runs one should be cautious to apply the same findings to the
latter stages of cloud evolution.

##  5 Discussion and Conclusions

Figure 23: Time it takes for clouds to increase (or decrease, in the case of
no-cooling or isotropic conduction) their cold gas mass by $1\%$, $5\%$,
$10\%$, and $50\%$. Where necessary, the time was linearly interpolated
between existing simulation outputs. All MHD simulations are variants of HD10.
General trends that have been described in detail in previous sections
stemming from the effects of magnetic fields, various cloud densities, and
differing temperature floors can be easily seen. The $x$-axis is logarithmic
up to $t/$$t_{\text{cc}}$$=10^{-0.5}$ and linear thereafter. The vertical
black markers show the final time reached for each run. Figure 24: Time it
takes for clouds to increase (or decrease, in the case of no-cooling or
isotropic conduction) their warm gas mass by $1\%$, $5\%$, $10\%$, and $50\%$.
Where necessary, the time was linearly interpolated between existing
simulation outputs. The bottom axis is logarithmic up to
$t/$$t_{\text{cc}}$$=10^{-0.5}$ and linear thereafter. The vertical black
markers show the final time reached for each run.

In this work we have performed high-resolution simulations of $500$ pc-scale
cold clouds in a hot ICM environment. We have gradually introduced an
increasingly more complex physical picture, by including radiative cooling,
small-scale reheating, magnetic fields with different topologies and
strenghts, and (anisotropic) thermal conduction, and investigated the impact
of each individually, with the aim of better understanding the evolution of
cold gas within a cluster environment.

Using a set of cloud parameters typical of fragmented filaments in the
realistic hot ICM, we confirm that radiative cooling is the dominant physical
process that determines the continued evolution of clouds on tens of Myr
timescales. Except for those simulations with deliberately unphysical models
(no cooling or isotropic thermal conduction), our cold cloud grows in mass for
all parameters probed here. However, as can be seen in Fig. 23, how fast the
cold cloud mass grows is strongly a function of the detailed physics being
modelled. The same is true for the warm gas mass, as summarized in Fig. 24.
Specifically we find that

  * •

The presence of radiative cooling breaks the scale-free nature of the cloud
crushing problem, by introducing an additional characteristic timescale which
depends on cloud density, temperature and metallicity.

  * •

Due to radiative cooling in the warm phase, cold clouds can significantly
increase their mass on cloud-crushing timescales, as the dynamical generation
of the warm phase via Kelvin-Helmholtz instabilities is the limiting factor
for growth. We find a drop-off in cloud growth as the cloud starts to become
co-moving with the wind, as mixing is driven mainly by Kelvin-Helmholtz
instabilities rather than pressure gradients induced by cooling.

  * •

As expected, initially denser clouds show faster mass growth due to the more
rapid condensation of gas from the warm to the cold phase. Lower density
clouds form lower density fragments post-shattering, and this results in
greater mixing of cold gas into a warm phase.

  * •

The exact minimum temperature to which the cloud can cool (the temperature
floor) splits the cloud evolution into two modes, which we link to theoretical
predictions of post-shattering clump lengthscales: above a certain temperature
($\sim 10^{4}$ K) the cloud ablates in the wind, with clumplets being formed
dynamically and a significant cloud “head” retained. Instead for low
temperature cooling, the cloud shatters due to cooling-induced pressure
instability, forming many fragments, and takes more time to start generating
warm gas. We therefore conclude that care must be taken to model low
temperature cloud cooling, that should be ideally tuned to observations and/or
well-motivated models of sub-cloud heating.

  * •

Adding magnetic fields has little impact on the early evolution of the cold
cloud mass but significantly reduces the amount of warm gas in the wake. This
is due to the fact that magnetic fields suppress the mixing of gas in the wake
of the cloud which occurs even for the weakest $B$-fields explored here ($1\
\mu$G). Higher $B$-field strengths are most efficient at suppressing cold
cloud mass growth as well as the amount of warm phase.

