# Resilience of Sloshing Cold Fronts against Subsequent Minor Mergers

Iraj Vaezzadeh,1, Elke Roediger,1 Claire Cashmore,1 Matthew Hunt1, John
ZuHone2, William Forman2, Christine Jones2, Ralph Kraft2, Paul Nulsen2,3,
Yuanyuan Su4, Eugene Churazov5,6
1E.A. Milne Centre, University of Hull, Cottingham Road, HU6 7RX, U.K.
2Smithsonian Astrophysical Observatory, Harvard-Smithsonian Center for
Astrophysics, 60 Garden St., Cambridge, MA 02138, USA
4Physics and Astronomy, University of Kentucky, 505 Rose street, Lexington, KY
40506, USA
3ICRAR, University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009,
Australia
5Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str 1, D-85741
Garching, Germany
6Space Research Institute (IKI), Profsoyuznaya 84/32, Moscow 117997, Russia
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Minor mergers are common in galaxy clusters. They have the potential to create
sloshing cold fronts (SCFs) in the intracluster medium (ICM) of the cluster.
However, the resilience of SCFs to subsequent minor mergers is unknown. Here
we investigate the extent to which SCFs established by an off-axis minor
merger are disrupted by a subsequent minor merger. We perform a suite of 13
hydrodynamic + N-body simulations of idealised triple cluster mergers in which
we vary the approach direction and impact parameter of the tertiary cluster.
Except for $\sim$1 Gyr after the first core passage of the tertiary cluster,
clear SCFs are present in all merger configurations. Subsequent head-on minor
mergers reduce the number of SCFs significantly, while subsequent off-axis
minor mergers only moderately reduce the number of SCFs. In particular, outer
($\gtrsim$500 kpc) SCFs are resilient. The results of this work indicate that
SCFs are easily formed in the course of a minor merger and are long-lived even
if a further minor merger takes place. SCFs thus should be ubiquitous, but
deriving the merger history of a given cluster based on its observed SCFs
might be more complex than previously thought.

###### keywords:

galaxies: clusters – galaxies: clusters: intracluster medium – galaxies:
clusters: general – methods: numerical

††pubyear: 2020††pagerange: Resilience of Sloshing Cold Fronts against
Subsequent Minor Mergers–LABEL:lastpage

##  1 Introduction

Galaxy clusters are the largest structures, at present, in the Universe which
are gravitationally collapsed. They are dominated by a dark matter halo which
is filled with the X-ray emitting intracluster medium (ICM) and galaxies. They
represent, for a host of reasons, an important probe of cosmological
parameters and astrophysical processes on large scales. According to the
current model of hierarchical structure formation, galaxy clusters grow via
mergers with other galaxy clusters and groups, and accretion of gas and dark
matter from the cosmic web (Kravtsov & Borgani, 2012). Mergers of galaxy
clusters are the most energetic events in the Universe since the Big Bang
(Sarazin, 2002), driving shocks and creating turbulence in the ICM.

Mergers can occur between similarly sized ($\sim$1:1) clusters (major mergers)
or between a main cluster and a smaller cluster or galaxy group (minor merger,
e.g. $\sim$1:3 or less). Minor mergers have been shown to induce contact
discontinuities (cold fronts) in the ICM of the primary cluster (Tittley &
Henriksen, 2005; Ascasibar & Markevitch, 2006; ZuHone & Roediger, 2016). In
contrast to a shock, a cold front is colder on the denser side of the
discontinuity and the ICM pressure is continuous across the front. Both cold
fronts and shocks appear in X-ray surface brightness maps as sharp edges and
are expected to be common in massive clusters because minor mergers are common
(Hallman et al., 2010). The first example of a cold front was discovered in
Abell 2142 (Markevitch et al., 2000; Vikhlinin et al., 2001), with many more
being discovered since (see e.g. Ghizzardi et al. 2010). Ghizzardi et al.
(2010) estimate, under the assumption of cold fronts in cool core (CC)
clusters being initiated by minor mergers, that a cluster undergoes one minor
merger (at least 1:10) per 3 Gyr. Most cold fronts come in two classes
depending on their originating mechanism: sloshing cold fronts (oscillation
class), merger cold fronts (remnant class) (Tittley & Henriksen (2005), see
Markevitch & Vikhlinin (2007) for a review). However, other mechanisms have
been proposed: shock-induced cold fronts (SICFs) (Birnboim et al., 2010)
formed by the collision of two shocks and CFs caused by colliding inflows of
gas (Zinger et al., 2018). The remnant/merger class cold fronts (hereafter
MCFs) form at the boundary between the gaseous atmospheres of the main cluster
and the ram-pressure stripped gas of the subcluster (Markevitch et al., 2000;
Vikhlinin et al., 2001; Tittley & Henriksen, 2005; Markevitch & Vikhlinin,
2007). Sloshing cold fronts (hereafter SCFs) can arise when an off-axis minor
merger perturbs the cool-core of the primary cluster (pulling it towards the
passing perturber), imparting angular momentum and causing the primary
cluster’s ICM to ‘slosh’ sub-sonically about the gravitational potential,
producing arc-like edges wrapped around the cluster core, initially at small
cluster-centric radii ($\lesssim 100$ kpc) (Tittley & Henriksen, 2005;
Ascasibar & Markevitch, 2006; Owers et al., 2011) which continue to grow to
large radii (Simionescu et al., 2012; Rossetti et al., 2013; Walker et al.,
2014).

The presence of cold fronts in galaxy clusters has been found to be almost
ubiquitous. Markevitch et al. (2003) found that 2/3 of their sample of 37
cool-core clusters (then termed cooling flow clusters) contained cold fronts.
Ghizzardi et al. (2010) subsequently found, from an X-ray flux-limited sample
of 32 clusters that 19 clusters contain at least one cold front (12 of which
are SCFs: Centaurus, A262, Perseus, A2199, 2A0335, A3558, A496, A1644, A2065,
A3562, A1795, and A576) and that they are readily found in systems undergoing
a merger in the plane of the sky.

The morphology and sharpness of cold fronts have been appreciated as a unique
tool to probe the microphysics of the ICM (e.g. viscosity, turbulence,
magnetic fields, etc., see ZuHone & Roediger 2016 for a review). The
interpretation of such cold fronts can be complicated by the unknown merger
history of the cluster. Binary merger simulations tailored to specific
clusters have been used to determine their cluster’s merger histories
(Roediger et al., 2011, 2012; Sheardown et al., 2018) as well as using these
merger histories and ICM flow patterns to study additional cluster physics.
However, if SCFs are not only long-lived, but also resilient to subsequent
minor mergers, this raises the question of how accurately a given cluster’s
merger history and current ICM motions can be derived from the observed
sloshing pattern. The study of the evolution of SCFs and how resilient they
are against subsequent mergers can help us to break the degeneracy.

Thus far, work in the literature has focused on idealised binary mergers to
explore the nature of gas sloshing in galaxy clusters, with no work that we
are aware of being focused on the effect of multiple mergers on SCFs.
Simulated binary mergers show that, left alone, the SCFs persist for extended
times (>10 Gyr) (Ascasibar & Markevitch, 2006; Roediger & Zuhone, 2012;
ZuHone, 2011). In this paper we show that the presence of SCFs is resilient to
subsequent minor mergers, i.e. SCFs exist throughout the evolution except for
$\sim$1 Gyr after the tertiary cluster’s infall (all times stated in this
paper are relative to the simulation start time). To this end, we run a suite
of twelve hydrodynamic + N-body triple merger simulations (in addition to a
binary merger for reference) exploring the effect that different trajectories
of a second minor merger have on the established SCFs.

