###### Abstract

Radio jets and the lobes they inflate are common in cool-core clusters and are
known to play a critical role in regulating the heating and cooling of the
intracluster medium (ICM). This is an inherently multi-scale problem, and much
effort has been made to understand the processes governing the inflation of
lobes and their impact on the cluster, as well as the impact of the
environment on the jet-ICM interaction, on both macro- and microphysical
scales. Developments of new numerical techniques and improving computational
resources have seen simulations of jet feedback in galaxy clusters become ever
more sophisticated. This ranges from modelling ICM plasma physics processes
such as the effects of magnetic fields, cosmic rays and viscosity to including
jet feedback in cosmologically evolved cluster environments in which the ICM
thermal and dynamic properties are shaped by large-scale structure formation.
In this review, we discuss the progress made over the last $\sim$ decade in
capturing both the macro- and microphysical processes in numerical
simulations, highlighting both the current state of the field as well as open
questions and potential ways in which these questions can be addressed in the
future.

###### keywords:

radio jets; active galactic nuclei; galaxy clusters; numerical modelling;
simulation techniques; plasma physics; magnetohydrodynamics; cosmic rays;
viscosity; thermal conduction

1 1 0 https://doi.org/ Recent Progress in Modelling the Macro- and Micro-
Physics of Radio Jet Feedback in Galaxy Clusters Recent Progress in Modelling
the Macro- and Micro-Physics of Radio Jet Feedback in Galaxy Clusters Martin
A. Bourne 1,2‡* and Hsiang-Yi Karen Yang 3,4,5,‡* Martin A. Bourne and Hsiang-
Yi Karen Yang Bourne, M. A.; Yang, H.-Y. K. Correspondence:
<EMAIL_ADDRESS>(M.A.B<EMAIL_ADDRESS>(H.-Y.K.Y) These
authors contributed equally to this work.

##  1 Overview

###  1.1 Jet Feedback in Galaxy Clusters: Observations and Theoretical
Motivation

Clusters of galaxies are the largest gravitationally collapsed objects in the
Universe. Because of their deep gravitational potentials, the virial
temperature of gas falling into clusters during hierarchical structure
formation is heated to $10^{7-8}$ K, producing Bremsstrahlung emission in
X-rays. Because the Bremsstrahlung emissivity of the intracluster medium (ICM)
is proportional to gas density squared, near the centre of many massive
clusters where the central density is high, the gas would cool significantly
due to the strong radiation – so-called “radiative cooling.” For clusters with
central cooling times much smaller than the Hubble time, i.e., the cool-core
(CC) clusters, it is predicted by the cooling-flow model (Fabian, 1994) that
there should be a significant inflow of gas toward the cluster centre,
triggering intensive star formation within the central brightest cluster
galaxies (BCGs). However, there is a lack of observational evidence for such
cooling flows, and the star formation rates (SFRs) of the BCGs are typically
10-100 times lower than those predicted by the cooling-flow model. This is
well-known as the “cooling-flow problem” of galaxy clusters (see McNamara and
Nulsen, 2007; Fabian, 2012, for a review).

The absence of cooling flows calls for some heating mechanisms to balance
radiative cooling in CC clusters, among which energetic feedback provided by
relativistic jets emerging from the central supermassive black holes (SMBHs)
is believed to be the most promising mechanism. This is motivated by the
prevalence of X-ray cavities or bubbles, which are low-density, X-ray dim
regions filled with radio-emitting plasma injected by the SMBH jets, in CC
clusters (e.g., Best et al., 2007; Mittal et al., 2009; Bîrzan et al., 2012),
as well as in many galaxy groups (e.g., Bîrzan et al., 2004; David et al.,
2011; Randall et al., 2011, 2015), which themselves represent a significant
fraction of local radio galaxy hosts (e.g., Best, 2004; Croston et al., 2008;
Ching et al., 2017). Another compelling piece of evidence for active galactic
nucleus (AGN) feedback is the observed correlation between the cavity power,
which is a proxy for the jet power, and the X-ray luminosity within the
cluster cores (Rafferty et al., 2006; Dunn and Fabian, 2008; Hlavacek-Larrondo
et al., 2012, 2022), suggesting that there is a global thermal balance between
heating and cooling in CC clusters. These observations have therefore
motivated a picture of a self-regulated AGN feedback loop: when a cluster has
increased X-ray luminosity, the ICM cooling rates and thus SMBH accretion
rates are enhanced, which triggers powerful ejections of relativistic jets,
heating the ICM and shutting off subsequent cooling and AGN activity (McNamara
and Nulsen, 2012). While the idea of an AGN feedback cycle is an attractive
solution to the cooling-flow problem, there is a huge dynamical range between
the scales of SMBH accretion discs ($\sim 1-10^{4}$ AU) and the cores of
galaxy clusters ($\sim 100$ kpc) (see Fig. 1 in Gaspari et al. (2020) for an
illustration of the multi-scale SMBH accretion and feedback processes
involved). The vast dynamical range involved has prohibited an all-in-one
simulation of AGN feedback, and hence the detailed processes of SMBH feeding
and feedback mechanisms remain one of the greatest unsolved problems in
astrophysics.

Because galaxy clusters are at the crossroads of cosmology and astrophysics,
the advantages of understanding AGN feedback in clusters are two-fold. On the
one hand, galaxy clusters are excellent laboratories for studying various
astrophysical phenomena, including radiative cooling, AGN feedback, magnetic
fields, shocks, turbulence, acceleration of cosmic rays (CRs), plasma physics,
etc. On the other hand, constraining these astrophysical mechanisms would
enable robust predictions of cluster observables and evolution histories that
are critical for inferences of cosmological parameters as well as the
formation and evolution of galaxies (see Kravtsov and Borgani, 2012; Allen et
al., 2011, for relevant reviews). In turn, the large-scale environments of
clusters and their evolution histories would also influence the operation of
the above astrophysical processes. AGN jet feedback in clusters is one great
example of this iterative process, where the theoretical understanding from
both the astrophysical and cosmological perspectives has emerged in the past
two decades. It is therefore our aim to review this exciting progress in this
article. AGN feedback is not limited to jets in clusters, additionally
manifesting itself via radiation and accretion disc driven winds (Fabian,
2012; King and Pounds, 2015; Morganti, 2017), and is expected to impact galaxy
formation over a wide range of redshifts by shaping galaxy properties,
regulating BH growth and driving BH-host galaxy scaling relations (Magorrian
et al., 1998; Ferrarese and Merritt, 2000; Kormendy and Ho, 2013; McConnell
and Ma, 2013; Sahu et al., 2019), which themselves may aid our understanding
of AGN physics. AGN feedback may even be important at the low mass end, with
recent studies of its affect on dwarf galaxies representing an area of growing
interest (e.g., Reines et al., 2013; Manzano-King et al., 2019; Mezcua, 2021;
Koudmani et al., 2022; Sharma et al., 2022). We refer interested readers to
additional reviews that provide comprehensive coverage of these topics (e.g.
McNamara and Nulsen, 2007; Fabian, 2012; McNamara and Nulsen, 2012; Kormendy
and Ho, 2013; King and Pounds, 2015; Harrison, 2017; Morganti, 2017; Harrison
et al., 2018; Hardcastle and Croston, 2020; Veilleux et al., 2020; Mezcua,
2021; Hlavacek-Larrondo et al., 2022).

###  1.2 Simulating Jets in Idealised Hydrodynamic Simulations

As early as the late 1970s and throughout the 1980s, groups were performing
simulations of jets to study their propagation and structural evolution (see
e.g.,Burns et al. (1991) for a review at the time). The pioneering works of
Rayburn (1977) and Norman et al. (1982) paved the way, performing the first
simulations of non-relativistic jets. Numerous works followed with a
particular emphasis on understanding radio observations of jets (Clarke et
al., 1989, 1992; Falle and Wilson, 1985; Smith et al., 1985; Wilson and
Scheuer, 1983), as well as numerical studies on the impact of resolution
(Koessl and Mueller, 1988), that included magnetic fields (Clarke et al.,
1986, 1989; Lind et al., 1989) and that were performed in 3D (Arnold and
Arnett, 1986; Clarke et al., 1990). The history of simulating jets is itself
an interesting topic, however, in this work, we restrict ourselves to more
recent advances in modelling jet feedback in galaxy cluster environments and
instead point the reader to reviews focusing on simulating jets in general
(Komissarov and Porth, 2021; Martí, 2019).

While much early work was motivated by radio observations of jets (e.g.,
Miley, 1980; Biretta et al., 1983), X-ray observations of galaxy clusters
using ROSAT (e.g., Boehringer et al., 1993; Carilli et al., 1994; Huang and
Sarazin, 1998) and later Chandra (e.g., Fabian et al., 2000; McNamara et al.,
2000) that show cavities111As highlighted by Fabian (2012), cavities had been
observed earlier with the Einstein telescope (Branduardi-Raymont et al., 1981;
Fabian et al., 1981), although their true nature had not been realised. in the
ICM provided further motivation to understand the interaction of radio jets
and the lobes they inflate with the surrounding hot ICM. Broadly speaking, two
approaches were used to study this scenario; one in which the jets were
simulated and the self-consistent inflation of the lobes studied (Clarke et
al., 1997; Rizza et al., 2000; Reynolds et al., 2001, 2002; Omma et al.,
2004), or for simplicity, one in which this process was pre-assumed and hot
bubbles were added or inflated “by hand” (Churazov et al., 2001; Brüggen and
Kaiser, 2001; Quilis et al., 2001; Brüggen et al., 2002). While the former
approach gives insights into how the jet itself interacts with the ICM and how
lobes are inflated, the latter is less computationally expensive. More complex
simulations have followed and contributed to an ever-growing literature, which
we discuss further throughout this review.

Apart from a few examples (Huško and Lacey, 2023a; Huško and Lacey, 2023b;
Huško et al., 2022), simulations that attempt to model the jets themselves are
performed using grid-based codes that allow for high resolution in low-density
regions, which are typical of jets. Traditionally such codes are Eulerian,
although novel refinement techniques have meant that moving-mesh (AREPO,
Springel, 2010) and meshless-finite-mass (GIZMO, Hopkins, 2015) codes have
also been used to simulate jets (e.g., Bourne and Sijacki, 2017; Weinberger et
al., 2017; Su et al., 2021). When it comes to launching a jet, some works
place finer control on how exactly the jet is injected into the simulation
domain, by defining the exact thermodynamic and kinetic state within the
launch region, i.e., the jet density, temperature and velocity (e.g., Krause,
2003; Krause et al., 2012; Hardcastle and Krause, 2013; Weinberger et al.,
2017). Other works simply add mass, momentum and/or energy to the injection
region and allow the jets’ thermodynamic properties to arise naturally from
this (e.g. Omma et al., 2004; Cattaneo and Teyssier, 2007; Dubois et al.,
2010; Gaspari et al., 2011; Li and Bryan, 2014; Yang and Reynolds, 2016;
Bourne and Sijacki, 2017). However, the ultimate result is a pair of fast low
density jets that drive bow shocks into the ICM (and potentially internal
shocks along the jets), thermalise, and inflate high-temperature lobes that
come into pressure balance with the ICM. These methods have been used to
perform a wide range of simulations, from high-resolution studies of lobe
inflation and energetics (e.g., Hardcastle and Krause, 2013, 2014; Bourne and
Sijacki, 2017; Weinberger et al., 2017; Ehlert et al., 2018; Bourne et al.,
2019; Yang et al., 2019; Perucho et al., 2022) to those that include SMBH
accretion models from which the jet power is calculated (e.g., Gaspari et al.,
2011, 2011, 2015; Prasad et al., 2015; Yang and Reynolds, 2016, 2016; Li et
al., 2017; Beckmann et al., 2019; Ehlert et al., 2022) and are performed on
Gyr time-scales to understand how jets can regulate the thermodynamic state of
the ICM. On top of this, additional physics relative to jet feedback and
galaxy clusters has been included in dedicated jet simulations such as
magnetic fields (Robinson et al., 2004; Li et al., 2006; Ruszkowski et al.,
2008; Dubois et al., 2009; Gaibler et al., 2009; O’Neill and Jones, 2010;
Mendygral et al., 2012; Sutter et al., 2012; Hardcastle and Krause, 2014;
Weinberger et al., 2017; Ehlert et al., 2018; Wang et al., 2020, 2021; Ehlert
et al., 2022), relativistic effects (Perucho et al., 2014; English et al.,
2016; Perucho et al., 2017; English et al., 2019; Yates-Jones et al., 2021;
Perucho et al., 2022), viscosity (Reynolds et al., 2005; Dong and Stone, 2009;
Guo, 2015; Kingsland et al., 2019), thermal conduction (Ruszkowski and
Begelman, 2002; Zakamska and Narayan, 2003; Voigt and Fabian, 2004; Parrish et
al., 2009; Bogdanović et al., 2009; Avara et al., 2013; Yang and Reynolds,
2016; Kannan et al., 2017; Beckmann et al., 2022) and CR physics (Guo and Oh,
2008; Mathews and Brighenti, 2008; Ruszkowski et al., 2017; Weinberger et al.,
2017; Bourne and Sijacki, 2017; Jacob and Pfrommer, 2017a, b; Prokhorov and
Churazov, 2017; Ehlert et al., 2018; Yang et al., 2019; Lin et al., 2023).

There are now many works that have coupled jet power to SMBH accretion rates
(e.g., Cattaneo and Teyssier, 2007; Dubois et al., 2010, 2012; Gaspari et al.,
2011, 2011, 2012; Yang et al., 2012; Li and Bryan, 2014; Li et al., 2017; Yang
and Reynolds, 2016; Prasad et al., 2015; Meece et al., 2017; Ehlert et al.,
2022). The physical connection between accretion and jet power, expected to
depend on the properties of the BH and its accretion flow (see e.g. Section 4
of Hlavacek-Larrondo et al., 2022, for a discussion), is still to be fully
understood. For simplicity, the simulations discussed here often assume a
fixed scaling between accretion rate, $\dot{M}$, and jet power, $\dot{E}_{\rm
Jet}$, of the form

$\dot{E}_{\rm Jet}=\epsilon\dot{M}c^{2},$ (1)

where $c$ is the speed of light and $\epsilon$ is a constant feedback
efficiency, although variable spin-dependent efficiencies have been included
in recently developed models (Beckmann et al., 2019; Talbot et al., 2021,
2022; Huško et al., 2022). Cattaneo and Teyssier (2007) presented some of the
earliest simulations that coupled a jet feedback scheme (Omma et al., 2004) to
a Bondi accretion model and therefore capture self-regulated jet feedback in a
galaxy cluster. The jets were injected using a combination of momentum and
thermal energy and while the feedback was able to prevent over-cooling of the
ICM it was overly effective and unable to preserve a CC. Numerous works
followed that similarly investigated injecting a combination of thermal energy
and momentum (Yang et al., 2012; Li and Bryan, 2014) as well as works that
considered purely kinetic jets (Gaspari et al., 2011, 2012; Prasad et al.,
2015; Yang and Reynolds, 2016, 2016) that were able to inhibit cooling and
match a range of observed cluster properties. Specifically, it has been found
that combining momentum-driven jets with cold-accretion models (Gaspari et
al., 2012, 2013, 2015; Li and Bryan, 2014; Li et al., 2017; Yang and Reynolds,
2016; Prasad et al., 2015; Meece et al., 2017; Ehlert et al., 2022), whereby
the SMBH grows due to accretion of cold gas, can regulate ICM cooling while
maintaining the properties of CC clusters, albeit in idealised cluster
environments. However, as discussed in the next section, reproducing
thermodynamic profiles and the CC/non-CC dichotomy is difficult to achieve in
full cosmological simulations.

