11institutetext: Institut de Recherche en Astrophysique et Planétologie
(IRAP), Université de Toulouse, CNRS, UPS, Toulouse, France
11email<EMAIL_ADDRESS>
# Dynamical properties and detectability of the magneto-thermal instability in
the intracluster medium

J. M. Kempf 11 F. Rincon 11 N. Clerc 11

###### Abstract

Context.  Amongst the many plasma processes potentially relevant to the
dynamics of the intracluster medium (ICM), turbulence driven at observable
scales by internal magnetised buoyancy instabilities such as the magneto-
thermal instability (MTI) stands out in the outskirts of the ICM, where the
background temperature decreases with the radius.

Aims.  We characterise the statistical properties of MTI turbulence in the ICM
and assess whether such large-scale magnetised plasma dynamics would be
detectable with the future X-ray calorimeter X-IFU on board ATHENA.

Methods.  We made use of scaling laws previously derived to phenomenologically
estimate the observable turbulent saturation levels and injection length of
MTI turbulence for different ICM thermodynamic profiles, and performed a
numerical magnetohydrodynamic simulation of the dynamics with Braginskii heat
and momentum diffusion. As a prospective exercise, we used the simulation to
virtually observe MTI turbulence through the X-IFU.

Results.  In bright enough regions amenable to X-ray observations, the MTI
drives mild turbulence up to $\sim 5\%$ and $\sim 100$ km/s (root-mean square
temperature fluctuation and velocity). However, the measurable integrated
temperature fluctuation and line-of-sight velocity fields, the latter being
essentially the azimuthal velocity component in cluster haloes, hardly exceed
$1\%$ and $10$ km/s, respectively (root-mean square). We show that such
moderate signals would be difficult to detect with upcoming X-ray telescopes.
We also find that MTI turbulence is anisotropic in the direction of gravity
and develops at scales ${\gtrsim}0.2$ Mpc. If the fluctuation intensities were
to be stronger than the current theoretical estimates, MTI fluctuations would
be detectable and their anisotropy discernible with the X-IFU.

Conclusions.  Finding direct signatures of magnetised plasma dynamics in the
ICM, even at observable scales typical of the fluid MTI, remains challenging.
This study only marks a first step in this direction. Several numerical and
observational strategies are discussed to make further progress in the future.

###### Key Words.:

galaxies: clusters: intracluster medium – instabilities – turbulence –
magnetohydrodynamics (MHD)
– methods: numerical – techniques: imaging spectroscopy

##  1 Introduction

Galaxy clusters are filled with hot and diffuse gas: $k_{\mathrm{B}}T{\sim}5\
\mathrm{keV},\ \rho{\sim}10^{-27}\ \mathrm{g/cm}^{3}$. This plasma, usually
referred to as the intracluster medium (ICM), radiates in the X-ray through
the combined emission of lines and Bremsstrahlung at a very high temperature.
It accounts for ${\sim}15\%$ of the total mass of clusters and is in rough
hydrostatic equilibrium within its dark matter potential well. Recent
observations however suggest that the ICM sustains subsonic turbulence
(Zhuravleva et al. 2018; Dupourqué et al. 2023), which could alter the
hydrostatic balance and the internal transport of chemicals and energy, and
hence shaping the evolution of clusters on cosmological timescales (Zhuravleva
et al. 2014). A departure from ideal hydrostatic equilibrium in the ICM could
also provide additional non-thermal pressure support (Eckert et al. 2019),
biasing the determination of cluster masses through X-ray and Sunyaev-
Zeldovich (SZ) measurements (Vazza et al. 2018; Angelinelli et al. 2020). The
assessment of such a systematic bias is critical for cosmology since the mass
distribution of galaxy clusters is used as a late-universe probe for the
cosmological parameters (see the reviews by Allen et al. 2011; Pratt et al.
2019).

The dynamics of the ICM remains observationally poorly constrained. The Soft
X-ray Spectrometer (SXS) calorimeter on board the Hitomi satellite made the
only direct measurement of a velocity field in the ICM of the Perseus cluster
core (Hitomi Collaboration et al. 2016, 2018). The instrument was able to
measure velocity gradients and turbulent broadening about $150$ km/s thanks to
both an emission line centroid shift and width measurements, respectively, but
only down to the SXS pixel resolution of $20$ kpc. So far, putting
observational constraints on ICM turbulence below this length scale is
exclusively achievable with the detection of ICM thermodynamic perturbations
(Hofmann et al. 2016) or thanks to indirect measurements of its kinematics
(Simionescu et al. 2019) through measurements of X-ray surface brightness
(Churazov et al. 2012; Zhuravleva et al. 2015) and SZ fluctuations (Zeldovich
& Sunyaev 1969; Mroczkowski et al. 2019). However, the next generation of
spatially resolved X-ray spectroscopy instruments, such as Resolve (Ishisaki
et al. 2018) or the X-ray Integral Field Unit (X-IFU; Barret et al. 2016,
2023) on board the future X-ray observatory ATHENA, and their unprecedented
spectral resolution ($7$ and $2.5$ eV below $7$ keV, respectively) will push
this limit downwards and provide us with an unprecedented opportunity to
characterise the ICM dynamics down to the kiloparsec scale for the closest
clusters.

The possible sources of turbulence in galaxy clusters are numerous and
diversely distributed across the ICM, ranging from active galactic nuclei
(AGN) feedback at the centre (Conroy & Ostriker 2008; Gitti et al. 2012) to
accretion of infalling baryons on the outer edge (Vazza et al. 2017; Iapichino
et al. 2017) and ram-pressure stripping in between (Domainko et al. 2006;
Ebeling et al. 2014; Li et al. 2023) for instance. Internal fluid
instabilities may also be important in this context. The ICM is stably
stratified against thermal convection (Cavagnolo et al. 2009; Ghirardini et
al. 2019) according to the usual Schwarzschild criterion on the entropy
gradient $\partial_{r}\log\left(T\rho^{1-\gamma}\right)>0$ (Schwarzschild
1906), with $\gamma$ being the adiabatic index. However, it is magnetised up
to $B{\sim}1{-}10\ \mathrm{\mu G}$ (Govoni & Feretti 2004; Ferrari et al.
2008; Botteon et al. 2022). ICM plasma is therefore thought to be in a
’dilute’ regime where the particles’ mean-free path is much larger (a dozen
orders of magnitude) than the particles’ Larmor radius. This ordering
introduces an anisotropy with respect to the direction of the local magnetic
field: heat and momentum collisionally diffuse primarily along the magnetic
field lines and barely across them (Braginskii 1965). In these conditions,
Balbus (2001, 2000) showed that the sign of the temperature (and no longer the
entropy) gradient dictates the stability of the stratified fluid. In
particular, a so-called magneto-thermal instability (MTI) is triggered when
the background temperature gradient is in the direction of the gravity, and
may excite turbulence. This magnetised buoyancy instability is to dilute
plasma what classical thermal convection is to usual gas. As seen from XMM-
Newton (Leccardi & Molendi 2008) and Chandra (Simionescu et al. 2011)
observations (sometimes combined for a SZ analysis, Shitanishi et al. 2018;
Ghirardini et al. 2019), all galaxy clusters exhibit a decreasing temperature
profile (though as of a certain radius for relaxed clusters) and should
therefore be unstable to the MTI, at least in the halo.

Magnetised buoyancy instabilities such as the MTI and its counterpart, the
Heat-flux driven Buoyancy Instability (HBI; triggered when the temperature
increases with radius, Quataert 2008), have been theorised two decades ago and
their potential relevance in the ICM has been largely emphasised since then
(Bogdanović et al. 2009; Balbus & Reynolds 2010; McCourt et al. 2012; Parrish
et al. 2012b; Kunz et al. 2012; Latter & Kunz 2012; Berlok & Pessah 2016b).
For example, they are thought to drive a substantial magnetic field
amplification through a dynamo, and thus making them a serious candidate
mechanism for cluster magnetisation. Magnetic seeds in galaxy clusters (e.g.
stemming from primordial magnetic fields, Durrer & Neronov 2013) would indeed
need to grow by several orders of magnitude to match the strength currently
observed in neighbour clusters, although strong enough primordial magnetic
seeds could also lead up to the present magnetic strengths through compression
only (Jedamzik & Pogosian 2020; see Donnert et al. 2018 for a review on
magnetic field amplification in galaxy clusters). Such magnetised buoyancy
instabilities remain however somewhat speculative since no observational
evidence of their presence has ever been detected. An additional complexity
stems from the conservation of the adiabatic invariant
$\mu\propto\varv_{\perp}^{2}/B$ (with $\varv_{\perp}$ being the perpendicular
component of the particle’s peculiar velocity with respect to the magnetic
field $\@vec{B}$), which drives pressure anisotropy when coupled to changes in
magnetic-field strength in high-beta and weakly collisional plasma. Such
pressure anisotropies can, when large enough, trigger mirror, firehose, ion-,
and electron-cyclotron whistler instabilities Schekochihin et al. (2005,
2010). The saturation of such micro-scale kinetic instabilities could severely
reduce and/or isotropise the heat conductivity (Riquelme et al. 2016; Komarov
et al. 2018) and hamper the development of the MTI in the ICM, if not
completely annihilate it (Drake et al. 2021); although, recent work by Berlok
et al. (2021) tends to show only modest consequences of these micro-
instabilities on the MTI saturation.

Overall, detecting the MTI, if it is present in the ICM and if detectable with
future X-ray missions, would be a very interesting step forward, not only from
a purely observational perspective to characterise turbulent transport or
magnetisation in the ICM, but also from a theoretical and astrophysical plasma
physics standpoint. It would open a direct observational window into the
physics of dilute and magnetised astrophysical plasmas on large fluid scales,
where previous work by Zhuravleva et al. (2019) could only rely on indirect
signatures of micro-scale plasma processes such as the enhanced collision rate
possibly due to the previously mentioned kinetic instabilities. In that paper,
the authors used deep Chandra observations of the Coma cluster to probe the
ICM dynamics through X-ray surface brightness fluctuations down to what should
be the viscous scale if only Coulomb collisions were responsible of momentum
transport. They could not detect any effect of the viscosity at this scale and
therefore deduced that the ICM plasma is either subject to enhanced collision
rates or to Braginskii anisotropic transport. The interest for such new
observational windows goes beyond the ICM because the same plasma regime is
also relevant in accretion flows around black holes for instance (Molokov et
al. 2007; Sharma et al. 2007). The current study is therefore intended to be a
first step towards bridging the gap between theoretical astrophysical plasma
studies and future X-ray astronomy observations.

The outline of this paper is as follows. In Section 2, we introduce different
thermodynamic profile models of the ICM, and we describe the
magnetohydrodynamics (MHD) model and the numerical methods used to simulate
MTI turbulence, as well as the post-processing pipeline developed to build
synthetic observations of the MTI through the X-IFU instrument. We use the
X-IFU specifications as defined before the recent design-to-cost exercise
(Barret et al. 2016, 2023) as a potential model of a future X-ray
spectrometer: it acts as our baseline instrumental configuration throughout
this paper. In Section 3, we characterise the phenomenological dynamical
properties of MTI-driven turbulence expected in the ICM, relying on scaling
laws derived recently by Perrone & Latter (2022a, b, respectively PL22a,b
hereafter) and on the ICM thermodynamic models previously introduced. The
objective of that section is to provide a first estimate of the expected
levels of measurable MTI turbulence in the ICM, taking into account various
observational effects such as the finite value of the ATHENA/X-IFU effective
collecting area or the cancellation of opposite-sign fluctuations present
along the line of sight. In Section 4, we present the numerical results from
our Braginskii-MHD simulation and discuss the consequences of the discrepancy
between numerical and real ICM plasma regimes, and the need for a rescaling of
the simulation to perform mock observations. As a prospective exercise that
may also prove of interest to theoreticians to fix ideas about the
observability of the MTI, we then present two synthetic observations of the
MHD simulation scaled in different ways. We qualitatively compare the output
velocity and thermodynamic fluctuation fields that can be reconstructed from
these observations with the raw input fields from the rescaled simulation.
Using one of the two synthetic observations, we observationally constrain the
anisotropic nature of the MTI. Our main conclusions are summarised in Section
5, where we also stress the caveats or the current work along with its
implications for future X-IFU observations. We finally extend the discussion
to other sources of anisotropic turbulence in the ICM.

##  2 Methods and models

In this section, we first present three different thermodynamic profile models
of the ICM, which will be used in Section 3 to phenomenologically discuss the
properties of MTI-driven turbulence in clusters. Based on the expected number
of photons from a 2-Ms observation with the X-IFU (which we use as a
prospective model of X-ray spectrometer), we also give a criterion to
determine an admissible range of radii at which measurement uncertainties due
to a lack of photons are expected not to be predominant over the instrumental
limitations themselves. We then describe the physical MHD model used to
numerically study the development of the MTI up to a state of sustained
turbulence. Finally, we present the methods used to build synthetic X-IFU
observations of a MHD simulation.

