# Velocity structure functions in multiphase turbulence: interpreting
kinematics of H$\alpha$ filaments in cool core clusters

Rajsekhar Mohapatra${}^{\href https://orcid.org/0000-0002-1600-7552\,1}$,
Mrinal Jetti${}^{\href https://orcid.org/0000-0003-0287-3246\,2}$, Prateek
Sharma${}^{\href https://orcid.org/0000-0003-2635-4643\,3}$ and Christoph
Federrath${}^{\href https://orcid.org/0000-0002-0706-2306\,1,4}$
1Research School of Astronomy and Astrophysics, Australian National
University, Canberra, ACT 2611, Australia
2 Department of Aerospace Engineering, Indian Institute Of Technology,
Chennai, Tamil Nadu 600036, India
3 Department of Physics, Indian Institute of Science, Bangalore, KA 560012,
India
4Australian Research Council Centre of Excellence in All Sky Astrophysics
(ASTRO3D), Canberra, ACT 2611, Australia
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>(MJ)E-mail<EMAIL_ADDRESS>(PS)E-mail<EMAIL_ADDRESS>(CF)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The central regions of cool-core galaxy clusters harbour multiphase gas, with
gas temperatures ranging from $10$ $\mathrm{K}$–$10^{7}$ $\mathrm{K}$.
Feedback from active galactic nuclei (AGNs) jets prevents the gas from
undergoing a catastrophic cooling flow. However, the exact mechanism of this
feedback energy input is unknown, mainly due to the lack of velocity
measurements of the hot phase gas, which has large thermal velocities.
However, recent observations have measured the velocity structure functions
($\mathrm{VSF}$s) of the cooler phases (at $10$ $\mathrm{K}$ and $10^{4}$
$\mathrm{K}$) and used them to indirectly estimate the motions of the hot
phase. In the first part of this study, we conduct high-resolution
($384^{3}$–$1536^{3}$ resolution elements) simulations of homogeneous
isotropic subsonic turbulence, without radiative cooling. We analyse the
second-order velocity structure functions ($\mathrm{VSF}_{2}$) in these
simulations and study the effects of varying spatial resolution, the
introduction of magnetic fields, and the effect of projection along the line
of sight (LOS) on the $\mathrm{VSF}_{2}$. In the second part of the study, we
analyse high-resolution ($768^{3}$ resolution elements) idealised simulations
of multiphase turbulence in the intracluster medium (ICM) from Mohapatra et
al. (2021a). We compare $\mathrm{VSF}_{2}$ for both the hot ($T\sim 10^{7}$
$\mathrm{K}$) and cold ($T\sim 10^{4}$ $\mathrm{K}$) phases. We also look for
the effect of projection along the LOS, the impact of different models of
idealised heating (volume-weighted vs mass-weighted), and magnetic fields on
the $\mathrm{VSF}_{2}$. For turbulence without radiative cooling, we observe a
steepening in the slopes of the $\mathrm{VSF}_{2}$ upon projection along the
LOS. In our runs with radiative cooling and multiphase gas, we find that the
$\mathrm{VSF}_{2}$ of the hot and cold phases have similar scaling. We note
that introducing magnetic fields steepens the $\mathrm{VSF}_{2}$ of the cold
phase. We also find that at large scales, projection along the LOS steepens
the $\mathrm{VSF}_{2}$ for the hot phase and mostly flattens it for the cold
phase. At small scales, the projected $\mathrm{VSF}_{2}$ are flatter for both
hot and cold phase gas.

###### keywords:

methods: numerical – hydrodynamics – magnetohydrodynamics – turbulence –
galaxies: clusters: intracluster medium

††pubyear: 2021††pagerange: Velocity structure functions in multiphase
turbulence: interpreting kinematics of H$\alpha$ filaments in cool core
clusters–C1

##  1 Introduction

The intracluster medium (ICM) refers to the hot X-ray emitting plasma in
galaxy clusters. In the central regions of relaxed clusters, the ICM is denser
and often has short cooling times ($\sim$ a few $100$ $\mathrm{Myr}$). These
regions are often multiphase and consist of hot ionised regions at $10^{7}$
$\mathrm{K}$ that are co-spatial with cooler atomic and molecular gas phases
at $\sim 10^{4}$ $\mathrm{K}$ and $10$ $\mathrm{K}$, respectively (Hu, 1992;
Conselice et al., 2001; Edge, 2001; McDonald et al., 2012; Werner et al.,
2013, 2014; Tremblay et al., 2018; Gendron-Marsolais et al., 2018; Lakhchaura
et al., 2018; Pulido et al., 2018; Babyk et al., 2019; Olivares et al., 2019;
Boselli et al., 2019; Rose et al., 2019; Vantyghem et al., 2019, 2021). The
cool and dense core is supposed to undergo a runaway cooling flow, till it
forms molecular gas (Fabian, 1994). However, feedback through active galactic
nuclei (AGNs) jets from the central supermassive blackholes (SMBH) of the
clusters injects energy back into the ICM. This energy heats up the core and
prevents a catastrophic cooling flow.

Many different mechanisms have been proposed to be a channel of energy
transfer from AGN jets to the ICM. These include sound waves (Tang & Churazov,
2018; Bambic & Reynolds, 2019), shocks (Yang & Reynolds, 2016), heating by
turbulent dissipation (Zhuravleva et al., 2014), heating by turbulent mixing
(Banerjee & Sharma, 2014; Hillel & Soker, 2016; Fujita et al., 2020), internal
gravity waves (Zhang et al., 2018), cosmic rays (CRs) (Pfrommer, 2013;
Ruszkowski et al., 2018), etc. Measuring velocities of the ICM plasma is
important to constrain these different heating methods and for the physical
understanding of the ICM.

Many observational studies have used both direct and indirect methods to
estimate turbulent velocities of the ICM, summarised in Simionescu et al.
(2019). Direct measurements of hot phase gas velocities are difficult due to
large thermal velocities of the plasma. The Hitomi space telescope (Hitomi
Collaboration, 2016) directly measured velocities of the hot-phase gas by
resolving the line-broadening of Fexxv and Fexxvi lines and revealed low
levels of turbulence. Recently, observations have obtained indirect estimates
of the level of turbulent velocities from X-ray brightness fluctuations
(Zhuravleva et al., 2014; Zhuravleva et al., 2018) and thermal Sunyaev-
Zeldovich (tSZ) fluctuations (Zeldovich & Sunyaev, 1969; Sunyaev & Zeldovich,
1970; Khatri & Gaspari, 2016), which represent line of sight (LOS) electron
pressure fluctuations. Li et al. (2020) (hereafter Li20) have used optical
data to obtain turbulent velocities of ICM gas in the atomic filaments in
several nearby clusters. They have analysed the first-order structure
functions of LOS velocity ($\mathrm{VSF}_{1}^{\mathrm{LOS,obs}}$) and find
them to be steeper than expected from Kolmogorov turbulence theory
(Kolmogorov, 1941, K41). Velocity structure functions ($\mathrm{VSF}$s) and
spatial power spectra are useful diagnostics since they represent the
variation of velocity with scale. Using $\mathrm{VSF}_{1}^{\mathrm{LOS,obs}}$,
Li20 show that the driving scale of turbulence in their sample of clusters
corresponds to the size of AGN-driven bubbles.