###  1.3 AGN Feedback in Cosmological Simulations

The simulations discussed above, typically performed in hydrostatic
atmospheres, place emphasis on understanding the detailed processes governing
lobe inflation and interaction with the ICM, which requires high spatial
(often achieved with dedicated lobe refinement techniques) and time
resolution. However, the study of cluster formation and evolution requires
simulations that capture structure formation processes over cosmic time, i.e.
cosmological simulations. Such simulations span a wide range of galaxy masses
and include various models to capture a wide array of physical processes
important to galaxy formation and we point the reader to reviews for a
detailed discussion (Somerville and Davé, 2015; Vogelsberger et al., 2020).
The inclusion of AGN feedback in cosmological simulations has been shown to be
an important ingredient to create realistic galaxy populations, particularly
at the high mass end (e.g., Di Matteo et al., 2005; Le Brun et al., 2014;
Schaye et al., 2015; Sijacki et al., 2015; Dubois et al., 2016; Weinberger et
al., 2018). There are a growing number of cosmological simulations that
capture the formation and evolution of galaxy clusters and aim to understand
the processes that shape their properties (e.g., Dubois et al., 2011; Le Brun
et al., 2014; Bahé et al., 2017; Barnes et al., 2017a, b, 2018; Hahn et al.,
2017; Rasia et al., 2015; Henden et al., 2018; Tremmel et al., 2019; Pakmor et
al., 2022). Such simulations have done remarkably well at matching a range of
observed cluster properties, however, achieving realistic thermodynamic
profiles in cluster cores and explaining the CC/non-CC dichotomy is
challenging, with several works concluding a need for modified AGN feedback
models (e.g., Genel et al., 2014; Barnes et al., 2017b; Henden et al., 2018).

Cosmological simulations, by the necessity of space and time resolution
constraints, implement simplified and/or low-resolution models of AGN feedback
that do not necessarily capture the processes of lobe inflation and their
interaction with the ICM in detail. In the most straightforward
implementation, feedback energy is injected thermally in resolution elements
local to the BH (e.g. Springel et al., 2005; Di Matteo et al., 2005; Booth and
Schaye, 2009; Schaye et al., 2015; McCarthy et al., 2017; Tremmel et al.,
2019). Modifications are often made to such models to circumvent the over-
cooling problem (see e.g., Booth and Schaye, 2009; Le Brun et al., 2014;
Bourne et al., 2015; Schaye et al., 2015, for a discussion), for example by
temporarily switching off radiative cooling (Tremmel et al., 2017, 2019), or
by storing energy until gas can be heated to a minimum temperature (Booth and
Schaye, 2009; Le Brun et al., 2014; Schaye et al., 2015; McCarthy et al.,
2017) or a fixed duty cycle has passed (Henden et al., 2018). Alternatively,
the energy can be injected as a kinetic outflow (Dubois et al., 2010, 2012,
2016; Weinberger et al., 2017, 2018; Davé et al., 2019). While some works
assume a single feedback mode that is the same irrelevant of the BH mass or
accretion rate (e.g., Schaye et al., 2015; McCarthy et al., 2017; Tremmel et
al., 2019), others implement separate “quasar” and “radio” modes (e.g.,
Sijacki et al., 2007, 2015; Dubois et al., 2012, 2016; Weinberger et al.,
2017, 2018; Davé et al., 2019), with the latter typically being used for low
accretion rates. One of the earliest such examples of dual AGN modes was
introduced by Sijacki et al. (2007), taking inspiration from early bubble
models of jet feedback (Churazov et al., 2001; Quilis et al., 2001; Dalla
Vecchia et al., 2004), by including the jet mode as thermal bubbles displaced
from the central SMBH. This model has subsequently been used in the Illustris
(Sijacki et al., 2015) and Fable (Henden et al., 2018) simulation suites. On
the other hand, Dubois et al. (2010), built on the work of Omma et al. (2004)
and Cattaneo and Teyssier (2007) to implement mass-loaded bipolar outflows
that aimed to capture a sub-relativistic accretion disc wind as jet-like
outflows and can self-consistently produce cavity like structures in the ICM.
This model has since been used in the HorizonAGN (Dubois et al., 2016) and
NewHorizon (Dubois et al., 2021) simulations, as well as other studies of
feedback in galaxy clusters (e.g., Beckmann et al., 2019, 2022). In a somewhat
similar vein, Davé et al. (2019) implement a jet feedback model for the low
accretion rate regime in the form of a slow, highly collimated bipolar
outflow, in the SIMBA suite of simulations, however, the jet is
hydrodynamically decoupled during its initial phase, meaning that it does not
interact with the inner parts of galaxies and clusters and instead couples its
energy on larger scales. Alternatively, some simulation suites, such as
IllustrisTNG (Weinberger et al., 2018), take a different approach and instead
of attempting to model jet feedback implement an efficient kinetic wind model
for high mass SMBHs in the low accretion rate regime that injects kinetic
energy into the local gas along a random direction (Weinberger et al., 2017),
which acts to effectively quench high mass galaxies.

###  1.4 The Importance of Macrophysics and Microphysics

Thanks to the advancement in computing power and numerical algorithms over the
past decade, numerical simulations of jet feedback in clusters have become
more and more sophisticated with realistic initial conditions from
cosmological contexts as well as with complex physical processes including
magnetic fields, CRs, plasma effects, etc. The latter is often dubbed as
“microphysics” in clusters because the gyro-radii of charged particles and CRs
in the magnetised ICM (on the order of $\sim$ AU for GeV CRs in $\mu$G
magnetic fields) are many orders of magnitude smaller than the size of
clusters ($\sim$ Mpc). Consequently, transport processes in the ICM, including
thermal conduction, viscosity, and CR propagation, are determined by plasma
physics that happen on the microscopic scales of the particle gyro-radii
(Section 3). As will be discussed in detail in later sections, recent
simulations have demonstrated that both the macrophysics from the large-scale
environments of clusters and the microphysics have crucial impacts on the AGN
feeding and feedback processes and the evolution of clusters, and thus they
will be the focus of this review. We also point out, however, that despite the
vast computational capabilities now available to the community, there is still
a need for robust, physically motivated analytic components and sub-grid
models to link between the scales above and below the resolution limit of
simulations.

The structure of this article is as follows. In Section 2, we will review the
important progress regarding the modelling of the macrophysical processes of
AGN feedback. We will first start by introducing the fundamental processes of
AGN lobe formation and lobe-ICM interaction (Section 2.1). This is followed by
discussions about the role of the cluster environment (Section 2.2) and open
questions and future opportunities (Section 2.3). In Section 3, important
findings regarding the microphysical processes will be summarized.
Specifically, in Section 3.1 we will discuss the roles of CRs in AGN feedback
and their observational signatures. We will then discuss the roles of plasma
physics, such as transport processes like conduction and viscosity in the ICM,
in Section 3.2. Open questions and future opportunities regarding the
microphysics are discussed in Section 3.3. Finally, Section 4 contains our
concluding remarks.

##  2 Modelling the Macrophysics

There is now significant literature on hydrodynamic and magnetohydrodynamic
(MHD) modelling of the large-scale processes driving the inflation of jet
lobes and their macroscopic evolution. Figure 1 provides a general
diagrammatic overview of the phases of lobe inflation and subsequent
evolution, which we discuss throughout this section. Specifically, in Section
2.1 we first consider the processes that are intrinsic to the jets and lobes,
including their energetics and processes through which they interact with the
ICM, while in Section 2.2 we additionally discuss the role of the cluster
environment, including cluster weather, and the role this has on shaping lobe
properties and cluster heating.

Figure 1: Diagram illustrating the general processes that occur during and
after lobe inflation. Panel A: A fast jet drives into the ambient medium,
forms a bow shock and inflates a hot lobe that expands into the ICM. The lobe
morphology can depend sensitively on the injected jet properties (e.g.,
content, velocity, geometry). The expanding shock wave results in a layer of
shocked ICM material surrounding the jet lobe. Panel B: As the lobe expansion
slows, which may or may not be accompanied by the jet switching off, the shock
driven into the ICM broadens into a sound wave that can detach from the lobes.
Panel C: Once the jet has ceased and as the lobe buoyantly rises through the
ICM, dense, low entropy material can be entrained and pulled up in the wake
and instabilities can lead to mixing of the lobe and ICM material. Sound waves
generated by the lobe expansion can continue to propagate to large distances
depending on the ICM viscosity. Panel D: This process continues at late times,
with mixing continuing to dilute the lobe material, although the rate at which
this occurs can depend sensitively on the ICM physical processes including
magnetic fields, viscosity and cluster weather (i.e. ICM and sub-structure
motions, see Section 2.2)

###  2.1 Bubbles, Shocks and Waves

####  2.1.1 Morphology, Direction and Energetics

Figure 2: Thin temperature projections illustrate how jet injection parameters
impact jet and lobe morphologies. All jets are $100$ Myr old with each row
illustrating the effect of changing one parameter, which from top to bottom
are jet power, half opening angle, velocity and resolution, respectively.
These quantities take fiducial values of $10^{46}$ erg s-1, $10^{\circ}$,
$15000$ km s-1 and $1.81\times 10^{5}$ M⊙, unless being varied. Colour bars
shown to the right of each row extend down to $10^{7}$ K, gas below this is
shown in black. (Figure 6 from Huško and Lacey (2023a), CC BY 4.0)

Observed jets have long been classified based on their radio morphology as set
out by Fanaroff and Riley (FR 1974), which broadly characterises them as
centre-brightened FR-I sources (e.g. 3C 449, Perley et al., 1979) that are
expected to form due to decelerating and becoming turbulent on small ($\sim$
kpc) scales222Frustrated and bent jets, such as wide/narrow angle tailed radio
galaxies, can provide morphological analogues to FR-I sources (e.g., Dehghan
et al., 2014; O’Brien et al., 2018; Murthy et al., 2022; Mukherjee et al.,
2018). (e.g., Komissarov, 1994; Rossi et al., 2008; Perucho et al., 2010;
Laing and Bridle, 2014; Tchekhovskoy and Bromberg, 2016), or edge-brightened
FR-II sources (e.g. Cygnus A, Carilli and Barthel, 1996) that are highly
collimated and expected to be relativistic along their entire length. Although
the differences are expected to result from the dynamics of the jets and
depend on both their power and how they interact with their environment, some
debate remains over the exact role of the central engine (see the review of
Hardcastle and Croston, 2020, for a fuller discussion), with a number of
works, both theoretical and observational, considering the relationships
between AGN properties, such as radio loudness, morphology and/or excitation,
and the properties of the accretion flow and/or BH (Wilson and Colbert, 1995;
Sikora et al., 2007; Tchekhovskoy et al., 2010, 2012; Best and Heckman, 2012;
Gendre et al., 2013; Mingo et al., 2014; Ineson et al., 2015; Mingo et al.,
2022). Several parameter studies that vary the jet (e.g., power, opening
angle, velocity) and environmental properties have been performed that can
recover both FR-I and FR-II sources (e.g., Krause et al., 2012; Tchekhovskoy
and Bromberg, 2016; Massaglia et al., 2016, 2019, 2022; Ehlert et al., 2018;
Li et al., 2018; Mandal et al., 2022; Yates-Jones et al., 2023), with
Massaglia et al. (2016) suggesting that simulating FR-I sources is challenging
and requires high-resolution 3D simulations to fully capture their turbulent
properties.

As discussed in Section 1.2, several methods have been developed to launch
jets and capture the lobe inflation process within the framework of galaxy
cluster evolution. It was found early on that to create a cocoon structure the
jet needs to be lower density than the surrounding medium (see discussion in
e.g., Ferrari, 1998; Krause, 2003; Weinberger et al., 2017, and references
therein). As outlined in Section 1.2, many studies now make use of fast
collimated outflows that efficiently thermalise through shocks to inflate
lobes of hot gas, however, a range of morphologies are still seen in
simulations depending on both numerical and physical choices for how the jet
is injected, as recently illustrated by Huško and Lacey (2023a) and shown in
Figure 2. The choice of using a cylindrical ($0^{\circ}$ opening angle) versus
conical injection, as well as the jet opening angle itself, can impact jet
properties (e.g., Falle, 1991; Krause et al., 2012; Huško and Lacey, 2023a;
Yates-Jones et al., 2023) and is a possible determinant of whether the jet has
FR-I or FR-II morphology (Krause et al., 2012; Yates-Jones et al., 2023). Jets
with a high momentum density, which occurs in slow/heavy jets333One can
achieve the same jet power by either having fast, light jets or slow, heavy
jets, with the latter having larger momenta than the former. are typically
elongated along the jet direction with narrow lobes, whereas light/fast jets
are found to be shorter and inflate wide lobes that have a notable expansion
perpendicular to the jet direction (e.g., Krause, 2003, 2005; Gaibler et al.,
2009; Guo, 2015, 2016; Weinberger et al., 2017; Su et al., 2021; Huško and
Lacey, 2023a; Weinberger et al., 2023; Yates-Jones et al., 2023). Although
additionally, Huško and Lacey (2023a) found that these trends can break down
for very slow outflows that are unable to inflate hot lobes nor generate
significant backflows (see third row of Figure 2). In terms of regulating
cluster cooling and heating, the recent study of Weinberger et al. (2023)
highlighted that the behaviour of very light jets inflating wider lobes makes
them more effective than heavy jets at removing low entropy material from
cluster cores and inhibiting cooling flows (see also, Su et al., 2021).