Throughout this work, we assume a universe with a $\Lambda$-CDM cosmology and
$H_{0}=70\ \rm{km/s/Mpc}$, $\Omega_{\mathrm{m}}=0.3$, $\Omega_{\Lambda}=0.7$.
We set the primordial helium and metal mass abundances to $Y=0.24$ and
$Z=0.06$ respectively. The virial radius $R_{200}$ is defined as the radius
within which the average matter density is $200$ times higher that the
critical density of the universe $\rho_{c}=3H^{2}/(8\pi G)$ where $H$ is the
Hubble parameter and $G$ the universal gravitational constant. Although this
radius varies from a cluster to another, we set it to a standard value of
$R_{200}{=}1.8$ Mpc for all clusters. The symbol
$\text{\ooalign{-\kern-3.00003pt-\cr$z$\cr}}_{0}$ (resp.  – $z$ ) denotes the
cosmological (resp. cosmological plus Doppler) redshift whereas the letter $z$
always represents a vertical coordinate in Cartesian geometry (in spherical
geometry, $r$ is the radial coordinate). $\@vec{\varv}$ and $\@vec{B}$
respectively stand for the velocity and magnetic field. The unit vector in the
direction of the local magnetic field is $\@vec{\hat{b}}=\@vec{B}/B$. The
pressure, density, temperature and entropy are respectively denoted by $p$,
$\rho$, $T$ and $S=T\rho^{1-\gamma}$ (the adiabatic index $\gamma$ equals
$5/3$ for mono-atomic perfect gas). Their respective fluctuations are always
preceded by the lower-case Greek $\delta$ and expressed in a non-dimensional
way: $\delta X=\left(X-\overline{X}\right)/\overline{X}$ where $X$ is one of
the previous thermodynamic quantity and $\overline{X}$ its average at iso-
gravity. The letter $\@vec{g}$ stands for the gravity and $g_{0}$ for its
intensity. The thermal diffusivity, the kinematic viscosity and the magnetic
resistivity are represented by $\chi$, $\nu$ and $\eta$ respectively. The
scale-height of a physical quantity $X$ is $H_{X}\equiv\left(\partial_{r}\log
X\right)^{-1}$. The symbol $<\cdot>$ denotes a spatial average, while the
root-mean square of the physical quantity $X$ is defined as
$\left.X\right|_{\mathrm{rms}}^{2}=<X^{2}>$ since we systematically assume
$<X>=0$ for turbulent fluctuations.

Figure 1: Density $\rho$ (top) and temperature $T$ (bottom) profiles as a
function of radius for three typical ICM models (Ghirardini et al. 2019;
McCourt et al. 2013; Simionescu et al. 2011). Only the range with decreasing
temperature is shown, until $R_{200}$. The grey shaded areas highlight radii
for which the X-IFU will collect less than two million photons in the 0.2-12
keV energy range during a 2-Ms observation.

###  2.1 Thermodynamic models of ICM

We introduce three different density and temperature profiles that will be
later used in Section 3 to show that the observable characteristics of MTI-
driven turbulence in galaxy clusters are robust against the choice of
thermodynamic model. The first density and temperature profiles (in green in
Fig. 1) comes from Ghirardini et al. (2019), in which universal thermodynamic
profiles are presented for typical cool core (CC) and non-cool core (NCC)
galaxy clusters as deduced from a joint X-ray and SZ analysis of a X-COP
sample with 12 different clusters. More specifically, we used the functional
forms from Sections 3.2 and 3.4 of their paper along with the best-fit
parameters given in their Table 3 for CC galaxy clusters. The reason behind
this choice is that dynamically relaxed CC clusters are a priori better suited
to detect MTI turbulence than their NCC counterparts whose dynamics at Mpc
scales is certainly dictated, or at least polluted, by merger events rather
than MHD instabilities. The second profiles (in blue in Fig. 1) are computed
from a 1D-spherical model by McCourt et al. (2013), which we have
independently reproduced. This model assumes hydrostatic equilibrium of the
plasma in a Navarro-Frenk-White gravitational dark matter potential (Navarro
et al. 1997) for a type III accretion history (McBride et al. 2009) with a
concentration parameter of 5 and no effect of thermal conduction. Finally, as
our closest cosmic neighbour, current X-ray missions have dedicated large
exposure times to very deep observations of the Perseus cluster (Fabian et al.
2003, 2006), which is a relaxed CC cluster (Simionescu et al. 2011). Perseus-
like thermodynamic profiles (in red in Fig. 1) are therefore well suited for
this study and we chose them to complete our set of ICM profiles. In practice,
we extracted the data corrected for clumping of the Perseus north-west arm,
available in Simionescu et al. (2011), and we fitted them with the same
functionals as those used in Ghirardini et al. (2019). The detailed form of
the three thermodynamic profiles can be found in Appendix A. For the three
models, only radii between $0.2R_{200}$ and $R_{200}$ are considered, as this
range ensures that all temperature profiles are decreasing with radius, thus
meeting the primary criterion for the MTI to be triggered. The chosen outer
radius is equal to the largest radius at which thermodynamic data were
available from observation in Simionescu et al. (2011); Ghirardini et al.
(2019).

All regions with decreasing temperature are not necessarily bright enough to
make their observation in X-ray possible in a reasonable amount of exposure
time, even with the large effective collecting area of the ATHENA/X-IFU
(Barret et al. 2016). The plasma emissivity is indeed proportional to the
squared density (but also non-trivially depends on the temperature and
metallicity; Rybicki & Lightman 1979). Assuming uniform temperature and
metallicity fields, the knowledge of the density profile determines the line-
of-sight integrated emissivity (i.e. the emission measure Eq. (14)) and
therefore the expected number of photons in a certain range of energy for a
given exposure time, when combined with the X-IFU effective area. We choose to
virtually observe bright enough cluster regions only, which would provide more
than $2\times 10^{6}$ photons in the $0.2-12$ keV range during a 2-Ms X-IFU
observation of a Perseus-like cluster. In our case, it consists in excluding
regions beyond $0.6R_{200}$ (grey shaded areas in Fig. 1-4-5). Below this
threshold (i.e. beyond the latter radius), we consider X-IFU observation
impossible because the uncertainties related to the Poisson noise due to a
lack of photons certainly exceed the instrumental limitations.

###  2.2 Numerical MHD model

We now detail the physical numerical MHD model used to study the MTI at
saturation. First, the equations and the numerical methods implemented to
evolve them are introduced. We then describe the initial hydrostatic
equilibrium along with the numerical parameters of our Braginskii-MHD
simulation of MTI turbulence.

####  2.2.1 Equations and numerical methods

The model relies on the usual system of compressible MHD equations in
conservative forms, where stratification and non-ideal effects are accounted
for; namely magnetic resistivity and anisotropic heat and momentum diffusion.
In practice, we use the same equations as in Parrish et al. (2012a) Eqs. (3-9)
with an additional magnetic resistive term and without the Bremsstrahlung
cooling term. This system of equations is closed thanks to the usual equation
of state for a perfect gas.

As a consequence of the ICM dilute plasma regime, the transport of heat and
momentum is anisotropic with respect to the direction of the local magnetic
field. The Braginskii heat flux and viscous stress tensor take the following
forms:

$\mathcal{Q}=-n_{\mathrm{e}}k_{\mathrm{B}}\chi\@vec{\hat{b}\hat{b}}\cdot{\@vec{\nabla}}T,$
(1)
$\Pi=-3\rho\nu\left(\@vec{\hat{b}\hat{b}}:\nabla\@vec{\varv}-\frac{1}{3}\nabla\cdot\@vec{\varv}\right)\left(\@vec{\hat{b}\hat{b}}-\frac{1}{3}\mathrm{I}\right),$
(2)

where we recall that $\@vec{\hat{b}}=\@vec{B}/B$ is the unit vector in the
direction of the magnetic field.

These Braginskii-MHD partial differential equations are solved in Cartesian
geometry using IDEFIX, a new finite-volume MHD code for astrophysical fluid
dynamics (Lesur et al. 2023). Amongst the vast choice of available Riemann-
solvers, we selected HLLD associated with a third-order reconstruction scheme
which proved to be robust enough. The physical quantities are integrated in
time using a third-order Runge-Kutta algorithm. We developed and included two
additional physical modules accounting for both the anisotropic heat and
momentum diffusions. Standard tests are presented in Appendix B, validating
the implementation of the Braginskii operators. All parabolic terms can be
integrated explicitly at each time step, but a Runge-Kutta-Legendre (RKL)
super time-stepping scheme is also available to speed up this integration. We
opted for the latter after making sure that the results we obtained with both
RKL and fully explicit integrations were identical for a reduced set of
numerical setups.

####  2.2.2 Simulation configuration and hydrostatic equilibrium

The Braginskii-MHD equations are integrated in a local cubic box of size
$L{=}1.5$, with lengths normalised by the sixth of the temperature scale-
height $H_{T}$ (i.e. $H_{T}/6{=}1$ in this setup). The box is thus $H_{T}/4$
long and spans $25\%$ of the temperature scale-height with 256 points
regularly spaced in each direction. Velocities are normalised by the thermal
velocity $\varv_{\mathrm{th},0}{=}\sqrt{T_{0}}$ at the bottom of the
atmosphere. The magnetic energy is normalised as a kinetic energy and magnetic
fields therefore by $\sqrt{\rho_{0}}\varv_{\mathrm{th},0}$. In these units,
times are normalised by $t_{0}{=}H_{T}/(6\varv_{\mathrm{th},0})$. The
simulation is integrated for $100t_{0}$ (roughly $8$ turnover times, see
Section 4). The numerical parameters of the setup are the thermal diffusivity
$\chi{=}1.5\times 10^{-2}$, the Prandtl number $\mathrm{Pr}{=}0.06$ and its
magnetic counterpart $\mathrm{Pm}{=}4$. We also define the magnetic Reynolds
number $\mathrm{Rm}=\left.\varv\right|_{\mathrm{rms}}/(k_{i}\eta)$, where
$\left.\varv\right|_{\mathrm{rms}}$ and $k_{i}$ are respectively the root-mean
square velocity and the integral scale that will be determined a posteriori
from the simulation.

In what follows, $(\@vec{e_{x}},\@vec{e_{y}})$ is the horizontal plane and
$-\@vec{e_{z}}$ the direction of the gravity $\@vec{g}=-g_{0}\@vec{e_{z}}$.
The local model of stratified atmosphere is initialised with the hydrostatic
equilibrium from Parrish et al. (2012a):

$T(z)=T_{0}\left(1-\frac{z}{H_{T}}\right),$ (3)
$g_{0}=\frac{2+\alpha}{H_{T}}T_{0},$ (4)
$\rho(z)=\rho_{0}\left(1-\frac{z}{H_{T}}\right)^{1+\alpha},$ (5)

where $T_{0}$ and $\rho_{0}$ are respectively the temperature and density at
the bottom of the atmosphere, and $\alpha$ a free non-dimensional parameter
controlling the level of stratification. We set $\alpha{=}2$, $H_{T}{=}6$,
$T_{0}{=}\rho_{0}{=}1$. All components of the initial velocity field follow a
random white noise with an amplitude of $10^{-4}$. Both horizontal components
of the magnetic field are initialised with an amplitude of $10^{-5}$. In the
horizontal directions, we chose periodic boundary conditions (BC) for all
fields and, in the vertical direction, we implemented a quasi-periodic BC as
described by Berlok & Pessah (2016a). This vertical BC allow us to hold both
the background hydrostatic equilibria and the background temperature profile
all along the simulation, while periodising the velocity, the magnetic field
and the thermodynamic perturbations. Thus, we do not need to ’sandwich’ the
MTI-unstable layer between two stable layers with isotropic heat conduction,
as usually done in such MTI configurations (PL22b; McCourt et al. 2011;
Parrish & Stone 2005).

###  2.3 Virtual X-IFU observations

We finally sketch how to construct synthetic X-IFU observations from our
numerical MHD simulation. The first step is to extract, from a Braginskii-MHD
simulation, 3D spatial $(x,y,z)$-representations of the density, temperature
and velocity fields adapted to a mock observation. The procedure is quite
technical, and thus described in details in Appendix C.1. The geometrical
configuration of the final box is always a parallelepiped rectangle, with
equal lengths in the plane of the sky $(\@vec{e_{y}},\@vec{e_{z}})$ but
elongated along the $\@vec{e_{x}}$-axis. The latter is assumed to be aligned
with the line of sight of the observation and the $\@vec{e_{z}}$-axis with the
cluster’s outward radial direction (see Fig. 2 for an illustrative sketch).
Each direction is sampled with $128$ different cells, which are therefore not
cubic but rather rectangular. Their volume is
$\ell_{x}\times\ell_{y}\times\ell_{z}$, with $\ell_{y}{=}\ell_{z}{<}\ell_{x}$.

Each cell is assumed to radiate in X-ray according to the APEC emission
spectrum for collisionally ionised diffuse gas (Smith et al. 2001), with no
galactic absorption considered: we checked that taking it into account with
the PHABS model and a hydrogen column density of $5\times 10^{20}\
\mathrm{cm}^{-2}$ leads to a decrease in flux only by a factor of two in the
$0.2-1$ keV range and to almost no flux loss beyond $1$ keV. A synthetic
observation is performed thanks to the E2E simulator SIXTE (Dauser et al.
2019), with a total exposure time of $2$ Ms. We refer to this procedure as the
forward problem.

Such a virtual observation can be reverse-engineered by fitting the mock
spectra in the $0.2-12$ keV range with the APEC model using the X-ray fitting
package XSPEC, after gathering pixels into bins, in order to reconstruct the
observed thermodynamic and velocity fields. This procedure is referred to as
the inverse problem.

The whole pipeline (sketched in Fig. 3) is somewhat idealised because no
background noise, no foreground absorption and no vignetting effects are
included. It is however sufficient for the purpose of this exploratory study.
More details about the technical work required to solve the forward and
inverse problems can be found respectively in Appendix C.2 and C.3.

Figure 2: Problem geometry and schematic of statistical fluctuation
cancellation for an observation of an ICM at Perseus redshift through the
X-IFU, which observes the plane of the sky $(\@vec{e_{y}},\@vec{e_{z}})$. The
coloured boxes can represent either temperature, density or velocity
fluctuations. The colour scale is arbitrary but blue and red tones are of
opposite signs, and the more opaque the stronger the fluctuation. The
dependency of the fluctuations with $y,z$ is overlooked for clarity.

##  3 Observational phenomenology of MTI turbulence in the ICM

This section aims at obtaining a first phenomenological estimate of the
measurable properties of MTI-driven turbulence in the ICM as a function of the
radial coordinate, in terms of both injection length and observable 2D
turbulent kinetic and buoyancy potential energies. For this purpose, we make
use of the three thermodynamic models presented in Section 2.1. Physical
quantities in this section are dimensional and either deduced from the local
values or from the local scale-heights of the aforementioned thermodynamics
models at a given radius. Although those models are global in essence, here
they are only used for their local properties (which however vary with
radius).