Motivated by these observational techniques, recent numerical studies have
also studied the $\mathrm{VSF}$s or velocity power spectra in similar
multiphase ICM environments. In Mohapatra & Sharma (2019), we looked at the
power spectra of density and velocity and studied the effects of radiative
cooling and thermodynamics of the ICM in idealised local box simulations of
size $(40$ $\mathrm{kpc})^{3}$. Hillel & Soker (2020) have analysed the jet-
ICM interaction in a $(40$ $\mathrm{kpc})^{3}$ octant of a galaxy cluster
core. They argue that bubbles inflated by AGN jets can drive turbulence over a
large range of scales, which sets the slope of the first-order velocity
structure function $(\mathrm{VSF}_{1})$. Wang et al. (2021) (hereafter W21)
have simulated a galaxy cluster in a $(1$ $\mathrm{Mpc})^{3}$ cube. They have
studied the $\mathrm{VSF}_{1}$ of the cold and hot phases and the coupling
between them. They show that the gas in the cold phase is magnetically
supported (see also Nelson et al. 2020) and follows a steeper
$\mathrm{VSF}_{1}$ compared to the hot phase, which is expected from free-
falling clouds.

However, Hillel & Soker (2020) use an adaptive mesh refinement (AMR) grid for
their simulations. This sets a variable inertial range of turbulence (due to
varying numerical viscosity), which would make it difficult to analyse the
exact slope of the structure functions. Their simulations are evolved for a
short time and hence do not have multiphase gas. The $\mathrm{VSF}_{1}$ of the
hot and cold phases could be different, as seen in W21. Cooling and the
presence of cold phase gas can also affect the scaling of velocity power
spectra at small scales, as seen in Mohapatra & Sharma (2019). On the other
hand, W21 use static mesh refinement, with higher resolution towards the
denser central region of the cluster. However, these also lead to the spatial
variation of numerical viscosity, which can affect the shape of the
$\mathrm{VSF}_{1}$. All three of these studies have also not looked into the
effect of projection along the LOS on the $\mathrm{VSF}_{1}$. ZuHone et al.
(2016) show that $\mathrm{VSF}_{1}^{\mathrm{LOS}}$ could be steeper than than
the three-dimensional $\mathrm{VSF}_{1}$. Hence to understand the
observational results and their implications, we need high resolution
simulations of multiphase turbulence (with converged slopes of power
spectra/$\mathrm{VSF}$s) in the ICM.

This study is divided into two parts. In the first part, we conduct idealised
simulations of homogeneous isotropic turbulence at subsonic rms Mach numbers
($\mathcal{M}$), without radiative cooling of the ICM. We study the slope of
structure functions and their convergence with resolution. We also quantify
the effects of magnetic fields and projection along the LOS on the
$\mathrm{VSF}$s. In the second part, we analyse the high resolution multiphase
ICM turbulence simulations from Mohapatra et al. (2021a) (hereafter M21). In
these runs we study the $\mathrm{VSF}$s of the hot and cold phases separately.
We compare $\mathrm{VSF}$s with and without magnetic fields and two different
types of idealised thermal heating models (namely heating $\propto\rho^{1}$ or
$\propto\rho^{0}$, where $\rho$ is the density of the gas). We also analyse
the effect of projection using different weightings along the LOS.

This paper is organised as follows. In section 2, we describe our setup for
part 1 (without cooling runs). In section 3, we present the results from our
idealised turbulence runs without cooling. Following this, in section 4, we
present the results from our analysis of multiphase simulations in M21. We
discuss our results and compare them to results from other studies in section
5. In section 6, we discuss the caveats of our study and future prospects. We
finally summarise and conclude in section 7.

##  2 Methods

###  2.1 Model equations

We model the ICM as a fluid using the compressible magnetohydrodynamic (MHD)
equations. The gas follows an ideal gas equation of state. Our model equations
are listed in section 2.1 of M21 (numbered eq. 1a to 1f). For the first part
of this study, we switch off radiative cooling and thermal heating. Thus, the
energy equation becomes (instead of eq. 1c in M21):

$\frac{\partial E}{\partial
t}+\nabla\cdot((E+P^{*})\mathbf{v}-(\mathbf{B}\cdot\mathbf{v})\mathbf{B})=\rho\mathbf{F}\cdot\mathbf{v},$
(1)

where $\rho$ is the gas mass density, $\mathbf{v}$ is the velocity,
$\mathbf{B}$ is the magnetic field,
$P^{*}=P+\frac{\mathbf{B}\cdot\mathbf{B}}{2}$ is the sum of thermal and
magnetic pressure, $P=\rho k_{B}T/(\mu m_{p})$, $\mathbf{F}$ is the turbulent
force per unit mass, $E$ is the total energy density, $\mu$ is the mean
molecular mass, $m_{p}$ is the proton mass, $k_{B}$ is the Boltzmann constant,
and $T$ is the temperature.

###  2.2 Numerical setup

We use a modified version of the FLASH code (Fryxell et al., 2000; Dubey et
al., 2008), with the HLL5R Riemann solver (Bouchut et al., 2007, 2010; Waagan
et al., 2011) for solving the MHD equations. We use a uniformly spaced 3D
Cartesian grid with $L_{x}=L_{y}=L_{z}=L=40$ $\mathrm{kpc}$, where $L$ is the
box size. We use periodic boundary conditions along all three directions. Our
default resolution is $768^{3}$ grid cells. In part 1, we also perform
simulations with $384^{3}$ and $1536^{3}$ resolution elements for checking
convergence.

###  2.3 Turbulence driving

We drive turbulence by using the stochastic Ornstein-Uhlenbeck (OU) process
with a finite auto-correlation timescale (Eswaran & Pope, 1988; Schmidt et
al., 2006; Federrath et al., 2010). We excite only large-scale modes in
Fourier space ($1\leq\absolutevalue{\mathbf{k}}L/2\pi\leq 3$, where
$\mathbf{k}$ is the wave vector) and remove the divergent component. We inject
power as a parabolic function of $\absolutevalue{\mathbf{k}}$, which peaks at
$\absolutevalue{\mathbf{k}}=4\pi/L$, corresponding to half of the box-size.
Turbulence at scales smaller than the driving scales ($\ell<L/3$) develops
self-consistently through a nonlinear cascade. We refer the reader to section
2.1 of (Federrath et al., 2010) for more details of the turbulence driving
method.

###  2.4 Initial conditions

We conduct new simulations only for part-1 of this study (turbulence without
cooling), which provide a benchmark to compare against the more complex
multiphase turbulence simulations. For consistency, we initialise all our
simulations with the same conditions as in M21, with $T=4\times 10^{6}$
$\mathrm{K}$, $n_{e}=0.086\,\mathrm{cm}^{-3}$. For our MHD runs, we set the
initial magnetic field with equal mean and rms components. We set the initial
plasma beta $\beta$ (ratio of thermal to magnetic pressure) to $100$, in line
with observations (Carilli & Taylor, 2002; Govoni & Feretti, 2004). We set the
mean field in the $z$ direction and the rms component as a power-law according
to small-scale dynamo growth following the Kazantsev spectrum (Kazantsev,
1968). The Fourier amplitude of the rms component $\mathrm{B}_{k}\propto
k^{1.25}$, for $2\leq k\leq 20$.

###  2.5 List of simulation models

We have conducted four new simulations without cooling for this study and
analyse three high-resolution simulations with cooling from M21. We list all
of them, including some of the relevant simulation parameters in table 1.