Some works have also considered if and how injecting magnetic fields with the
jets can influence lobe morphology and dynamics (e.g., Gaibler et al., 2009;
Hardcastle and Krause, 2014; Weinberger et al., 2017; Ehlert et al., 2018;
Massaglia et al., 2019; English et al., 2016). Gaibler et al. (2009) found
that injected fields, amplified in the jet head, can stabilise against Kelvin-
Helmholtz instabilities and generate a cleaner lobe structure. Works that
consider magnetic fields with moderate to high values of plasma $\beta$
(defined as the ratio between thermal and magnetic pressure, $\beta\equiv
P_{\rm th}/P_{\rm B}$) find, as expected, that they have limited impact on the
overall lobe dynamics (Hardcastle and Krause, 2014; Weinberger et al., 2017;
Ehlert et al., 2018; English et al., 2016) although they can affect the long-
term stability of lobes by suppressing instabilities that would otherwise mix
the jet and ICM material (e.g., Weinberger et al., 2017; Ehlert et al., 2018).
On the other hand, Massaglia et al. (2019) show that strong fields ($\beta=3$)
can impact morphology, leading to the jet becoming distorted due to non-
axisymmetric modes. In terms of observational constraints, lobe magnetic
fields have been measured using inverse-Compton observations of FR-II sources
(e.g., Kataoka and Stawarz, 2005; Croston et al., 2005; Ineson et al., 2017;
Harwood et al., 2016), with Ineson et al. (2017) finding observed field
strengths to be below equipartition in all of their sources with a median
ratio of $0.4$ and hence very low values of $\beta$ would seem unlikely,
although Croston et al. (2005) do find some magnetically dominated sources.
Another important factor that many simulations of jet feedback on galaxy
cluster scales neglect is the effect of relativity, both in terms of the jet
dynamics and in correctly accounting for high-temperature gas. For jet speeds
approaching the speed of light, the effects of relativity are expected to play
an important role in shaping lobe properties and increase the efficiency and
volume over which jets can heat the ICM (Perucho et al., 2014, 2017, 2022).
Additionally, the behaviour that fast-light jets inflate wider lobes that
interact more isotropically with the ICM is found to extend into the
relativistic regime (English et al., 2016; Yates-Jones et al., 2023), while
simulations that consider CR-dominated jets also produce “fat” lobes (see
Section 3.1).

The mechanisms that launch a jet and determine its direction occur on scales
of the accretion disc or smaller444A binary companion can also result in jet
precession (Begelman et al., 1980; Krause et al., 2019), although the
mechanisms governing this are also not resolved in large-scale cluster
simulations. (e.g., Tchekhovskoy et al., 2010, 2011; Sądowski and Narayan,
2015; Liska et al., 2018, 2019; Lu et al., 2023), which is far below the
resolution limit of galaxy cluster simulations. As such, ab-initio modelling
of the jet direction evolution is not possible. Traditionally, simulations of
jets in galaxy clusters assume a fixed jet direction potentially with some
small angle precession (e.g. Falceta-Gonçalves et al., 2010; Gaspari et al.,
2012; Li and Bryan, 2014; Yang and Reynolds, 2016; Bourne and Sijacki, 2017;
Martizzi et al., 2019), although some works alternatively randomly re-
orientate the jet direction by hand (e.g., Cielo et al., 2018), to improve
coupling between the jet and ICM. Several subgrid models have been developed
recently, which while differing in the exact assumptions made regarding the
subgrid accretion disc properties, can track BH spin evolution (e.g., Dubois
et al., 2014; Fiacconi et al., 2018; Bustamante and Springel, 2019; Beckmann
et al., 2022; Huško et al., 2022). Dubois et al. (2014) additionally used the
SMBH spin to determine the radiative efficiencies and jet direction, while
Beckmann et al. (2019) further developed this model to use spin-dependent jet
efficiencies derived from GRMHD simulations (McKinney et al., 2012). In a
similar vein, Talbot et al. (2021) coupled a jet feedback model (Bourne and
Sijacki, 2017) to the accretion disc model of Fiacconi et al. (2018), assuming
spin-dependent Blandford-Znajek jet efficiencies also derived from GRMHD
simulations (Tchekhovskoy et al., 2010) as well as the back reaction of the
jet on the SMBH evolution. While the models of Dubois et al. (2014) and
Fiacconi et al. (2018) both assume a thin $\alpha$-disc, Huško et al. (2022)
assume a thick disc, which typically leads to slower spin alignment
timescales. While such methods are not necessary for modelling single feedback
events, simulations that capture self-regulated feedback on Gyr timescales and
include spin evolution find that the jets can re-orient and inject energy more
isotropically within the cluster (Beckmann et al., 2019; Huško et al., 2022).

The energetics governing lobe inflation has been an important area of study in
particular due to its consequence on how the feedback heats the ICM.
Observational measurements of the lobe pressure and volume are often used to
constrain lobe and jet energetics which show a correlation between the
estimated lobe power and ICM cooling rate (Fabian, 2012; Hlavacek-Larrondo et
al., 2012, 2022). These lobe powers have also been compared to estimated Bondi
(1952) accretion rates to constrain the efficiency, i.e. $\epsilon$ in
Equation 1, of the central engine (Allen et al., 2006; Merloni and Heinz,
2007; McNamara et al., 2011; Russell et al., 2013; Nemmen and Tchekhovskoy,
2015), with some studies requiring $\epsilon>1$. Setting aside the
uncertainties in estimating lobe powers and Bondi accretion rates from
observations (e.g. Russell et al., 2013), high efficiencies could indicate
that modes other than Bondi are needed to feed the BH, such as cold accretion
(see e.g., McNamara and Nulsen, 2012, for further discussion, including
limitations of the Bondi model). Alternatively, these results could be
evidence that jets can extract spin energy from the BH via the Blandford-
Znajek mechanism (Blandford and Znajek, 1977), which allows efficiencies over
$100\%$ (Tchekhovskoy et al., 2011, 2012; McKinney et al., 2012). In any case,
considering the lobe inflation process in more detail, as highlighted by
McNamara and Nulsen (2007), under the assumption that the jet kinetic energy
is efficiently thermalised and the lobe inflation proceeds in pressure
equilibrium with the ICM, the total energy needed to inflate the lobe is given
by the lobe enthalpy,

$H=E_{\rm lobe}+PV=\frac{\gamma}{\gamma-1}PV,$ (2)

where $\gamma$ is the adiabatic index of the lobe gas, $E_{\rm lobe}$ is the
internal energy of the lobe and $PV$ is the product of the lobe pressure and
volume. The resulting ratio of injected jet energy to lobe energy to work done
is $E_{\rm jet}$:$E_{\rm lobe}$:$PV=1$:$1/\gamma$:$(\gamma-1)/\gamma$, which
for a non-relativistic gas ($\gamma=5/3$, which is commonly assumed in
simulations) suggests the lobes should retain $60\%$ of the injected energy
with the remaining $40\%$ going into the ICM via work done. However, whether
such behaviour is seen in reality depends on what dominates the lobes (e.g.,
relativistic or non-relativistic particles, magnetic fields etc) and hence
determines the effective value of $\gamma$, and on the suitability of assuming
that lobes are inflated in pressure equilibrium with their environment, i.e.
how explosive the feedback is. With respect to the latter point, Tang and
Churazov (2017) performed idealised simulations of spherically symmetric AGN
feedback events with varied energies and duration, finding that shorter, more
explosive, injection events result in larger fractions of energy (up to $\sim
88\%$) going into shocks, while for longer, gentler, injection events the
fraction of injected energy ending up in shocks is negligible. As such, the
lobe enthalpy varies from $\sim 0$ for instantaneous injection up to roughly
the injected energy for infinitely long injection episodes. Hydrodynamic
simulations of jets in a cluster environment performed by Hardcastle and
Krause (2013) found that while the lobe energy is dominated by the thermal
component, there are non-negligible kinetic and potential energy
contributions. They additionally find that the ratio of lobe energy to energy
gain by the ICM achieves a roughly constant value of $\sim$ one after an
initial lobe inflation phase where the ratio can be larger, highlighting the
non-negligible kinetic energy content of the lobe and the fact that they can
be over-pressured compared to the ICM. Follow-up work of English et al. (2016)
presented relativistic jet simulations both with and without MHD for a range
of jet velocities and powers. All jets studied inflate lobes that experience
an initial phase of being over-pressured (up to a factor $\sim 5$) before lobe
pressures rapidly decline, and while low-power jets tend to come into rough
pressure equilibrium with the ICM, lobe pressure ratios for their highest
power jets never drop below $\sim 2$ and begin to increase again as the lobes
move to larger radii. They additionally find that the shock-to-lobe energy
ratios attain a roughly constant value of $\sim 1.5$, consistent with the
results of earlier MHD simulations by Hardcastle and Krause (2014) and
slightly higher than values found in pure hydrodynamic simulations of
Hardcastle and Krause (2013). Other jet simulations have additionally found
ratios of $\gtrsim 1$, with varying levels of lobe kinetic energy, for both
purely hydrodynamical simulations (Bourne and Sijacki, 2017; Guo et al., 2018;
Bourne et al., 2019; Bourne and Sijacki, 2021; Huško and Lacey, 2023a) and
those including magnetic fields (Hardcastle and Krause, 2014; Weinberger et
al., 2017; Ehlert et al., 2018). Bourne and Sijacki (2021) find that due to
the lobes initially being over-pressured, the instantaneous $PV$ and lobe
enthalpy (calculated from equation 2) underestimate the integrated $PdV$ work
done during lobe inflation and total jet energy, respectively. Perucho et al.
(2017) additionally highlight the importance of modelling relativistic jets,
which can result in larger lobe pressures by at least a factor of two.

####  2.1.2 The Lobe-ICM Interaction

Figure 3: An illustration of jet-driven shocks within the central $100$ kpc of
a simulated cluster. The simulation includes self-regulated AGN feedback
assuming jet power coupled to SMBH accretion of cold gas with an efficiency
parameter of $\epsilon=1\%$ (as defined in equation 1). The jets are assumed
to have a small angle precession ($\theta=0.15$) with a 10 Myr period around
the $z$-axis in the simulation. The left panel shows the energy dissipation
rate while the right-hand panel shows shock Mach numbers, illustrating that
stronger shocks and hence higher dissipation rates are seen closer to the jet
but that overall the shocks are typically quite weak. (Figure 1 from Li et al.
(2017), ©AAS, reproduced with permission)

While it is clear that shocks play a critical role in lobe inflation, they can
also make an important contribution to heating the ICM. While the jet itself
drives a bow shock into the ICM, the initial rapid expansion of the jet lobes
can also drive lateral shocks. As the lobe expansion slows the shocks become
weaker and potentially broaden into sound waves, providing a mechanism through
which energy can be communicated isotropically to the ICM (e.g. Reynolds et
al., 2001; Brüggen et al., 2007; Prasad et al., 2015; Yang and Reynolds, 2016;
Bourne and Sijacki, 2017; Li et al., 2017; Weinberger et al., 2017; Guo et
al., 2018; Martizzi et al., 2019; Huško et al., 2022). It is also worth noting
that, similar to typical observations of jet feedback in galaxy clusters
(Fabian et al., 2006; Forman et al., 2007; Croston et al., 2011; Sanders et
al., 2016), many modern simulations find that shocks are often not very strong
(e.g. Brüggen et al., 2007; Sternberg and Soker, 2009; Li and Bryan, 2014; Li
et al., 2017; Bourne and Sijacki, 2021; Ehlert et al., 2018; Huško and Lacey,
2023a). Li et al. (2017) explicitly tracked the numerical dissipation within
shock cells and found that heating due to shocks exceeded that due to
turbulence by $\sim$ an order of magnitude and, as shown in Figure 3 taken
from Li et al. (2017), that the vast majority of shocks were weak. Other works
such as Yang and Reynolds (2016) and Martizzi et al. (2019) estimate the
dissipation rates from the expected entropy jump for weak shocks

$ds=\simeq\frac{2\gamma k_{\rm B}}{3(\gamma+1)^{2}\mu m_{\rm
H}}(\mathcal{M}^{2}-1)^{3},$ (3)

finding that strong shocks can play an important role within the jet cone
while weak shocks can dominate heating outside the cones, however, Yang and
Reynolds (2016) find that weak shocks alone are unable to completely offset
cooling. Bourne and Sijacki (2021), who perform simulations in a
cosmologically evolved cluster, make use of more conservative criteria to
define shocks to avoid misclassification due to galaxy formation processes
(see Schaal and Springel, 2015; Schaal et al., 2016, for details). They also
find that weak shocks are an important component of the energy budget, albeit
with somewhat lower levels of direct dissipation and instead highlight the
importance of compressive heating at shock fronts.

As well as shocks, simulations also exhibit sound/compression waves driven by
lobe expansion. These sound waves “detach” from the lobes once the expansion
speed drops below the ICM sound speed and have the potential to travel
isotropically to large distances (Ruszkowski et al., 2004a, b; Sternberg and
Soker, 2009; Dubois et al., 2010; Bourne and Sijacki, 2017, 2021; Guo et al.,
2018; Huško et al., 2022). Bambic and Reynolds (2019) recently presented a
detailed study of sound wave production by momentum-driven jets in an ICM
atmosphere (see also Section 3.2.3 and Figure 7), finding that constructive
interference can lead to $\gtrsim 25\%$ of the jet energy being converted to
sound waves, exceeding the level expected from spherical blast waves ($\sim
10\%$) (Tang and Churazov, 2017). The rate of heating and volume over which
sound waves are energetically important, though, depend on the microphysics of
the ICM (see Section 3.2.3) that dictates the cluster viscosity (Ruszkowski et
al., 2004a, b; Sijacki and Springel, 2006; Zweibel et al., 2018).