We first introduce the physics of the MTI and of its saturation mechanism. We
then present scaling laws for the injection length and for the root-mean
square of the 3D velocity and density fluctuation fields of MTI-induced
turbulence, theoretically derived by PL22a,b. We finally assess, from an
observational point of view, how much the intensity of the measurable 2D
fluctuations is affected by the statistical cancellation of 3D fluctuations
with opposite signs present along the line of sight. These first results will
provide guidance on where to point the X-IFU if we were to detect features of
magnetised turbulence induced by the MTI in ICM haloes.

Figure 3:  Schematic diagram of the pipeline to build a X-IFU synthetic
observation and to reconstruct the physical fields from such an observation:
the forward and inverse problems are solved separately. The tool and/or model
used at each step is indicated below the respective boxes.

###  3.1 Physics of the MTI

The simplest MTI-favourable configuration consists in a dynamically negligible
horizontal magnetic field threading a plasma layer with temperature gradient
and gravity both pointing downwards. When a blob of fluid is subsonically
displaced from its initial equilibrium position, it is in pressure equilibrium
with its new surrounding but remains thermally connected to its initial
neighbours thanks to the dragging of the magnetic field lines. Provided that
conduction parallel to the magnetic field is fast enough, the perturbation is
then isothermal. A rising blob will then be hotter (and less dense) than the
neighbouring fluid and keep rising through buoyancy, triggering the MTI.

There are three ways for this instability to be inhibited, namely through
viscous damping, magnetic tension and entropy stratification in a stable
atmosphere (with respect to the Schwarzschild criterion), which are specific
of short (for the two first effects) and long wavelengths respectively. On
scales in between, the MTI can fully develop up to a state of saturated
turbulence in the ICM on timescales ${\sim}$Gyr, shorter than the Hubble time,
with a maximal growth rate $\omega_{T}=\sqrt{g_{0}/H_{T}}$. The MTI way of
saturating remains however unclear. PL22a,b argue that the instability
saturates in such a way that the energy injected is roughly balanced by
anisotropic thermal diffusion at all scales, leading to theoretical scaling
laws that they were able to numerically verify for the kinetic and buoyancy
potential energies as well as for the injection length. But other authors
(Parrish et al. 2012b; McCourt et al. 2013) claim that MTI saturation is best
described by a standard convection-like mixing-length theory, which would lead
to other dependencies of the kinetic energy on the local or global atmospheric
model. It could be that the former saturation mechanism is best suited to
describe MTI turbulence on scales smaller than typical ICM scale-heights,
while the latter mixing-length theory would describe MTI turbulence on larger
scales, with a transition regime occurring in regions where the instability
length scale becomes of the order of the temperature scale-height. In this
paper, we make intensive use of the scaling laws from PL22a,b, therefore
fostering a local approach to MTI turbulence, which we subsequently find to be
justified for ICM radii sufficiently small where X-ray emissivity is high
enough to make observations possible (see Section 3.2); however, global models
are needed to eventually settle this debate.

###  3.2 Length scale of the MTI

The scaling relation derived by PL22a,b for the injection length of MTI-driven
turbulence reads:

$\ell_{i}\approx 1.1\frac{(\chi\omega_{T})^{\frac{1}{2}}}{N},$ (6)

with $N$ the Brunt-Väisälä frequency of a perfect gas and $\chi$ the Spitzer
thermal diffusivity111In PL22a,b, the thermal diffusivity absorbs an
additional $\left(\gamma-1\right)/\gamma$ factor with respect to the classical
expression Eq. (8). In the current work, this physical prefactor is included
in the numerical prefactor of the scaling laws Eqs. (6)-(9)-(11). No
suppression factor is considered.  (Spitzer 1962) respectively defined
according to:

$N=\sqrt{\frac{g_{0}}{\gamma H_{S}}},$ (7) $\chi=4.98\times
10^{31}\left(\frac{k_{\mathrm{B}}T}{5\
\mathrm{keV}}\right)^{\frac{5}{2}}\left(\frac{n_{\mathrm{e}}}{10^{-3}\
\mathrm{cm^{-3}}}\right)^{-1}\ \mathrm{cm^{2}s^{-1}},$ (8)

where $n_{\mathrm{e}}$ is the electron number density. In PL22a,b, Eq. (6) is
given without the numerical prefactor but the tabulated data are sufficient to
recover it (its value was confirmed by the corresponding author, priv. comm.).
The MTI injection length is not always small compared to $H_{T}$, especially
beyond $0.6R_{200}$ and this may prevent us from using local models to study
MTI dynamics in these regions. We consider that the use of local simulations
remains justified to model MTI dynamics at radii below $0.6R_{200}$, as
$\ell_{i}{\lesssim}0.5H_{T}$ there for the three ICM models. The regions where
this criterion is no longer fulfilled are located at large cluster radii,
where the previously introduced criterion on the minimal number of photons
required for a given exposure time is not met anyway (grey shaded areas in
Figs. 1-4-5). This further justifies our local approach to MTI turbulence, and
the use of the scaling laws from PL22a,b. We note that the expected injection
length of MTI-driven turbulence at $0.6R_{200}$ is ${\sim}0.7$ Mpc, which is
not negligible compared to the injection length typically associated with
turbulence driven by mergers, accretion, shocks, etc. In the next subsection,
we assess the levels of turbulence induced by the instability.

Figure 4: Theoretical injection length of turbulence induced by the MTI (top)
and the same quantity divided by the local temperature scale-height (bottom)
as a function of the radius for three models of ICM. The colour encoding is
the same as in Fig. 1. The dash-dotted black line is the X-IFU field of view
at Perseus redshift. Again, the grey shaded areas highlight radii at which a
X-IFU observation will not provide enough photons. These radii also roughly
correspond to regions where the hypothesis of locality for MTI-driven
turbulence may break down (and so would the scalings Eqs. (6)-(9)-(11)).

###  3.3 Turbulence levels of the MTI

To assess the expected levels of turbulence driven by the MTI in the ICM, we
use the scaling relations for the root-mean square velocity
$\left.\varv\right|_{\mathrm{rms}}$ and density fluctuations
$\left.\delta\rho\right|_{\mathrm{rms}}$ derived in PL22a,b:

$\left.\varv\right|_{\mathrm{rms}}^{2}\approx
0.024\frac{\chi\omega_{T}^{3}}{N^{2}},$ (9)
$\frac{g_{0}^{2}}{2N^{2}}\left.\delta\rho\right|^{2}_{\mathrm{rms}}\left(\frac{N}{\omega_{T}}\right)^{\frac{3}{2}}\propto\left(\frac{\chi\omega_{T}^{3}}{N^{2}}\right)^{\frac{3}{4}}.$
(10)

While the former relation is provided as such in their paper and dimensionally
consistent, the latter is provided without the prefactor, which must be
dimensional in this case because of the weaker (to the power $3/4$) dependency
of the buoyancy potential energy on $\chi\omega_{T}^{3}/N^{2}$ making the
right-hand side of Eq. (10) not the dimension of a specific energy. This means
that the turbulent saturation of this quantity is controlled by one or several
other physical processes as resistivity or viscosity for instance (Perrone,
priv. comm.). Such residual dependencies on both the magnetic and the usual
Prandtl numbers ($\mathrm{Pm}=\nu/\eta$ and $\mathrm{Pr}=\nu/\chi$
respectively) have been numerically highlighted at least in 2D (PL22a).
Further theoretical and numerical work would be needed to overcome this
limitation and to derive the full dependencies. Absent such a fully self-
consistent scaling law, we chose to dimensionalise the numerical prefactor,
which we determined to be $0.003$ thanks to the data available in PL22b, by
the root-square of the thermal velocity $\varv_{\mathrm{th}}$ at
$k_{\mathrm{B}}T{=}5$ keV. This is the easiest way of making this scaling
dimensionally consistent without using any MTI-related quantities, which would
otherwise introduce unwelcome dependencies of the potential energy on MTI-
related parameters that are not seen in the results from PL22a,b. The scaling
relation for the buoyancy potential energy finally reads:

$\frac{g_{0}^{2}}{2N^{2}}\left.\delta\rho\right|^{2}_{\mathrm{rms}}\left(\frac{N}{\omega_{T}}\right)^{\frac{3}{2}}\approx
28\
\mathrm{\left(cm/s\right)}^{\frac{1}{2}}\times\left(\frac{\chi\omega_{T}^{3}}{N^{2}}\right)^{\frac{3}{4}}.$
(11)

Figure 5: Intensity levels of MTI turbulence. Dashed line: root-mean square of
the 3D turbulent velocity field as a function of the radius in the ICM. Full
line: root-mean square of the 3D turbulent line-of-sight velocity (top) and
temperature (bottom) fluctuations. Dotted line: same but for the estimated 2D
projected quantities after integration along the line of sight. The dash-
dotted black lines are X-IFU detectability limits (determined in Section 4)
for a typical $2$-Ms observation of a Perseus-like cluster at $0.25R_{200}$.
The colour encoding follows Fig. 1.

Eqs. (9)-(11) translate into moderate values of the root-mean square velocity
$\mathcal{O}(100\ \mathrm{km/s})$ (dashed lines on the top plot in Fig. 5) and
temperature fluctuations222MTI-induced turbulence is expected to be subsonic.
If so, the density and temperature fluctuations are anti-correlated and the
same scaling law can be used for both fluctuation fields, as
$\left.\delta\rho\right|_{\mathrm{rms}}=\left.\delta T\right|_{\mathrm{rms}}$.
$\mathcal{O}(5\%)$ (full lines on the bottom plot), in the observable regions
of interest. According to PL22a,b, the MTI turbulent velocity field at
saturation exhibits an anisotropy with respect to the direction of gravity;
with the vertical (i.e. radial) component of the velocity being larger than
its horizontal (i.e. azimuthal) counterparts. For illustrative purposes, we
consider here that the root-mean square of the line-of-sight velocity field
(full lines on the top plot in Fig. 5) is smaller than the root-mean square
velocity itself by a factor ${\sim}2.5$ (this value will be further justified
by the simulation in Section 4), rather than the usual $\sqrt{3}$ factor for
isotropic turbulence. Indeed, X-ray spectrometers like the X-IFU have only
access to the line-of-sight velocity because it is deduced from the
measurement of the Doppler shift of spectral lines. The relative contribution
of the horizontal component of the velocity field to the line-of-sight
velocity increases as the X-IFU points further away from the cluster centre,
as illustrated in Fig. 2. Indeed, at large ICM radii, the projection of the
vertical (i.e. radial) velocity on the line-of-sight direction is weak,
especially at locations where the plasma emissivity is high. This geometrical
effect has two unfavourable consequences for the possible detection of the
MTI. On the one hand, the anisotropy of the kinetic energy components makes
the observed line-of-sight velocity, which mostly correlates with the
horizontal velocity in cluster haloes, weaker than its purely vertical
counterpart and therefore harder to measure. On the other hand, the horizontal
velocity does not correlate with density fluctuations; unlike the vertical
velocity, due to the convective nature of the instability. Combined together,
these two effects might limit the opportunity to detect relevant signatures of
the MTI in the ICM. A naive solution to observe the vertical component of the
velocity field would be to look at the ICM arbitrarily close to its centre.
However, this is not possible since we are aiming at regions with decreasing
background temperature in relaxed CC clusters, and central regions are likely
more disturbed by AGNs.

###  3.4 Observation along the line of sight

The values of the root-mean square fluctuations shown in Fig. 5 (full lines)
are for the 3D line-of-sight velocity and the 3D temperature fluctuation
fields. However, these 3D fields are not available as such from an observation
but their 2D counterparts integrated along the line of sight are because the
ICM is optically thin. More precisely, the observed line-of-sight velocity is
often assimilated to the emission-weighted velocity (Roncarelli et al. 2018;
Cucchetti et al. 2019) and the observed temperature to the spectroscopic
temperature (Mazzotta et al. 2004), respectively defined as:

$\varv_{\mathrm{ew}}=\left.\int\rho^{2}\varv_{\mathrm{los}}ds\right/\int\rho^{2}ds,$
(12)
$T_{\mathrm{s}}=\left.\int\rho^{2}T^{0.25}ds\right/\int\rho^{2}T^{-0.75}ds,$
(13)

where $s$ is the line-of-sight coordinate (Fig. 2). Accordingly and for later
use, we also define the emission measure which tracks the total plasma
emissivity:

$EM=\int\rho^{2}ds.$ (14)

When observing or integrating along the line of sight, fluctuations with
opposite signs will statistically cancel each other, especially when the
average size of turbulent eddies is small with respect to the length over
which significant emission takes place (which is precisely the length over
which the fields should be integrated). We now aim at quantifying the
reduction in signal due to this observational effect, illustrated in Fig. 2.