##  2 Method

###  2.1 Simulations

We run the simulations using the hydrodynamic + N-body code, FLASH v4.3
(Fryxell et al., 2000). FLASH is a modular multi-physics Eulerian adaptive
mesh refinement (AMR) hydrodynamic code. The use of an N-body solver with high
particle resolution allows us to accurately model effects like dynamical
friction which captures the evolution of collisionless components (DM,
galaxies), and the decay of the subcluster orbits due to dynamical friction
and tidal stripping.

Our simulations run in a three-dimensional domain of 13 Mpc3 in size, with
periodic (‘wrap-around’) boundary conditions, chosen due to the required
initial separation of the clusters and to ensure that boundary effects do not
affect the mergers. Our simulations run for $\sim$10 Gyr with snapshots
produced every 50 Myr. For simplicity, we do not take account of cosmological
expansion and assume a flat $\Lambda$CDM cosmology with $h=0.7$. For the
analysis we make use of the Python based library, yt (Turk et al., 2011).

The AMR allows us to place refinement in areas of interest and save
computational effort in areas of less interest. As such we refine on particle
density: when the number of particles in a block of 163 grid cells exceeds
1100, the block is refined; similarly if the number of particles drops below
550 the block will be derefined. We achieve a maximum spatial resolution of
3.7 kpc within, typically, the inner 250 kpc for the primary cluster and the
inner 125 kpc for the secondary and tertiary clusters. Within the primary
region of interest (innermost 4 Mpc of the simulation domain), the lowest
level of resolution is 25.4 kpc; the lowest resolution in the domain is 101
kpc. The simulations use $5\times 10^{6}$ particles for the primary and
$5\times 10^{5}$ for the secondary and tertiary clusters.

####  2.1.1 Initial Conditions

As we are not attempting to recreate a particular system, the clusters need
not be tuned to any specific X-ray observation or temperature profile and can
instead be generic. Radial profiles of key cluster parameters can be seen in
Figure 1. All clusters used in the simulations are designed to be in
hydrostatic equilibrium with spherical symmetry and self-gravity, using the
method of ZuHone (2011). We refer to the primary, secondary and tertiary
clusters as clusters A, B and C, respectively. Where appropriate, clusters B
and C are also referred to as first and second infaller, respectively.

The ICM is modelled as an ideal gas using the standard $\beta$ profile of
Cavaliere et al. (1976), cut at $r_{\mathrm{cut}}=2783$ kpc to satisfy
convergence:

$\rho_{\mathrm{ICM}}(r)=\rho_{0}\left(1+\left(\frac{r}{r_{\mathrm{c}}}\right)^{2}\right)^{-\frac{3\beta}{2}}\left(1+\left(\frac{r}{r_{\mathrm{cut}}}\right)^{3}\right)^{-1}$
(1)

where $r_{c}$ is core radius. Mohr et al. (1999) find (based on a detailed
study of an X-ray flux limited sample of 45 clusters) the approximate value of
$\beta$ to be $\sim$0.64 for clusters and as such $\beta=0.64$ is the value we
adopt.

The total density of the clusters is set to a spherical Hernquist profile
(Hernquist, 1990) which matches the Navarro-Frenk-White profile within
$0.1r_{200}$ but has the advantage of dropping off more steeply in density
which converges to a finite total mass (Springel et al., 2005):

$\rho_{\mathrm{total}}(r)=\frac{M_{\mathrm{total}}}{2\pi}\frac{a}{r(r+a)^{3}}$
(2)

where $M_{\mathrm{total}}$ is the total mass and $a$ is the scale radius (a
description of which can be found in Donnert (2014)) with $a=566$ kpc for
cluster A and $a=230$ kpc for clusters B and C (clusters B and C are identical
in all respects). Note that this is the total density of the cluster (DM and
ICM) with the particle density of the cluster being set as the difference
between this total Hernquist profile and the ICM $\beta$-profile.

As we initialise cluster C at a large distance from cluster A (to control when
it reaches pericentre) we embed all clusters in a warm-hot intergalactic
medium (WHIM)-like atmosphere to truncate the atmospheres of the in-falling
clusters and to avoid unrealistic sizes. We model the WHIM-like atmosphere
with a uniform density, temperature and pressure with values of
$\rho=4.67\times 10^{-29}$ g cm-3, $T=2.17$ keV, $P=1.65\times 10^{-13}$ erg
cm-3, these parameters equal the ICM parameters of cluster A at $r=2062$ kpc,
which can be seen in Figure 1. As our WHIM-like atmosphere is slightly warmer
than the typical range found in the literature ($10^{5}-10^{7}\mathrm{~{}K}$,
Davé et al. 2001) due to technical reasons, we do not draw conclusions
concerning absolute temperatures within the simulations. We have tested the
binary merger simulation with a selection of WHIM parameters and find that our
conclusions would not be affected by this choice. The masses of the clusters
are chosen to be representative of an average sized cluster for cluster A and
a small cluster for clusters B and C.

Figure 1: Radial profiles of the enclosed mass, gas density, gas temperature and gas pressure of cluster A (left) and clusters B and C (right). The density, temperature and pressure at which the WHIM-like atmosphere are set can be seen in the profiles of cluster A.  Table 1: Cluster parameters Cluster | $M_{200}(M_{\odot})$ |  $r_{\mathrm{200}}$(kpc) | $N_{p}$ | Particle Mass$(M_{\odot})$
---|---|---|---|---
A | $5\times 10^{14}$ | 1637 | $5\times 10^{6}$ | $1.30\times 10^{8}$
B | $5\times 10^{13}$ | 760 | $5\times 10^{5}$ | $1.21\times 10^{8}$
C | $5\times 10^{13}$ | 760 | $5\times 10^{5}$ | $1.21\times 10^{8}$

####  2.1.2 Setup parameters

The two clusters that constitute the initial binary merger (A and B) are set
with a distance equalling the sum of their respective $\,r_{\mathrm{200}}$
radii. We motivate this separation based on the findings of Vitvitska et al.
(2002) that, in cosmological simulations, the velocity of an in-falling
subcluster is $\sim 1.1\,V_{\mathrm{c}}$ at $r_{\mathrm{vir}}$ (for which we
use $\,r_{\mathrm{200}}$ as a proxy), where
$V_{c}^{2}=GM_{\mathrm{vir}}/R_{\mathrm{vir}}$ is the circular velocity of the
Hernquist profile; for cluster A, $V_{\mathrm{c}}=1150$ kms-1. Vitvitska et
al. (2002) also revealed that the tangential velocity dispersion of a merger
is dependent on the mass ratio of the primary cluster to the secondary. They
find that the average tangential velocity for minor mergers, is
$v_{\perp}=0.71\,V_{\mathrm{c}}$, corresponding to 817 kms-1 in our
simulations. We control the pericentre distance (the closest approach of the
two cores) of the simulations through the use of a tangential velocity
component in accordance with these findings. We find that the parameters found
in Table 2 (between 0 kms-1 (head-on) and 817 kms-1) produce clear sloshing
fronts.

A mass ratio of R$=$1:10 is shown to create clear SCFs in the host cluster
(ZuHone, 2011) and is thus chosen as the mass ratio between the merging
clusters. For the second infaller (cluster C) we use ad hoc tangential
velocities to produce, for each approach approach, an on-axis merger and one
off-axis merger to each side of cluster A. We adopt a naming convention in
with head-on mergers are named HeadX, clockwise mergers ClockX and anti-
clockwise mergers AntiClockX. The parameters used for cluster C in each
simulation can be found in Table 3. The off-axis infalls of cluster C
typically have a pericentre distance of $\sim$100 kpc.

A further simulation probing the effect of a subsequent merger out of the
plane of the initial merger has also been performed. In this simulation (which
we call LOS1) cluster C is initialised with the parameters found in Table 4.
Cluster C has an off-axis approach despite only having a radial velocity
component due to the drift of cluster A after cluster B’s infall.