The without-cooling runs are listed in the upper part of the table and the
with-cooling runs in the lower part, indicated by ‘f’ in the label. Most of
these runs use hydrodynamic equations (HD), by setting $\mathbf{B}=0$
throughout the domain. The runs with MHD are denoted by ‘mag’ in the label.
The default resolution is $768^{3}$, indicated by ‘HR’ (high resolution) in
the label. The $384^{3}$ and $1536^{3}$ resolution runs are labelled ‘LR’ (low
resolution) and ‘VHR’ (very high resolution), respectively. We use these
simulations for checking the convergence of our $\mathrm{VSF}$s. For the with-
cooling runs, the number after ‘f’ denotes the parameter $f_{\mathrm{turb}}$,
which is the ratio between the turbulent energy injection rate and the
radiative cooling rate. All our runs have $f_{\mathrm{turb}}=0.10$. In these
runs, ‘f0.10HR’ and ‘f0.10magHR’ have mass-weighted thermal heating with
$Q_{\mathrm{mw}}\propto\rho$ and the ‘f0.10vwHR’ run has
$Q_{\mathrm{vw}}\propto\rho^{0}$, where $Q$ is the thermal heating rate
density. We have described these methods in detail in section 2.2.2 of M21. We
show the rms Mach number of the hot phase gas ($\mathcal{M}_{\mathrm{hot}}$)
in column 5. For the without-cooling runs, this column just denotes the rms
Mach number ($\mathcal{M}$). The hot phase is heated more in the volume
weighted run, so even though the rms velocity in the hot phase is similar to
the mass-weighted run, the Mach number is smaller.

Table 1: Simulation parameters for different runs

Label | Resolution | $\mathrm{f}_{\mathrm{turb}}$ | Thermal heating | $\mathcal{M}_{\mathrm{hot}}$ | Magnetic fields | Radiative cooling |  $t_{\mathrm{end}}$ $(\mathrm{Gyr})$
---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
Without cooling runs |  |  |  |  |  |  |
LR | $384^{3}$ | NA | N | $0.26\pm 0.01$ | N | N | $1.042$
HR | $768^{3}$ | NA | N | $0.27\pm 0.01$ | N | N | $1.042$
magHR | $768^{3}$ | NA | N | $0.17\pm 0.01$ | Y | N | $2.083$
VHR | $1536^{3}$ | NA | N | $0.27\pm 0.01$ | N | N | $1.042$
With cooling runs (from M21) |  |  |  |  |  |  |
f0.10HR (fiducial) | $768^{3}$ | $0.10$ | mass-weighted | $0.91\pm 0.05$ | N | Y | $1.003$
f0.10vwHR | $768^{3}$ | $0.10$ | volume-weighted | $0.23\pm 0.01$ | N | Y | $1.003$
f0.10magHR | $768^{3}$ | $0.10$ | mass-weighted | $0.59\pm 0.05$ | Y | Y | $1.003$

Notes: Column 1 shows the simulation label. We list the without-cooling runs
in upper half of the table and the with-cooling runs in the lower half. The
default resolution of all the runs is $768^{3}$ cells, indicated by the label
$\mathrm{HR}$. We use $\mathrm{LR}$ and $\mathrm{VHR}$ to denote $384^{3}$ and
$1536^{3}$ cells, respectively (also shown in column 2). For our second set of
runs (turbulence with cooling), the number following ‘f’ in the label denotes
$\mathrm{f}_{\mathrm{turb}}$ (also shown in column 3), which is the ratio of
turbulent energy input rate to the radiative cooling rate. Column 4 lists the
type of thermal heating (volume-weighted or mass-weighted) implemented in the
with-cooling simulations, which is by default set to mass-weighted. In column
5, we denote the steady state rms Mach number ($\mathcal{M}_{\mathrm{hot}}$)
of the hot phase gas ($T>10^{7}$ $\mathrm{K}$). In columns 6 and 7, we denote
whether magnetic fields and cooling are switched on, respectively. Runs with
magnetic fields are labelled by $\mathrm{mag}$ and runs with cooling are
labelled with $\mathrm{f}$. Finally, in column 8, we show the end time of each
simulation in $\mathrm{Gyr}$.

##  3 Results: turbulence without cooling

In this section, we describe the results of our turbulence without cooling
simulations. These control simulations are useful to understand the effects of
magnetic fields and projection along the LOS on the structure functions in
homogeneous isotropic turbulence. These will help us interpret our results
from the more complex multiphase turbulence simulations.

###  3.1 Velocity structure functions

According to the K41 model of homogeneous, isotropic, incompressible HD
turbulence, the turbulent statistics in the inertial range, defined by the
scales $L_{\mathrm{driv}}\gg\ell\gg\ell_{\eta}$ attain a universal form
uniquely determined by the energy transfer rate ($\epsilon$). Here
$\ell_{\eta}$ is the dissipation scale and $L_{\mathrm{driv}}$ is the driving
scale of turbulence. Most of the scaling relations of K41 theory, such as
slopes of power spectra and structure functions in turbulent statistics are
valid in this range of scales.

For homogeneous and isotropic turbulence the longitudinal $\mathrm{VSF}$s
depend only on the separation $r$. In this study, we define the first, second
and $p^{\mathrm{th}}$ order $\mathrm{VSF}$s as 111Note that the absolute value
sign in eq. 2d is absent in fluid dynamics literature (Pope, 2000), but
astrophysical studies (such as Li20,W21) use the definition of $\delta v_{r}$
in eq. 2d. But we have mainly studied the $\mathrm{VSF}_{2}$, where this
difference is irrelevant. We also define the $\mathrm{VSF}_{2}$ as the trace
of the second order velocity structure function tensor and do not decompose it
into transverse and longitudinal components. However, we have verified that
these components have similar scaling with $r$.
$\displaystyle\mathrm{VSF}_{1}(r)=\langle\delta v_{r}\rangle,$ (2a)
$\displaystyle\mathrm{VSF}_{2}(r)=\langle\delta v_{r}^{2}\rangle,$ (2b)
$\displaystyle\mathrm{VSF}_{p}(r)=\langle\delta
v_{r}^{p}\rangle,\text{respectively, where}$ (2c) $\displaystyle\delta
v_{r}=|\mathbf{v}(\mathbf{x}+\mathbf{e}_{1}r,t)-\mathbf{v}(\mathbf{x},t)|.$
(2d) $\mathbf{e}_{1}$ is a generalised unit vector and $\langle\rangle$
represents the ensemble average. In the inertial range, the $\mathrm{VSF}$s
are expected to scale as: $\mathrm{VSF}_{p}=C_{p}\epsilon^{p/3}r^{p/3},$ (2e)
where $C_{p}$ is a constant. For $p$ different from 3, there are corrections
to this scaling because of intermittency (She & Leveque 1994).

The first-order velocity structure function ($\mathrm{VSF}_{1}$) has been
recently studied in both observational (Li20) and numerical studies (Hillel &
Soker 2020; W21). The second-order velocity structure function
($\mathrm{VSF}_{2}$) is simply related to the velocity correlation function

$\mathrm{VCF}(r)=\langle\mathbf{v}(\mathbf{x}+\mathbf{e}_{1}r,t)\cdot\mathbf{v}(\mathbf{x},t)\rangle$
(3a) for isotropic homogeneous turbulence, for which the two are related as
(e.g., see Pope 2000)
$\mathrm{VCF}(r)=2v_{\mathrm{rms}}^{2}-2\mathrm{VSF}_{2}(r),$ (3b) where
$v_{\mathrm{rms}}$ is the rms velocity. The velocity power spectrum is the
Fourier transform of the velocity correlation function VCF, and describes the
distribution of kinetic energy with scale. Therefore, we focus on the second
order statistics for the remainder of the study but return to
$\mathrm{VSF}_{1}$ in section 5.4.