Hints about the behaviour of the root-mean square and dispersion of the
emission-weighted velocity and of the second-order velocity structure function
after integration along the line of sight can be gained from previous works by
Churazov et al. (2012), Zhuravleva et al. (2012), ZuHone et al. (2016), Clerc
et al. (2019), and Mohapatra et al. (2022). Many different models have been
developed, accounting for the two-dimensionality of the observable fields or
the variation of the background thermodynamic fields over the field of view
for example. The present phenomenological discussion does not require such
sophisticated derivations. So we chose the simplest model, namely the root-
mean square velocity along a single line of sight (Eq. (5) from Clerc et al.
2019, which is identical to Eq. (A9) in Zhuravleva et al. 2012), to infer the
decrease in signal due to fluctuation-cancellation along the line of sight. In
both equations, the brackets denote an average over independent realisations
of a 1D velocity field with the same underlying power spectra. In our case,
the assumption made when using this simple model is therefore to assume that
the root-mean square of a single realisation of an emission-weighted 2D
velocity field is equivalent to the root-mean square of independent
realisations of a 1D velocity field, postulating the same underlying power
spectra for all processes. Although this hypothesis may not hold in general,
we deem the estimates obtained from it sufficiently enlightening as a zeroth-
order assessment of the signal reduction due to the cancellation of
fluctuations along the line of sight. Using this toy-model for the different
density profiles (with the MTI turbulent energy spectra that will be later
presented in Section 4, Fig. 8), we obtain first estimates of the root-mean
square emissivity-weighted velocity after integration along the line of sight
for MTI-induced turbulence. In the case of temperature fluctuations, the
spectroscopic weighting scheme introduces an additional complexity with
respect to the simple emissivity weighting. However we checked, thanks to
simulated turbulent fields, that the levels of fluctuation-cancellation of
temperature fluctuations are similar for both weighting schemes. We therefore
use the same toy-model Eq. (5) from Clerc et al. 2019 to anticipate the loss
of signal due to the cancellation of fluctuations along the line of sight for
both the emissivity-weighted velocity field and the spectroscopic temperature
fluctuations. These forecasts are shown as dotted lines in Fig. 5.

The results of this section are well summarised in Fig. 5. The latter overall
shows that the root-mean square of the observable spectroscopic temperature
fluctuations (dotted lines on the bottom panel) is below the root-mean square
of the respective 3D temperature fluctuation field (full lines on the same
plot). The only effect responsible for that is the cancellation of opposite-
sign fluctuations when observing along the line of sight. The situation is
slightly less straightforward in the case of the velocity field. The MTI-
driven turbulent 3D velocity field has a pristine root-mean square intensity
deduced from Eq. (9) (dashed lines on the top panel in Fig. 5). However, only
the line-of-sight component of the velocity field is accessible to an
observation. In cluster peripheries, the line of sight is mostly aligned with
the azimuthal direction rather than the radial direction (at least wherever
the plasma emissivity is higher). As shown in PL22a,b, the MTI saturates in
such a way that the vertical component of its velocity field carries most of
the total kinetic energy. As a result, the root-mean square of the 3D line-of-
sight velocity field (before integration along the line of sight, full lines
on the top plot) is ${\sim}2.5$ smaller than the pristine root-mean square of
the full 3D velocity field (dashed lines on the top plot in Fig. 5). Then, the
cancellation of fluctuations along the line of sight brings further down the
root-mean square of the emissivity-weighted velocity field (dotted lines on
the top panel in Fig. 5). Our analysis suggests that the estimated ratio
between the actual root-mean square of a 3D MTI-like fluctuation field and its
measurable line-of-sight integrated 2D counterpart ranges from 2.6 to 5.6
according to the density profile used and the ICM radial coordinate. Keeping
this phenomenology in mind, we assess in Section 4 whether such moderate
levels of fluctuations
($\left.\varv_{\mathrm{ew}}\right|_{\mathrm{rms}}{\sim}10\ \mathrm{km/s},\
\left.\delta T_{\mathrm{s}}\right|_{\mathrm{rms}}{\sim}0.5\%$ at $0.25R_{200}$
in a Perseus-like cluster) would be detectable with our baseline instrumental
X-IFU configuration (the corresponding detectability limits are shown in Fig.
5), by modelling the observable ICM magnetised dynamics more quantitatively
thanks to a Braginskii-MHD simulation and to X-IFU synthetic observations.

##  4 Quantitative numerical study with virtual observations

Armed with the previous phenomenological estimates, we now characterise more
precisely the statistical properties and the spatial structure of MTI dynamics
in the ICM and assess to what extent these characteristics may be recovered
from an observation through the X-IFU. For this purpose, we first present
diagnostics derived from a local Braginskii-MHD numerical simulation of MTI
turbulence. We then discuss the reason why the saturation levels seen in the
simulation are not representative of those found in the real ICM by comparing
the numerical against ICM plasma regimes, and we rescale the turbulent
fluctuation fields of the simulation accordingly. Finally, we perform two
X-IFU mock observations of a snapshot of the rescaled MHD simulation in the
stage of developed turbulence. We evaluate the quality of the reconstructed
fields for each run and we bring to light the inherent anisotropy of MTI-
induced turbulence in the second mock observation.

Figure 6: Time evolution of key MTI quantities. Top: Potential (in green),
vertical (in dark purple), and horizontal (in gold) kinetic energy densities
and vertical (in dark blue) and horizontal (in light blue) magnetic energy
densities. Bottom: Kinetic energy anisotropy $\delta_{K}=\sqrt{2K/K_{h}}$.
Figure 7: Snapshots of MTI turbulent fields at $t=100t_{0}$. Top: Vertical
$(x,z)$ cuts at $y=0.75$. Bottom: Horizontal $(x,y)$ cuts at $z=0.75$. From
left to right: Norm of the velocity and magnetic fields (in code units) and
temperature fluctuations (in $\%$ of the background temperature $T(z)$, Eq.
(3)). Magnetic field lines are superimposed as black lines on the temperature
fluctuations.

###  4.1 Simulation of MTI-driven turbulence

The time evolution of key quantities related to the MTI are shown in Fig. 6.
In the top panel, the average volume density of the potential, kinetic and
magnetic energies during the simulation, respectively defined as
$U=g_{0}^{2}/(2N^{2}){\times}{<\rho\delta\rho^{2}}>$, $K={<\rho\varv^{2}>}/2$
and $M={<B^{2}>}/2$, are presented. In the linear regime, the MTI displays an
exponential growth phase before saturating in a turbulent state. The
anisotropy between the vertical and horizontal components of the kinetic
energy is quantified by the bias $\delta_{K}=\sqrt{2K/K_{h}}$, with
$K_{h}=K_{x}+K_{y}$ (bottom panel in Fig. 6), which is higher during the
linear growth phase than at saturation, where it settles around 2.5. This is
the value we used to convert from $\left.\varv\right|_{\mathrm{rms}}$ to
$\left.\varv_{\mathrm{los}}\right|_{\mathrm{rms}}$ in Fig. 5 introducing the
qualitative phenomenology of MTI observations. This strong anisotropy is a
well-known feature of the MTI (PL22a,b). The instability is indeed driven by
the buoyancy and tends to fight its physical origin, namely the temperature
gradient. The latter is brought to isothermality via preferentially vertical
convection-like fluid motions though the anisotropic conductive heat flux also
comes into play in this process. We note that Braginskii-MHD equations with
anisotropic viscosity support a fluid version of the firehose instability
(Kunz et al. 2012). We do not see such a small-scale instability develop in
our simulation, certainly on account of the numerical dissipation at the grid
level and of the low magnetic Reynolds number of the simulation.

Although the magnetic field initially grows as a result of the MTI, the
magnetic field evolution shows no sign of an exponential kinematic dynamo
phase after the MTI saturates because of the value of the magnetic Reynolds
number, which is about $5.4$ in our simulation. This is below the critical
magnetic Reynolds number $\mathrm{Rm_{c}}=35$ identified by PL22b for the
small-scale fluctuation dynamo to initiate a kinematic growth with MTI
turbulence. Therefore, the magnetic field strength here never reaches
equipartition with the turbulent kinetic energy; unlike in the higher-
resolution fiducial run from PL22b which has $\mathrm{Rm}=110$.

Fig. 7 shows key MTI turbulent fields at the very end of the simulation (at
$t{=}100t_{0}$): vertical and horizontal snapshots of the velocity and
magnetic field norms, and of the fluctuation temperature field. Because the
magnetic and velocity fields are divergence-free, the vertical anisotropy
related to the different components of the magnetic and kinetic energies
reflects into the shape of the turbulent structures which are more elongated
along the vertical direction than in the horizontal directions (leftmost and
centre slices in Fig. 7). Such an anisotropy is also seen in the temperature
fluctuation field (rightmost slices). Another interesting feature of
anisotropic heat diffusion is the formation of almost isothermal magnetic
field lines with possibly strong temperature gradients across them (rightmost
slices in Fig. 7). Observing such temperature fluctuations in the ICM could
thus indirectly shed light on the magnetic field geometry, if anisotropic heat
flux is active in such plasma.

In Fig. 8, we look at the shell-integrated spectral energy volume densities at
saturation (at $t{=}100t_{0}$). In the case of the kinetic energy, this
quantity is:

$E_{K}(k)=\sum_{k\leq|\@vec{k}|\leq k+\Delta
k}\frac{1}{2}\sum_{i=x,y,z}\left|\mathcal{F}[\sqrt{\rho}\varv_{i}](\@vec{k})\right|^{2},$
(15)

where $\sqrt{\rho}\varv_{i}$ needs to be replaced by $B_{i}$ for computing the
spectra of the magnetic energy. For the potential energy, it is:

$E_{U}(k)=\sum_{k\leq|\@vec{k}|\leq k+\Delta
k}\frac{g_{0}^{2}}{2N^{2}}\left|\mathcal{F}\left[\sqrt{\rho}\delta\rho\right](\@vec{k})\right|^{2}.$
(16)

$\mathcal{F}$ is the Fourier transform of the field between brackets and
$\@vec{k}=k_{x}\@vec{e_{x}}+k_{y}\@vec{e_{y}}+k_{z}\@vec{e_{z}}$ the
wavevector. Taking the Fourier transform of the previous fields is justified
here for the $x$ and $y$ coordinates because the BC are periodic in these
directions. This is also justified in the vertical $z$ direction here thanks
to the quasi-periodic BC which periodises the velocity and magnetic field and,
in the case of the thermodynamic fields, their fluctuations rather than the
fields themselves. All fields present a typical phenomenological picture of
developed turbulence in which energy cascades from larger to smaller scales in
spectral space, though on a relatively small range of scales here. The large-
scale flow is characterised by the injection scale $\ell_{i}$, defined as

$\ell_{i}=\frac{2\pi}{k_{i}}=2\pi\frac{\int k^{-1}E_{K}dk}{\int E_{K}dk}.$
(17)

In the MHD simulation, we have $k_{i}\approx 28L^{-1}$, meaning that the
injection length is about $22\%$ of the box length. The turnover time
$\ell_{i}/\varv_{\mathrm{rms}}$ is therefore ${\sim}13t_{0}$.

Figure 8: Energy spectra of the potential (green), kinetic (purple) and
magnetic (yellow) energy volume densities.

###  4.2 Rescaling of the simulation

We aim at making virtual observations at $0.25R_{200}$ in a Perseus-like
cluster but the dimensional parallel heat diffusivity used in the simulation
$\chi=1.5\times 10^{-2}\varv_{\mathrm{th},0}H_{T}/6$ is roughly two orders of
magnitude less than the value expected from the Spitzer diffusivity Eq. (8) at
the same radius. We therefore anticipate, from the scaling relation Eq. (9),
that the raw dimensional root-mean square velocity in our MHD simulation is
one order of magnitude less than what would be found in the ICM. The same
argument holds for the potential energy. For instance, Eq. (9) is equal to
$14$ km/s when used with the simulation parameters given in 2.2.2 and then
dimensionalise with the thermal velocity $\varv_{\mathrm{th},0}$ of the
Perseus cluster at $0.25R_{200}$. But Eq. (9) can also directly be used with
the physical parameters deduced from Perseus thermodynamic profiles at the
same radius as done in Section 3, Fig. 5, in which case we find
$\left.\varv\right|_{\mathrm{rms}}{\sim}55$ km/s. Consequently, to make a
virtual observation, the fluctuation fields of the simulation first need to be
rescaled according to Eqs. (9)-(11) to match the expected intensity of MTI-
induced turbulent fluctuations in the realistic ICM diffusion regime.
Similarly, the scaling law Eq. (6) predicts the physical injection length of
MTI turbulence in the ICM, which is not the one that we see in the simulation
(when the lengths are dimensionalised by $H_{T}/6$) since the simulation is
not in the right regime. Eq. (6) thus provides a new length scale from which
the box size of the MHD simulation can be dimensionalised, by matching the
expected physical injection length with the one found in our simulation.

This physically motivated rescaling is always conducted before any synthetic
observation is performed in the next section. A peculiar rescaling is fully
characterised by the final injection length $\ell_{i}$ and root-mean square of
the emission-weighted velocity
$\left.\varv_{\mathrm{ew}}\right|_{\mathrm{rms}}$ and spectroscopic
temperature fluctuations $\left.\delta T_{\mathrm{s}}\right|_{\mathrm{rms}}$.

Figure 9:  The fiducial synthetic observation OBSfid based on the Braginskii-
MHD simulation rescaled with the values of Eqs. (6)-(9)-(11) at $0.25R_{200}$
in a Perseus-like cluster. From left to right: emission measure fluctuation
$\delta EM$, spectroscopic temperature fluctuation $\delta T_{\mathrm{s}}$ and
emission-weighted velocity $\varv_{\mathrm{ew}}$ maps. Top: line-of-sight
integrated input fields from the rescaled MHD simulation. Bottom: output
fields reconstructed from the synthetic observation OBSfid. The colourbars are
set according to the input quantities. The values of the thermodynamic
fluctuation fields are given in percents of the average background quantities
(see Eq. (36)).  Figure 10:  The custom synthetic observation OBScustom (at
$0.25R_{200}$ with Perseus density profile), based on the optimally custom-
rescaled Braginskii-MHD simulation. From left to right: emission measure
fluctuation $\delta EM$, spectroscopic temperature fluctuation $\delta
T_{\mathrm{s}}$ and emission-weighted velocity $\varv_{\mathrm{ew}}$ maps.
Top: line-of-sight integrated input fields from the rescaled MHD simulation.
Bottom: output fields reconstructed from the synthetic observation OBScustom.
The colourbars are set according to the input quantities. The values of the
thermodynamic fluctuation fields are in percents of the average background
quantities (see Eq. (36)).