Several early simulations assumed that lobes can be simply modelled as low-
density bubbles of hot gas that rise buoyantly through the cluster atmosphere
(Churazov et al., 2001; Brüggen and Kaiser, 2001; Quilis et al., 2001; Brüggen
et al., 2002; Dalla Vecchia et al., 2004). However, in omitting the inflation
phase these simulations miss the potential impact that the jet momentum has on
bow shock production and the dynamics of the lobes, which may, at least
initially, propagate on timescales shorter than the buoyant rise time (Omma et
al., 2004; Bourne and Sijacki, 2017; Ehlert et al., 2018; English et al.,
2019). Observations of clusters can exhibit multiple old cavities that go out
to large radii (Forman et al., 2007; Diehl et al., 2008; Fabian et al., 2011;
Randall et al., 2011, 2015). Early simulations of bubbles were found to be
disrupted by Rayleigh-Taylor and Kelvin-Helmholtz instabilities on relatively
short timescales unless additional physics such as physical viscosity
(Reynolds et al., 2005; Sijacki and Springel, 2006) or magnetic fields (Jones
and De Young, 2005; Dursi, 2007; Ruszkowski et al., 2007; Dursi and Pfrommer,
2008; O’Neill et al., 2009) were included. However, Sternberg and Soker (2008)
showed that the formation of a dense layer of gas around jet-inflated lobes
can inhibit the development of such instabilities, thus allowing bubbles to
travel further and live longer. This being said, simulations of wide jets
(large opening angle) have found that vortices can develop that effectively
mix lobe material with the ICM (Gilkis and Soker, 2012; Hillel and Soker,
2016, 2017), with Hillel and Soker (2016) finding that it is more effective
than both turbulence and shock heating (see also, Yang and Reynolds, 2016; Li
et al., 2017). On the other hand, simulations that make use of narrow jets
($0^{\circ}$ opening angle) find that mixing of jet material into the ICM via
Kelvin-Helmholtz instabilities is sub-dominant (Bourne and Sijacki, 2017;
Weinberger et al., 2017; Martizzi et al., 2019) and that the level of mixing
reduces with increasing jet power (Ehlert et al., 2018; Bourne and Sijacki,
2021). Bourne and Sijacki (2017) highlight the impact of resolution on mixing,
finding that lower resolution results in increased levels of mixing, citing
the existence of larger Kelvin-Helmholtz vortices in this regime, and
additionally highlight that appropriate refinement of the jet lobes is
required to avoid excessive numerical mixing (see also, Weinberger et al.,
2017). Martizzi et al. (2019) highlight that the level of mixing can depend
upon the hydro-solver implemented, which can impact how well mixing
instabilities are resolved (see also, Ogiya et al., 2018). Further, the
inclusion of magnetic fields has the potential to inhibit mixing (e.g., Jones
and De Young, 2005; Dursi, 2007; Ruszkowski et al., 2007; Dursi and Pfrommer,
2008; O’Neill et al., 2009; Bambic et al., 2018; Weinberger et al., 2017;
Ehlert et al., 2018). In addition, the ability to effectively mix lobe and ICM
material depends on the constitution of the lobe material (Yang et al., 2019)
and the plasma properties of the ICM (see Section 3.2.2).

Observations of galaxy clusters can show multiple cavities at a range of radii
representing different generations of jet lobes that have risen buoyantly
through the ICM (Dunn et al., 2006; Forman et al., 2007; Diehl et al., 2008;
Fabian et al., 2011; Randall et al., 2011). Such rising bubbles can displace
significant amounts of ICM material and increase its gravitational potential
energy (Hardcastle and Krause, 2013, 2014; Weinberger et al., 2017; Yang et
al., 2019; Bourne and Sijacki, 2021; Huško and Lacey, 2023b). It was argued
early on that displaced material would fall back behind a rising cavity,
driving turbulence and ultimately converting its potential energy into heat, a
process known as cavity heating (Churazov et al., 2001, 2002; Bîrzan et al.,
2004). Yang and Reynolds (2016) point out that the exact heating rate is
dependent on the ICM viscosity and find that in any case kinetic energy within
the wake, which would be the source of cavity heating, accounts for only a
small fraction of the injected energy. Instead, Yang and Reynolds (2016) find
that clusters can exhibit large circulation patterns whereby gas is lifted out
of the cluster core and effectively heated by shocks and mixing within the jet
cone, before flowing back to the cluster core. The role of jet-driven gas
circulation in galaxy clusters has been highlighted by several works (e.g.,
Hillel and Soker, 2016; Soker et al., 2016; Guo et al., 2018; Chen et al.,
2019) with Chen et al. (2019) suggesting that if low entropy material can
effectively mix with higher entropy material at large radii before returning
to the cluster core, this process can act somewhat akin to a heat pump. Other
simulations have found that the rising bubbles can lift low entropy material
to larger radii, which can result in the formation of cold filaments and
clumps, either through direct uplift or by stimulating condensation (e.g.,
Revaz et al., 2008; Gaspari et al., 2012; Li and Bryan, 2014; Brighenti et
al., 2015; Guo et al., 2018; Beckmann et al., 2019; Wang et al., 2019, 2020;
Huško and Lacey, 2023b; Huško et al., 2022; Zhang et al., 2022). Such material
can display properties similar to cold gas observed in galaxy clusters (e.g.
Salomé et al., 2006; McDonald et al., 2012; Hamer et al., 2014, 2016; Fabian
et al., 2016; Russell et al., 2017). Recent MHD simulations show that
efficient coupling, facilitated by magnetic fields, can promote angular
momentum transfer between hot gas stirred by jets and cold material, which
along with the effects of magnetic braking can have a critical impact on cold
gas angular momentum (Wang et al., 2020, 2021; Ehlert et al., 2022). Ehlert et
al. (2022) show that this in turn has an impact on cold gas morphology as it
promotes the formation of transient cold discs and radial filaments as opposed
to the long-lived discs seen when magnetic fields are not included. This cold
gas can play an important role in the jet life-cycle as a source of future
fuel, with many simulations that implement cold accretion models to power
momentum-driven jets (e.g., Gaspari et al., 2012, 2013; Li and Bryan, 2014; Li
et al., 2017; Yang and Reynolds, 2016; Prasad et al., 2015; Meece et al.,
2017) finding good agreement between simulated and observed CC cluster
properties.

Assuming the lobe lifetimes are sufficiently long, they can excite g-modes
within the cluster atmosphere (Churazov et al., 2002; Omma et al., 2004; Zhang
et al., 2018). It has been argued that such modes can decay into volume-
filling turbulence, which then dissipates to heat the ICM (Zhuravleva et al.,
2014, 2016). However, idealised hydrodynamic simulations performed by Reynolds
et al. (2015) found that only a small fraction ($\lesssim 10\%$) of the
injected energy ends up as turbulence (similar results were also found when
including magnetic draping, Bambic et al., 2018). Other simulations that
specifically include jets have found that the level of turbulence produced is
subdominant compared to the total injected jet energy and often located in the
jet vicinity itself (e.g, Yang and Reynolds, 2016; Bourne and Sijacki, 2017;
Hillel and Soker, 2017; Weinberger et al., 2017; Li et al., 2017; Prasad et
al., 2018). On the other hand, Martizzi et al. (2019) found that jet-driven
turbulence can dominate the heating budget in the central regions of the
cluster, although this is only when the jet is active, and its contribution
becomes negligible at large radii. The inclusion of additional physical
processes may also affect the turbulent contribution to the heating budget.
Simulations of Beckmann et al. (2019), that include spin-driven jets find that
continuous evolution of the jet direction can drive turbulence over a larger
volume, while recent results from Wang et al. (2021) found that the inclusion
of magnetic fields can promote jet-driven ICM turbulence due to tighter
coupling between jets and the ICM. Some simulations that include “cluster
weather” (Bourne and Sijacki, 2017; Lau et al., 2017; Ehlert et al., 2021),
find that large-scale turbulence and bulk motions, such as that driven by
orbiting substructures or cosmic accretion, are required in combination with
AGN feedback to match both the line of sight velocities and velocity
dispersions measured from Hitomi observations of the Perseus cluster (Hitomi
Collaboration et al., 2016, 2018). Specifically, while simulated AGN feedback
could produce the required levels of velocity dispersion, it was unable to
produce other features such as the large velocity shear. Although turbulence
may be energetically unimportant, it can play other roles. While the motions
of rising lobes can move metals and create elongated distributions (e.g.,
Roediger et al., 2007; Gaspari et al., 2011, 2011; Duan and Guo, 2018), jet
induced turbulence has been found to additionally effectively redistribute the
metals (Gaspari et al., 2011, 2011). Additionally, Gaspari et al. (2013)
showed that subsonic turbulence can promote thermal instability and provide a
supply of low angular momentum cold gas to feed the central black hole through
“chaotic cold accretion” (see also, McCourt et al., 2012; Gaspari et al.,
2015, 2017; Wang et al., 2019, 2020).

###  2.2 The Role of Environment and Cluster Weather

Having focused on the general macroscopic evolution of jet lobes in a galaxy
cluster, in this section we consider the role that the environment can play in
shaping lobe properties and how they interact with the ICM. At the simplest
level, the ICM can be modelled as a smooth, hydrostatic atmosphere with some
radially dependent density and temperature profiles, with both the absolute
density (e.g., Yates et al., 2018; Perucho et al., 2022) and profile slope
(e.g., Hardcastle and Krause, 2013, 2014; English et al., 2019; Massaglia et
al., 2022; Mandal et al., 2022) being able to impact jet morphology,
propagation and radio emission properties. Yates-Jones et al. (2021) further
showed that offsetting the jet launch location from the cluster centre and
varying the jet direction can impact the jet properties, with the jets
encountering a higher density medium typically being shorter and having hot
spots with higher surface brightness. Building on this, the ICM is not a
simple single-phase gas but rather contains both a hot X-ray emitting plasma
as well as cold filaments and disc structures (e.g., Salomé et al., 2006;
McDonald et al., 2012; Hamer et al., 2014, 2016; Fabian et al., 2016; Russell
et al., 2017). Small-scale simulations ($\lesssim 1$ kpc) of jets show that
their interaction with the cold dense clouds or discs can impact jet
morphology and dynamics as well as how effectively the jet couples to the
surrounding medium (e.g., Wagner and Bicknell, 2011; Wagner et al., 2012;
Mukherjee et al., 2018, 2021; Talbot et al., 2021, 2022). While in cluster
scale simulations, the interaction of jets with cold gas can help thermalise
the jets (Prasad et al., 2015; Meece et al., 2017), inhibit their propagation
(Bourne et al., 2019) and in the case of light jets deflect them, making their
interaction with the ICM more isotropic (Ehlert et al., 2022).

Observations suggest that the ICM is turbulent (e.g., Schuecker et al., 2004;
Sanders et al., 2011; Zhuravleva et al., 2012; Pinto et al., 2015; Walker et
al., 2015; Hitomi Collaboration et al., 2016, 2018; Hofmann et al., 2016) and
while some simulations suggest that jets can drive local turbulence (Yang and
Reynolds, 2016; Martizzi et al., 2019; Gaspari et al., 2012, 2013; Prasad et
al., 2018; Yang et al., 2019; Wang et al., 2021; Ehlert et al., 2022), other
processes such as those arising from structure formation and sloshing are
expected to drive widespread turbulence and bulk motions within the ICM (e.g.,
Ruszkowski and Oh, 2011; Vazza et al., 2012, 2017; ZuHone et al., 2013; Bourne
and Sijacki, 2017; Bourne et al., 2019; Bourne and Sijacki, 2021; Lau et al.,
2017; Valdarnini, 2019; Bennett and Sijacki, 2022), while tangled magnetic
fields may also drive MHD turbulence (Ehlert et al., 2018, 2019, 2021). Such
“cluster weather” could play an important role in lobe dynamics, with a number
of studies attempting to mimic its effects by applying small perturbations to
spherically symmetric environments. For example Krause (2005) included small
density perturbations to a King atmosphere, finding that this results in
differences in back-flow locations and small asymmetries in jet lengths.
O’Neill and Jones (2010) performed MHD simulations of jets in a cluster
environment containing a tangled magnetic field and an ambient medium on which
fluctuations ($\pm 10\%$) in density have been superimposed to match observed
ICM pressure variations. However, while these fluctuations could perturb the
jets, they were insufficient to significantly alter the jet dynamics and
energy only primarily couples to the ICM within the jet cone. On the other
hand, some works instead induce ICM gas motions, for example Bourne and
Sijacki (2017) used hydrodynamic simulations to consider the long-term
evolution of jets and their lobes in a cluster stirred by orbiting
substructures, finding that the resulting turbulent and bulk motions can
deflect the lobes and increase mixing of lobe and ICM material when compared
to a hydrostatic atmosphere. Ehlert et al. (2018) study the dynamics of jets
in an ICM atmosphere including a turbulent magnetic field, finding that while
magnetic draping can suppress instabilities the presence of turbulence can
reduce the effects of this draping and in the buoyant rise phase disrupt the
jet lobes, while Ehlert et al. (2022) find that the turbulence can deflect
bubbles, assuming the jet is sufficiently light and provide a more isotropic
distribution of lobe positions, akin to observations of multiple generations
of lobes in some galaxy clusters (e.g., Forman et al., 2007; Diehl et al.,
2008; Chon et al., 2012; Fabian et al., 2011; Ubertosi et al., 2021). However,
as we discuss later in this section, only a handful of studies include high
resolution jets in cosmologically evolved cluster environments (Heinz et al.,
2006; Morsony et al., 2010; Mendygral et al., 2012; Bourne et al., 2019;
Bourne and Sijacki, 2021; Yates-Jones et al., 2023), and, to the best of our
knowledge, just one includes self consistently evolved magnetic fields
(Mendygral et al., 2012).

Observed radio jets can exhibit bent morphologies (e.g. Patnaik et al., 1986;
Dehghan et al., 2014; O’Brien et al., 2018). Ram pressure due to crosswinds,
arising from relative motion between jets and the ICM, is expected to lead to
such bending and result in the formation of narrow-angle-tailed (NAT) and
wide-angle-tailed (WAT) radio galaxies (Morsony et al., 2013; Gan et al.,
2017; Jones et al., 2017; O’Neill et al., 2019; O’Dea and Baum, 2023). Further
to this, cluster-merger-driven shocks can also impact jet morphology. Gan et
al. (2017) compared hydrodynamic and magnetically dominated jets when
subjected to crosswinds and shocks to mimic cluster weather, finding that
while the hydrodynamic jets were unable to survive, the magnetic jets can be
significantly bent in the presence of a cross-wind but not solely by a shock
front. O’Neill et al. (2019) simulated a shock front passing a NAT radio
galaxy, finding that the shock action compresses the jets and that turbulence
increases in the jet tails, which in turn amplifies the magnetic fields and
promotes mixing. Nolting et al. (2019a, b) extended this work to consider the
interaction of merger-like shock fronts impacting generic jet pairs. Shocks
moving parallel to the jet axis can significantly affect the jet dynamics,
with the post-shock flow potentially impeding or reversing the jet and
destroying the cocoon (Nolting et al., 2019a). Similarly, when the jet axis is
in the plane of the shock front, the jet lobes are disrupted and in the case
of active jets, the action of the shock bends them to form a NAT radio galaxy
(Nolting et al., 2019b).

To self-consistently capture large-scale turbulence driven by structure
formation, full cosmological simulations are necessary (e.g., Genel et al.,
2014; Schaye et al., 2015; Dubois et al., 2016; McCarthy et al., 2017; Henden
et al., 2018; Springel et al., 2018). Models have been developed to capture
the low accretion state associated with jet feedback, for example, by adding
bubbles of hot gas to mimic radio lobes (e.g., Sijacki and Springel, 2006;
Sijacki et al., 2007, 2015), or driving bi-polar kinetic outflows (e.g.,
Dubois et al., 2010, 2012; Davé et al., 2019), that can regulate cooling and
star formation within galaxy clusters. However, as discussed in Section 1.3,
to capture representative volumes of the Universe over cosmic timescales,
these models by necessity have to make simplifying assumptions (e.g. low
velocities, high mass loading and/or hydrodynamic decoupling) and sacrifice
resolution in the injection region and/or jet lobes. While the focus of such
simulations is on capturing the global effects of the feedback on resolvable
scales as opposed to studying lobe inflation and subsequent interaction with
the ICM in intricate detail, they are still able to reproduce observational
features, such as X-ray rims and cavities (Sijacki and Springel, 2006; Sijacki
et al., 2007; Dubois et al., 2010), and the effects of cluster weather
displacing lobes and promoting mixing (Sijacki et al., 2008) that are seen in
the high resolutions simulations we discuss below. Additionally, cosmological
simulations of galaxy clusters have been used, for example by Vazza et al.
(2021), to study how cluster dynamics can impact the evolution of relativistic
electron populations and magnetic fields produced by radio galaxies (see also,
Vazza et al., 2023a, b), finding that cluster motions interact with radio
lobes, affecting their dynamics, constituent magnetic fields, and radio
properties.