####  3.1.1 Effect of resolution

Figure 1: The compensated and normalised second-order velocity structure
function $(\mathrm{VSF_{2}})$ in the absence of cooling for three different
resolutions LR, HR, VHR, averaged over five eddy turnover time scales. The
errorbars indicate the temporal variation. The lines show the compensated fits
to the $\mathrm{VSF}$s in the inertial range. The corresponding ‘m’ values
indicate the slopes of the uncompensated fits. The bottleneck effect dominates
the inertial range in the LR run. The slopes of the $\mathrm{VSF}_{2}$ fits
for the HR and VHR runs, are in close agreement with K41 theory, albeit
somewhat steeper due to intermittency corrections (She & Leveque, 1994). Note
that the errorbars appear large since we show the compensated
$\mathrm{VSF}_{2}$ with a small $y$ range in the plot.

Since we do not include explicit viscosity in our simulations, the viscous
scale ($\ell_{\eta}$) is set by the spatial resolution in our simulations. The
scale $\ell_{\eta}\propto\Delta x$, where $\Delta x$ is the size of a grid
cell. The velocity power spectra and $\mathrm{VSF}$s flatten close to
$\ell_{\eta}$ due to the bottleneck effect, which is seen in both numerical
simulations and experiments (although it is more prominent in the power
spectra than in VSFs; e.g., Schmidt et al., 2006; Donzis & Sreenivasan, 2010).
Hence in simulations we need to have enough spatial resolution to obtain
convergent slopes of the $\mathrm{VSF}$s in the inertial range.

In fig. 1, we show the $\mathrm{VSF}_{2}$ compensated by the K41 slope
($r^{2/3}$) for three different resolutions. For these runs, the driving scale
is the same, at $r/L=0.5$, whereas $\ell_{\eta}$ decreases with increasing
resolution. The $\mathrm{VSF_{2}}$ flattens at the large scales, which
corresponds to the driving scale of turbulence. At smaller scales, it shows a
subtle rise, around $r/L\approx 0.02$ for the VHR run and $r/L\approx 0.04$
for the HR run. This rise is due to the bottleneck effect. In the LR run, the
rise due to the bottleneck effect is close to the driving scale. Hence, the
apparent slope of the $\mathrm{VSF}_{2}$ in the inertial range is shallower.
The steep exponential dip in the $\mathrm{VSF_{2}}$ at small scales indicates
the dissipation scale.

We fit power-law functions to the $\mathrm{VSF_{2}}$, $0.04<r/L<0.1$ for the
HR run and $0.02<r/L<0.1$ for the VHR run. The convergent slope ($\sim 0.69$)
of the power-law fit is slightly steeper than the slope of the fit to our HR
run, which is the standard resolution for all our simulations. The slight
steepening from the K41 slope of $2/3$ in the VHR run may be due to the
effects of intermittency (She & Leveque, 1994). We use this resolution study
to make sure that our turbulence with cooling runs from M21 have enough
separation of scales between $L_{\mathrm{driv}}$ and $\ell_{\eta}$.

####  3.1.2 Effect of magnetic fields

Figure 2: The compensated and normalised $\mathrm{VSF_{2}}$ of the turbulence
without cooling runs with HD (HR) and MHD (magHR). The vertical bars indicate
$1$-$\sigma$ error margins. We fit power-laws to the inertial range. The
slopes to the uncompensated $\mathrm{VSF}$s are indicated as ‘m’ values. The
$\mathrm{VSF}_{2}$ is shallower in the MHD simulation and is closer to the IK
scaling ($r^{0.5}$) than the GS scaling ($r^{2/3}$). Note that the errorbars
appear large since we show the compensated $\mathrm{VSF_{2}}$ with a small $y$
range in the plot.

In fig. 2, we show the compensated $\mathrm{VSF}_{2}$ of the HR and magHR
runs. The $\mathrm{VSF}_{2}$ becomes shallower for the magHR run. The slopes
are slightly steeper than the scaling relation proposed by Iroshnikov (1964)
and Kraichnan (1965), where $\mathrm{VSF}_{2}\propto\ell^{0.5}$ (also known as
IK scaling). They are shallower than the scaling relation proposed by
Goldreich & Sridhar (1995), where $\mathrm{VSF}_{2}\propto\ell^{2/3}$ (GS
scaling). The magHR run has small $\mathcal{M}\sim 0.17$), rms Alfven Mach
number (the ratio of rms velocity and rms Alfven velocity;
$\mathcal{M}_{A}\sim 1.56$) and $\beta\approx 100$. In this parameter regime,
MHD turbulence is nearly incompressible. The small scale eddies could be
anisotropic, with different sizes perpendicular and parallel to the direction
of the large-scale magnetic field. According to Boldyrev (2005), the scaling
of the $\mathrm{VSF}_{2}$ is expected to lie in between the IK and GS scaling,
and the exact exponent is determined by the anisotropy of turbulent
fluctuations. Similar to the HD runs, there maybe a slight steepening from the
theoretical scaling due to the effect of turbulent intermittency, which has
been proposed in Grauer et al. (1994); Politano & Pouquet (1995); Müller &
Biskamp (2000). More recently, Grete et al. (2020, 2021) using parameters
similar to ours, have found the kinetic energy power spectrum to scale as
$\propto k^{-4/3}$, which is also shallower than both $k^{-5/3}$ (GS) and
$k^{-3/2}$ (IK) scaling222The M0.50P1.00isoA1.00H simulation from Grete et al.
(2020) is comparable to our magHR run, but note that they use $\gamma=1$ and
the magnetic field only has a mean component..

####  3.1.3 Effect of projection

Figure 3: The $\mathrm{VSF}_{2}$ of the 3D velocity field (HR and magHR) and
the projected 2D emission-weighted line of sight velocity fields
($\mathrm{VSF}_{2}^{\mathrm{LOS}}$, labelled HR proj and magHR proj) for
turbulence without cooling runs, with and without magnetic fields. The lines
indicate fits to the inertial range of the $\mathrm{VSF}$s. The projected
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$ along the $x-$, $y-$ and $z-$directions have
been added. The slopes of these fits are indicated by ‘$m$’. Upon projection,
the $\mathrm{VSF}$s decrease in amplitude and become steeper, approximately by
a factor of $r^{0.7}$ for both the runs.

In this section, we study the effect of projection on the $\mathrm{VSF}_{2}$,
since we can only directly measure the projected LOS velocities through
observations. Projection can generally have two different effects –
cancellation of velocities along the LOS, and smaller apparent projected
distances compared to the actual 3D distances. The first removes power from
small scales, since smaller eddies (being more numerous) are likely to have
more cancellation along the LOS and this leads to the steepening of the
$\mathrm{VSF}$. On the other hand, the smaller apparent projected distance
shifts power from large scales to small scales, which can lead to flattening
of the $\mathrm{VSF}$.