Table 2: Initial binary merger parameters. Both clusters have a $y$-position of 0, $\mathbf{v}_{\|}$ represents velocity in the $x$-direction and $\mathbf{v}_{\perp}$ represents velocity in the $y$-direction. Cluster | x-position (kpc) |  $\mathbf{v}_{\|}$ (km/s) |  $\mathbf{v}_{\perp}$ (km/s)
---|---|---|---
1 | 0 | 0 | 0
2 | 2397 | -966 | -299
Table 3: Positions and impact parameters for the second infaller in each simulation. The positions and velocities of the primary cluster and first infaller are constant in every case. Simulation |  $x$ (Mpc) |  $y$ (Mpc) |  $\mathbf{v}_{\|}$ (km/s) |  $\mathbf{v}_{\perp}$ (km/s)
---|---|---|---|---
Head1 | 0 | 5 | -100 | -32.7
Head2 | 5 | 0 | -100 | 0
Head3 | 0 | -5 | 100 | -75
Head4 | -5 | 0 | 100 | -100
Clock1 | 0 | 5 | -100 | 0
Clock2 | 5 | 0 | -100 | -175
Clock3 | 0 | -5 | 100 | -225
Clock4 | -5 | 0 | 100 | 0
AntiClock1 | 0 | 5 | -100 | -200
AntiClock2 | 5 | 0 | -100 | 32.7
AntiClock3 | 0 | -5 | 100 | 32.7
AntiClock4 | -5 | 0 | 100 | -200
Table 4: Position and impact parameters for the second infaller in the supplementary simulation LOS1. The $x$ and $y$ positions are both zero. The positions and velocities of the primary cluster and first infaller are the same as in Table 2. Simulation |  $z$ (Mpc) |  $\mathbf{v}_{\|}$ (km/s) |  $\mathbf{v}_{\perp}$ (km/s)
---|---|---|---
LOS1 | 4500 | -100 | 0

###  2.2 Cold Front Identification

To quantify the number of cold fronts present in cluster A along a given
direction we create a ray between two specified points in the simulation
domain. We perform the process described below separately along both the $x$
and $y$ directions.

Ghizzardi et al. (2010) found that the surface brightness discontinuities in
their sample span an angular extent of at least 30∘. We cast 50 rays using the
yt ray object (each consisting of 750 points, 1.3 kpc apart) to form 30∘
sectors of a circle of radius 1 Mpc about the centre of cluster A (defined by
the minimum of its gravitational potential) which can be seen in Figure 2. We
then take the average of all these rays to produce a single temperature within
the 30∘ sector.

We scan along the averaged temperature profile (seen in Figure 3) and mark
changes in temperature of more than 2$\%$ between neighbouring points
(corresponding to a scale of 1.3 kpc). We repeat this process in the opposite
direction to mark steep gradients on the opposite side of the core). In some
cases not all points along a steep drop will be marked as CF points in the
initial sweeps due to small scale fluctuations. To counter this, and ensure
that each CF is continuous, we perform a rolling average of every three points
along the profile (corresponding to a scale of $\sim$4 kpc). This smoothing
step has the consequence of effectively setting the minimum separation between
CFs as $\sim$4 kpc. We define the start point of each CF as the first point in
the profile that has met the 2$\%$ change criterion and the end point as the
last in a set of points after the smoothing. The average CF position is simply
the average of the start and end position of the CF.

To avoid spurious detections we remove detections with a temperature ratio
between the start and end points of <1.15 (based on the temperature ratio of
1.15 found in Abell 2219 by Canning et al. 2017). Examples of identified SCFs
can be seen in Figure 2. The algorithm makes no distinction as to the origin
and nature of the front counted, and therefore will detect MCFs when the
infalling subcluster travels through a detection sector, we therefore refer to
detections while the infalling subcluster’s atmosphere could be causing the
identification simply as CFs rather than SCFs. Several parameters affect the
number of identified cold fronts, such as limits on gradient, minimum allowed
temperature ratio or rolling average length/minimum separation. We tested a
range of these parameters on the binary merger run and visually confirmed the
CF detections in temperature slice images such as Figure 2 as well projected
temperature weighted by density squared such as in Figure 5. Relaxing the
gradient limit and temperature ratio limit results in more CFs being
identified, but such weaker CFs would not be easily identifiable in
observations. Our conclusions regarding the resilience of SCFs do not depend
on the exact choice of these parameters.

Figure 2: Temperature slice of the binary merger simulation at 10 Gyr centred
on cluster A. To calculate the averaged radial temperature profile (see Figure
3), we extract and then average the profiles along the black lines that sweep
30∘ sectors along the $x$\- and $y$-axes, with the resulting averaged ray
being represented in red on this plot. The vertical green, orange and red
lines denote, respectively, the start, average and end positions of identified
CFs. Figure 3: The averaged temperature profile in the $x$-direction of the
binary merger at 10 Gyr. Profiles along individual rays are plotted in pale
orange, with their average in black. The vertical green, orange and red lines
denote, respectively, the start, average and end positions of identified CFs
that have met the temperature ratio threshold (>1.15). Note that the $x$-axis
has been adjusted so the gravitational potential minimum (i.e. the cluster
centre) is at 0 Mpc.

##  3 Results

###  3.1 Sloshing process

We briefly describe the mechanism which causes the ICM sloshing in the
simulations, making reference to Figure 4, which shows the binary merger
simulation. Note that the colour scale has been made constant for all
simulations so that comparisons may be more readily drawn. This has the
consequence that saturation occurs at certain times/locations such as when gas
is shock heated, but the features we are interested in are unaffected. In all
simulations, the sloshing patterns caused by the initial binary merger between
cluster A and cluster B are formed in the same way.

Cluster B approaches the core of cluster A, pulling the dark matter and ICM
peaks of cluster A equally towards cluster B. The ICM peak of cluster A is
subject to ram pressure, unlike the DM peak, and as such is no longer in
equilibrium with the DM-peak. When cluster B passes its pericentre position
($\sim$200 kpc) at $\sim$1.5 Gyr, the gas core of cluster A is no longer
subject to this ram pressure, and begins to fall from this maximum separation
back to the gravitational potential minimum, coinciding with the DM peak. As
the gas core falls back, it ’overshoots’ the DM peak and travels to a new
maximum position on the opposite side. In this fashion, the gas continues to
oscillate about the DM peak of cluster A, creating the characteristic
’sloshing’ fronts staggered about the core of cluster A. Cluster B passes
south of cluster A, attracting its DM core to the south, thereby initiating
the sloshing in a north-south direction. Cluster B’s off-axis infall
trajectory is such that it imparts angular momentum to cluster A and causes
cluster A to ’swirl’ in a clockwise direction after first pericentric passage
($\sim$1.5 Gyr) which gives the resultant spiralling SCFs a clockwise
direction.

Cluster B reaches first apocentre at $\sim$2.75 Gyr; at this stage the ICM in
the core of cluster A is clearly sloshing about the perturbed potential and
creating SCFs. Second pericentric passage occurs at $\sim$4.1 Gyr; other than
the CFs that are directly hit by the infalling subcluster, the overall pattern
of SCFs remains resilient to this second pericentric passage. The sloshing
fronts then continue to travel outwards, for at least 8.5 Gyr (until
$t_{\mathrm{max}}$), from the core, with subsequent pericentric passages of
cluster B not affecting their evolution. Several SCFs are visible at
$t_{\mathrm{max}}$, forming a clear (though not perfectly continuous) spiral
shape out to $\sim$800 kpc, which is $\sim$0.5$\,r_{\mathrm{200}}$. SCFs are
clearly present at $\sim$5 Gyr in Figure 4, which is the approximate time that
cluster C will reach first pericentre.