Brüggen et al. (2005) presented some of the earliest simulations to consider
radio mode feedback in a cosmologically evolved cluster by injecting thermal
energy off-centre from the cluster core to mimic the inflation of jet lobes in
a dynamic environment. However, Heinz et al. (2006) presented the first
example of high-resolution bipolar jets in a dynamic cluster environment,
using non-radiative, hydrodynamic simulations that included $10^{46}$ erg s-1
jets in the centre of a cosmologically evolved, $7\times 10^{14}$ M⊙
cluster555While the simulations performed by Heinz et al. (2006) were non-
radiative, the simulations used to produce the initial conditions did include
radiative cooling and star formation (Springel et al., 2001).. ICM motions
were able to effectively move the lobes and redistribute material, helping to
make the energy injection into the ICM more isotropic and replenish higher-
density material at small radii for later jets to couple to. Mock radio and
X-ray images exhibited lobe and jet morphologies and features as seen in real
observations. A further parameter study using the same cluster but varying jet
power and lifetime was performed by Morsony et al. (2010). As in Heinz et al.
(2006), the ICM motions effectively displace, disrupt and mix the jet lobes,
with more powerful jets being able to affect the ICM at larger radii.
Interestingly, Morsony et al. (2010) found that single AGN outbursts can
potentially be realised as multiple independent lobe structures that could
otherwise be interpreted as multiple jet outbursts. MHD simulations performed
by Mendygral et al. (2012) studied $6\times 10^{44}$ erg s-1 jets in a non-
radiative cosmologically evolved cluster that hadn’t undergone any recent
mergers. They found that the magnetic field lines are swept up by and drape
over the inflated lobes and that despite the dynamically relaxed nature of the
system, ICM motions can still impact the lobe morphologies and dynamics, with
mock radio images exhibiting morphologies akin to WAT radio galaxies with bent
jets.

Figure 4: Volume rendering of jet lobes (green) and cold material (red) within
a cosmologically evolved galaxy cluster. The top, middle and bottom rows show
low, medium (labelled “Fid”) and high-power jets, respectively. The small
panels show the evolution of the jet lobes for two different viewing angles
(rotated by $90^{\circ}$ about the $z-$axis with respect to each other), while
the large panels additionally overlay the gas velocity field. Overall, jet
lobes can be displaced, disrupted and mixed by cluster weather and cold
structures, with lower-power jets being more susceptible. (Figure 2 from
Bourne and Sijacki (2021), CC BY 4.0)

Later work by Bourne et al. (2019) presented radiative hydrodynamic moving-
mesh simulations of a $4\times 10^{44}$ erg s-1 jet pair in a low-redshift
cosmologically evolved cluster that exhibited a massive cold central disc
structure. Mock X-ray images show classic cavities and bright rim structures,
with both the cold gas and cluster weather affecting the lobe structure and
location, with the latter acting to displace lobes from their initial
trajectory. Shocks are initially driven into the ICM during lobe inflation and
similar to results in hydrostatic atmospheres (e.g., Hardcastle and Krause,
2013; English et al., 2016; Bourne and Sijacki, 2017; Weinberger et al.,
2017), $\sim 40\%$ of the jet energy goes into $PdV$ work. However, at later
times the cluster weather can effectively mix jet and ICM material, with the
aid of an in-falling substructure that directly impacts one of the lobes.
Bourne and Sijacki (2021) extended this work to study different jet powers and
perform a more detailed analysis of ICM heating. Figure 4, taken from their
paper illustrates the evolution of different power jets and the lobes that
they inflate when subjected to cluster weather. Overall, they found that weak
shocks and weather-aided mixing dominate the heating budget, and that while
high-power jets drove stronger shocks that were more efficient at thermalising
energy, lower-power jets and the lobes they inflate simply move material
around and were more susceptible to weather-aided mixing (see also, Ehlert et
al., 2018). As well as recovering cavities and rims in mock X-ray
observations, mock radio images show morphologies ranging from FR-I at low
power to FR-II at high power (see also, Massaglia et al., 2016; Ehlert et al.,
2018).

Both FR-I and FR-II jet morphologies were also realised in the simulations of
Yates-Jones et al. (2023), who presented the initial results of the
CosmoDRAGoN project, the first simulations to study relativistic jets in
cosmologically evolved clusters. The project performed a parameter study by
simulating fixed power jets in a cosmologically evolved cluster (taken from
the 300 Project (Cui et al., 2018)) varying both velocity (from non-
relativistic to relativistic) and jet opening angle. They produce mock radio
surface brightness maps which illustrate both core-brightened FR-I and edge-
brightened FR-II sources. As in previous works performed in idealised cluster
environments (Krause et al., 2012), they recover the result that FR-I sources
can arise from wider jets. They further find that slower jets transition from
being FR-II to FR-I sources sooner and additionally highlight that even their
most powerful jets eventually transitions to FR-I provided it has a wide
enough opening angle. Although to a lesser extent than found in some of the
simulations discussed above, Yates-Jones et al. (2023) also find that cluster
weather can affect lobe morphology, particularly low power (FR-I) sources once
the jet has been switched off and the lobe enters the buoyant rise phase (see
also, Bourne and Sijacki, 2021).

Vernaleo and Reynolds (2006) raised the problem that if jet direction remains
unchanged, the low-density path cleared by an early jet episode remains, and
later feedback events can escape largely uninhibited, resulting in inefficient
coupling between the jet and the ICM. As discussed in Section 2.1.1, various
approaches have been taken to improve coupling between the jets and ICM, and
to distribute energy more isotropically (e.g. jet precession/reorientation,
large opening angles, light jets). Processes driving both precession and jet
re-orientation are still somewhat uncertain and whether or not accretion discs
can do this on short timescales is debated (Nixon and King, 2013), with SMBH
binaries being an alternative method to drive precession (Begelman et al.,
1980; Krause et al., 2019). In any case, the simulations discussed in this
section, which include dynamic cluster environments, provide an alternative
natural relief to the problem, by removing the low-density channels,
redistributing the lobes, and promoting mixing and distributing the energy
isotropically. In displacing lobes, cluster weather may also provide a
mechanism to explain observations of multiple lobe generations having quite
different trajectories (Dunn et al., 2006; Forman et al., 2007; Diehl et al.,
2008; Fabian et al., 2011, 2022).

As already mentioned in Section 2.1, magnetic fields can play a potentially
important role in lobe dynamics and interactions. Observations of galaxy
clusters show they have an extended magnetic field with a strength of a few to
tens $\mu$G and plasma $\beta\sim 100$ (Carilli and Taylor, 2002; Feretti et
al., 2012). Simulations have shown that bubbles moving above the Alfvén speed
can “sweep up” field lines and form a draping layer, in which the magnetic
field is amplified. This can suppress instabilities along the direction of the
field lines, reduce mixing and slow the bubble due to magnetic tension (e.g.,
Jones and De Young, 2005; Dursi, 2007; Ruszkowski et al., 2007; Dursi and
Pfrommer, 2008; O’Neill et al., 2009; Bambic et al., 2018; Ehlert et al.,
2018). Provided the Alfvén speed in the draping layer is sufficiently large
compared to the buoyancy or shear velocities, Rayleigh-Taylor and Kelvin-
Helmholtz instabilities with wavelengths up to an order of magnitude larger
than the layer thickness can be suppressed (Dursi, 2007). While the formation
of the draping layer is found to require that the coherence scale of the
magnetic field is larger than the bubble (Ruszkowski et al., 2007; O’Neill et
al., 2009; Pfrommer and Jonathan Dursi, 2010), O’Neill et al. (2009) also
highlighted that the dynamical effect of the draping layer can be reduced by
tangled magnetic fields and that effective amplification requires that the
scale on which the fields are tangled is greater than the bubble. Further, the
existence of turbulence in the ambient medium can perturb the lobe surface,
inhibit magnetic draping and potentially reduce its effectiveness at
suppressing mixing (Ehlert et al., 2018). 3D simulations of jets in tangled
magnetic fields have also shown that field lines carried with the jet lobes
can align as they are stretched and amplified within the wake of rising
bubbles (Ruszkowski et al., 2007; O’Neill and Jones, 2010; Mendygral et al.,
2012; Ehlert et al., 2018). As mentioned in Section 2.1.2 already, ICM
magnetic fields can promote coupling between the jet and ambient medium
leading to the generation of ICM turbulence (Wang et al., 2021), while cold
gas angular momentum can also be affected due to magnetic braking and
increased coupling between the hot and cold gas phases (Wang et al., 2020,
2021; Ehlert et al., 2022), which directly impacts cold gas morphology and has
implications for SMBH feeding. On the other hand, Bambic et al. (2018) found
that the magnetic tension can inhibit g-modes decaying into volume-filling
turbulence. Finally, it is becoming ever more important to include realistic
magnetic fields in cluster simulations to explore microphysical processes such
as anisotropic thermal conduction (Section 3.2.1), anisotropic viscosity
(Section 3.2.2) and CR transport mechanisms (Section 3.1).

###  2.3 Open Questions and Future Opportunities

As is clear from the examples given in previous subsections, there is a
burgeoning collection of jet models implemented in a range of codes currently
used in the literature. While many of these models, which include self-
regulated feedback, are now able to successfully prevent over-cooling within
galaxy clusters, the mechanisms through which this is achieved are not
necessarily agreed upon. Some processes such as the generation of weak shocks
are common between simulations, albeit with different relative importance to
the heating budget. On the other hand, processes such as mixing and the
generation of turbulence appear to be dependent on the jet injection model,
resolution and/or refinement techniques and physics included (e.g. MHD,
viscosity, cluster weather, etc) and as such are not firmly settled. This
being said, several works are now finding that the process is expected to be a
gentle continuous one (Yang and Reynolds, 2016; Meece et al., 2017; Bourne and
Sijacki, 2021; Ehlert et al., 2022). Martizzi et al. (2019) showed that the
Riemann solver used can affect the jet and lobe evolution depending on how
diffusive the solver is and on how well small-scale physical processes, such
as mixing, are captured. Additionally, the recent study of Weinberger et al.
(2023) argue that differences between codes, and the differences between model
parameters within a code make bigger differences to results than having high
resolution or numerical convergence. As such, while we can accept that jet
feedback regulates the thermodynamics of the ICM, the exact mechanisms are not
agreed upon and further constraints need to be placed on jet feedback models
based on observations. This could be achieved by making more use of mock X-ray
and radio images and comparing them to observations of lobe morphologies,
distributions and lifetimes (e.g., Shin et al., 2016). Future X-ray missions
with a larger effective area and hence higher signal-to-noise ratio, such as
AXIS666https://axis.astro.umd.edu/(Mushotzky et al., 2019), could potentially
reveal faint cavities at larger radii and increase the sample size of cavities
for a more comprehensive statistical study.

The role of environment is still an open question, only a few works include
high-resolution jet feedback in clusters with self-consistently driven weather
(Heinz et al., 2006; Morsony et al., 2010; Mendygral et al., 2012; Bourne et
al., 2019; Bourne and Sijacki, 2021), even fewer include cluster weather and
magnetic fields (Mendygral et al., 2012). A parameter space study of different
cluster environments, with a range of dynamical states and over cosmologically
interesting timescales needs to be considered to fully understand the role
that cluster weather can play in displacing lobes, mixing lobe material and
the role that magnetic fields can play in these process, i.e. to what levels
of ICM turbulence are the fields still able to inhibit mixing? In relation to
the dynamical state of the system, recent works suggested cluster mergers play
an important role in driving the CC/NCC distribution (Burns et al., 2008;
Rasia et al., 2015; Hahn et al., 2017; Chadayammuri et al., 2021), although
other recent work found little difference in the dynamical state of CC versus
NCC clusters (Barnes et al., 2018). As highlighted by Bourne and Sijacki
(2021), if CC clusters are more dynamically quiet, one could expect that lobes
are more likely to survive, conversely if NCCs are dynamically active, it
could be that lobes are more likely to be disrupted, such behaviour could
additionally explain why lobes and bubbles are prevalent in CC clusters (Dunn
and Fabian, 2006; Fabian, 2012), although there is also an observational bias
towards being able to detect cavities in CC vs NCC clusters due to higher
X-ray photon counts in the former (e.g. Panagoulia et al., 2014; Hlavacek-
Larrondo et al., 2022). The upcoming XRISM777https://xrism.isas.jaxa.jp/en/
mission (XRISM Science Team, 2020), a replacement for the ill-fated
Hitomi888https://www.isas.jaxa.jp/en/missions/spacecraft/past/hitomi mission,
as well as more distant missions such as Athena999https://www.the-athena-x-
ray-observatory.eu/en(Croston et al., 2013), will provide unprecedented
measurements of the ICM velocity structure and turbulence. Such measurements,
combined with observations of jet lobes should allow us to begin to address
the interplay between cluster weather and AGN feedback. It is also possible
that such observations, in combination with future simulations that include
both ICM motions and models that track the evolution of SMBH/accretion disc
angular momentum and jet direction, could play a role in untangling the
apparent degeneracy between intrinsic jet direction and cluster weather in
determining lobe spatial distributions.