In fig. 3, we show the $\mathrm{VSF}_{2}$ of the 3D velocity field (HR and
magHR) and the projected LOS velocity ($\mathrm{VSF}_{2}^{\mathrm{LOS}}$),
normalised by the square of the 3D rms velocity $v_{\mathrm{rms}}^{2}$. The
LOS velocity ($v_{\mathrm{LOS}}$) is weighted by emission (using to
$Z_{\odot}/3$ emission table from Sutherland & Dopita (1993)). For example,
$v_{x}$ projected along the $x$ direction is constructed as follows:

$v_{x,\mathrm{emw-
proj-x}}(y,z)=\frac{\int_{-L/2}^{L/2}v_{x}\mathcal{L}\mathrm{d}x}{\int_{-L/2}^{L/2}\mathcal{L}\mathrm{d}x},$
(4)

where $\mathcal{L}=n_{e}n_{i}\Lambda(T)$, $n_{e}$ and $n_{i}$ are electron and
ion densities, respectively and $\Lambda(T)$ is the cooling function333Note
that although we use an emission weight during projection, the runs in section
3 do not have any radiative cooling, with the energy equation given by eq. 1.
Runs in section 4 have radiative cooling, with the energy equation given by
eq. 5. from Sutherland & Dopita (1993). We project the velocity along all
three directions and sum their second order structure functions to improve
statistics. The amplitude of $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ close to the
driving scale ($r/L\sim 0.5$) is slightly smaller than $\mathrm{VSF}_{2}$. At
scales smaller than the driving scale, the $\mathrm{VSF}_{2}^{\mathrm{LOS}}$
is also steeper than $\mathrm{VSF}_{2}$, roughly by a factor of $r^{0.7}$.
This implies that cancellation along the LOS is the dominant effect. ZuHone et
al. (2016) derive a scaling relation for the projected
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$, and expect them to be steeper by a factor
of $r$ compared to the 3D $\mathrm{VSF}_{2}$. The slopes of our
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$ are shallower than their predictions444We
have also tested different projection weights, such as mass and volume, by
replacing $\mathcal{L}$ with $\rho$ and unity in eq. 4, respectively, and find
the steepening of the slope of the $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ to be by
$r^{0.7}$.. Esquivel & Lazarian (2005) use a 3D-Gaussian cube with a 3D
spectral index of $-11/3$ (which corresponds to K41 spectral density). Upon
projection, they also find the projected structure function to be steeper than
the 3D structure function, but the steepening is less than the factor of $r$
(see their fig. 3).

##  4 Results: turbulence with cooling

In this section we analyse the high-resolution turbulence with cooling runs
from M21. In these simulations, we have implemented both the radiative cooling
of the ICM and an idealised feedback heating loop, where the energy lost due
to cooling is injected back into the gas in the form of turbulent kinetic
energy and thermal heat. We implement two different models of thermal heat
feedback, mass-weighted feedback ($Q_{\mathrm{mw}}$) and volume-weighted
feedback ($Q_{\mathrm{vw}}$). The details of our feedback loop implementation
and our model setup are outlined in section 2.2.2 of M21. The energy equation
for these runs is thus given by:

$\frac{\partial E}{\partial
t}+\nabla\cdot((E+P^{*})\mathbf{v}-(\mathbf{B}\cdot\mathbf{v})\mathbf{B})=\rho\mathbf{F}\cdot\mathbf{v}+Q-\mathcal{L},$
(5)

where $Q$ is the thermal heating rate density and $\mathcal{L}$ is the
radiative cooling rate density. The cooling rate $\mathcal{L}=0$ for
$T<10^{4}$ $\mathrm{K}$.

For this part of the study, we have analysed three simulations with
$f_{\mathrm{turb}}=0.1$, which is the ratio between the turbulent energy
injection rate and the radiative cooling rate. The simulation parameters for
these runs are listed in table 1. The volume-weighted probability distribution
functions (PDFs) of the mach number, temperature and pressure for these runs
are shown in figure 12 of M21. In this study, we focus on the turbulent
statistics of the velocity field of these simulations.

Figure 4: First row: integrated total emission along the line of sight (here
the $x$ direction) for gas in the hot phase ($T>10^{7}$ $\mathrm{K}$) for our
turbulence with cooling runs: f0.10HR, f0.10vwHR, f0.10magHR at $t=0.911$
$\mathrm{Gyr}$. Second row: emission weighted line of sight velocity
$v_{\mathrm{LOS}}$ projection maps ($v_{x,\mathrm{hot},\mathrm{emw-proj},x}$)
for the hot phase at the same time snapshot. Third row: similar to first row,
but showing emission from the cold phase gas ($10^{4}$ $\mathrm{K}<T<2\times
10^{4}$ $\mathrm{K}$). Fourth row: similar to second row, but showing
$v_{\mathrm{LOS}}$ of gas in the cold phase ($v_{x,\mathrm{cold},\mathrm{emw-
proj},x}$). The colourbars at the right of each row represent the surface
brightness and the LOS velocities along the $x$ direction, respectively. We
notice more small-scale features in all the projection plots of the f0.10vwHR
run. The weak emission from the hot phase of f0.10HR is due to the lack of gas
above $10^{7}$ $\mathrm{K}$.

###  4.1 2D projection maps

In fig. 4, we show the projections of the mock X-ray emission from the hot gas
and the net emission from the cooler H$\alpha$ emitting gas for our three runs
at $t=0.911$ $\mathrm{Gyr}$. We also present maps of the LOS velocity of gas
emitting in these two wavelength ranges. We describe each of the rows of fig.
4 in the following paragraphs.

####  4.1.1 Mock X-ray emission

In the first row of fig. 4, we show the logarithm of LOS ($x$ direction)
integrated total emission from gas in the X-ray emitting hot phase ($T>10^{7}$
$\mathrm{K}$). The net hot phase emission corresponds to the X-ray surface
brightness measurements in Churazov et al. (2012); Zhuravleva et al. (2014);
Zhuravleva et al. (2018).

Among the different runs in our study, we observe X-ray emission among all
sightlines for all three of our runs. But the net emission from the f0.10HR
run is smaller than the rest, since it has less gas in the hot phase above the
cutoff temperature of $10^{7}$ $\mathrm{K}$ (refer to fig. 12 of M21). Even
though the three runs have similar amplitude of density fluctuations in the
hot phase (fig. 7 of M21), the fluctuations in emission (approximately
$\propto\rho^{2}\sqrt{T}$ for Bremsstrahlung emission from the hot phase) seem
to be the largest for f0.10HR run and the smallest for the f0.10vwHR run. This
could be attributed to the differing nature of fluctuations in the hot phase.
In section 3.4 of M21, we showed that runs with small $f_{\mathrm{turb}}$ and
$\mathcal{M}_{\mathrm{hot}}\ll 1$ show isobaric density fluctuations in the
hot phase, whereas runs with larger values $f_{\mathrm{turb}}$ and
$\mathcal{M}_{\mathrm{hot}}\gtrsim 1$ show adiabatic density fluctuations in
the hot phase. For the f0.10HR run with transonic
$\mathcal{M}_{\mathrm{hot}}=0.91$, the density fluctuations are expected to be
more adiabatic, whereas for the f0.10vwHR run with subsonic
$\mathcal{M}_{\mathrm{hot}}=0.23$, they are expected to be more isobaric.
Adiabatic density fluctuations are expected to produce larger fluctuations in
emission ($\propto\frac{\gamma+3}{2}\delta\rho/\rho$) compared to isobaric
ones of the same amplitude ($\propto\frac{3}{2}\delta\rho/\rho$). The
f0.10magHR run has $\mathcal{M}_{\mathrm{hot}}=0.59$ and shows trends in
between the two extremes. The variations in emission are weak and at small
scales in f0.10vwHR and occur at a larger scale for f0.10magHR and f0.10HR. We
attempt to explain this feature later in this subsection.

####  4.1.2 LOS velocity maps - hot phase

In the second row of fig. 4, we show the projected LOS velocity of the hot
phase gas, weighted by emission ($v_{x,\mathrm{hot},\mathrm{emw-proj},x}$). We
construct this quantity by setting $\mathcal{L}=0$ for $T<10^{7}$ $\mathrm{K}$
in eq. 4. The LOS velocity of the hot phase could be measured by future high-
resolution X-ray spectrographs such as
XRISM555https://heasarc.gsfc.nasa.gov/docs/xrism (and Hitomi Collaboration
2016 in the recent past). These can give us useful information about turbulent
velocities in the hot phase and their variation with scale.