###  4.3 Virtual observations of MTI-induced turbulence with the X-IFU

We now consider the final state (after ${\sim}8$ turnover times) of the MHD
simulation examined in the previous sections, and virtually observe its
rescaled dynamics through the X-IFU according to the methodology described in
Section 2.3. Solving the forward problem, we obtain a mock observation, which
we can then reverse-engineer by solving the inverse problem to reconstruct the
emission measure, temperature and velocity fields. In the next sections, we
display the maps derived from the synthetic observations. We qualitatively
(and quantitatively in the Appendix D) compare them to the respective pristine
input data from the rescaled Braginskii-MHD simulation, numerically integrated
along the line of sight.

Figure 11: Diagnostic tools for probing the anisotropic structure of turbulent
fields. From left to right: second-order horizontal and vertical structure
functions and their ratios for the emission measure fluctuations (top) and
emissivity-weighted velocity (bottom). Full (resp. dashed) lines correspond to
the quantities computed from the rescaled MHD simulation on a X-IFU–like
(resp. Voronoi) grid with $12$ independent MTI turbulent fields. The dots
represent the same ratios, but derived from a single rescaled field of the MHD
simulation used as input to OBScustom (top maps in Fig. 10). They are
naturally noisier than those computed from the statistics of 12 independent
fields, but also more anisotropic in average in the case of the emission
measure fluctuations. Stars show the ratios as derived from the fields
reconstructed from OBScustom (bottom maps in Fig. 10). The $y$-axis of the
leftmost and centre panels showing the structure functions are in arbitrary
units but similar for each row, so that the horizontal and vertical structure
functions can be directly compared.

####  4.3.1 Two synthetic observations

We virtually point the X-IFU at $0.25R_{200}$ in a mock galaxy cluster with
Perseus-like thermodynamic profiles ($\rho{\sim}1.6{\times}10^{-27}\
\mathrm{g/cm^{3}},\ T{\sim}6.2\ \mathrm{keV},\ H_{\rho}{\sim}0.3\
\mathrm{Mpc},\ H_{T}{\sim}2.8\ \mathrm{Mpc}$) and at redshift
$\text{\ooalign{-\kern-3.00003pt-\cr$z$\cr}}_{0}{=}0.0179$ (the corresponding
comoving distance is $77$ Mpc). At this cosmological redshift, the field of
view of the X-IFU is about $\mathrm{fov_{X-IFU}}=120$ kpc (see Fig. 4). The
mock observation and field reconstruction pipelines are run two times for
different input fluctuation fields:

  * •

In the fiducial observation $\mathrm{OBSfid}$, we take the fluctuation fields
from the Braginskii-MHD simulation in the phase of developed turbulence at the
very end of the simulation, and rescale their intensities and length scales
according to the phenomenological discussion from Section 3, Fig. 5. This
translates into $\ell_{i}{\approx}315$ kpc (that is
$2.6\mathrm{fov_{X-IFU}}$),
$\left.\varv_{\mathrm{ew}}\right|_{\mathrm{rms}}{\approx}10$ km/s and
$\left.\delta T_{\mathrm{s}}\right|_{\mathrm{rms}}{\approx}0.5\%$. This run
will allow us to formally determine the X-IFU detectability limits that we
introduced in the phenomenological discussion from Section 3, Fig. 5. We see a
posteriori that the fluctuation fields reconstructed from this synthetic
observation are too noisy to derive precise physical diagnostics of
turbulence. This is why we performed a second virtual observation.

  * •

Theoretical and observational uncertainties related to the estimates from the
phenomenological discussion from Section 3 necessarily remain because of all
the assumptions that we made. We therefore lead the following formal exercise
to better understand what physical information could be deduced from a mock
X-IFU observation if the fluctuation fields of the MTI ended up being
detectable: we perform a second virtual observation $\mathrm{OBScustom}$, in
which the properties of the fluctuation fields are optimally customise. We set
$\ell_{i}{\approx}84$ kpc,
$\left.\varv_{\mathrm{ew}}\right|_{\mathrm{rms}}{\approx}50$ km/s and $\delta
T_{\mathrm{s}}|_{\mathrm{rms}}{\approx}5\%$, which ensure that the fluctuation
levels are above the X-IFU detectability limits identified in the previous
run. OBScustom corresponds to a kind of best-case scenario in which the
intensities of the temperature fluctuation and velocity fields are
respectively increased by factors of $10$ and $5$ with respect to the fiducial
synthetic observation OBSfid.

In each case, we simulated an observation of $2$ Ms. Although such long
observations are not currently forecasted with the ATHENA/X-IFU, we stand by
this value for the needs of this formal and prospective exercise. We note that
Roncarelli et al. (2018) used similar exposure times. 2 Ms of observation
leads to $N_{\mathrm{ph}}\sim 15\times 10^{6}$ photons in the $0.2-12$ keV
range in both cases. Such a total number of photons is quite low given the
very large exposure time used, the reason being that we are looking at the
cluster periphery where the MTI is expected to grow, rather than its centre as
was done, for instance, in Roncarelli et al. (2018) and Cucchetti et al.
(2019). Given the small number of photons, pixels need to be gathered into
bins using the Voronoi tessellation method (Cappellari & Copin 2003) to obtain
satisfying spectral fits. We found that grouping pixels into regions with
${\sim}4{\times}10^{4}$ photons in average was sufficient in this respect.
This leads to ${\sim}340$ Voronoi regions in total. The extracted spectra are
then fitted with the APEC model using the X-ray fitting package XSPEC.

####  4.3.2 X-IFU maps of integrated and reconstructed quantities

To assess the quality of the 2D fields inferred from a synthetic observation,
they should be compared to the actual input 3D physical quantities integrated
along the line of sight. We therefore computed the numerical counterparts of
the physical quantities Eqs. (12)-(13)-(14) from the rescaled MHD simulation
on a 2D X-IFU–like grid divided into Voronoi regions:

$\varv_{\mathrm{ew}}=\left.\sum_{i}\rho_{i}^{2}\varv_{x,i}\right/\sum_{i}\rho_{i}^{2},$
(18)
$T_{\mathrm{s}}=\left.\sum_{i}\rho_{i}^{2}T_{i}^{0.25}\right/\sum_{i}\rho_{i}^{2}T_{i}^{-0.75},$
(19) $EM=\sum_{i}\rho_{i}^{2}\ell_{x},$ (20)

where $i$ runs over all cells pertaining to a given Voronoi region, therefore
summing on cells with close enough position in the sky but possibly very
different positions along the line of sight. In the context of subsonic
turbulence, we are equally interested in the simulated and observed
thermodynamic fluctuations $\delta T_{\mathrm{s}}$ and $\delta EM$, whose
cumbersome derivation is deferred to Appendix E.

Fig. 9 shows X-IFU Voronoi-tesselated maps from the synthetic observation
OBSfid, physically motivated by Section 3, Fig. 5. The emission measure
fluctuation (leftmost maps) and emission-weighted velocity (rightmost maps)
fields are partially recovered, although they look very noisy with respect to
their respective input maps. In the case of the spectroscopic temperature
fluctuations (middle panels), the reconstructed signal is overwhelmed by the
observational noise due to both a lack of photons and instrumental
limitations. From the quantitative analysis of the biases and standard
deviations of the reconstructed output fields with respect to the true input
fields presented in Appendix D, we know that the X-IFU detectability limits
for a 2-Ms observation at $0.25R_{200}$ of the Perseus cluster are about
$\left.\varv_{\mathrm{ew}}\right|_{\mathrm{rms,min}}{\sim}20\ \mathrm{km/s}$,
$\left.\delta T\right|_{\mathrm{rms,min}}{\sim}2\%$. We note that, even on the
ideal input maps from the rescaled MHD simulation (top panels), very few
turbulent structures are seen because the MTI injection length at
$0.25R_{200}$ in a Perseus-like cluster is $315$ kpc, which is $2.6$ times
larger than the X-IFU field of view.

Because of both weak fluctuation intensities and large injection length,
OBSfid cannot be used to satisfactorily derive usual physical diagnostics like
turbulent spectra or structure functions. We therefore performed the synthetic
observation $\mathrm{OBScustom}$ with optimally customised parameters to
further check our procedures for building mock observation and for
reconstructing physical fluctuation fields. We stress that these parameters
are not directly motivated by PL22a,b scaling laws. They will however allow us
to estimate to which extent the anisotropic structure of MTI turbulence may be
constrained with the perturbation fields available from a X-IFU observation.
Fig. 10 shows that the fluctuation fields reconstructed from OBScustom (for
which the root-mean square of the input fluctuation fields now exceed the
expected observational noise) are visually well recovered.

Altogether, we now know to which extent the capabilities of the X-IFU would
allow us to retrieve, or not, thermodynamic and velocity fluctuations
according to their expected intensities. More specifically, the X-IFU
detectability thresholds are
$\left.\varv_{\mathrm{ew}}\right|_{\mathrm{rms,min}}{\sim}20\ \mathrm{km/s}$,
$\left.\delta T\right|_{\mathrm{rms,min}}{\sim}2\%$ for a 2-Ms observation of
a Perseus-like cluster at $0.25R_{200}$. In the next section, we assess
whether a signature of MTI turbulence, namely its anisotropic structure with
respect to the direction of the temperature gradient, could be observationally
detected if the fluctuation intensities turned out to be higher (by factors of
5 to 10) than the physical estimates used to scale OBSfid. For this purpose,
we use the synthetic observation OBScustom.

###  4.4 Anisotropic structure functions

We now investigate the detectability of MTI-induced anisotropy in the best-
case scenario, represented by the synthetic observation OBScustom. To this
end, we introduce the horizontal and vertical second-order structure functions
and their ratio:

$\mathcal{SF}_{X}^{h}(y)=<\left[X\left(\@vec{c}+y\@vec{e_{y}}\right)-X\left(\@vec{c}\right)\right]^{2}>,$
(21)
$\mathcal{SF}_{X}^{v}(z)=<\left[X\left(\@vec{c}+z\@vec{e_{z}}\right)-X\left(\@vec{c}\right)\right]^{2}>,$
(22)
$\mathcal{R}_{X}(r)=\left.\mathcal{SF}_{X}^{h}(r)\right/\mathcal{SF}_{X}^{v}(r),$
(23)

where $X$ is either $\delta EM$ or $\varv_{\mathrm{ew}}$, the spatial average
runs over the centres $\@vec{c}$ of the Voronoi regions. We set $y{=}z{=}r$ in
the last equation. We ignore the spectroscopic temperature fluctuations as, in
our setup, the physical information they contain is redundant with, and of
worse quality than, that found in the emission measure fluctuations. In
isotropic turbulence, scalar structure functions does not depend on the
particular direction along which they are computed and the quantity
$\mathcal{R}_{X}$ is thus equal to unity at all scales (statistically speaking
at least) for any isotropic scalar field $X$. This ratio provides somehow a
measurement of the anisotropic structure of a turbulent field (Rincon 2006).
$\mathcal{R}_{X}>1$ indicates that the vertical injection length $\ell_{v}$ is
greater than the horizontal injection length $\ell_{h}$ (corresponding to
vertically elongated structures) because the second-order horizontal and
vertical structure functions $\mathcal{SF}_{X}^{h,v}$ are expected to tend
towards $0$ for $y,z{\ll}\ell_{h},\ell_{v}$, towards the same value
${<}X^{2}{>}$ when $y,z{\gg}\ell_{h},\ell_{v}$, and to increase in-between.

This new diagnostic is used to characterise the anisotropy of the emission
measure fluctuation and emissivity-weighted velocity fields from the best-case
synthetic observation OBScustom. All relevant quantities related to this
diagnostic are presented in Fig. 11. Dots and stars on the rightmost panels
show the ratio $\mathcal{R}_{X}$ as derived from the input fields of this mock
observation (top maps in Fig. 10), and from the reconstructed fields (bottom
maps in Fig. 10), on the corresponding Voronoi grid; whereas the horizontal
and vertical structure functions $\mathcal{SF}_{X}^{h,v}$ are plotted in the
leftmost and centre panels respectively. In all cases but the structure
functions of the emission-weighted velocity field, data computed from the
input fields of the virtual observation (dots) and from solving the inverse
problem (stars) are in broad agreement. This is actually expected when the
statistical and systematic errors (gathered into the dispersion $\sigma_{X}$
in this work) are small enough with respect to the quantity $X$ itself, as
previously demonstrated by ZuHone et al. (2016), Roncarelli et al. (2018), and
Cucchetti et al. (2019) for isotropic second-order structure functions. In the
case of the reconstructed emissivity-weighted velocity field from OBScustom,
the statistical dispersion $\sigma_{\varv_{\mathrm{ew}}}$ with respect to the
true input field is of the order of the root-mean square of the true input
field. The structure functions computed from the reconstructed emissivity-
weighted velocity field (green stars on the leftmost and centre bottom plots
in Fig. 11) are therefore uniformly biased with an additional
$\sigma^{2}_{\varv_{\mathrm{ew}}}$ term with respect to the structure
functions computed from the true input field (green dots). This effect is
independent of the direction in which the structure function is computed, so
it does not affect the ratio of the structure functions itself.

We overall see that the emission measure fluctuation field is relatively
anisotropic $\mathcal{R}_{\delta EM}{\gtrsim}2$ while the emission-weighted
velocity field is rather isotropic
$\mathcal{R}_{\delta\varv_{\mathrm{ew}}}{\sim}1$. To check that these specific
features are robust against the variance associated with a single realisation
of a MTI turbulent field, we accumulated better statistics by computing the
same ratios and structure functions (dashed lines in Fig. 11), on the same
Voronoi grid, from $3$ different snapshots of the simulation at saturation,
separated by at least one turnover time. Each snapshot represents, after
integration along the line of sight, $4$ independent realisations of a MTI
turbulent field since $\ell_{i}/L{\sim}22\%$ at saturation. We still get
$\mathcal{R}_{\delta EM}{\sim}2$. To further check that these specific
features does not originate from binning X-IFU pixels into bigger Voronoi
regions, we also computed $\mathcal{R}_{X}$ and $\mathcal{SF}_{X}^{h,v}$ on
the original X-IFU grid (full lines in Fig. 11). In this case, the fields from
the simulation are first degraded at X-IFU pixel resolution and the average in
Eqs. (21)-(22) runs over the centres of X-IFU pixels. Binning pixels into
Voronoi regions does not alter the degree of anisotropy seen in the fields,
although it slightly changes the shape of the ratio $\mathcal{R}_{X}$.