The focus of this review is on the role of AGN feedback in the form of jets,
often dubbed the “radio mode” (Fabian, 2012), which is expected to be the
dominant mechanism for regulating heating and cooling in the ICM. This being
said, accreting SMBHs can also release energy in the form of radiation and
wide-angled winds that have the potential to impact their environment. Several
models employed in cosmological simulations have been developed that invoke
separate modes of feedback — “quasar” vs “radio” — during periods of high and
low accretion (in terms of the Eddington rate), respectively, to differentiate
between radiatively efficient vs radiatively inefficient accretion (e.g.,
Sijacki et al., 2007; Dubois et al., 2012; Sijacki et al., 2015; Weinberger et
al., 2018; Davé et al., 2019). However, as highlighted by Qiu et al. (2019),
studies on the effects of radiative feedback in CCs are lacking. To begin to
remedy this Qiu et al. (2019) performed simulations invoking both kinetic
outflows and radiative feedback in an idealised Perseus-like cluster. Further,
while they invoke a transition accretion rate between radiatively efficient
and inefficient modes, they allow kinetic feedback to occur simultaneously
with radiative feedback at high accretion rates finding that to prevent over-
cooling of the ICM kinetic outflows must be active across all accretion rates.
Such works suggest there is ample opportunity to further study the role and
impact of different feedback channels in shaping CC clusters. In addition, it
has been argued by Hardcastle and Croston (2020), that the “traditional”
paradigm of clearly separated quasar and radio modes distinguished by whether
accretion is radiatively efficient or inefficient, and their role in galaxy
formation is not so clear cut in reality, citing, for example, populations of
high power radio jets occurring via radiatively efficient accretion at high
Eddington ratios (e.g., Hine and Longair, 1979; Laing et al., 1994; Mingo et
al., 2014; Hardcastle and Croston, 2020). Additionally, simulations of radio-
mode feedback typically do not explicitly implement different models for low-
power/FR-I jets and high-power/FR-II jets. However, their environmental
dependence (Comerford et al., 2020) and potential evolutionary sequence
(Macconi et al., 2020) could be used to constrain future models of accretion
and/or feedback. As such, not only is there scope to consider distinct
feedback regimes and how their efficiency couples to the accretion mode, but
as in Qiu et al. (2019) to further develop and explore models for determining
how feedback should be channelled, potentially simultaneously, through
different mechanisms (i.e. via radiation, winds and/or jets). Additionally,
self-consistent lobe inflation, its influence on SMBH accretion, as well as
the interaction of jets with the interstellar medium of the host galaxies
require high-resolution simulations on galaxy scales (e.g., Antonuccio-Delogu
and Silk, 2010; Wagner and Bicknell, 2011; Wagner et al., 2012; Mukherjee et
al., 2018, 2021; Talbot et al., 2021, 2022). These simulations could provide
valuable insights and inform future SMBH accretion and jet feedback models on
cluster and cosmological scales.

Finally, as highlighted in the introduction, clusters sit at the crossroads of
cosmology and astrophysics. In representing the high mass end of the halo mass
function and the culmination of hierarchical structure formation, galaxy
clusters have the potential to be used as probes of cosmology (Allen et al.,
2011; Kravtsov and Borgani, 2012), with a number of current/future/proposed
survey missions (e.g. DES101010https://www.darkenergysurvey.org/(The Dark
Energy Survey Collaboration, 2005),
SPT-3G111111https://www.anl.gov/hep/spt3g(Benson et al., 2014),
eRosita121212https://www.mpe.mpg.de/eROSITA(Merloni et al., 2012), Advanced
ACTPol131313https://act.princeton.edu/(Henderson et al., 2016),
Euclid141414https://sci.esa.int/web/euclid(Laureijs et al., 2011), Vera C.
Rubin Observatory’s LSST151515https://rubinobservatory.org/(LSST Dark Energy
Science Collaboration, 2012), Athena${}^{\ref{note:athena}}$,
SKAO161616https://www.skao.int/en(Braun et al., 2015), Simons
Observatory171717https://simonsobservatory.org/(Ade et al., 2019),
AXIS${}^{\ref{note:axis}}$, CMB-S4181818https://cmb-s4.org/(Abazajian et al.,
2016), CMB-HD191919https://cmb-hd.org/(Sehgal et al., 2019),
ngVLA202020https://ngvla.nrao.edu/(McKinnon et al., 2019)) aiming to leverage
this opportunity. In order to do this one typically needs to be able to
accurately measure galaxy cluster masses, however, as their mass is dominated
by dark matter, this cannot be done directly. One solution is to use cluster
observables (e.g., X-rays properties, Sunyeav-Zel’dovich (SZ; (Sunyaev and
Zeldovich, 1972)) flux, radio emission, weak lensing) to estimate cluster
masses, which requires a thorough understanding of the appropriate cluster
mass scaling relations. Given that AGN feedback plays such a pivotal in
shaping the ICM properties that give rise to these observables, it is
imperative to continue developing cosmological simulations of structure
formation that include robust modelling of jet feedback as well as additional
baryonic physics, such as magnetic fields, CRs, conduction and viscosity, in
order to fully understand cluster observable scaling relations, how they
evolve with redshift, and understand any biases in derived masses (see e.g.,
Barnes et al., 2017a; Henden et al., 2019, for recent examples). Furthermore,
in redistributing material within galaxy groups and clusters, AGN feedback can
directly impact the matter power spectrum, knowledge of which is necessary to
derive cosmological parameters from weak lensing observations (see discussions
in Chisari et al., 2019; van Daalen et al., 2020; Amon and Efstathiou, 2022,
and associated references). While existing cosmological simulations typically
agree that feedback causes suppression of the matter power spectrum, the
relevant levels and scales at which this occurs are model dependent (e.g., van
Daalen et al., 2011; Chisari et al., 2018, 2019; Springel et al., 2018; van
Daalen et al., 2020), with ample scope remaining to constrain the relative
importance of jet feedback in groups and clusters.

##  3 Modelling the Microphysics

Much of our understanding of AGN feedback in galaxy clusters has come from the
extensive literature based on ideal hydrodynamic/MHD simulations reviewed in
the previous section. However, some of the important physical processes, such
as thermal conduction, viscosity, and CRs, have been largely neglected, and
their effects on cluster feedback are relatively poorly understood. As
mentioned in § 1.4, these physical processes, including conduction and
viscosity coefficients as well as the propagation of CRs, are determined by
plasma physics that happen on microscopic scales compared to the sizes of
galaxies and clusters, i.e., the “microphysics”. In this section, we will
review the impact of these microphysical mechanisms on AGN feedback in
clusters. We will discuss the roles of CRs in Section 3.1 and the roles of ICM
plasma physics in Section 3.2.

###  3.1 Roles of Cosmic Rays

####  3.1.1 Motivations for Considering Cosmic Rays

Effects of CRs on the formation and evolution of galaxies and galaxy clusters
in the cosmological context have received growing recognition in recent years.
In particular, state-of-the-art cosmological simulations have shown that
baryonic feedback processes from stars and SMBHs play a vital role in
reproducing the observed luminosity functions of galaxies (e.g., Di Matteo et
al., 2005; Dubois and Teyssier, 2010; Stinson et al., 2013; Le Brun et al.,
2014; Schaye et al., 2015; Sijacki et al., 2015; Dubois et al., 2016;
Weinberger et al., 2018; Martin-Alvarez et al., 2022). For lower-mass
galaxies, starburst-driven galactic outflows are key sources of feedback for
suppressing SFRs; however, the physical mechanisms for launching galactic
outflows had been elusive. Only recently have simulations shown that outflows
with mass loading factors comparable to observed levels can be driven when the
effects of CRs are included (Booth et al., 2013; Salem and Bryan, 2014;
Ruszkowski et al., 2017). These studies further showed that the properties of
the galactic outflows, SFRs within the galaxies, as well as the circumgalactic
medium (CGM), all sensitively depend on the detailed CR transport processes on
microscopic scales (Butsky and Quinn, 2018; Ji et al., 2020; Buck et al.,
2020). These results point to the importance of CRs in the processes of
structure formation, and the necessity for understanding how CRs propagate and
interact with the magnetised medium.

While on the scales of galaxies, CRs are in rough energy equipartition with
thermal gas, magnetic fields, and turbulence, on the scales of galaxy clusters
CRs produced via structure formation shocks suffer strong collisional losses
in the ICM (Jubelgas et al., 2008). Therefore, in general, CRs are not
dynamically dominant in galaxy clusters. In particular, the amount of CRs
within galaxy clusters has been severely constrained by the non-detection of
gamma-ray radiation of clusters to be less than $10-20$ per cent (Ackermann et
al., 2010; Ahnen et al., 2016; Brunetti et al., 2017). Although CRs are not
energetically dominant for clusters as a whole, locally they could still play
an important role, e.g., within AGN jets for which the energy composition is
largely unknown.

The primary motivation for considering the effects of CRs in the context of
AGN feedback can be traced back to early observations of cluster radio
bubbles. Assuming that the AGN jet-inflated bubbles are in approximate
pressure equilibrium with the ambient ICM, it was found that for some of the
cluster radio bubbles, the total bubble pressure inferred from the ICM
($\equiv P_{\rm ext}$) is much greater than the internal pressure from
radiating CR electrons (CRe) estimated from the radio observations ($\equiv
P_{\rm int}$) (Morganti et al., 1988; Hardcastle et al., 1998; Dunn and
Fabian, 2004; Rafferty et al., 2006; Bîrzan et al., 2008). Generally speaking,
the AGN bubbles could be supported by magnetic fields, thermal gas, CR protons
(CRp), and CRe. Magnetic pressure typically is in rough equipartition with the
radiating CRe and hence likely subdominant. The low X-ray surface brightness
within the AGN bubbles puts stringent constraints on the amount of thermal gas
with a temperature of several keV (Nulsen et al., 2002). Therefore, these
observations suggest that the radio bubbles with high $P_{\rm ext}/P_{\rm
int}$ ratios are likely supported by either ultra-hot thermal gas or non-
radiating CRp. While feedback from thermally dominated bubbles inflated by
kinetic-energy-dominated jets is more extensively studied (Section 2), the
effects of CR-dominated bubbles are less well understood.

####  3.1.2 Impact of Cosmic Rays on AGN Feedback Processes

In terms of AGN feedback, jet-inflated bubbles filled with thermal gas versus
CRs are expected to exhibit distinct properties that could potentially affect
the long-term evolution of CC clusters. First, as CRs are relativistic charged
particles, they have a softer equation of state (EoS) than thermal gas, i.e.,
with an adiabatic index of $\gamma_{\rm cr}=4/3$ (in the ultra-relativistic
limit; see Enßlin et al. (2007) for a more general formalism) rather than
$\gamma_{\rm gas}=5/3$ as typically assumed for ideal, monatomic gas. This
means that the enthalpy (given by equation 2) of CR-dominated lobes would be
$4PV$ compared to $2.5PV$ for a non-relativistic fluid (McNamara and Nulsen,
2007), suggesting that CR dominated lobes would do less work on their
environment. The different adiabatic index would alter the compressibility of
the fluid when CRs are mixed with the thermal gas. Second, because of the
additional pressure support from CRs, AGN bubbles filled with CRs are
generally less dense and more buoyant, which may uplift the ICM more
efficiently than thermal bubbles. Last but not least, the CRs are expected to
provide heat to the ICM via both collisional heating (including Coulomb and
hadronic processes) as well as collisionless heating (through the CR streaming
instability; more details below). All of the above have motivated the
investigation of the effects of CRs in the context of AGN feedback.

Regarding modelling the effects of CRs in fluid simulations, there have also
been substantial developments over the past decade. In particular, the CR
hydrodynamic/MHD framework has matured and become one of the primary numerical
tools for studying the roles of CRs in galaxies and clusters (see review by
Zweibel (2013, 2017)). In this framework, the CRs are treated as a second
fluid, and an additional equation for evolving the CR energy density is solved
together with the standard fluid equations, with an extra term $\nabla P_{\rm
cr}$ in the momentum equation describing the pressure force from CRs. The
underlying assumption behind this fluid treatment is that the CRs are well
scattered by self-excited waves in the magnetised plasma via the streaming
instability (Kulsrud and Pearce, 1969; Wentzel, 1974), i.e., the “self-
confinement picture” of CR transport, or by waves as part of the cascade of
MHD turbulence, i.e., the “extrinsic turbulence picture.” In the self-
confinement picture, the CRs can stream relative to the gas down their
pressure gradients along magnetic fields with the Alfvén velocity,
$\mathbf{u}_{\rm s}=-{\rm sgn}(\hat{\mathbf{b}}\cdot\nabla P_{\rm
cr})\mathbf{u}_{\rm A}$, where $\mathbf{u}_{\rm A}$ is the Alfvén velocity and
$\hat{\mathbf{b}}$ is the unit vector of the magnetic field. Therefore, the
transport of CRs can be generally prescribed as advection with the gas plus
flux terms associated with the streaming and diffusion of CRs. When CRs
stream, there is also net energy transfer from the CRs to the thermal gas via
Alfvén waves, which corresponds to the collisionless CR streaming heating
term, $-\mathbf{u}_{\rm A}\cdot\nabla P_{\rm cr}$. In the extrinsic turbulence
picture, on the other hand, the CRs are scattered by forward-propagating and
backwards-propagating waves. Under the assumption of balanced turbulence, the
transport of CRs from these waves cancels out and thus CRs essentially advect
with the gas, and streaming heating vanishes. CR diffusion in the simulations
represents not only diffusion due to gyroresonant scattering but also CR
transport due to unresolved fluctuations in the magnetic field. Regardless of
the CR transport model considered, CRs could heat the ICM via collisional
processes such as hadronic and Coulomb interactions. In terms of numerical
implementations of the CR hydrodynamic equations, it was recognized in earlier
simulations that the CR streaming fluxes would cause spurious oscillations
near the extrema of CR pressure distributions unless very stringent simulation
timesteps are used (Sharma et al., 2010). More recently, this numerical
difficulty has been overcome by introducing a two-moment method for solving an
additional equation for CR fluxes (Jiang and Oh, 2018; Thomas and Pfrommer,
2019). With the above advancement in treating CR physics in large-scale
simulations, significant progress has been made in terms of understanding the
impacts of CRs on AGN feedback in clusters.

CR heating of the ICM has long been considered a possible source to suppress
radiative cooling in CC clusters, as the CRs injected by the AGN jets could
diffuse or stream outside the bubbles and interact with the ICM via
collisional and collisionless processes. While early simulations were not able
to demonstrate stabilization of the CCs partially due to relatively simple
setups as well as the omission of streaming heating (Boehringer and Morfill,
1988; Loewenstein et al., 1991; Rephaeli and Silk, 1995; Colafrancesco et al.,
2004; Jubelgas et al., 2008; Pfrommer et al., 2007), it was then realised that
CR streaming heating via Alfvén waves could be a viable mechanism for heating
the ICM, and the amount of CR heating could be sufficient to counteract
radiative cooling and stabilize the CCs (Guo and Oh, 2008; Enßlin et al.,
2011; Fujita and Ohira, 2011, 2013). In addition to providing heat to the ICM,
the CRs also affect the gas dynamics differently from thermal gas. Comparing
AGN bubbles inflated by kinetic-energy dominated jets versus CR-energy
dominated jets, the former tends to be more elongated, while CR-dominated
bubbles tend to be more oblate due to their lower momenta (Guo and Oh, 2008;
Sijacki et al., 2008; Guo and Mathews, 2011; Yang et al., 2019), as shown in
Figure 5. While AGN bubbles inflated by kinetic-energy dominated jets are
easily deformed by fluid instabilities due to the large shear velocity
relative to the ambient medium, CR jet-inflated bubbles are generally more
stable (Sijacki et al., 2008; Yang et al., 2019). The morphology of the CR-
filled bubbles can thus more easily explain the “fat” bubbles with smooth
surfaces seen in observations of X-ray cavities, such as the young bubbles
near the centre of the Perseus cluster. The larger cross-sections of CR
bubbles, in addition to the buoyancy of the CRs, result in more efficient
uplift of the ICM. The outward mass transfer driven by CR bubbles alone could
bring the fast-cooling gas away from the core and help reduce cooling near the
cluster centre (Sijacki et al., 2008; Mathews and Brighenti, 2008; Weinberger
et al., 2017; Yang et al., 2019). It is worth noting that the dynamical
effects arising from having CR-energy dominated jets are in many aspects
similar to the light jets discussed in Section 2.1.1 because of their lower
momenta compared to kinetic-energy dominated, heavy jets. However, because of
the differences in the EoS and heating mechanisms between CRs and thermal gas,
the detailed heating processes driven by CR jets and light jets may still be
different.