###  3.2 Evolution of Triple Mergers - Qualitative Description

To draw meaningful comparisons between simulations, we first qualitatively
describe the evolution of the triple mergers based on visual inspection. We
use slices of temperature and projections of temperature weighted by density
squared (to better represent observational images) to analyse the simulations.
All images are viewed with the same colour scale and field of view, which in
some cases causes saturation. It is the effect of a subsequent minor merger on
the SCFs established by the initial off-axis binary minor merger that we seek
to determine. As all simulations proceed in much the same way until cluster
C’s first pericentric passage (see Figure 4 until $\sim$5.3 Gyr), we begin
analysis of the triple mergers at this stage. We have made the decision to
keep the position of cluster C constant for each approach direction, and as
such the first pericentre time (5.3-5.9 Gyr) and distance (50-100 kpc) of
cluster C varies across the simulations due to the drift of cluster A after
the infall of cluster B.

Due to the difference in evolution between the head-on mergers and those
clockwise and anti-clockwise with respect to the initial binary merger, we
describe these evolutions separately below. In this description, we will refer
to the movie (Figure 5) and Figures 6-10 in the body of the text which show
the simulations aligned on the 2nd infaller’s pericentre time. The appendix
features movies of each subset of simulations at different scales as well as a
movie of all simulations at a scale of 500 kpc and an image of the final time
of all simulations.

Figure 4: Movie (the movie can be viewed at https://youtu.be/fZYljFAQfBA) of
projections of temperature with density squared weighting through the $x-y$
plane of the binary merger simulation. The top panel is 500 kpc on a side, and
the bottom panel is 2.5 Mpc on a side. The arrow denotes the trajectory of the
infaller (note this is not to scale and serves only as a guide). Figure 5:
Movie (the movie can be viewed at https://youtu.be/97yOHejlpyI)of projections
of temperature with density squared weighting through the $x-y$ plane for all
simulations. Each panel is 2.5 Mpc on a side. The rows are comprised,
respectively, of the head-on, clockwise and anti-clockwise simulations; the
columns represent approach direction of cluster C. The leftmost column is the
binary merger and supplementary simulation (discussed in Sec 4.1.1) with the
colour scale for all simulations. The innermost arrow in each panel represents
the trajectory of the first infaller, the dotted circle in LOS1 indicates a
trajectory towards the viewer of the second infaller, and the outer arrow
represents the trajectory of the second infaller (note that neither arrow is
to scale, and should serve only as a guide to the panels). The annotated
arrows bear this meaning in all subsequent figures. It is clear from row one
that a second infaller with zero impact parameter will disrupt the cool-core
of the system; however, the large scale sloshing fronts remain. There is no
obvious difference between the different approach directions. Figure 6:
Projections of temperature with density squared weighting through the $x-y$
plane for all simulations just prior to first pericentre of cluster C. The
phase of each simulation has been shifted such that the 2nd infaller reaches
first pericentre simultaneously in Figures 6-10. Each panel is 2.5 Mpc on a
side. The innermost white circle common to all panels marks the boundary of
the region we define as small scale (less than 0.12$\,r_{\mathrm{200}}$
($\sim$200 kpc)). The outermost white circle common to all panels marks the
boundary of the region we define as intermediate (between 0.12 and
0.3$\,r_{\mathrm{200}}$). Beyond this outermost circle is the large scale
region (greater than 0.3$\,r_{\mathrm{200}}$ ($\sim$500 kpc)). These
annotations share the same meaning across Figures 6-10.  Figure 7: Projections
of temperature with density squared weighting through the $x-y$ plane for all
simulations just after first pericentre of cluster C. Explanations of the
annotations can be found in Figures 5 and 6. In all simulations, the sloshing
is highly disturbed during this phase between cluster C’s first pericentre and
first apocentre. Although some SCFs can be detected, they lack the
characteristic spiral pattern. MCFs can be detected at the boundary of cluster
C’s atmosphere with the atmopshere of cluster A. Figure 8: Projections of
temperature with density squared weighting through the $x-y$ plane for all
simulations at first apocentre of cluster C. Explanations of the annotations
can be found in Figures 5 and 6. We see in the head-on simulations that the
MCF and slingshot tail of cluster C is less pronounced than in the off-axis
mergers due to the loss of much of its cool gas during its core crossing. We
see new sloshing initiated in the core of cluster A in the off-axis mergers
which is not present in the head-on mergers. Figure 9: Projections of
temperature with density squared weighting through the $x-y$ plane for all
simulations at the approximate time of second pericentre of cluster C.
Explanations of the annotations can be found in Figures 5 and 6. The
simulations are now out of phase with regard to the second pericentre of
cluster C due to the different configurations. We see that in the head-on
mergers cluster C has been stripped of all of its cool gas before reaching
second pericentre and is therefore not visible in this image. In the case of
the off-axis mergers cluster C still carries some cool gas to second
pericentre, but this has little effect on the overall evolution of the system
other than to cause small-scale instabilities. Figure 10: Projections of
temperature with density squared weighting through the $x-y$ plane for all
simulations during the ‘relaxation phase’. Explanations of the annotations can
be found in Figures 5 and 6. This figure shows the state of each simulation
during the ‘relaxation phase’; due to the difference in phase between the
simulations in this figure, it represents the approximate maximum simulation
time. There is a clear lack of a CC in the head-on mergers, which is likely
the reason that no small scale SCFs have been generated. We see small scale
SCFs in the cores of cluster A in the off-axis simulations initiated by
cluster C, and large scale fronts initiated by cluster B. There is a lack of
continuity between these as the intermediate scale fronts are, in some cases,
destroyed, and in other cases disturbed.

####  3.2.1 General overview of triple mergers

First pericentric passage of cluster C occurs between 5.3-5.9 Gyr. We will
refer, throughout our analysis, to specific moments (such as first pericentric
passage etc.) which can be seen in Figures 6-10. SCFs extending up to
$\sim$400 kpc caused by cluster B are well established by the time of cluster
C’s first pericentric passage (Figure 6). We see strong disruption or even
destruction of SCFs during the epoch from first pericentric passage until
first apocentre (6.5-7 Gyr) of cluster C (Figure 7). By the time of cluster
C’s first apocentre (Figure 8), these SCFs initiated by the initial merger are
re-established to varying degrees in the different simulations. SCFs continue
to emerge from the core of cluster A for off-axis infalls of cluster C, likely
generated due to the core motion initiated by cluster C as opposed to a
continuation of the sloshing caused by cluster B. We do not distinguish
between SCFs generated by cluster B and cluster C as we are interested in the
number of observable SCFs after two mergers. Cluster C’s second pericentric
passage (Figure 9) has little effect on the evolution of the system. We define
the epoch post second pericentric passage (7.5-8 Gyr, (Figure 10)) as the
‘relaxation phase’, which roughly corresponds to the final 2-2.5 Gyr of the
simulations. Once the simulations enter the ‘relaxation phase’ we typically
see between 3-5 SCFs in the non-head-on mergers and 1-3 in head-on mergers. To
quantify the differences in cold front structure between simulations, we
define three categories of CFs according to their position as a fraction of
cluster A’s $\,r_{\mathrm{200}}$. These categories are as follows: small
scale: less than 0.12 $\,r_{\mathrm{200}}$ ($\sim$200 kpc); intermediate:
between 0.12 and 0.3 $\,r_{\mathrm{200}}$; large scale: greater than 0.3
$\,r_{\mathrm{200}}$ ($\sim$500 kpc). These regions have been annotated on
Figures 6-10.