We are therefore able to reliably identify MTI-induced anisotropy in the
emission measure fluctuation field with an ideal 2-Ms observation of a
Perseus-like cluster at $0.25R_{200}$, if the actual levels of MTI turbulent
fluctuations at saturation were to be five to ten times higher than the
current theoretical estimates from Section 3, Fig. 5. Finally, we note that
our technique is broadly applicable to the detection of anisotropic dynamics
beyond the case of the MTI.

##  5 Conclusions and discussion

###  5.1 Summary and main conclusions

The current work was intended as a first step towards bridging the gap between
theoretical astrophysical plasma studies and future X-ray observations with
the X-IFU on board the future European X-ray observatory ATHENA, given the
instrumental specifications targeted before the recent design-to-cost exercise
(the discussion is extended to present X-ray telescopes later in this
section). We characterised the dynamical properties of the MTI, a magnetised
buoyancy instability suspected to operate at fluid scales in the ICM (as
opposed to micro-scale kinetic instabilities) and conducive to the anisotropic
conductive heat flux, and we assessed whether it would be detectable with the
X-IFU.

Our main conclusion comes from the combination of the results of Sections 3
and 4: while the raw physical estimates for MTI fluctuations suggest that they
should be detectable, we have shown that several physical and observational
factors conspire in a non-trivial, cumulative way to put the actual
observational MTI signal close to the X-IFU detectability limits. More
specifically, fluctuation fields from a Braginskii-MHD simulation, rescaled in
a physically motivated way according to the results of Section 3 for a
Perseus-like cluster at $0.25R_{200}$, and used as inputs of the 2-Ms
synthetic observation OBSfid cannot be satisfactorily recovered (but only
marginally so and this was not obvious in the first place). Yet this virtual
observation corresponds in many ways to an ideal scenario because of the very
large exposure time, as well as the absence of background noise and of any
other sources of turbulence but the MTI. Our analysis suggests that MTI-
induced turbulence, if it is indeed described by the physical scaling laws of
PL22a,b in cluster haloes, would not be detectable in the real ICM, even with
the upcoming generation of X-ray spectrometers such as the X-IFU. We
identified several effects conspiring to hamper the detectability of MTI
turbulence in ICM regions amenable to X-ray observations:

  * •

The levels of MTI turbulence at saturation, as computed from the scaling laws
derived by PL22a,b, are only moderate for radii between $0.2$ and $0.6R_{200}$
with $\left.\varv\right|_{\mathrm{rms}}{=}\mathcal{O}(100\ \mathrm{km/s})$,
$\left.\delta T\right|_{\mathrm{rms}}{=}\mathcal{O}(5\%)$ for the three
different ICM models of thermodynamic profiles.

  * •

The contribution of the azimuthal (i.e. horizontal) component over the radial
(i.e. vertical) component of the velocity field is predominant on the
observable line-of-sight velocity in the halo of galaxy clusters, where the
MTI can develop. The former is significantly smaller than the latter because
of the inherent anisotropy of MTI turbulence with respect to the direction of
gravity, the root-mean square of the line-of-sight velocity field is then
$\left.\varv_{\mathrm{los}}\right|_{\mathrm{rms}}{=}\mathcal{O}(50\
\mathrm{km/s})$. This reduction is stronger than in the case of isotropic
turbulence.

  * •

The root-mean square of the measurable turbulent fluctuations are also lowered
because of the statistical cancellation of opposite-sign fluctuations present
along the line of sight due to the expected MTI injection scale (${\sim}300$
kpc) being smaller than the typical length over which significant emission
takes place (${\gtrsim}$Mpc). This effect, which has been statistically
estimated thanks to Eq. (5) from Clerc et al. (2019), leads to
$\left.\varv_{\mathrm{ew}}\right|_{\mathrm{rms}}{=}\mathcal{O}(10\
\mathrm{km/s})$, $\left.\delta T\right|_{\mathrm{rms}}{=}\mathcal{O}(1\%)$. It
is likely to intrinsically limit the ability of current and future X-ray
observations to probe small-scale fluctuations of MTI and other types of
turbulence, and to measure small-scale spectral energy density below some
critical threshold.

  * •

Finally, there is no robust cross-correlation between the horizontal velocity
and the density fluctuations in MTI-induced turbulence, which could otherwise
have been exploited if present.

We made some assumptions and used some simplifications when we
phenomenologically quantified the observable dynamical properties of the MTI:
uncertainties related to the previous estimates necessarily remain. An
additional synthetic observation OBScustom was therefore performed to
anticipate to what extent the conclusions would differ if the estimations of
the observable MTI turbulent levels, as expected from Section 3 and Fig. 5,
are too weak in reality. We have shown that, when the intensity of the
fluctuations is higher than the X-IFU detectability limits, the inherent
vertically anisotropic structure of MTI-driven turbulence can be detected in
the emission measure fluctuation field as recovered from a 2-Ms observation
with the X-IFU. Obviously though, this characteristic is neither a strong
observational signature of, nor specific to, the MTI since many other
processes could drive anisotropic turbulence in the ICM.

More broadly, our study confirms that directly identifying plasma effects in
the internal magnetised dynamics of the ICM remains a difficult observational
task given the intrinsic performance of X-ray instrumentation. We further
discuss the possible detection of anisotropic MTI dynamics at large fluid
scales in 5.3. Such an enterprise may benefit from the advent of large radio
telescopes such as the SKA to combine observations of the ICM dynamics with
observations of its magnetic geometry, enabling the use of cross-diagnostics
to identify magnetised plasma structures in the ICM.

###  5.2 Validity of the scaling laws and magnetic feedback

The phenomenological estimates of the measurable length scales and intensities
of MTI turbulent fluctuations depicted in Section 3 are subservient to the
validity of the scaling laws derived in PL22a,b. Though we do believe that
these relations hold in regions of galaxy clusters amenable to X-ray
observations, on dimensional grounds the scaling for the buoyancy potential
energy Eq. (10) is currently incomplete and might be subject to revision
before the root-mean square of the density and temperature fluctuations can be
self-consistently computed. Furthermore, additional work would be required to
determine whether these scaling laws are robust against the development of
kinetic instabilities triggered by pressure anisotropy. For instance, a well-
known feature of electron-scale instabilities is the reduction and/or
isotropisation of the heat flux (Riquelme et al. 2016; Komarov et al. 2018;
Drake et al. 2021), which would lower the MTI injection scale according to Eq.
(6). The intensity of the MTI thermodynamic and velocity fluctuations would
also suffer from such a reduction of the Spitzer conductivity, if we assume
Eqs. (9)-(11) to hold as such in the presence of micro-scale kinetic
instabilities, thereby further compromising the detection of any relevant MTI
signal from an X-IFU observation.

At this stage, it is also relevant to wonder whether (and, if so, how) the
structure of MTI turbulence depends on the feedback of the magnetic field on
the flow through the Lorentz force when equipartition between magnetic and
kinetic energies is reached. This is indeed a difference between the fiducial
simulation of PL22b and ours because of the higher viscosity and resistivity
we used in our simulation, and which might bias our conclusions on the
possible anisotropy of the relevant fluctuation fields. From the PL22b
fiducial simulation, it seems that the magnetic feedback tends to make the
flow even more anisotropic at equipartition. This peculiarity may, at the same
time, lower the intensity of the observable line-of-sight velocity in the
outskirts of the ICM, and exacerbate the anisotropy of the density
fluctuations, which we were already able to detect with an ideal X-IFU
synthetic observation of the Braginskii-MHD simulation yet not at
equipartition. In any case, determining whether the MTI drives a dynamo in the
ICM requires a clear picture of the ICM effective collisionallity. Such a
picture is still currently missing but is under intense theoretical and
numerical investigation (Rincon et al. 2016; St-Onge & Kunz 2018; St-Onge et
al. 2020; Squire et al. 2019, 2023; Arzamasskiy et al. 2023).

As a final note, we emphasise that the vertical and horizontal injection
lengths of MTI-induced turbulence may be subject to different scaling laws,
although we assumed Eq. (6) for both. This could also bias the degree of
anisotropy of realistic MTI turbulent fields, with respect to those used in
this study.

###  5.3 Future observational strategies

We found that directly detecting features of MTI turbulence in the ICM would
be very demanding, even with the X-IFU. Yet what we learnt from trying to do
so and what this implies for future observations are worth discussing. A
possible observational strategy for the detection of MTI-driven turbulence
could be to aim the instrument at the Perseus cluster and build a mosaic of
X-IFU maps with a shorter exposure time for each observation. Given the large
injection length expected for MTI-driven turbulence, ${\sim}34$ Voronoi
regions (instead of ${\sim}340$ currently) seem enough to satisfactorily map
the X-IFU field of view without degrading the spatial resolution of the
observed field too much. Assuming the same cumulated exposure time and signal-
to-noise ratio requirement, ten observations of $200$ ks each would roughly
lead to the same total number of Voronoi regions with an average global
effective field of view about three times that of X-IFU. The ratio between MTI
injection length and such a global field of view would then be about
${\sim}0.9$, which is quite close to the ratio of $0.7$ used in the virtual
observation $\mathrm{OSBcustom}$, partially justifying the approach adopted
for prospective purposes in the last synthetic observation, at least in terms
of the injection scale (providing similar arguments to justify the boost of
the velocity and thermodynamic fluctuations is more challenging, as discussed
in 5.2). In addition, this strategy would offer some flexibility in the choice
of the pointing configuration, which could then be optimised for the detection
of MTI turbulent features, as explored by ZuHone et al. (2016) with Hitomi in
the case of isotropic turbulence.

We have shown that the MTI-driven anisotropy is better inscribed in the
emission measure (i.e. density) fluctuations than in the emissivity-weighted
velocity field, in the periphery of galaxy clusters at least. The detection of
such density perturbations does not require X-ray spectrometers. However, we
argue that current XMM-Newton, Chandra, and XRISM observatories would not
collect enough photons with a reasonable exposure time to probe the
anisotropic dynamics of the faint cluster haloes. The next generation of X-ray
telescopes such as the ATHENA/X-IFU will increase the photon collecting area
by a factor ${\sim}10$ relative to current X-ray missions. They will therefore
provide key observations of dim ICM outskirts, precisely in those regions most
favourable to the development of the MTI.

###  5.4 Competing sources of anisotropic turbulence in the ICM

The MTI is not the only source of anisotropic turbulence in the ICM. We now
discuss the role of stable entropy stratification and accretion in this
context.

Using idealised high-resolution simulations of compressible or solenoidal
stratified turbulence, Mohapatra et al. (2020, 2021) were able to numerically
relate the ratio between the root-mean square of the vertical
$\left.\varv_{z}\right|_{\mathrm{rms}}$ and the horizontal
$\left.\varv_{h}\right|_{\mathrm{rms}}$ velocity with the Froude number. The
latter is defined as
$\mathrm{Fr}=\left.\varv\right|_{\mathrm{rms}}/(\ell_{i}N)$ and compares the
timescales associated with the turbulence and with the stratification.
Injecting the scaling laws from PL22a,b, we find $\mathrm{Fr}\approx
0.15\omega_{T}/N$ for MTI-driven turbulence, which is about ${\sim}0.1$ in
regions of galaxy clusters amenable to X-IFU observations. This estimation
lies right in the range that Mohapatra et al. (2020, 2021) identified as the
more susceptible to the effects of stratification. In their simulations, the
turbulent forcing is however isotropic (or, in some cases, perpendicular to
the direction of gravity): MTI-induced turbulence exhibits a different
structure and may not be subject to the exact same physics in the presence of
strong entropy stratification. At this stage, it is still unclear whether the
stable stratification will isotropise MTI turbulence, in the stratification
regime relevant to the ICM $(\mathrm{Fr}{\sim}0.1)$, or whether it will stay
vertically anisotropic, on account of its convective nature. Some clues can
however be gained from our simulation and from the most stratified MTI run in
PL22b, which used $N{\approx}0.8\omega_{T},\ \mathrm{Fr}{\sim}0.2$ and
$N{=}\omega_{T},\ \mathrm{Fr}{\sim}0.15$, respectively, and are therefore not
too far from real ICM stratification regimes. These simulations suggest that
the MTI vertical anisotropy remains active despite the strong entropy
stratification. Further progress on this question would require additional
high-resolution numerical simulations of strongly stratified MTI turbulence,
in the regime relevant to the ICM.

On the other hand, Simonte et al. (2022) and Vazza et al. (2018) argue that
local simulations of stratified turbulence are too idealised and that, in real
clusters, global accretion drives, preferentially, radial turbulence with
enough momentum and energy to overcome the effect of the stable
stratification. Accretion could thus be another relevant source of vertical
anisotropy in the ICM. However, with this turbulent forcing being compressible
with Mach numbers $\gtrsim 0.3$, the velocity field is no longer divergence-
free and the anisotropy between the radial and azimuthal kinetic energies may
reflect less on the structure of the turbulent eddies.

High-resolution global simulations including MTI and other magnetised plasma
effects, internal stirring mechanisms, and accretion-driven turbulence are
needed to analyse, more precisely, the competition between different sources
of turbulence in the ICM and to draw a clearer picture of the ICM internal
dynamics in the outskirts of galaxy clusters. Such global MHD simulations
could also be virtually observed through the X-IFU, and their internal
dynamics reconstructed, to assess what turbulent components dominate and could
be identified with a mock observation. Previous larger-scale cosmological
style simulations with anisotropic heat conduction failed to shed light on any
distinctive MTI flow in cluster peripheries (Ruszkowski et al. 2011). PL22a,b
however clearly showed that MTI saturation is very dependent on the level of
parallel thermal diffusivity used. Care should therefore be taken to simulate
the right anisotropic diffusive regime and to control perpendicular numerical
dissipation as much as possible if one wants to detect any MTI flow in such
simulations of structure formation.