Figure 5: Comparisons of simulation results with varied jet composition and
assumptions for modeling CR transport. The simulations model a single outburst
of jet activity (jet power of $5\times 10^{45}$ erg s-1 and jet duration of 10
Myr) in a Perseus-like hydrostatic atmosphere. Rows from top to bottom show
the results from kinetic-energy dominated jets (KIN), CR dominated jets (CR),
and CR dominated jets with diffusion and heating (CRdh). The morphology of
jet-inflated lobes tend to be elongated for the KIN case, whereas bubbles
inflated by CR dominated jets are wider. Comparisons between the bottom two
panels show that CR heating is more efficient and the amount of cold gas is
less for the CRdh case, as the CRs can diffuse outside the bubbles and heat
the ICM due to hadronic and Coulomb interactions. (Figure 3 from Yang et al.
(2019), ©AAS, reproduced with permission)

While steady-state models have shown that CR heating could be a viable
mechanism for stabilizing the CCs (Fujita and Ohira, 2013; Jacob and Pfrommer,
2017a, b), recent 3D CR hydrodynamic/MHD simulations have brought rich
information about the interplay between the CRs and the ICM. Ehlert et al.
(2018) performed 3D MHD simulations of a single AGN outburst with varied jet
parameters and demonstrated that CRs could reproduce observed diverse
morphologies of radio lobes and provide sufficient heating to the ICM. They
identified jet energy as the critical parameter for determining the bubble
morphology and CR distribution, while jet luminosity is responsible for
setting the Mach numbers of shocks. Using 3D MHD simulations of self-regulated
AGN jets dominated by CRp, Ruszkowski et al. (2017) showed that the key to
successful CR feedback is the CR transport mechanisms, such as CR diffusion
and streaming. With these CR transport processes, CRs could then diffuse or
stream outside the low-density bubbles and get into contact with the denser
ambient ICM, allowing CR heating to operate (shown in the middle and bottom
rows of Figure 5). Without CR transport (i.e., if only CR advection is
considered), the amount of CR heating would be insufficient, and cooling
catastrophes would still occur (Ruszkowski et al., 2017; Wang et al., 2020).
It was also found that, due to the significant outward mass transfer driven by
CR bubbles, it would take longer for the cluster to contract again and trigger
another cycle of AGN activity. As a result, the feeding of the central SMBH
and hence the AGN activity tends to be more episodic when the cluster is
regulated by CR feedback (Ruszkowski et al., 2017), whereas the AGN activity
tends to be more quasi-continuous for a similar setup with kinetic-jet
feedback (Yang and Reynolds, 2016). This implies that the composition of AGN
jets is a major source of uncertainties in simulation predictions for cluster
evolution as well as SMBH growth.

Most of the simulations of CR feedback have focused on CRp-dominated jets;
however, observational constraints on the composition of cluster radio bubbles
suggest that the distribution is bimodal (Dunn and Fabian, 2004; Rafferty et
al., 2006; Croston et al., 2018). For some bubbles, the internal pressure from
radiating CRe is much lower than the external pressure of the ICM ($P_{\rm
ext}/P_{\rm int}\gg 1$), implying significant pressure support from thermal
gas or non-radiating particles. For other bubbles with $P_{\rm ext}/P_{\rm
int}\sim 1$, the CRe pressure is enough to keep the bubbles in pressure
equilibrium with the surrounding ICM. The latter case has been recently
investigated by Lin et al. (2023). By comparing the feedback effects of CRp
and CRe-dominated jets, they found that, although CRe cool much faster than
CRp due to IC and synchrotron radiation, the CRe-dominated bubbles quickly
become thermally dominated $\sim 20$ Myr after injection. Consequently, the
CRe bubbles would not deflate, but they behave similarly to thermal bubbles
both in terms of dynamics and ICM heating. This also implies that, for bubbles
inflated by CRe jets, there should be a transition of its composition from
$P_{\rm ext}/P_{\rm int}\sim 1$ to $P_{\rm ext}/P_{\rm int}\gg 1$. Future
measurements of this pressure contrast as a function of radius for cluster
radio bubbles may be a viable way to probe the intrinsic composition of AGN
jets/bubbles.

While the above simulations have focused on the properties of the hot ICM,
more recent studies have looked into how CR-jet feedback could help suppress
SFRs and quench central galaxies. Using the FIRE (Feedback In Realistic
Environments) simulations of idealised clusters of $10^{12}-10^{14}\
M_{\odot}$ with toy models for AGN feedback, Su et al. (2020) showed that both
turbulent stirring and CR heating are efficient mechanisms for maintaining a
stable, low-SFR halo over billion-year timescales, in which the core density
and cooling rates are suppressed due to the non-thermal pressure support from
turbulence or CRs. Later simulations of bi-polar jet feedback by Su et al.
(2021) further performed a thorough parameter study on the relevant jet
parameters, including the jet energy composition (kinetic, thermal, CR, or
magnetic), jet widths, jet precession, injected mass fluxes, and duty cycles.
They showed that CR-dominated jets can most efficiently quench the central
galaxy, whereas kinetic jets are less efficient unless they have wide opening
angles or precession angles. They attributed the efficient CR feedback to
three key factors. First, the injected CRs form wider cocoons that suppress
inflows and provide pressure support to the gas, resulting in longer cooling
times. Second, as shown in recent simulations of thermal instabilities (Ji et
al., 2020; Butsky et al., 2020), the cooling gas would remain diffuse due to
the pressure support from CRs, slowing down the “precipitation” of cold gas
that would be otherwise accreted onto the central black hole. Lastly, because
CRs can diffuse and stream through the gas, they tend to generate more gentle
outflows rather than explosive ones expected for a pressure-driven blastwave.
As a result, CR feedback is less likely to overheat the cluster cores compared
to injections in other energy forms.

####  3.1.3 Observational Signatures and Constraints

Any successful CR feedback model must satisfy observational constraints of
galaxy clusters. The CRs injected from the AGN jets are expected to produce
non-thermal emission that may be detected using multi-wavelength observations.
For CRe, they could radiate via synchrotron emission when they interact with
magnetic fields as well as via inverse-Compton (IC) scattering of ambient
photons (e.g., the cosmic microwave background, CMB) — this is often called
“leptonic emission.” For CRp, their inelastic collisions with ambient nuclei
could produce neutral pions, which decay into gamma rays — the so-called
“hadronic interaction.” The hadronic process could also produce charged pions,
which decay into charged muons and subsequently to secondary electrons and
positrons, together with the production of neutrinos. The secondary particles
could again emit due to the synchrotron and IC radiation.

Radio observations near the cluster cores could potentially provide
constraints on the amount and distributions of CRs injected by AGN. Indeed,
some clusters are known to host radio mini-halos (e.g., Gitti et al., 2007;
Giacintucci et al., 2014; Richard-Laferrière et al., 2020), which are faint,
diffuse radio structures extending $\sim 50-300$ kpc, characterized by steep
radio spectra with typical spectral indices of $\alpha>1$ (as defined in
$S_{\nu}\propto\nu^{-\alpha}$, where $S_{\nu}$ is the flux density, and $\nu$
is the frequency). The physical origin of the radio mini-halos is not yet
fully understood. Generally speaking, because the cooling times of CRe are
typically shorter than the time it takes to transport them to the extent of
the observed mini-halos, these CRe would need to be produced in situ or be re-
accelerated from an old seed population. In terms of the emission mechanisms,
the mini-halos could be produced by CRe either via hadronic interactions
between CRp and the ambient ICM (e.g., Pfrommer and Enßlin, 2004; Fujita et
al., 2007; Zandanel et al., 2014) or via turbulence reacceleration, e.g., by
sloshing motions (Giacintucci et al., 2011; ZuHone et al., 2013) or AGN
activity (Bravi et al., 2016)). For the latter scenario, Jacob and Pfrommer
(2017b) have investigated in detail the predicted radio emission from a
steady-state model of CR plus conductive heating. From comparisons with an
observed sample of mini-halos, they showed that CC clusters hosting mini-halos
cannot be heated by CRs but CCs without mini-halos can. To this end, they
proposed a scenario where CC clusters with recent AGN injections should host
radio “micro-halos”; after the CRs diffuse or stream out of the cores, CR
heating becomes insufficient and mini-halos light up due to secondary
particles produced via hadronic interactions. More simulations on this front
are needed in order to verify this picture as well as to understand the source
of the seed CRe and the interplay between AGN feedback and sloshing motions in
clusters (e.g., ZuHone et al., 2021a, b). Recent radio observations at low
frequencies by LOFAR212121https://lofar-surveys.org have also enabled
constraints on the interaction between the CRs and the ICM. For example,
Brienza et al. (2021) found signatures of old AGN jet-inflated bubbles that
retain “mushroom-like” structures over hundreds of Myr timescales. The bubbles
are not thoroughly mixed with the ambient ICM, suggesting suppression of fluid
instabilities and CR diffusion at the bubble surface, probably under the
influence of magnetic fields.

As mentioned in Section 3.1.1, the non-detection of gamma-ray emission from
galaxy clusters has strongly limited the amount of hadronic CRs to be within a
few per cent compared to the thermal pressure for clusters as a whole
(Ackermann et al., 2010; Ahnen et al., 2016). Cosmological simulations by
Vazza et al. (2013) investigated quasar-, jet-, and radio-mode feedback and
found successful models that could reproduce X-ray observations of the cluster
gas and yield gamma-ray emission below the upper limits by Fermi. Other
simulations including CR-dominated jets (Ruszkowski et al., 2017; Yang et al.,
2019) showed that, for CR heating to balance radiative cooling, only a small
amount of CR pressure support is required. In their successful self-regulated
models, CR streaming is a critical ingredient as it would act to remove energy
from CRs and heat the gas, reducing the amount of CR pressure support to
levels consistent with the gamma-ray constraints. Recent simulations by
Beckmann et al. (2022) have also investigated the roles of CR-dominated jets
in terms of ICM heating and observable signatures. Consistent with previous
works (Ji et al., 2020; Butsky et al., 2020), they found that CRs can modify
the development of thermal instabilities and help maintain gas in the hot and
warm phases. However, they found that their simulations including CR-dominated
jets would produce gamma-ray emission in excess of current observational
limits due to the formation of an extended, CR-pressure supported warm nebula,
while AGN jets with lower CR fractions ($\sim 10\%$) are allowed and could
successfully halt strong cooling flows. The discrepancies in the results have
not been fully understood yet, but here we list two possible reasons that
might play a role. First, the simulations in Ruszkowski et al. (2017) used the
mass dropout technique for the cold gas, while Beckmann et al. (2022) retain
the cold gas in the simulations. Since the gamma-ray emission produced by
hadronic interactions is proportional to the product of the CRp number density
and the gas density, it is conceivable that the latter simulation would
produce greater gamma-ray emission if the contribution from the cold gas is
significant. Another difference between these simulations is that, in contrast
to typical AGN feedback prescriptions that connect the jet properties with the
black hole accretion rates, Beckmann et al. (2022) additionally modelled the
influence of the black hole spin on the jet directions, as discussed in
Section 2.1.1. To this end, their jets are more randomly oriented and could
not penetrate and deposit heat to larger radii compared to previous self-
regulated simulations with jet precession along a fixed direction. This might
have caused less effective feedback and the CR-pressure supported warm nebula
that produces the excess gamma-ray emission. More detailed studies are needed
in order to pin down this issue. Regardless, the observational gamma-ray
limits will provide crucial constraints on the amount of CRs allowed in the
cluster cores.

###  3.2 Roles of Plasma Physics of the Intracluster Medium

While a lot of our understanding of AGN feedback has been gained via fluid
simulations, the ICM is in fact a weakly collisional, magnetised plasma. The
Coulomb mean free path of the ionized gas in the ICM is (Spitzer, 1962)

$\lambda_{\rm mfp}\equiv\frac{3^{3/2}(k_{\rm B}T)^{2}}{4\pi^{1/2}n_{\rm
e}e^{4}\ln\Lambda}\approx 23\ {\rm kpc}\left(\frac{T}{10^{8}\ {\rm
K}}\right)^{2}\left(\frac{n_{\rm e}}{10^{-3}\ {\rm cm}^{-3}}\right)^{-1},$ (4)

where $n_{\rm e}$ is the electron number density and $T$ is the temperature of
the ICM. Typical values for $\lambda_{\rm mfp}$ range from $\sim 0.1$ kpc near
cluster cores to $\sim 10$ kpc in cluster outskirts. The ratio between the
collisional mean free path and the size of the system is thus $\lambda_{\rm
mfp}/L\sim 0.1-10^{-3}$. For regions in clusters with $\lambda_{\rm mfp}\ll
L$, the ICM could be safely approximated as a collisional fluid; for other
regions where $\lambda_{\rm mfp}\lesssim L$, the ICM is instead weakly
collisional. In addition, as discussed in Section 2.2, the ICM is magnetised
with typical magnetic field strengths on the order of $1-10\ \mu$G (Carilli
and Taylor, 2002). Even though the magnetic field is not dynamically dominant
(plasma beta $\beta\equiv P_{\rm th}/P_{\rm B}\sim 100$), it significantly
constrains the motions of the charged particles within the ICM as the Larmor
radius of their gyro motions ($r_{\rm g}$) is typically more than ten orders
of magnitude smaller than the collisional mean free path, i.e., $r_{\rm
g}\ll\lambda_{\rm mfp}\lesssim L$. As a result, transport processes such as
viscosity and thermal conduction in this weakly collisional, magnetised ICM
are expected to be anisotropic along magnetic field lines, whereas the
perpendicular transport is significantly suppressed. Also, the parallel
transport coefficients along field lines are mediated by plasma physics that
occur on the microscopic scales of the particles’ gyroradii, i.e., the
microphysics. Over the past decade, there has been substantial progress in
terms of our understanding of the microphysical plasma processes. These
developments and the plasma properties of the ICM have been summarized in a
recent review article (Kunz et al., 2022). Here we will focus on the
discussion about how these plasma effects could alter the standard
hydrodynamic picture of AGN feedback as described in the previous sections.