  * •

The combined effects of magnetic pressure and magnetic tension act to keep the
hot and cold/warm phases separate at the boundary layer and reduce the mixing
efficiency of the Kelvin-Helmholtz instabilities. Interestingly, the
suppression of cloud growth is most pronounced for the field line
configuration that is initially perpendicular to the wind. This stems from the
fact that these $B$-fields drape around the head of the cloud, increasing
their stabilizing effect in the surface layer. Also, the wake region becomes
highly disordered, which reduces the effective surface area of the cloud
relative to the case where the $B$-fields are initially aligned with the flow.

  * •

If isotropic, thermal conduction can deposit sufficient heat from the hot ICM
into the cold and warm phases to prevent gains in the cold mass. In this case,
the cloud evolution follows a destruction mode evolution which more closely
resembles the traditional no-cooling cloud crushing solution, although at a
rate around an order of magnitude slower.

  * •

If the conduction is anisotropic the cloud instead grows similarly to the non-
conduction hydrodynamic and MHD runs because the the efficiency of heat
transport via electrons is greatly reduced. The ICM is then unable to
efficiently heat the cold cloud. Thermal losses via radiative cooling in the
warm gas dominate again, albeit on a slower timescale than without conduction.

  * •

This has important implications for both the ICM and other environments:
simulations which either do not include magnetic fields or else do not
explicitly model the conductive heat flux will not accurately account for
changes to the heating efficiency caused by the specific orientation of the
magnetic field lines. The presence of a magnetic field in any orientation,
even if at the lower end of galaxy cluster values, will prevent conduction
from destroying cold clouds.

  * •

Anisotropic thermal conduction can give early cold gas production a boost, if
the magnetic field is aligned with the flow. Otherwise, the cold gas mass
evolution behaves very similarly to the non-conducting case. In both cases,
the amount of warm gas is significantly increased even in comparison to the
non-conducting, and even the non-magnetic case, as warm gas is created both
through mixing and through hot gas that looses thermal energy though
conduction.

In general, we find that both modes of evolution (whether ablation from the
surface of the cloud, or shattering of the original cloud into smaller
cloudlets) impact the long-term evolution of the total cold gas mass. Another
important driver, not surprisingly, is the total amount of warm gas produced
in and around the cloud. This, in turn, depends on the mixing efficiency in
the wake of the cloud, as well as on the efficiency of reheating such mixed
gas via thermal conduction. We conclude that adding magnetic fields and
anisotropic thermal conduction does not change the overall behaviour of cold
clumps, but the relevant timescales do change, as well as the morphology of
the mixed wake.

The growth of the clouds and the further fragmentation of the original parent
clump may have important implications for the nature of the cold clump
accretion Gaspari et al. (2013) onto the central BCG. The shattering and/or
ablation of cold clouds, as well as the growth in the cold mass, could alter
the evolution of the cluster as a whole. Our results are also an important
consideration when attempting to explain the so-called cooling flow problem,
whereby feedback processes from the central AGN in the galaxy cluster are
theorised to heat cold gas to observed levels, and prevent cooling of hot gas.
We find that the cold gas mass in our simulations can grow by a factor 10
within a few $100$ Myr (although other cluster processes that we have not
explicitly modelled would be expected to also affect cloud evolution on these
timescales). Any proposed heating mechanism within clusters should be able to
explain the observed cooling times and gas temperatures whilst also taking
this extra cold mass into account.

We note that the cluster parameter space is very large, and so it is difficult
in practice to build a wholly comprehensive picture of cloud evolution in
these systems through simulations that cover the entire range of free
parameters. Instead, we have presented detailed studies for a typical fragment
as those seen in the cluster-scale hydrodynamic simulations of Beckmann, R. S.
et al. (2019) and the magnetised version of the same cluster discussed in
Beckmann et al. (2022a); Beckmann et al. (2022b). For our simulated cloud, we
have found robust general trends governing cloud evolution under increasingly
complex physics, such as magnetic fields and thermal conduction, and have
shown that even with magnetic fields and thermal conduction, the continued
evolution of individual cold clouds cannot be ignored when studying the
cooling flow problem of galaxy clusters.