###### Acknowledgements.

The authors are grateful to CNES for financial support provided by Didier
Barret for the internships of J. M. K. during which a major part of this work
was conducted. J. M. K. is thankful for the organisation of the SIXTE
workshops by the development team, allowing him a smooth start with the
software. The work benefited from very fruitful exchanges with Lorenzo
Perrone, Henrik Latter, Simon Dupourqué and Alexeï Molin. The authors also
thank Geoffroy Lesur for giving them access to a beta version of IDEFIX and
for his support in the integration of the Braginskii physical modules. This
work was granted access to the HPC resources of CALMIP under allocation
2021/2022-P16006 and the authors thank the associated support team. We are
finally grateful for the generous hospitality of the Wolfgang Pauli Institute,
Vienna, where preliminary results were discussed during the 2022 Plasma
Kinetics workshop.

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##  Appendix A Thermodynamic profiles

All thermodynamic profiles presented in Section 2 and used to derive the
properties of MTI-induced turbulence in Section 3 can be parametrised as
(Vikhlinin et al. 2006; Ghirardini et al. 2019):

$\frac{T(x)}{T_{500}}=T_{1}\frac{\frac{T_{\mathrm{min}}}{T_{1}}+\left(\frac{x}{r_{\mathrm{cool}}}\right)^{a_{\mathrm{cool}}}}{1+\left(\frac{x}{r_{\mathrm{cool}}}\right)^{a_{\mathrm{cool}}}}\frac{1}{\left(1+\left(\frac{x}{r_{t}}\right)^{2}\right)^{\frac{d}{2}}},$
(24)
$\rho(x)=\rho_{1}\frac{\left(\frac{x}{r_{\mathrm{c}}}\right)^{-\frac{\alpha}{2}}}{\left(1+\frac{x^{2}}{r^{2}_{\mathrm{c}}}\right)^{\frac{3\beta}{2}-\frac{\alpha}{4}}}\frac{1}{\left(1+\frac{x^{\theta}}{r_{\mathrm{s}}^{\theta}}\right)^{\frac{\epsilon}{2\theta}}},$
(25)

with $x=r/R_{500}$, $T_{500}=5$ keV and where we simply set $R_{500}{=}1.1$
Mpc for all clusters. All the other quantities are parameters to fit. For the
universal thermodynamic profiles from Ghirardini et al. (2019), the best-fit
parameter values from that paper for CC clusters are directly used. Fits are
performed for Perseus temperature and density profiles (Simionescu et al.
2011), but also for the 1D-spherical accretion model of McCourt et al. (2013)
since the data resulting from the numerical integration of this model are not
perfectly robust against numerical derivation (which we need to obtain the
local scale-height of the different atmospheres). In the case of the density,
the fit is not performed on the data but rather on their logarithm, so that
the weak tail of the density profiles are well fitted too. The best parameter
values are given in Table 3 for the different profiles. We do not fit the data
from the different entropy profiles. Instead, we assume perfect hydrostatic
equilibrium to deduce the entropy scale-height from the temperature and
density scale-heights:

$H_{S}=\frac{H_{\rho}H_{T}}{H_{\rho}-\left(\gamma-1\right)H_{T}}.$ (26)

Table 1: Set of parameters for the temperature and density profiles of the
different thermodynamic models of ICM.

Temperature model | $T_{1}$ | $T_{\mathrm{min}}$ | $r_{\mathrm{cool}}$ | $a_{\mathrm{cool}}$ | $r_{\mathrm{t}}$ | $d$
---|---|---|---|---|---|---
Ghirardini et al. (2019) | 1.3 | 0.29 | 0.061 | 0.74 | 0.40 | 0.66
McCourt et al. (2013) | 0.38 | 0.49 | 0.47 | 3.0 | 2.7 | 9.5
Simionescu et al. (2011) | 1.4 | 0.00 | 0.12 | 10 | 1.3 | 1.7

Density model | $\rho_{1}\left(\mathrm{in}\ 10^{-27}\mathrm{g/cm^{3}}\right)$ | $r_{\mathrm{c}}$ | $\alpha$ | $\beta$ | $r_{\mathrm{s}}$ | $\epsilon$
---|---|---|---|---|---|---
Ghirardini et al. (2019) | 0.020 | 0.041 | 0.80 | 0.49 | 1.2 | 4.7
McCourt et al. (2013) | 22 | 0.26 | 3.4 | 0.83 | 2.0 | 20
Simionescu et al. (2011) | 7.3 | 0.00051 | 0 | 0.90 | 1.4 | 11

333 We enforced $\theta=3$ in all cases and $\alpha=0$ in the case of Perseus
for convergence purposes (Shi et al. 2016; Dupourqué et al. 2023).

##  Appendix B Validation of the Braginskii operators in IDEFIX

We implemented the parabolic operator for anisotropic heat diffusion Eq. (1)
with the centred asymmetric scheme described in Sharma & Hammett (2007,
Section 2.1), though without harmonic mean nor slope limiter. The Braginskii
viscosity Eq. (2) is implemented in a similar way, again without any slope
limiter. The implementation of these operators in IDEFIX is validated thanks
to the methodology and setups described in Parrish et al. (2012a, Sections
3,4). The linear growth rates of both MTI and HBI diagonal modes that we
numerically measured are compared with the theoretical values expected from
the dispersion relation.

For the MTI setup, the tests are run in 2D ($\@vec{e_{y}},\@vec{e_{z}}$) at
either low or high resolution ($64^{2}$ and $256^{2}$, respectively the stars
and the dots in Fig. 12) in a square box of size $L=0.1$. Lengths are
normalised by the third of the temperature scale-height, velocities and
magnetic fields respectively by the thermal velocity $\varv_{\mathrm{th,0}}$
at the bottom of the atmosphere and by $\sqrt{\rho_{0}}\varv_{\mathrm{th,0}}$.
The atmosphere is initialised according to the hydrostatic equilibrium
described by Eqs. (3-5) with $\alpha=1$ and $H_{T}{=}3$. We initially seed
both components of the velocity field with a fundamental divergence-free
diagonal eigenmode $\propto\exp\left(\sigma
t+i\left[k_{y}y+k_{z}z\right]\right)$ with an amplitude of $10^{-4}$ and where
$k_{y}=k_{z}=2\pi/L$. The kinematic viscosity $\nu$ is based on the thermal
diffusivity through the Prandtl number
$\mathrm{Pr}=\left\\{0;0.01;0.06\right\\}$ (respectively the blue, green and
red curves and points in Fig. 12).

The runs are initialised with a purely horizontal magnetic field
$B_{y,0}=10^{-5}$. In this configuration, the dispersion relation is:

$\sigma\left(\sigma+\frac{2}{5}\chi k_{y}^{2}\right)\left[\sigma+3\nu
k_{y}^{2}\left(1-\frac{k_{y}^{2}}{k^{2}}\right)\right]+\sigma
N^{2}\frac{k_{y}^{2}}{k^{2}}+\frac{2}{5}\chi\omega_{T}^{2}k_{y}^{2}\frac{k_{y}^{2}}{k^{2}}=0.$

Regarding the HBI setup, the runs are performed in a box with $L=0.2$,
$H_{T}=2$ and with the same hydrostatic equilibria as Parrish et al. (2012a)
in the corresponding local HBI runs (Section 4.1 in that paper). The magnetic
field is initialised vertically with $B_{z,0}=10^{-5}$ while the initial
velocity field is the same than in the MTI runs. The dispersion relation is
then:

$\sigma\left(\sigma+\frac{2}{5}\chi k_{z}^{2}\right)\left[\sigma+3\nu
k_{z}^{2}\left(1-\frac{k_{z}^{2}}{k^{2}}\right)\right]+\sigma
N^{2}\frac{k_{y}^{2}}{k^{2}}-\frac{2}{5}\chi\omega_{T}^{2}k_{z}^{2}\frac{k_{y}^{2}}{k^{2}}=0.$

Figure 12:  Comparison between the MTI (top) and HBI (bottom) linear growth
rates numerically measured and theoretically computed thanks to the dispersion
relations. The initial velocity perturbation is a fundamental divergence-free
diagonal mode $(k_{y}=k_{z}=2\pi/L)$ with amplitude $10^{-4}$. Different
colour tones are for the different Prandtl numbers. Runs with resolution
$64^{2}$ and $256^{2}$ are represented by stars and dots.

We successfully tried other 2D configurations with higher-order harmonic modes
as well as 3D setups with the gravity set artificially along the
$\@vec{e_{x}}$\- or $\@vec{e_{y}}$-axis to check that the operators are
correctly implemented in all directions.

##  Appendix C Building X-IFU mock observations

In this Appendix, we compile technical details about the post-processing
observational pipeline. The first subsection is specific to our MHD simulation
since it describes how to convert its outputs to a $(x,y,z)$-spatial
representation of the density, temperature and velocity fields suited to mock
observations. The second and third subsections are more generic in the sense
that they indicate how to solve the forward and inverse problems, once the
spatial representation of the different fields has been built.

###  C.1 Designing boxes adapted to synthetic observations

In this section, we elaborate on how to build, from the Braginskii-MHD
simulation, a spatial $(x,y,z)$-representation of the density $\rho$,
temperature $T$ and velocity $\varv_{\mathrm{los}}$ adapted to a mock
observation through the X-IFU with SIXTE. First, the physical box size $L$ is
set by matching the numerical injection length in the simulation with the
chosen dimensional integral length of MTI-driven turbulence, which is either
picked from Eq. (6) for OBSfid or customised for OBScustom. In the same
manner, the fluctuation amplitudes from the MHD simulation at $t{=}100t_{0}$
are rescaled in such a way that their root-mean square match the requested
values, deduced from Section 3, Fig. 5 for OBSfid or again custom-picked for
OBScustom.

Next, we extract the fluctuation fields of the rescaled MHD simulation in the
parallelepiped rectangle extending over the whole box along the line of sight
(i.e. the $\@vec{e_{x}}$-axis), centred on the centre of the box in the
$(\@vec{e_{y}},\@vec{e_{z}})$ plane and extending over
$1.1\times\mathrm{fov_{X-IFU}}$ at least (about 130 kpc at
$\text{\ooalign{-\kern-3.00003pt-\cr$z$\cr}}_{0}=0.0179$) in this plane. In
practice, this translates into a parallelepiped rectangle with $256\times
n^{2}$ cells from the MHD simulation, where $n=\lceil 256\times
1.1\times\mathrm{fov_{X-IFU}}/L\rceil$ is the number of extracted cells in one
direction $\@vec{e_{y}}$ or $\@vec{e_{z}}$ of the sky plane. But virtually
observing an ICM-like resolved and extended X-ray source with SIXTE requires
that the field of view of a single observing pixel contains enough individual
source points. For instance, in the work of Roncarelli et al. (2018), one
X-IFU pixel encompassed ${\sim}8$ cells from their simulation. We find that
${\sim}4$ simulation cells per X-IFU pixel are enough to properly run our
observational pipeline. It is possible that the number of cells extracted from
the MHD simulation in a single direction of the sky plane is less than
${\sim}2$ per X-IFU pixel, depending on the dimensional injection length
chosen to set the size of the simulation box. In such cases (in which both
OBScustom and OBSfid lie), the extracted fluctuation fields need to be
interpolated on a new grid with $128$ cells in each direction, so that
${\sim}4$ cells lie in the field of view of one X-IFU pixel as illustrated in
Fig. 13. This is why the final cells with volume
$\ell_{x}\times\ell_{y}\times\ell_{z}$ are no longer cubic but rectangular
with $\ell_{y}=\ell_{z}<\ell_{x}$.

For both OBSfid and OBScustom, the length $L$ over which the cells are stacked
in the line-of-sight direction (which is the physical length of the simulation
box) may be smaller than the length scale over which significant emission
takes place at $0.25R_{200}$ in Perseus. We therefore rescale the integrated
emissivity in the box in such a way that it matches the expected value from
the integration of the squared density over the full line of sight, that is
the total plasma emissivity (i.e. the emission measure) at $0.25R_{200}$ in
Perseus. This rescaling does not affect the root-mean square of the
fluctuations after integration over the full line of sight themselves because
the latter is self-consistently computed thanks to the toy-model Eq. (5) in
Clerc et al. (2019) (dotted lines in Fig. 5), and the amplitude of the
fluctuation fields rescaled accordingly.

At this stage, we stress that only the fluctuations of the thermodynamic
fields, and not the global (background plus fluctuation) thermodynamic fields,
are extracted from the MHD simulation and rescaled. The case of the velocity
is different from the case of the thermodynamic fields since there is no bulk
motion in the simulation and the field fluctuates around a zero average value.
So we still need a proxy for the background temperature and density profiles
by which the thermodynamic fluctuations will be weighted before being
virtually observed. We chose the following local expression:

$X(z)=X_{0}\left(1-\frac{z}{H_{X}}\right),$ (27)

with $X{=}T,\rho$ and where $H_{X}$ is derived from the Perseus thermodynamic
profiles presented in 2.1. With this choice, we are actually neglecting the
cluster curvature by assuming a purely vertical dependency of the
thermodynamic background profiles. With this hypothesis, the error made on the
background temperature and density is less than $1\%$ when looking at
$0.25R_{200}$ of a cluster at Perseus redshift. However this error can grow up
to $10\%$ for a similar density profile at
$\text{\ooalign{-\kern-3.00003pt-\cr$z$\cr}}_{0}\gtrsim 0.08$. At higher
redshift, the methodology later described in Appendix E to isolate the
fluctuations from the global fields can no longer be used as such, and should
be modified to account for the cluster curvature.