####  3.2.2 Head on mergers

A movie of the evolution of the head-on mergers from cluster C’s first infall,
at two scales, is presented in Appendix A Figure 14. When cluster C passes
through the main cluster head-on, the SCFs, established by the previous
merger, are thoroughly disturbed and hard to detect for the next $\sim$2 Gyr
until the ‘relaxation phase’. SCFs can be picked out during this ‘messy’ phase
between the second infaller’s first pericentre and second apocentre, but they
lack the characteristic spiral pattern as they are so disturbed. Detection of
any SCFs between first pericentric passage ($\sim$5.5 Gyr) and apocentre
($\sim$6.6 Gyr) of cluster C is difficult (Figure 7). However, by the time of
first apocentre (Figure 8) it becomes possible to see SCFs that had already
made their way out of the inner $\sim$200 kpc. Due to cluster C hitting the
core of cluster A in a head-on fashion, cluster C loses much of its initial
cool gas, hence we see a less pronounced MCF for cluster C beyond first
pericentric passage.

Subsequent pericentric passages ($\sim$7.8 Gyr and $\sim$9 Gyr) of cluster C
have a small impact on the evolution of SCFs (Figure 9), making them appear
more turbulent and less orderly; however, they continue to grow outwards until
$t_{\mathrm{max}}$. By approximately second apocentre ($\sim$8.3 Gyr) the
turbulence and highly disturbed morphology have decreased sufficiently so that
cold fronts are detectable beyond $\sim$200 kpc. During the ’relaxation phase’
it is clear that the cool core of the primary cluster has been disrupted by
the head-on merger. We confirm this by viewing slices of entropy, which
confirm that the entropy in the core of cluster A in these simulations is
greater than in the off-axis cases. This approximately isentropic core is
likely responsible for the lack of new SCFs emerging after pericentric passage
of cluster C. The structure of the SCFs during the ’relaxation phase’ (Figure
10), until $t_{\mathrm{max}}$, is disturbed, but a staggered spiral pattern
can be seen outside the inner $\sim$200 kpc, albeit broken in several places.
There are fewer SCFs at $t_{\mathrm{max}}$ in all four head-on simulations
than in the case of a single binary merger due to the lack of a CC resulting
in no SCFs emerging from the core (Figure 10).

####  3.2.3 Clockwise mergers

We now turn to the simulations in which cluster C approaches cluster A in a
clockwise direction with respect to our standard viewing direction, in which
cluster B has a clockwise infall. A movie of the full evolution of the
clockwise mergers at two scales is presented in Appendix A Figure 15. First
pericentric passage ($\sim$5.6 Gyr) disturbs the SCFs considerably and the
potential is offset which causes violent gas motions in the core (Figure 7).
Clock1 and Clock4 have a closer approach at first pericentric passage than
Clock2 and Clock3. This is due to the infall of cluster B on cluster A causing
the system to drift, thereby misaligning the starting position of cluster C
which we keep fixed for each approach direction. This closer pericentric
distance has the effect of causing more disruption to the established SCFs
than is seen in Clock2 and Clock3. SCFs can be detected during this complex
phase after first pericentre, but the spiral pattern is unrecognisable in the
immediate aftermath of first pericentric passage (Figure 7). SCFs emerge from
the core of cluster A as cluster C reaches first apocentre ($\sim$1 Gyr after
first pericentric passage, Figure 8). The axis of sloshing is perpendicular to
the infall direction of cluster C, strongly indicating that the SCFs emerging
from the core beyond this time are initiated by cluster C as opposed to
cluster B. The sloshing pattern is broken in several places, and the SCFs are
prone to instabilities and turbulence, with several exhibiting Kelvin-
Helmholtz instabilities (KHIs).

Cluster C has maintained much of its cool gas to first apocentre ($\sim$6.8
Gyr), and thus can be seen to exhibit a MCF and slingshot tail (Sheardown et
al., 2019) at first apocentre ($\sim$1.5 Mpc from the core). Cluster C carries
very little cool gas into the core during its second pericentric passage
($\sim$8 Gyr, Figure 9); its second pericentric passage disturbs the potential
causing the SCFs to shift positions, but they remain intact and continue to
grow. During the ‘relaxation phase’ (Figure 10) there is a staggered SCF
structure in the inner 300 kpc (400 kpc in the case of Clock1) and the large
scale SCFs at $\sim$500 and $\sim$750 kpc are still present, though they
suffer varying levels of disruption and instability across the four
simulations (in particular, in Clock1 the largest scale SCF is punctured by
the infall of cluster C). In all cases there are fewer SCFs at
$t_{\mathrm{max}}$ than in the binary simulation, but there are still clear
SCFs in the core and at large radii; clear SCFs at intermediate radii are
absent.

####  3.2.4 Anti-clockwise mergers

A movie of the full evolution of the anti-clockwise mergers at two scales is
presented in Appendix A Figure16. For anti-clockwise infalls of cluster C
(contrasting cluster B’s clockwise infall with respect to our standard viewing
direction), first pericentric passage ($\sim$5.6 Gyr) disrupts the SCFs
established by the binary merger (Figure 7). There are SCFs in the core region
as cluster C reaches first apocentre ($\sim$6.7 Gyr, Figure 8), seemingly
initiated by the passage of cluster C as the sloshing motion is perpendicular
to the infall direction of cluster C. As cluster C reaches first apocentre
(Figure 8), the characteristic spiral pattern of the SCFs has appeared (though
it is of course more fractured than we see in the binary merger) due to the
swirling ICM combined with the sloshing gas core. Cluster C has carried much
of its cool gas to first apocentre and therefore exhibits an MCF and slingshot
tail at this radius ($\sim$1.5 Mpc from the core).

Cluster C does not carry much cool gas to second pericentre ($\sim$8 Gyr,
Figure 9), and as such only its gravitational influence is felt when it
reaches second pericentre. Its second pericentric passage displaces the
potential causing more sloshing along the axis perpendicular to the infall
direction; however, this does not disrupt the evolution of the fronts to a
large degree. During the ’relaxation phase’ (Figure 10) after second
pericentric passage there is a clear spiral structure to the SCFs within the
inner 200-400 kpc as well as large scale fronts at $\sim$700 kpc (in the case
of AntiClock1, these large scale fronts look very weak). In all cases there
are fewer SCFs during the ’relaxation phase’ and at $t_{\mathrm{max}}$ than in
the binary simulation, though similarly to the clockwise set of simulations,
there are still fronts at small and large radii, with intermediate fronts
being those that do not survive the merger with cluster C.

###  3.3 Automated CF Counting

The visual inspection presented above indicated that the infall of cluster C
reduced the number of SCFs compared to the binary merger at all times and that
the head-on infalls of cluster C cause the strongest reduction. In this
section, we verify this visual impression with an automated algorithm to count
CFs (see 2.2 for the description of the algorithm). We count in 30∘ sectors
along the $x$ and $y$ axes, and as such any CF with an angular extent large
enough that it lies in two sectors will be counted in both tallies. The
algorithm makes no distinction as to the origin and nature of the front
counted, and therefore will detect MCFs when the infalling subcluster travels
through a detection sector.

Calibration of the algorithm suggests that periods during which a subcluster
travels through the detection sector will see increased detection of merger
class fronts. Usually, a MCF is identified in clusters as the leading edge of
the cold gaseous atmosphere of an infalling subcluster. When a subcluster
travels through the detection sector, our algorithm detects not only this edge
but several behind it as well which we will consider MCFs. This could be
mitigated by running the algorithm only along the direction perpendicular to
the trajectory of the infaller. However, as we have two infallers this will
quickly become impractical, and thus analysis of the triple mergers is limited
to the post-second pericentric passage (‘relaxation’) phase of the triple
merger simulations to increase confidence that the detections are purely SCFs.