####  3.2.1 Influence of Thermal Conduction on AGN Feedback

Thermal conduction has long been proposed as one of the heating mechanisms in
CC clusters, as it could potentially channel heat from the reservoir of
thermal energy in cluster outskirts toward the cluster cores (e.g., Zakamska
and Narayan, 2003; Voigt and Fabian, 2004). While balancing radiative cooling
by conductive heating alone would require fine-tuning (e.g., Bregman and
David, 1988; Zakamska and Narayan, 2003), it may provide partial heating to
the CCs and relieve the burden on the AGN. Indeed, some theoretical models
have considered conductive heating together with AGN jet heating (e.g., Guo
and Oh, 2009; Jacob and Pfrommer, 2017a). However, the fact that thermal
conduction occurs anisotropically in the magnetised ICM has introduced further
complications. In particular, it is found that anisotropic conduction in the
conditions of CCs would trigger the heat-flux driven buoyancy instability
(HBI; (Quataert, 2008)), which would act to reorient magnetic field lines in
the direction perpendicular to the temperature gradients. In CCs where the
temperature gradients are primarily radial, the HBI would then wrap the field
lines in the azimuthal direction, which would shut off conductive heat fluxes
from cluster outskirts and potentially worsen the cooling-flow problem
(Parrish et al., 2009).

Fortunately, later simulations have found that the HBI could likely be
circumvented by turbulent motions in the ICM (Ruszkowski and Oh, 2010), which
could originate from a variety of sources, including g-modes excited by galaxy
motions (Ruszkowski and Oh, 2011) and AGN jet-driven turbulence (Yang and
Reynolds, 2016). These works found that the turbulent motions can efficiently
randomize the field lines and resume heat fluxes to an effective Spitzer
fraction (the suppression factor compared to the full Spitzer value) of $\sim
1/3$, consistent with that for fully tangled magnetic fields. In this case,
both conductive heating and AGN heating contribute to counteracting radiative
cooling in the CCs. The amount of conductive heating, though, is likely to be
subdominant compared to direct AGN heating, shown by both idealised cluster
simulations (Yang and Reynolds, 2016) as well as cosmological simulations
(Kannan et al., 2017). Recently, simulations by Beckmann et al. (2022) that
include black-hole-spin driven AGN jets found, instead, that AGN jet-driven
turbulence is only able to randomize field lines close to the cluster cores,
but the HBI could still operate outside $\sim 50$ kpc and isolate the CCs from
conductive heating. To this end, they concluded that conductive heating plays
a negligible role in regulating radiative cooling in CCs. Again, the
difference in the above results could be due to different implementations of
the AGN feedback prescriptions (see discussion in Section 3.1.3). In addition,
the field-line wrapping effect of the HBI would likely be washed out if other
sources of ICM turbulence from cosmic accretion or galaxy motions are included
in the simulations. All of these simulations including anisotropic conduction
have assumed full Spitzer conductivity along the field lines. If the
conductive coefficient in the ICM is significantly suppressed due to
microphysical plasma processes, as suggested by recent particle-in-cell (PIC)
simulations (e.g., Roberg-Clark et al., 2016, 2018; Komarov et al., 2018),
then the contribution from conductive heating would be even more inhibited.

####  3.2.2 Influence of Viscosity on AGN Feedback

While AGN jet-inflated bubbles in purely hydrodynamic simulations are easily
deformed and disrupted by Rayleigh-Taylor and Kelvin-Helmholtz instabilities
(see discussion in Section 2.1.2), the observed AGN bubbles are generally more
regular and do not exhibit clear signs of instabilities. For instance, the
young X-ray cavities near the centre of the Perseus cluster have smooth
surfaces, and the more evolved ghost cavity in the northwest direction does
not break up but shows a flattened morphology (Fabian et al., 2005). These
observed properties of the AGN bubbles have posed challenges to purely
hydrodynamic modelling of jet-inflated bubbles and called for the
consideration of additional mechanisms to suppress the hydrodynamic
instabilities such as magnetic fields (e.g., Jones and De Young, 2005; Dursi,
2007; Ruszkowski et al., 2007; Dursi and Pfrommer, 2008; O’Neill et al., 2009)
or viscosity (Reynolds et al., 2005; Sijacki and Springel, 2006). In addition,
the morphology of the ${\rm H}_{\alpha}$ filaments behind the northwest ghost
bubble in Perseus suggests that they may be dragged up by the buoyantly rising
bubble (Fabian et al., 2003). These observations have motivated the
consideration that the ICM may have a non-negligible level of viscosity.

Indeed, viscous simulations by Reynolds et al. (2005) have demonstrated that
isotropic viscosity could act to suppress the fluid instabilities and prevent
the bubbles from disruption. Additionally, the streamlines behind the
simulated bubbles are consistent with the coherent structure of the observed
${\rm H}_{\alpha}$ filaments. Subsequent work by Sijacki and Springel (2006)
also showed that the properties of AGN jet-inflated bubbles, including their
morphology, maximum distance from the cluster centre, and survival time,
depend sensitively on the level of ICM viscosity. However, this is not the
full story because viscosity is expected to be anisotropic in the magnetised
ICM, and the parallel viscosity coefficient mediated by microphysical plasma
processes should be considered when modelling the evolution of AGN bubbles. To
this end, simulations by Dong and Stone (2009) studied the buoyant evolution
of initially static bubbles including anisotropic viscosity along magnetic
fields of different initial geometries. They found that anisotropic viscosity
can efficiently suppress fluid instabilities in the direction parallel to the
magnetic field while having little effect on instabilities that develop
perpendicular to the field. As a result, the fate of the bubbles is sensitive
to the assumed geometry of the magnetic field. While the bubbles can be
stabilized by viscosity along initially horizontal fields or toroidal fields
confined to the bubble interior, the bubbles are deformed in the case with
vertical field geometry.

Figure 6: Impact of assumptions about ICM viscosity on the evolution of AGN
jet-inflated bubbles. Cases A-D show simulations with no viscosity, isotropic
viscosity with full Spitzer values, anisotropic viscosity with full Braginskii
values, and anisotropic viscosity limited by microinstabilities, respectively.
While the hydrodynamic instabilities are suppressed by viscosity in cases B
and C, when the parallel viscosity along magnetic field lines is suppressed by
microinstabilities (case D), the viscosity is strongly limited and the bubbles
are deformed as in the inviscid case (A). This illustrates the importance of
modelling the ICM microphysics in AGN simulations. (Figure 1 from Kingsland et
al. (2019), ©AAS, reproduced with permission)

Kingsland et al. (2019) further investigated the effects of anisotropic
viscosity on AGN jet-inflated bubbles in a more realistic tangled magnetic
field geometry taking into account microphysical plasma effects learned from
recent PIC simulations. Specifically, for a weakly collisional, magnetised
plasma like the ICM, the viscosity originates from pressure anisotropy that
arises from the conservation of the adiabatic invariants on timescales much
greater than the inverse of the ion gyrofrequency (Chew et al., 1956;
Schekochihin et al., 2005). Recent plasma simulations have found that when the
pressure anisotropy becomes greater than thresholds that are on the order of
$1/\beta$ (where $\beta$ is the plasma beta value), microinstabilities
including the firehose and mirror instabilities would be triggered and pin the
pressure anisotropy at the marginal-stability thresholds, effectively
suppressing the parallel viscosity coefficients (see Kunz et al., 2022, and
references therein). Incorporating these latest findings into Braginskii-MHD
simulations, Kingsland et al. (2019) found that, for anisotropic viscosity
with full Braginskii values (Braginskii, 1965), the integrity of the bubbles
could be preserved because viscosity along tangled field lines could suppress
the instabilities in multiple orientations on the bubble surface (see case C
in Figure 6). However, when suppression of the parallel viscosity coefficients
by the microinstabilities is considered, the suppression is so strong that the
bubbles are deformed just as in the inviscid case (case D in Figure 6). This
is because the plasma beta value in the bubble interior could be as high as
$10^{4}$, which dramatically limits the amount of pressure anisotropy and
viscosity in the vicinity of the bubbles. Therefore, they concluded that
Braginskii/anisotropic viscosity is unlikely to be the primary mechanism for
preserving the coherence of AGN bubbles, but other mechanisms (e.g., magnetic
fields) are still required to reproduce the morphology of observed bubbles.
This further emphasized the importance of modelling the ICM plasma properties
as the bubble evolution has a direct influence on their ability to uplift the
ICM and where the bubbles deposit heat provided by the AGN.

Finally, since the pressure anisotropy in the ICM provides an anomalous
“effective” viscosity (Kunz et al., 2022), there could be heating associated
with parallel viscous dissipation of gas motions (so-called gyroviscous
heating). By assuming the pressure anisotropy is pinned at the marginal-
stability thresholds in the turbulent ICM, Kunz et al. (2011) showed that this
mechanism could provide heating rates comparable to the radiative cooling
rates. Recent PIC simulations have further studied the detailed process of how
particles can be gyroviscously heated by large-scale turbulent fluctuations
via magnetic pumping (Ley et al., 2023). More studies would be required in
order to integrate these microphysical phenomena into large-scale turbulence
models of the ICM for a full assessment of gyroviscous heating of the ICM.

####  3.2.3 Heating by Sound-Wave Dissipation

In X-ray images of the Perseus cluster, there are ripple-like structures that
were interpreted as sound waves driven by central episodic AGN outbursts
(Fabian et al., 2003; Sanders and Fabian, 2007). Motivated by this
observational finding, early simulations have shown that viscous dissipation
of sound waves could be a viable mechanism for heating the ICM (Ruszkowski et
al., 2004a, b), although some studies found that their energy content may be
small due to viscous damping (Sijacki and Springel, 2006). Sound-wave heating
is an attractive solution to the cooling-flow problem because it could heat
the ICM isotropically as the waves propagate from the cluster centre outward.
The question of whether sound-wave dissipation could be a significant
contributor to ICM heating could be broken down into two: how much of the
injected energy by the AGN can be stored in the form of sound waves, and how
the waves propagate and dissipate in the ICM. We will summarize the progress
along these lines below.

Figure 7: Illustration of sound waves generated by jets. The left-hand panel
shows the jet and cocoon temperature and density structure, with key features
labelled. The right-hand panel shows jet entropy and the acoustic flux
density, with structure in the latter illustrating the production of sound
waves within shocked ICM material. (Figure 1 from Bambic and Reynolds (2019),
©AAS, reproduced with permission)

Several recent simulations have been conducted in order to quantify how much
of the injected AGN energy can be converted into sound waves. Using
simulations of spherical injections into a uniform medium with varied injected
energy and duration, Tang and Churazov (2017) found that the fraction of AGN
energy that goes into sound waves is dependent on the duration of the
injection. When the duration is long, in which the solution approaches the
“slow piston” limit, the energy of the sound waves is close to zero. For
instantaneous outbursts (duration approaches zero), the energy fraction of
sound waves is $\lesssim 12\%$. Bambic and Reynolds (2019) further performed a
parameter study using axisymmetric simulations of an AGN outburst with bipolar
jet geometries. Figure 7 is taken from their paper and illustrates the
generation of sound waves by a jet. They found that up to $\sim 25\%$ of the
injected energy could be stored in the form of sound waves for optimal
parameter combinations. Using jet parameters informed by previous self-
regulated AGN feedback simulations, Wang and Yang (2022) found that the
production efficiency of compressional waves (including weak shocks and sound
waves) is $\sim 9\%$ for a single AGN energy injection. However, this
production efficiency drops to less than $\sim 3\%$ in self-regulated feedback
simulations. This is because, with repeated AGN outbursts, shocks are almost
continuously generated and their energy dominates that of the weak shocks. To
this end, shock dissipation together with destructive interference of sound
waves would act to reduce the amount of energy in compressional waves.
Considering the above effects, they concluded that sound-wave heating may be a
subdominant source of heating in CC clusters. However, Wang and Yang (2022)
have assumed powerful AGN injections via bipolar jets. Using viscous
simulations of a single spherical, gentle (longer-duration) AGN outburst,
Choudhury and Reynolds (2022) found that $\sim 20\%$ of the injected power can
be carried away by sound waves. Overall, the above results suggest that the
production efficiency of sound waves is $\sim 3-25\%$, depending on how the
feedback energy is injected (e.g., geometry, duration) and how the waves
generated by repeated events interfere with each other. Finally, the above
estimates are derived based on hydrodynamic treatments of the ICM, and there
may be additional plasma effects that need to be taken into account in terms
of sound-wave generation. For instance, Kempski et al. (2020) have performed
Braginskii-MHD simulations including CRs, and showed that the phase shifts
between CR pressure and density fluctuations would generate instabilities when
the CR fraction is greater than a threshold proportional to the plasma beta.
Therefore in an ICM with high plasma beta values, this mechanism could
potentially be important for exciting sound waves.

Another question to address is the propagation and dissipation of sound waves
in the ICM. Pioneering work by Fabian et al. (2005) showed that nearly
adiabatic acoustic waves would damp within one wavelength of their source and
overheat the cluster core if thermal conductivity and viscosity are at their
full Braginskii values. If this were true, it would pose challenges to the
interpretation of observed ripples as propagating sound waves. Assuming
thermal conduction is completely suppressed and viscosity is reduced to $10\%$
of the Braginskii value, heating due to sound-wave dissipation could then
balance radiative cooling for a power-law spectrum of waves. More recently,
Zweibel et al. (2018) revisited this topic by additionally taking into account
the self-limiting nature of electron thermal conduction, differences between
electron and ion temperatures, and plasma effects from kinetic theories.
Though the kinetic effects somewhat suppress damping and mitigate the problem
of wave propagation, the conclusions are reinforced that, in order for
acoustic waves to propagate to large radii and heat the CC, drastic
suppression of the transport coefficients is needed. Such reduced transport
could come from increased effective collisionality of the ICM due to magnetic
field fluctuations on small scales driven by plasma instabilities (e.g.,
Roberg-Clark et al., 2016; Kunz et al., 2014). However, future works are still
required to fully assess the generation and damping of acoustic waves under
the influence of plasma instabilities as well as realistic sources and
geometries.

###  3.3 Open Questions and Future Opportunities

Following the discussion in Section 3.1, it is obvious that constraining the
composition of AGN jets and bubbles is one of the key questions to address
since different jet compositions would have different dynamical and thermal
impacts on the ICM, which would then affect the long term evolution of
clusters as well as SMBH accretion histories. One of the proposed ways to
distinguish AGN bubbles dominated by ultrahot thermal gas and CRs is by using
the SZ effect (Sunyaev and Zeldovich, 1972), which arises due to the IC
scattering of the CMB photons by electrons in the ICM. It was shown by