Figure 13:  Schematic of the different grids used in the observational
pipeline. The relative sizes of the grids are consistent with respect to each
other: there is almost one cell from the MHD simulation grid (full grey lines,
left), after rescaling of the box as in OBScustom, in a X-IFU pixel (black
dashed lines, left) and there are ${\sim}4$ cells of the finer grid (light
blue lines, right), refined from the MHD simulation grid within the dotted red
square and used for the synthetic observation, in a pixel of X-IFU. For the
sake of clarity, the absolute sizes of the grids are however not to scale. For
instance, there should be 3168 pixels in X-IFU’s hexagon (in our baseline
instrumental configuration) instead of ${\sim}30$ on the schema, in which
X-IFU border pixels are not squared whereas they should be as well in reality.

###  C.2 Forward problem

The purpose of the observation simulator SIXTE is essentially to create a
photon list from a catalogue of point sources in the sky, according to the
specifications of the instrument chosen for the virtual observation. A
straightforward way to simulate the mock observation of an extended and
resolved astrophysical source is to regard it as a collection of independent
point sources More specifically, the initial spatial $(x,y,z)$-representation
of the density, temperature and velocity fields need to be converted to a
hybrid $(y,z,E)$-representation of the ideal spectra as a function of the sky
position $(y,z)$, and where the energy variable $E$ ranges uniformly from
$0.2$ to $12$ keV, covering the X-IFU energy range with a $1$-eV resolution.
Each cell of this spatial representation is given a set of coordinates on the
sky plane $(\@vec{e_{y}},\@vec{e_{z}})$ according to the position of its
centre in the mock cluster (there are 128 cells stacked in the line-of-sight
direction $\@vec{e_{x}}$, all with the same sky coordinates). Its X-ray line
and Bremsstrahlung emission is modelled thanks to the APEC emission spectrum
for collisionally-ionised diffuse gas (Smith et al. 2001). After summing the
spectra along the line of sight $\@vec{e_{x}}$, we recover a hybrid $(y,z,E)$
representation by placing each integrated spectrum at its corresponding sky
position. This catalogue of point sources is the input to the SIXTE simulator,
which eventually leads to a mock observation of the ICM. In this process, the
important instrumental and geometrical effects taken into account are the PSF,
the response function, the detector geometry and its pixel filling-factor
(${\sim}97\%$). Cross-talk and pile-up effects are simulated as well but
marginal in our case, hence not accounted for in the post-processing step.
More details about the SIXTE software and the response matrices of our
baseline instrumental X-IFU configuration can respectively be found in Dauser
et al. (2019)444https://www.sternwarte.uni-erlangen.de/sixte/ and in Barret et
al. (2016)555http://x-ifu.irap.omp.eu/resources/for-the-community. Our setup
does not simulate the vignetting. Therefore, the same response function can be
used for all group of pixels in the post-processing step regardless of their
off-axis position.

The APEC model takes the metal abundance $Z$, the electron temperature
$k_{\mathrm{B}}T$, the redshift  – $z$ and the norm $\mathcal{N}$ as
parameters. While the temperature is a direct output of the MHD simulation,
the chemical abundance is not self-consistently computed and we simply assume
a uniform metal abundance $Z=0.3Z_{\odot}$ for the modelling of the spectra,
with $Z_{\odot}$ the solar value measured by Anders & Grevesse (1989). The
line-of-sight velocity field $\varv_{\mathrm{los}}=\varv_{x}$ is encoded in
the redshift according to:

$\text{\ooalign{-\kern-3.00003pt-\cr$z$\cr}}=(1+\text{\ooalign{-\kern-3.00003pt-\cr$z$\cr}}_{0})\sqrt{\frac{1+\frac{\varv_{x}}{c}}{1-\frac{\varv_{x}}{c}}}-1,$
(28)

with $c$ the speed of light in vacuum. The norm is expressed as:

$\mathcal{N}=10^{-14}\frac{n_{\mathrm{e}}n_{\mathrm{H}}\mathcal{V}}{4\pi
d_{c}^{2}(\text{\ooalign{-\kern-3.00003pt-\cr$z$\cr}})},$ (29)

where $n_{\mathrm{e}}$ and $n_{\mathrm{H}}$, the electron and hydrogen number
densities, are deduced from the density given the primordial abundances
considered, $d_{c}(\text{\ooalign{-\kern-3.00003pt-\cr$z$\cr}})$ is the
comoving distance and $\mathcal{V}{=}\ell_{x}\ell_{y}\ell_{z}$.

###  C.3 Inverse problem

Solving the inverse problem is to reverse-engineer the virtual observation in
order to recover the thermodynamic and velocity fields. We recall that, for
each observation, we divide the X-IFU grid into ${\sim}340$ Voronoi regions,
from which a spectrum is extracted. These spectra are then fitted with the
APEC model using the X-ray fitting package XSPEC. The spectral fit is done by
minimising the Cash statistic (Cash 1979) and without any a priori knowledge
of the fitting parameters but the cluster cosmological redshift
$\text{\ooalign{-\kern-3.00003pt-\cr$z$\cr}}_{0}$. Since no vignetting effect
is simulated in the forward problem, the same Ancillary Response File (ARF) is
used for all Voronoi regions irrespective of their positions. At this step,
the pixel filling-factor from the X-IFU must be taken into account in the ARF
used for the fit in XSPEC, meaning that the response of all energy channels
are $97\%$ of the response from the ARF used by SIXTE to generate the mock
observation.

##  Appendix D Quantitative analysis of bias and dispersion

Table 2: Bias and dispersion of the output fields recovered from virtual
observations with respect to their input counterparts from the rescaled MHD
simulation.

Observation | $t_{\mathrm{exp}}\ (\mathrm{Ms})$ | $N_{\mathrm{ph}}\times 10^{6}$ | $b_{EM}\ (\%)$ | $b_{T_{\mathrm{s}}}\ (\%)$ | $b_{\varv_{\mathrm{ew}}}\ (\mathrm{km/s})$ | $\sigma_{EM}\ (\%)$ | $\sigma_{T_{\mathrm{s}}}\ (\%)$ | $\sigma_{\varv_{\mathrm{ew}}}\ (\mathrm{km/s})$
---|---|---|---|---|---|---|---|---
OBSfid | 2 | 14.2 | -1.25 | -0.19 | -2.15 | 2.32 | 1.64 | 19.2
OBScustom | 2 | 14.9 | -1.39 | -0.57 | -12.0 | 2.65 | 2.67 | 49.0

666 Column 1: name of virtual observation. Column 2: exposure time. Column 3:
total number of photons obtained. Column 4-6(7-9): average bias (dispersion)
of the emission measure, spectroscopic temperature and emission-weighted
velocity fields respectively. The values of the thermodynamic biases and
dispersions are given in percents of their respective input quantities.

The bias and standard deviation of the output fields coming from solving the
inverse problem can be quantified, with respect to their input counterparts
from the rescaled MHD simulation, by:

$b_{X}=<(X_{\mathrm{obs}}-X_{\mathrm{sim}})/X_{\mathrm{sim}}>,$ (30)
$\sigma_{X}^{2}=<\left(X_{\mathrm{obs}}-X_{\mathrm{sim}}\right)^{2}/X_{\mathrm{sim}}^{2}>,$
(31)

where $X$ is either the emission measure $EM$ or the spectroscopic temperature
$T_{\mathrm{s}}$. In the case of the emission-weighted velocity
$\varv_{\mathrm{ew}}$, the bias and dispersion are rather defined according
to:

$b_{\varv_{\mathrm{ew}}}=<(\varv_{\mathrm{ew,obs}}-\varv_{\mathrm{ew,sim}})>,$
(32)
$\sigma_{\varv_{\mathrm{ew}}}^{2}=<\left(\varv_{\mathrm{ew,obs}}-\varv_{\mathrm{ew,sim}}\right)^{2}>.$
(33)

$<{\cdot}>$ is the spatial average running over all the Voronoi regions. The
subscripts $\mathrm{obs}$ and $\mathrm{sim}$ respectively stand for the
observed and true input quantities. All these quantities are tabulated in
Table 6 for the different synthetic observations. We do not necessarily seek
to compare the quality of the reconstructed fields between the two different
observations since they use different input physical fields anyway. We are
rather aiming at validating the post-processing pipeline by checking that
there is no significant bias or dispersion of the reconstructed fields with
respect to the input quantities, independently for each observation. We are
also looking for a quantitative criterion on the input fluctuation fields to
predict whether they will be visually well recovered (as in OBScustom, Fig.
10) or not (as in OBSfid, Fig. 9) with a X-IFU observation.

The first row in Table 6 shows the biases and standard deviations of the
reconstructed fields with respect to the input quantities in the case of
OBSfid: both the thermodynamic and velocity fields are very well reconstructed
with no significant biases and acceptable dispersions. More quantitatively, we
find dispersions $\sigma_{T_{\mathrm{s}}}{\sim}2\%,\
\sigma_{\varv_{\mathrm{ew}}}{\sim}20$ km/s for the observed temperature and
velocity field. This proves the robustness of our observational post-
processing chain. Yet the fluctuation fields reconstructed from this synthetic
observation are not necessarily visually satisfactorily retrieved, as seen in
Fig. 9. The root-mean square of the input temperature fluctuation and velocity
fields after integration along the line of sight $\left.\delta
T_{\mathrm{s}}\right|_{\mathrm{rms}}{\sim}0.5\%,\
\left.\varv_{\mathrm{ew}}\right|_{\mathrm{rms}}{\sim}10\mathrm{km/s}$ are
smaller than the respective standard deviations
$\sigma_{T_{\mathrm{s}}}{\sim}2\%,\ \sigma_{\varv_{\mathrm{ew}}}{\sim}20$ km/s
of the reconstructed temperature and velocity fields with respect to the input
quantities. We deduce that having $\left.\delta
X\right|_{\mathrm{rms}}\gtrsim\sigma_{X}$, with $X=EM,T_{\mathrm{s}}$ or
$\left.\varv_{\mathrm{ew}}\right|_{\mathrm{rms}}\gtrsim\sigma_{\varv_{\mathrm{ew}}}$
are necessary (albeit maybe not sufficient) conditions to satisfactorily
detect the fluctuation fields from a synthetic observation. In this case,
$\sigma_{X}$ can therefore be seen as a detectability threshold due to the
inherent noise associated with a X-IFU observation: the fluctuation fields
will be detected (i.e. visually well recovered) if their intensities exceed
the noise level associated with the observation. This is why, in the
phenomenological discussion of Section 3, Fig. 5, we used
$\sigma_{T_{\mathrm{s}}}{\sim}2\%$ and $\sigma_{\varv_{\mathrm{ew}}}{\sim}20$
km/s as the typical X-IFU detectability limits for a 2-Ms observation of the
Perseus cluster at $0.25R_{200}$. The case of the emission measure needs more
caution because this extensive physical quantity is proportional to the
squared density: when the root-mean square of the emission-measure and
spectroscopic temperature fluctuations are small enough, they relate according
to $\left.\delta EM\right|_{\mathrm{rms}}{\sim}2\left.\delta
T_{\mathrm{s}}\right|_{\mathrm{rms}}$. In the case of the emission measure
fluctuation field, the factor two puts the root-mean square $\left.\delta
EM\right|_{\mathrm{rms}}{\sim}1\%$ closer to the corresponding detectability
limit $\sigma_{EM}{\sim}2\%$. This explains why the emission measure
fluctuations are, to some extent, visually better recovered than those of
spectroscopic temperature in the case of the mock observation OBSfid (bottom
maps in Fig. 9).

The biases and dispersions of the velocity and thermodynamic fields
reconstructed from the observation OBScustom are given in the second row of
Table 6. In the case of the thermodynamic fields, they are very similar to
those found for the previous mock observation and prove once again the
robustness of the reconstruction. However, the output velocity field is
slightly biased towards negative values and more dispersed than in the first
virtual observation. This is a consequence of the root-mean square of the
input velocity field being larger in this synthetic observation: a similar
behaviour was seen when the Mach number from an input hydrodynamic simulation
was increased in Roncarelli et al. (2018).

##  Appendix E Computation of the thermodynamic fluctuations in the Voronoi
regions

In this respect, we first derive the average thermodynamic quantity
$\overline{X}$ at fixed $z_{i}$ (which represents an iso-gravity line when the
cluster curvature is neglected, see Appendix C.1):

$\overline{X}(z_{i})=\frac{1}{n_{y}\left(z_{i}\right)}\sum_{j}X(y_{j},z_{i}),$
(34)

where $X$ is either the spectroscopic temperature $T_{\mathrm{s}}$ or the
emission measure $EM$. $j$ runs on all the X-IFU pixels of the $y$-axis at a
given vertical position $z_{i}$ on the 2D grid and $n_{y}$ is the number of
such pixels (which depends on $z_{i}$ because of X-IFU’s hexagonal shape). The
same Voronoi region can therefore be accounted for more than once in the
average of a single row. The next step is to compute the average quantity
$<X>$ in a single Voronoi region:

$<X>=\frac{1}{n_{\mathrm{pix}}}\sum_{i}\overline{X}(z_{i}),$ (35)

where $n_{\mathrm{pix}}$ is the number of pixels in the Voronoi regions
considered and $i$ runs on all those pixels, whose vertical position is
$z_{i}$. We finally obtain the thermodynamic fluctuation $\delta X$ in a given
Voronoi region according to:

$\delta X=\frac{X-<X>}{<X>}.$ (36)

We emphasise that this procedure is greatly simplified by the fact that the
cluster curvature can be neglected. Ideally, it should return the same
fluctuation field that we would get if we were to directly integrate the
thermodynamic fluctuations themselves along the line of sight (not weighted by
the background quantities, as done for the velocity field actually). We
checked whether this procedure is robust or not in this sense for diverse
fluctuation fields that we parametrise with different intensities and
injection lengths. In the parameter ranges relevant to the present study, we
find very little influence of these diverse parameters on the robustness of
the method, which proves to be overall satisfying.