Comparing these projected velocity maps among our simulations, we observe that
their amplitude is largest for the f0.10HR run, smaller for f0.10magHR and
smallest for the f0.10vwHR run. Similar to the hot phase mock X-ray emission
maps, we find more small-scale structures in the f0.10vwHR run. The difference
in the amplitudes of the emission weighted velocities can be attributed to
cancellation along the LOS. The hot phase emission is mostly uniform and
volume-filling at large scales for the f0.10vwHR run. But for the f0.10HR run,
the hot-phase emission has non-uniform spatial distribution and peaks strongly
in localised regions. Hence cancellation of turbulent eddies along the LOS
could play a stronger role upon projection for the f0.10vwHR run.

####  4.1.3 Mock optical emission

In the third row of fig. 4, we show the logarithm of the LOS integrated total
emission from gas in the H$\alpha$ emitting cold phase ($10^{4}$
$\mathrm{K}<T<2\times 10^{4}$ $\mathrm{K}$). This mock cold phase emission is
comparable to the emission from atomic filaments, which have been studied in
many recent observations (Gendron-Marsolais et al., 2018; Olivares et al.,
2019; Boselli et al., 2019).

Analysing the cold phase emission in our simulations, we find that the
emission from the cold phase gas has a large area filling fraction for all
runs. The emission peaks are filamentary for the f0.10HR and f0.10magHR runs.
But for the f0.10vwHR run, the emission mainly comes from smaller point
sources. The size of these cool clouds corresponds to the density of gas with
$10^{4}$ $\mathrm{K}<T<2\times 10^{4}$ $\mathrm{K}$ in these runs (see the
density PDFs of these simulations in fig. A1). The f0.10vwHR run has denser
gas in this temperature range, which correspond to smaller clouds. In
comparison, the f0.10HR and f0.10magHR runs have rarer low temperature gas,
which corresponds to more fluffy cool clouds.

####  4.1.4 LOS velocity maps - cold phase

In the fourth row of fig. 4, we show the projected LOS velocity of the cold
phase gas, weighted by emission ($v_{x,\mathrm{cold},\mathrm{emw-proj},x}$).
We construct this quantity by setting $\mathcal{L}=0$ for $T>2\times 10^{4}$
$\mathrm{K}$ and $T<10^{4}$ $\mathrm{K}$ in eq. 4. These correspond to the LOS
velocity maps obtained from integral field unit spectroscopy (IFU) maps of the
cold phase gas in observations such as Gendron-Marsolais et al. (2018); Sarzi
et al. (2018); Tremblay et al. (2018); Boselli et al. (2019) which were
studied in Li20.

In our runs, we observe similar trends in the LOS velocity maps of hot and
cold phases. The LOS velocities are smaller for the f0.10vwHR run compared to
the other two runs. The small dense clouds in the f0.10vwHR run correspond to
the small-scale features in the emission-weighted LOS velocities. These small-
scale features are also seen in the emission and LOS velocities of the hot
phase gas, which implies that cooling affects the morphology and kinematics of
the hot phase as well, especially at small scales.

###  4.2 Cold and hot gas velocity PDFs

Figure 5: The volume PDF of LOS velocity for gas in the hot (solid lines) and
cold phases (dash-dotted lines) for our three runs, averaged over 28 snapshots
from $t=0.651$ $\mathrm{Gyr}$ to $t=1.003$ $\mathrm{Gyr}$. The three panels
show the PDFs along the three directions ($x$, $y$ and $z$). The shaded
regions around the PDFs denote the temporal variation in the PDF. The hot
phase velocities follow a normal distribution. The cold phase PDFs are flatter
or distorted near the peak, which could correspond to the bulk motion of the
cold clouds and not their dispersion velocities, which could be much smaller.
The rms velocity for the hot and cold phases for different runs are as
follows: f0.10HR $h:221\pm 15$ $\mathrm{km/s},\ c:205\pm 18$ $\mathrm{km/s}$;
f0.10vwHR $h:218\pm 12$ $\mathrm{km/s},\ c:167\pm 11$ $\mathrm{km/s}$;
f0.10magHR $h:185\pm 14$ $\mathrm{km/s},\ c:183\pm 20$ $\mathrm{km/s}$.

In this subsection, we discuss the distribution of gas velocities in the hot
and cold phases in our simulations. In fig. 5, we show the volume PDFs of
these quantities for the three different directions. The solid lines represent
hot gas PDFs and the dash-dotted lines represent the cold gas PDFs. The
velocity dispersions of the gas in different phases are denoted in the caption
of fig. 5.

All the hot phase velocities show a smooth normal distribution. The widths of
these PDFs are similar for all three directions for both hydro runs, which
indicates isotropy. The cold phase velocity PDFs are somewhat distorted near
the peaks. This could be due to the large-scale bulk motion of the clouds,
which dominate the velocity dispersion in the PDFs. One can interpret them as
a sum of many off-centre narrower Gaussians.

The difference between the widths of the PDFs is correlated to the ratio of
densities of the hot and cold phases
$\chi=\rho_{\mathrm{cold}}/\rho_{\mathrm{hot}}$ (see density PDF in fig. A1).
We observe that the f0.10magHR run has equal velocity dispersion for both the
hot and cold phases. Since the amplitude of velocity dispersion is dominated
by the contribution of large-scale modes, we infer that the hot and cold
phases are coupled at large scales for our MHD run, which also has the
smallest $\chi$. The smaller amplitude of velocity dispersion in this run is
due to the conversion of turbulent kinetic energy into magnetic energy.

For the f0.10HR and f0.10vwHR runs, the cold phase has smaller velocity
dispersion compared to the hot phase, implying that the cold phase has slower
large-scale motion compared to the hot phase in these runs. The difference in
the velocity dispersion of hot and cold phases is larger for the f0.10vwHR
run, which also has a larger $\chi$.

###  4.3 Velocity structure functions of multiphase gas

In this subsection, we study the $\mathrm{VSF}$s of the hot and cold phases in
our simulations, which are the main focus of our study. We compare between the
$\mathrm{VSF}$s of the two phases and their variation across our different
simulations. We also study the effect of projection along the LOS on the
$\mathrm{VSF}$s.

####  4.3.1 Hot and cold gas velocity structure functions

Figure 6: The $\mathrm{VSF}_{2}$ of the hot ($T>10^{7}K$, unfilled data
points) and cold ($T<2\times 10^{4}K$, filled data points) phase gas, in our
three turbulence with cooling runs. The $\mathrm{VSF}_{2}$ are normalised by
$v_{\mathrm{rms}}^{2}$. The error bars denote the temporal variation in the
$\mathrm{VSF}_{2}$. The subscripts $\mathrm{h}$ (hot) and $\mathrm{c}$ (cold)
in the legends denote the phase of the gas. We fit separate power-law scaling
relations to the $\mathrm{VSF}_{2}$ of the two phases. We denote the scaling
exponents by $m$ and the phases by the subscripts $h$ and $c$, and list them
in the bottom right part of each panel. For the hydro runs, the
$\mathrm{VSF}_{2}$ of the cold phase has a smaller amplitude than the
$\mathrm{VSF}_{2}$ of the hot phase and is slightly steeper. But for the MHD
run, the $\mathrm{VSF}_{2}$ of the cold phase has the same amplitude at large
scales but follows a much steeper scaling relation at smaller scales. We
discuss the possible implications of these findings in the main text.

Figure 6 shows $\mathrm{VSF}_{2}$ of the hot ($T>10^{7}K$) and cold
($T<2\times 10^{4}K$) phase gas in the three different with cooling runs. The
$\mathrm{VSF}_{2}$ are normalised by the square of the rms velocity of all gas
($v_{\mathrm{rms}}^{2}$). The unfilled data points represent the hot phase
$\mathrm{VSF}_{2}$ and the filled data points represent the cold phase
$\mathrm{VSF}_{2}$. Similar to the turbulence without cooling runs in section
3, the $\mathrm{VSF}_{2}$ for both hot and cold phases increases with
increasing $r$ and flattens close to the driving scale ($r/L\approx 0.5$).