####  3.3.1 Binary Merger

The results of the CF counting algorithm performed on the binary merger
simulation are shown in Figure 11. First pericentric passage occurs at
$\sim$1.5 Gyr, at which point we see the number of detections increase. We
confirm through visual inspection that this is a combination of both SCFs in
the core and MCFs as cluster B is travelling roughly along the $x$-axis and
thus through the $x$-axis aligned detection sector. A natural consequence of
this is that detection along the $y$-axis more accurately captures the number
of SCFs during this period between the first and second pericentric passages.
Second pericentric passage occurs at $\sim$4.1 Gyr, at which point we see an
inversion in the number of CFs detected along the $x$ and $y$ axes. This
inversion appears to be due to cluster B’s trajectory through its approach to
second pericentre and beyond being along the $y$-axis. Beyond second apocentre
($\sim$4.8 Gyr), we see the number of detected CFs settle to approximately 5-7
along each axis. The variation during this phase is due to the large scale CFs
not being consistently detected (see section 2.2 for reasons), with little
variation seen in the detection at small and intermediate radii (<500 kpc or
0.3$\,r_{\mathrm{200}}$). The large CF $\sim$750 kpc north and west of the
centre is detected along the $y$-axis but not the $x$-axis. There is also a CF
$\sim$500 kpc east of the core which is sporadically detected; we note that
the gradient in the temperature profile is consistently detected, but the
temperature ratio across the front does not always fulfil the detection
criterion.

Figure 11: The number of CFs identified in the Binary Merger simulation at 0.5
Gyr intervals. The blue and red lines represent CFs (with a temperature ratio
of at least 1.15) identified in a 30∘ sector centred on the $x$ and $y$ axes,
respectively. Criteria for CF identifications are described in Section 2.2.
Figure 12: The number of detected CFs in all simulations. The black line shows
the number of detected CFs in the binary merger simulation, and the coloured
lines show results from triple mergers. We see that the number of CFs detected
in the binary merger is, in general, greater than in the triple mergers. We
see a clear trend that the number of CFs is lower in all triple mergers in
which cluster C has collided with the main cluster core head-on. In all off-
axis mergers, the number of CFs is only mildly reduced, generally by less than
a factor of two.

####  3.3.2 Head-on Mergers

We have seen from visual inspection (see Section 3.2.2) that the head-on
mergers disrupt the cool core of the system, and as such we see no SCFs
detected in the inner $\sim$250 kpc during this phase (7.5-10 Gyr), with the
exception of Head1. Consistent detections occur, therefore, at or beyond
$\sim$200 kpc (0.12$\,r_{\mathrm{200}}$) towards $t_{\mathrm{max}}$. Large
scale CFs, at $\sim$800 kpc, are only detected at fortuitous moments when they
lie at shallow angles to the detection sector. Detection of large scale CFs
along $y$ is found to be more reliable than detection along $x$, raising the
interesting question of whether it is the initial merger that sets the
direction of the large scale sloshing, regardless of the second infaller’s
direction. We see the number of detected CFs is consistently lower in all
head-on simulations when compared to the simple binary merger. The number of
CFs with at least a temperature ratio of 1.15 only drops to zero along _both_
$x$ and $y$ at a few timesteps. We therefore conclude that even a zero-impact
parameter merger is not sufficient to destroy all CFs established by a
previous merger, but only those at small cluster-centric radii. Intermediate
($\sim$0.12-0.3 $\,r_{\mathrm{200}}$) and large scale CFs can be detected
consistently during the ‘relaxation phase’.

####  3.3.3 Clockwise and Anti-clockwise Mergers

We have seen from visual inspection that an off-axis infall of cluster C does
not significantly disrupt the cool core of the system. As such, there is cold
gas in the core region that is able to slosh and form small scale SCFs. The
number of detected SCFs in the off-axis mergers ($\sim$3-5) is generally
greater than the number detected for the head-on mergers, but is lower than in
the binary merger; however, there is no time at which _no_ SCFs are detected.
This leads us to conclude that the number SCFs of at least a temperature ratio
of 1.15 is resilient to subsequent off-axis minor mergers, though we have seen
from visual inspection that the small scale SCFs at $t_{\mathrm{max}}$ are
initiated by the infall of cluster C.

##  4 Discussion

Our simulations indicate that once sloshing due to a 1:10 minor merger is
established, a subsequent minor merger with the same mass ratio is not able to
destroy SCFs long-term. It can only disrupt them short-term for about 1Gyr,
between the second merger’s first pericentric passage to its first apocentre.
In this section, we discuss potential limits and consequences of the results.

###  4.1 Simulation setup

####  4.1.1 Parameter space sampling

The merger parameter space for three clusters is vast. We have only sampled a
small part of this parameter space, namely a constant mass ratio between the
three clusters, varying only the impact parameter of the second infalling
cluster in the orbital plane of the first merger. We have performed a
supplementary simulation (LOS1) in which cluster C’s infall is perpendicular
to the initial merger plane and off-axis. The state of this simulation at
$t_{\mathrm{max}}$ can be seen in Figure 13; SCFs are visible in a spiral
pattern, and the simulation is qualitatively similar to the clockwise and
anti-clockwise simulations. It is not trivial to distinguish this simulation
from the main suite of simulations presented in this paper, lending further
weight to the assertion that SCFs should be ubiquitous.

Even head-on second minor mergers did not erase SCFs established by the first
minor merger that had made their way to large radii. A more massive second
infaller may erase these SCFs, but such an event will be rarer than the low-
mass case studied here. The timing of the second merger might also play a role
in the distribution of SCFs. More complete sampling of the parameter space
necessitates a separate study due to the size of the parameter space.

Figure 13: Projected temperature weighted by density squared of the LOS1
simulation at $t_{\mathrm{max}}$.The image is 2.5 Mpc on a side. The arrow
represents the trajectory of the first infaller, and the dotted circle
represents the trajectory of the second infaller (note that neither annotation
is to scale, and should serve only as a guide to the infall directions). Clear
SCFs are present in the characteristic spiral pattern, and it is not trivial
to distinguish this snapshot from the clockwise or anticlockwise mergers
presented in this paper.

####  4.1.2 WHIM-like atmosphere and Boundaries

We wish to keep the simulations as idealised as possible, and as such we must
impose certain constraints which have expected consequences. We have tested
the simulations using different combinations of boundary conditions and
cluster profiles. We find that embedding clusters in a WHIM-like atmosphere
and in a domain with periodic boundary conditions produces the most realistic
results. As previously stated, the WHIM-like atmosphere acts to pre-truncate
the atmospheres of the infalling clusters (B and C), and therefore prevents
them from carrying an unrealistic amount of cool gas into the merger. Clusters
are expected to accrete gas from the cosmic web along filaments; cluster A
slowly accretes matter from the WHIM-like atmosphere isotropically. Our
simulation suite has also been run with open (‘outflow’) boundaries in which
the gradients of all variables are set to zero at boundaries, allowing matter
to flow in or out of the domain. We find that running the simulation with open
boundary conditions for such a long simulation time exacerbates the
aforementioned isotropic accretion effect as matter is allowed to flow into
the domain, leading to more compression and heating of the ICM. Even in this
scenario, the conclusion of SCF resilience stands. We therefore implement
periodic (‘wrap-around’) boundaries which effectively act as a universe in
miniature and mitigate the accretion of gas onto the cluster.

####  4.1.3 Physical effects

It is expected that KHIs will develop along CFs and disrupt them (ZuHone et
al., 2010; Roediger & Zuhone, 2012) in the absence of magnetic fields
(Lyutikov, 2006) or viscosity (ZuHone et al., 2015) to dampen them. ZuHone et
al. (2010) showed that viscosity suppresses instabilities and turbulence, thus
suppressing mixing, which in our case would act to reduce heat transfer to the
cool core of cluster A. This lack of core heating due to mixing is offset in
our simulations by the absence of radiative cooling. However, radiative
cooling would help to preserve the cool core and therefore potentially affect
the nature of SCFs at small radii. ZuHone et al. (2011) found that increased
magnetic field strength in the ICM leads to the cool core gas being more
resilient to effects of sloshing, and smoother cold fronts. It is clear that
both magnetic fields and viscosity would only make SCFs more resilient to
destruction by turbulence and KHI and our conclusions would therefore not be
altered by inclusion of these effects.