## Acknowledgements

FJ developed code, ran simulations, analysed data, interpreted results and
wrote the manuscript. RSB initiated the project, developed code, analysed
data, interpreted results and wrote the manuscript. DS and YD interpreted
results, provided discussion and edited the manuscript. We would like to thank
the referee, Prof. Peng Oh, for a constructive referee report that improved
the paper. FJ and RSB thank Newnham College, Cambridge, for financial support.
This work was supported by the ERC Starting Grant 638707 “Black holes and
their host galaxies: co-evolution across cosmic time" and by STFC. This work
was performed using resources provided by the Cambridge Service for Data
Driven Discovery (CSD3) operated by the University of Cambridge Research
Computing Service (www.csd3.cam.ac.uk), provided by Dell EMC and Intel using
Tier-2 funding from the Engineering and Physical Sciences Research Council
(capital grant EP/P020259/1), and DiRAC funding from the Science and
Technology Facilities Council (www.dirac.ac.uk). DiRAC is part of the National
e-Infrastructure.

## Data Availability

The data used in this work may be shared on reasonable request to the
corresponding author.

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##  Appendix A Relative Contribution of Shattering vs Wind

Figure 25: Here we plot our fiducial HD0.1 against a no-wind version HD0.1_nw,
for which the only difference is that the wind speed is set to $0$ km/s, such
that the run models a clump that is initially stationary within the ICM. No
significant difference in the cold gas mass growth is seen in the shattering-
dominated timescale $\lesssim 0.5$ $t_{\text{cc}}$. Beyond this the wind
contribution to mixing starts to dominate and the no-wind run lags behind in
mass growth.

To investigate the relative contributions to mixing and subsequent cold mass
growth from shattering versus wind-mixing, we run a version of our flagship
cloud HD0.1 with the cloud stationary, in the no-wind run HD0.1_nw. As
expected the mass growth rate post-shattering is reduced in the absence of a
wind, as the degree of mixing via fluid instabilities is reduced. The mass
growth rate during shattering is similar, and this gives us a measure of the
shattering timescale where cloud collapse and shattering dominates over the
contribution from the wind. From Fig 25 we see that this shattering timescale
is approximately $0.5$ $t_{\text{cc}}$.

##  Appendix B Convergence

Figure 26: The change in cold mass against $t_{\text{cc}}$ for a range of
different resolutions for the same simulation run MHD_1N, with respect to the
minimum resolution run which has level_max equal to 10. While there is an
initial paucity of old gas at early times with a higher resolution, this is
reversed at around 5$t_{\text{cc}}$. This indicates that if the resolution
were to be increased further, the growth effect we observe should be amplified
even more.

In this paper we test a significant range of magnetic field strengths. The
computational expense becomes quite large for high-strength fields, where the
Alfvén velocity becomes large and the time-step correspondingly small, and
also when conduction is modelled. A sensible resolution must therefore be set
to make our production runs feasible. Here we would like to test the
convergence of our key results when adopting different resolutions for one
representative run. The “resolution parameter” that we use in ramses is the
maximum refinement level level_max, which gives the number of refinements we
allow the grid to make. In a maximally refined grid there will be $8^{\Delta
x-1}$ leaf (terminal) cells, where $\Delta x\equiv$ level_max. We test the
same simulation MHD_1N (see Section 4.2) with a range of values for level_max
in order to determine the convergence behaviour of our simulations, whilst
keeping the minimum refinement level level_min, which sets the number of cells
in the coarse grid, constant. The runs detailed here have a box size of ($195$
kpc)3, so the maximum spacial resolution is $195/2^{\Delta x-1}$. This is the
same as for all non-MHD runs performed in this paper. The MHD and conduction
runs have a box-size ($303$ kpc)3 (and also used a level_min value of $6$ as
opposed to the value of $5$ used for the hydro runs). We plot the change in
cold mass for a range of different maximal resolutions below with respect to
the lowest resolution simulation. We find that for times less than
5$t_{\text{cc}}$there is less cold gas when the resolution of the simulation
is increased, however the trend is reversed later on and we expect that our
lower resolution simulations are giving us a lower bound on the long-term
growth of the cold phase.

  *[ICM]: intracluster medium
  *[BCG]: brightest cluster galaxy
  *[SMBH]: supermassive black hole
  *[AGN]: active galactic nucleus
  *[CGM]: circumgalactic medium