We fit separate power-law functions to the scaling range of the
$\mathrm{VSF}_{2}$ for the hot and cold phases. We have denoted the exponents
of these functions as $m_{h}$ and $m_{c}$ in the bottom right side of the
figure panels for the fits to the hot and cold phase $\mathrm{VSF}_{2}$,
respectively. We have also listed these scaling exponents in table B1.

##### f0.10HR

For the f0.10HR run, the hot phase $\mathrm{VSF}_{2}$ is steeper than the K41
scaling of $r^{2/3}$. It is also steeper than the turbulence without cooling
run (HR) in section 3.1. The steepening of the $\mathrm{VSF}_{2}$ could be due
to transonic turbulence in the hot phase ($\mathcal{M}_{\mathrm{hot}}=0.91$),
where the slopes of the $\mathrm{VSF}_{2}$ could be approaching the Burgers
turbulence slopes (Burgers, 1948) of $r^{1}$, which is valid for compressible
supersonic turbulence (Federrath, 2013). Our scaling exponents are consistent
with the $\mathrm{VSF}_{2}$ slopes in the simulations of Schmidt et al. (2008)
and the $\mathrm{VSF}_{1}$ slope in the transonic regime of Federrath et al.
(2021) (see fig. 2 of their paper)666See fig. 11 for a direct comparison of
the $\mathrm{VSF}_{1}$ slope with Federrath et al. (2021)..

The $\mathrm{VSF}_{2}$ for the cold phase has a smaller amplitude at the
driving scale, consistent with the smaller velocity dispersion for the cold
phase in fig. 5. At scales smaller than the driving scale, the cold phase
$\mathrm{VSF}_{2}$ has a slightly steeper slope compared to the hot phase
$\mathrm{VSF}_{2}$.

##### f0.10vwHR

For the f0.10vwHR run, the hot phase $\mathrm{VSF}_{2}$ is much shallower than
the K41 scaling. Since this run has a small $\mathcal{M}_{\mathrm{hot}}=0.23$,
we expect the velocity field in this run to have significant contribution from
gas motions due to thermal instability. These velocities are generated when
the gas separates into hot and cold phases due to thermal instability, even in
the absence of external turbulent driving. They are supposed to have a flat
$\mathrm{VSF}_{2}\propto r^{0}$ (with no structure across scales)777The
velocities due to thermal instability are expected to have
$\mathrm{VSF}_{2}\propto r^{0}$, since the growth rate of thermal instability
is independent of scale ($t_{\mathrm{TI}}$ is defined in eq. 6b of M21) in the
absence of thermal conduction. So both large and small scale isobaric modes
grow at the same rate, as seen in fig. 2 of Sharma et al. (2010). Das et al.
(2021) consider isochoric modes, which grow more slowly, but are not relevant
for the temperatures and length scales of cool cores.. Such velocities
increase the amplitude of small-scale velocity fluctuations, which flatten the
$\mathrm{VSF}_{2}$.

The cold phase $\mathrm{VSF}_{2}$ has a much lower amplitude compared to the
hot phase $\mathrm{VSF}_{2}$. This behaviour is expected since the value of
$\chi$ is large, i.e., the cold phase is much denser than the hot phase.
Although the amplitude of turbulent acceleration is the same for the two
phases, the cold/dense phase with a small volume filling factor has a smaller
turbulent velocity. Similar to the f0.10HR run, the cold phase
$\mathrm{VSF}_{2}$ is slightly steeper than the hot phase $\mathrm{VSF}_{2}$.

The trends in the $\mathrm{VSF}_{2}$ of the cold phase for both of our
hydrodynamic runs (f0.10HR and f0.10vwHR) are similar to the results of Gronke
et al. (2021) (see their fig. 19). Using a setup similar to ours, they show
that the cold phase $\mathrm{VSF}_{2}$ has a smaller amplitude compared to the
$\mathrm{VSF}_{2}$ of the rest of the gas but has similar scaling with $r$.

##### f0.10magHR

For the f0.10magHR run, the hot phase $\mathrm{VSF}_{2}$ scales with $r$ as
$r^{0.5}$. The rms Alfvén Mach number of the hot phase
$\mathcal{M}_{A,\mathrm{hot}}\sim 0.92$. So the $\mathrm{VSF}_{2}$ of the hot
phase is slightly flatter than the magHR run (without cooling MHD simulation
in section 3.1.2) and is closer to the IK scaling. The hot phase
$\mathrm{VSF}_{2}$ could also flatten due to the effect of thermal instability
induced small-scale velocities, as we discussed in the paragraphs above.

The cold phase $\mathrm{VSF}_{2}$ has the same amplitude as the hot phase
$\mathrm{VSF}_{2}$ at large scales. The large-scale velocities of the cold and
hot phases are coupled due to magnetic fields. But at scales smaller than the
driving scale, the cold phase $\mathrm{VSF}_{2}$ is much steeper than the hot
phase $\mathrm{VSF}_{2}$ (more than twice as steep).

The scaling of the $\mathrm{VSF}_{2}$ in this run can be attributed to the
effect of magnetic fields. By Alfvén’s theorem of flux freezing, the magnetic
field lines are frozen into the gas as it condenses out of the hot phase into
to the small-scale cold clouds. The amplification of the magnetic field due to
flux freezing depends on $\chi$ as $\chi^{2/3}$. The cold phase is dominated
by magnetic pressure, with $\beta\lesssim 0.1$ (see fig. 13 of M21) and the
gas is constrained to move along the magnetic field lines. This effect becomes
stronger at small scales. It reduces the gas velocities at these scales and
makes the $\mathrm{VSF}_{2}$ steep.

####  4.3.2 Projected hot and cold gas velocity structure functions

Figure 7: The second-order velocity structure function of the 3D velocity
field ($\mathrm{VSF}_{2}$) and the projected LOS velocity field
($\mathrm{VSF}_{2}^{\mathrm{LOS}}$, denoted by ‘proj’ in the legend) for our
turbulence with cooling runs. We have summed the
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$ along our three LOS directions $x$, $y$ and
$z$. In the left panel, we show these quantities for the hot phase gas and in
the right panel for the cold phase gas. All the $\mathrm{VSF}_{2}$ are
normalised by $v_{\mathrm{rms}}^{2}$. We fit broken power-law functions to the
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$ of both hot and cold phases, and their
slopes are denoted by $m_{\mathrm{proj}}$. The scaling exponents at large and
small scales, are denoted by the subscripts $l$ and $s$, respectively and are
shown in the bottom right of each panel. The break scales are shown as the
vertical lines. We have also listed them in table B1. _Left panel:_ For the
f0.10HR run, the $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ is slightly flatter than
the $\mathrm{VSF}_{2}$ for the hot phase gas. It steepens slightly for the
f0.10magHR run and becomes significantly steeper for the f0.10vwHR run upon
projection. _Right panel:_ For the cold phase, the
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$ for the f0.10HR and f0.10magHR is shallower
than the $\mathrm{VSF}_{2}$. But for the f0.10vwHR run, it is roughly parallel
to the $\mathrm{VSF}_{2}$ at large scales and shallower at small scales. We
discuss these findings in detail in the main text.