###  4.2 Ubiquity

Previous work has shown that in relaxed, CC clusters, CFs are likely to be
detected (Markevitch et al., 2003; Ghizzardi et al., 2010). Our simulations of
triple mergers with an off-axis second infaller have shown that small and
large scale SCFs are present in systems having undergone this type of merger
history. SCFs thus should still be ubiquitous. Should the merger history of a
cluster be such that a head-on collision has disrupted the CC, the simulations
have shown that large scale SCFs are still present. This result sheds some
light on systems such as A2142 and A1763 (discussed in sections 4.3.2 and
4.3.4, respectively) which lack a strong cool-core but do exhibit cold fronts.
Of course, since these SCFs are at large radii, they present an observational
challenge due to the reduced X-ray emission in cluster outskirts. However,
with the next generation of X-ray telescopes (including the recently launched
eRosita), our work suggests we may find that even non-CC clusters host large
scale SCFs. Indeed, a systematic search of archival data may also yield more
examples of cold fronts in NCC clusters. We have also seen that MCFs and SCFs
can coexist at many times during a cluster’s life.

###  4.3 Example Clusters

Having a single temporal snapshot of a system that is evolving over several
Gyr presents a significant challenge to our ability to reconstruct a formation
history. Roediger et al. (2011) deduced the ages of inner SCFs and thus merger
history in simulations; Walker et al. (2018) used this same principle to
calculate the age of large scale SCFs in the Perseus Cluster. Our simulations
indicate that large-scale SCFs indeed hold clues about longer-term cluster
growth. Detailed analysis on how to distinguish binary and triple mergers and
how to determine further merger history will be presented separately (paper in
prep.). In this section, we qualitatively examine some examples from the
literature and consider whether their features could indicate a triple merger
scenario. We will consider the projections of temperature weighted by density
squared (Figure 5) used in the previous analysis to make qualitative
comparisons with images found in the literature, noting that confirmation of
the scenarios we hypothesise will require further analysis.

####  4.3.1 RXJ2014.8-2430

RXJ2014.8-2430 ($z=0.138$, $M_{500}=5.38\times 10^{14}M_{\odot}$ (Pratt et
al., 2010), $\,r_{\mathrm{200}}=1750$ Mpc (Croston et al., 2008)) is a strong
cool core cluster. Walker et al. (2014) analysed _ROSAT_ , _Chandra_ and _XMM-
Newton_ data to reveal two CFs in the central 200 kpc
($\sim$0.46$\,r_{\mathrm{200}}$) and one to the south at 810 kpc
($\sim$0.46$\,r_{\mathrm{200}}$). In our simulations, $\,r_{\mathrm{200}}$ of
the primary is 1637 kpc, and the largest scale CF in the binary merger
simulation is $\sim$800 kpc ($\sim$0.49$\,r_{\mathrm{200}}$) at 10 Gyr. The
authors suggest that the lack of intermediate fronts indicates that the large-
scale CFs arise due to an earlier merger than the inner CFs. Our simulations
suggest that this scenario is entirely plausible. In particular, several
configurations, in which the first and second minor mergers have a non-zero
impact parameter produce SCFs which, given appropriate rotation, match this
pattern. For the large scale CF to have had time to reach
$\sim$0.46$\,r_{\mathrm{200}}$, we suggest an off-axis minor merger with an
age of 7.5-8.5 Gyr.

####  4.3.2 A2142

Abell 2142 ($z=0.0909$, $M_{200}=1.25\times 10^{15}M_{\odot}$,
$\,r_{\mathrm{200}}=2.16$ Mpc (Munari et al., 2015)) is an intermediate
(between CC and non-CC) cluster which hosts two CFs at small radii, one CF at
intermediate radii and one at large radii. The small/intermediate fronts are
at $\sim$300 kpc ($\sim$0.139$\,r_{\mathrm{200}}$) NW of the centre, $\sim$100
kpc ($\sim$0.046$\,r_{\mathrm{200}}$) south of centre (Markevitch et al.,
2000; Owers et al., 2011), and at 12 kpc ($\sim$0.0056$\,r_{\mathrm{200}}$)
from the gas density peak which connects to the southern front (Johnson,
2011). Rossetti et al. (2013) discovered a fourth CF $\sim$1 Mpc
($\sim$0.46$\,r_{\mathrm{200}}$) to the southeast of the centre spanning an
angular extent of 70∘, corresponding to 1.2 Mpc. Rossetti et al. (2013)
compared the images of A2142 to the simulations of Roediger & Zuhone (2012)
and Roediger et al. (2012) and suggested that the fronts in A2142 could have
been initiated by an intermediate (i.e. mass ratio of 1:2 or 1:3) merger
event. There is, as in RXJ2014.8-2430, a relative lack of intermediate CFs,
suggesting that the large scale CFs are formed by an early off-axis merger,
and the small radius CFs are formed by a more recent merger. Eckert et al.
(2017) present a study of the infalling group ($\sim$1.3 Mpc north of the
core), which (by virtue of it exhibiting a slingshot tail in its wake) could
be approaching for a second pericentric passage. The motion of this infalling
group is perpendicular to the direction of sloshing, which could imply it
caused the small scale sloshing in A2142, but it is unlikely that the CF at 1
Mpc from the core could have had time to reach this radius if initiated by the
infalling group. This poses an interesting challenge to the assumption that
sloshing fronts cannot be generated in isentropic gas (which a NCC
approximates). We see in our simulations with relatively close pericentre
distances (e.g. Clock4) that the cool core is weaker than in simulations with
a larger pericentre distance. It could, therefore, be the case that the
infaller responsible for the small scale SCFs in A2142 were caused by a close
passing infaller which has significantly disrupted the core but not so much
that it is isentropic. Einasto et al. (2018) showed that A2142 is embedded in
a supercluster and suggest that it is the result of several past and ongoing
merger events. Interestingly, the axis of the supercluster (the suggested
accretion direction) lies perpendicular to the infalling group of Eckert et
al. (2017). Given the large scale SCF at $\sim$1 Mpc, and the ongoing
accretion in A2142, this suggests sloshing is indeed resilient to merger
activity. Venturi et al. (2017) found a two-component radio halo in A2142,
with the inner component being bounded by the two innermost CFs, and the
larger, more diffuse radio emission extending to the largest CF. It is
possible that these two components trace separate merger events in A2142.

####  4.3.3 A426 - Perseus

Perseus ($z=0.017284$ (Collaboration, 2018); $M_{200}=6.65\times
10^{14}M_{\odot}$, $r_{\mathrm{200}}=1.79$ Mpc (Simionescu et al., 2011)) is a
relaxed cluster with SCFs in an anti-clockwise spiral pattern originating
$\sim$10 kpc ($\sim$0.0056$\,r_{\mathrm{200}}$) from the centre and
terminating with a SCF at $\sim$100 kpc from the centre (Churazov et al.,
2003). Churazov et al. (2003) state that the chain of galaxies to the west of
the BCG (NGC 1275) indicates an ongoing merger with an infaller from the west.
Simionescu et al. (2012) found a surface brightness edge at $\sim$700 kpc
($\sim$0.39$\,r_{\mathrm{200}}$) east of the cluster centre which they connect
to the western edge of the inner SCFs anti-clockwise via a feature north of
the centre. Walker et al. (2020) discovered two large scale fronts (1.2 Mpc
($\sim$0.67$\,r_{\mathrm{200}}$) and 1.7 Mpc ($\sim$0.95$\,r_{\mathrm{200}}$))
which they suggest are caused by sloshing initiated $\sim$8.7 Gyr ago. The
large scale fronts in this case are at considerably larger radii than in our
simulations, and so we are hesitant to make a direct comparison with a
particular simulation. We speculate that the old age of the sloshing combined
with continuous accretion onto the cluster over many Gyr could move SCFs out
to radii as large as observed in Perseus. However, the apparent lack of CFs at
intermediate radii does suggest that this is not a single merger event. Zhang
et al. (2020) proprosed that the large scale CFs here are in fact shock
induced cold fronts (SICFs) caused by a runaway merger shock encountering the
accretion shock in the outskirts of the cluster.

####  4.3.4 A1763

Abell 1763 ($z=0.2312$, $M_{200}=1.77\times 10^{15}M_{\odot}$,
$\,r_{\mathrm{200}}=1.62$ Mpc (Rines et al., 2013)) is an NCC cluster lying
south-west of Abell 1770, with a filament connecting the two clusters. Abell
1763 exhibits an elongation along the NE-SW axis, consistent with a merger
along the filamentary direction. The system features a double peaked ICM X-ray
emission, with the eastern peak coinciding roughly with the BCG (Douglass et
al., 2018). Haines et al. (2018) identify two $\sim 10^{14}M_{\odot}$
subclusters (A1763-g7 and A1763-g9) infalling on Abell 1763 at z=0.237, with
A1763-g9 lying $\sim$750 kpc ($\sim$0.46$\,r_{\mathrm{200}}$) SW of the BCG
and A1763-g7 lying $\sim$1.2 Mpc ($\sim$0.74$\,r_{\mathrm{200}}$) SW of the
BCG. Douglass et al. (2018) observed a spiral pattern in the X-ray surface
brightness extending clockwise outward from the western X-ray peak to
$\sim$850 kpc north of the core. The authors suggest that A1763-g7 is the
subcluster responsible for the sloshing spiral, and is returning for an anti-
clockwise 2nd pericentric passage. We see in our simulations that for an
$\sim$0.5$\,r_{\mathrm{200}}$ SCF to be present, the sloshing must have been
initiated several Gyr ago, by which time the responsible subcluster has
merged/ reaches only apocentres of $\lesssim\,$0.3$\,r_{\mathrm{200}})$.
Therefore we suggest that A1763-g7 is not responsible for the $\sim$850 kpc
SCF, though it is likely responsible for the small-scale SCFs in A1763.

##  5 Conclusions

We have performed a suite of 13 hydrodynamic + N-body simulations exploring
the effects of different triple cluster merger configurations on gas sloshing
in the ICM established by a prior off-axis binary minor merger. We have varied
the approach direction and impact parameter of the third cluster to ascertain
the resilience of already established sloshing. We have not yet fully sampled
the parameter space, and hence sensitivity to mass ratio and timing of the
second merger are not yet explored.

We showed that a second merger leads to a short-term disruption of the SCFs
after its first pericentric passage; however during the ‘relaxation phase’
(after its second pericentric passage) we see that the SCFs are present and
thus have long-term resilience. We note that it is not trivial to distinguish
between the resultant systems at $t_{\mathrm{max}}$. Our main conclusions are
as follows:

1\. Cold fronts can always be found in systems that have undergone two minor
mergers in sequence, provided the second infalling cluster has a non-zero
impact parameter.

2\. Sloshing cold front patterns are resilient even to on-axis mergers, with
SCFs being temporarily obscured by the second infaller’s first pericentric
passage and reappearing by the second infaller’s second pericentric passage.

3\. Systems containing small ($\sim$0.12$\,r_{\mathrm{200}}$) and large
($\sim$0.5$\,r_{\mathrm{200}}$) scale cold fronts with few or no intermediate
fronts are likely to have undergone two off-axis minor mergers in sequence.

4\. Clusters with a spiral pattern that can be traced from small to large
radii with no gaps are likely to have only undergone a single off-axis binary
minor merger, with the pericentre time being calculable from the radius of the
largest cold front.

## Acknowledgements

We thank the anonymous referee for their comments that have improved this
paper. The software used in this work was in part developed by the DOE NNSA-
ASC OASCR Flash Center at the University of Chicago. IV acknowledges the
support of University of Hull Astrophysical Data Science Cluster and access to
Viper, the University of Hull High Performance Computing Facility, as well as
informative conversations with Prof. Brad Gibson. WF acknowledges support from
the Smithsonian Institution, the Chandra High Resolution Camera Project
through NASA contract NAS8-03060, and NASA Grants 80NSSC19K0116, GO1-22132X,
and GO9-20109X. CC is funded by the Science and Technology Facilities Council
through the University of Hull’s Consolidated Grant ST/R000840/1.

## Data Availability

The simulation data and code used for analysis from this study are available
from IV upon reasonable request.

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##  Appendix A Appendix A

Figure 14: Movie (the movie can be viewed at https://youtu.be/q-CqnaGtg0E)of
projections of temperature with density squared weighting through the $x-y$
plane for the head-on simulations with the binary merger on the left column
for reference. All panels are centred on the minimum potential of the domain
(the centre of cluster A). The top row panels are 500 kpc on a side and the
bottom row panels are 2.5 Mpc on a side. The individual simulations have been
staggered such that the first pericentre of cluster C occurs at the same time
in the movie. The arrows have the same meaning as in previous figures. We
clearly see that the head-on merger has disrupted the cool-core of cluster A
but that cold fronts are still visible outside the core region. Figure 15:
Movie (the movie can be viewed at https://youtu.be/ptxtqtfvlSU) of projections
of temperature with density squared weighting through the $x-y$ plane for the
clockwise simulations with the binary merger on the left column for reference.
All panels are centred on the minimum potential of the domain (the centre of
cluster A). The top row panels are 500 kpc on a side and the bottom row panels
are 2.5 Mpc on a side. The individual simulations have been staggered such
that the first pericentre of cluster C occurs at the same time in the movie.
The arrows have the same meaning as in previous figures. We see that the
clockwise mergers do not disrupt the cold fronts that have already travelled
out of the core region of cluster A and that new sloshing is initiated by
cluster C. Figure 16: Movie (the movie can be viewed at
https://youtu.be/fNv38D89Phk) of projections of temperature with density
squared weighting through the $x-y$ plane for the anti-clockwise simulations
with the binary merger on the left column for reference. All panels are
centred on the minimum potential of the domain (the centre of cluster A). The
top row panels are 500 kpc on a side and the bottom row panels are 2.5 Mpc on
a side. The individual simulations have been staggered such that the first
pericentre of cluster C occurs at the same time in the movie. The arrows have
the same meaning as in previous figures. We see that the anti-clockwise
mergers do not disrupt the cold fronts that have already travelled out of the
core region of cluster A and that new sloshing is initiated by cluster C.
Figure 17: Movie (the movie can be viewed at https://youtu.be/0bICdEy1F3M) of
projections of temperature with density squared weighting through the $x-y$
plane for all simulations. Each panel is 0.5 Mpc on a side and the simulation
times have been altered such that the simulations simultaneously show cluster
C’s first pericentric passage. The rows are comprised, respectively, of the
head-on, clockwise and anti-clockwise simulations; the columns represent
approach direction of cluster C. The leftmost column is the binary merger and
supplementary simulation (discussed in Sec 4.1.1) with the colour scale for
all simulations. The arrows and dotted circle have the same meaning as in
previous figures. Figure 18: Projection of temperature with density squared
weighting through the $x-y$ plane for all simulations at the maximum
simulation time for each simulation. Each panel is 2.5 Mpc on a side and the
annotations have the same meaning as in previous figures.