In this subsection, we show the effect of projection along the LOS on the
$\mathrm{VSF}_{2}$ of the hot and cold phases. In fig. 7, we show the
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$ and the $\mathrm{VSF}_{2}$ for the hot (left
panel) and cold phases (right panel). The $\mathrm{VSF}_{2}^{\mathrm{LOS}}$
are constructed using emission-weighted LOS velocities, snapshots of which are
shown in the second and fourth rows of fig. 4. We have summed the
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$ along $x-$, $y-$ and $z-$ axes and
normalised them by the 3D $v_{\mathrm{rms}}^{2}$, so that we can directly
compare them with the 3D $\mathrm{VSF}_{2}$ and look for the effects of
projection. We have also averaged them temporally over 28 snapshots from
$t=0.651$ $\mathrm{Gyr}$ to $t=1.003$ $\mathrm{Gyr}$. In all cases, the
projected $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ increase with increasing
$r_{\mathrm{proj}}$ and peak close to the driving scale, where
$r_{\mathrm{proj}}$ is the separation between two points along the projection
plane. The amplitude of the $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ is slightly
smaller than the $\mathrm{VSF}_{2}$ at the driving scale for all of our runs
because power at large scales is shifted to smaller projected separations. In
contrast to the $\mathrm{VSF}_{2}$, the $\mathrm{VSF}_{2}^{\mathrm{LOS}}$
flattens at small scales inside the scaling range and is better fit by a
broken power-law function $\mathrm{SBP}(r_{\mathrm{proj}})$, which is
described in eq. B7. The flattening of the $\mathrm{VSF}_{2}^{\mathrm{LOS}}$
at small scales is due to insufficient cancellation of the small-scale eddies
upon projection, especially for the cold phase. We discuss the effects of
projection in detail for these two phases in the next two paragraphs.

##### Effect of projection on the hot phase VSFs

Here we discuss the $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ for the hot phase gas.
Although the amplitudes of the normalised $\mathrm{VSF}_{2}$ are approximately
equal at the driving scale, the amplitudes of the normalised
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$ are smaller and vary for our three runs.
These trends are similar to the variation in the amplitudes of the projected
LOS hot phase velocity maps, shown in the second row of fig. 4. The hot phase
gas in the f0.10HR run is clumpy and has strong localised emission features
surrounding the cold/dense strongly emitting gas (see fig. 4). Observations of
cool-core halos (e.g., Fabian et al. 2003, Werner et al. 2014), multiphase
galactic outflows (e.g., Strickland et al. 2004), and ram pressure-stripped
tails of galaxies moving through the ICM (e.g., Sun et al. 2021) show similar
spatial correlation between different temperature phases. Along any particular
LOS, we expect to find only few such bright patches of hot phase gas mostly
around cold clouds. Due to smaller number of eddies from these patches, the
LOS velocities do not cancel out upon projection. In contrast, the hot phase
gas in the f0.10vwHR run is more volume-filling and shows variations only at
small scales (see middle columns of fig. 4). In this case, we have larger
number of cancelling eddies during projection and as a result the
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$ is steeper than the $\mathrm{VSF}_{2}$. For
the f0.10magHR run, the emission features lie in between the two extremes, and
it shows weak cancellation of LOS velocity upon projection. This leads to
steeper $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ than $\mathrm{VSF}_{2}$.

##### Effect of projection on the cold phase VSFs

Here we discuss the $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ and its relation with
the $\mathrm{VSF}_{2}$ for gas in the cold phase. These two quantities are
plotted in the right panel of fig. 7. In all our runs, the
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$ becomes flatter than the $\mathrm{VSF}_{2}$
and has larger values at small scales. The flattening effect of projection is
much stronger at small scales because the cold phase gas is distributed into
small-sized clouds (see the third row of fig. 4). Since this phase is not
volume-filling, we find few cold clouds along any particular LOS during
projection. Hence the cancellation of eddies upon projection along the LOS is
small. In addition to that, the distance between any of these cold clumps in
the projected plane is also smaller than the real 3D distance
($r_{\mathrm{proj}}\leq r$). This can make the
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$ larger at small $r_{\mathrm{proj}}$. These
effects make the $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ shallower than the
$\mathrm{VSF}_{2}$, especially for the f0.10magHR run which has a steep 3D
$\mathrm{VSF}_{2}$. For the fiducial f0.10HR run, the
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$ is slightly flatter than the
$\mathrm{VSF}_{2}$ at large scales. The f0.10vwHR run, which has the smallest
cool clouds among our three runs, shows the flattest
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$ slopes at small scales. But at large scales,
the $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ is roughly parallel to the
$\mathrm{VSF}_{2}$.

##  5 Discussions

In this work, we have two key aims: (i) to study the relation between the 3D
$\mathrm{VSF}$s of hot and cold phases and (ii) to understand the effect of
projection along the LOS on these $\mathrm{VSF}$s. Our findings are important
to understand the relation between the $\mathrm{VSF}$s studied in
observational studies (e.g., Li20) of the atomic gas and future X-ray
observations of the hot phase gas (such as with XRISM) and the 3D velocity
structure of the hot ICM gas.

In the first part of this study, we studied the effect of magnetic fields and
the effect of projection on the $\mathrm{VSF}$s in homogeneous idealised
turbulence simulations without cooling. In the second part, we analysed three
high resolution multiphase ICM simulations with global energy balance from
M21. Here we discuss the effects of important physical elements such as
cooling and magnetic fields, and our methods such as nature of thermal
feedback energy on our results. We also discuss the effect of projection for
both parts of our study and finally compare our results with other simulations
and observations.

###  5.1 Effect of cooling physics and thermal feedback models

The $\mathrm{VSF}_{2}$ in part 1 of our study follow K41 scaling for the HD
run and closer to IK scaling for the MHD run, which are according to
theoretical expectations. In part 2 of this study, cooling and thermal
instability clearly affect the the scaling and amplitude of the
$\mathrm{VSF}_{2}$ in both hot and cold phases.

In addition to separating the gas into hot and cold phases, thermal
instability also introduces small velocities in the gas during phase
separation. These velocities are expected to have a flat $\mathrm{VSF}_{2}$,
similar to white noise, since thermal instability in absence of thermal
conduction has the same growth rate at all scales. For turbulence at small and
intermediate $\mathcal{M}_{\mathrm{hot}}$, such as in f0.10vwHR and f0.10magHR
runs, this flattens the $\mathrm{VSF}_{2}$ of the hot phase.

Our two different feedback models lead to different density distributions and
different velocity dispersion in the hot and cold phases. The f0.10vwHR run
has denser cold phase gas (larger $\chi$) than the f0.10HR run, whereas their
hot phase density is similar. At scales larger than the size of the cool
clouds, we expect the turbulent cascade to be established in the hot phase.
Then at intermediate scales, the hot phase gas drags the cold clouds along
with it. This imprints the velocity structure of the hot phase into the cold
phase, making their $\mathrm{VSF}_{2}$ scale similar to the $\mathrm{VSF}_{2}$
of the hot phase. The small scale eddies maybe less efficient at dragging cold
clouds of comparable size, which can explain why the cold phase
$\mathrm{VSF}_{2}$ are slightly steeper than the hot phase $\mathrm{VSF}_{2}$.
The effect of this drag acceleration, which should depend on scale, is weaker
if the density contrast between the two phases is higher, as seen in the
f0.10vwHR run. A similar effect is also seen in the simulations of Gronke et
al. (2021), where a larger density contrast between hot and cold phases leads
to weaker mixing between them. Gronke et al. (2021) also find the cold phase
$\mathrm{VSF}_{2}$ to be roughly parallel to the $\mathrm{VSF}_{2}$ of rest of
the gas but slightly smaller in amplitude.

####  5.1.1 Driving turbulence only in the hot phase

To understand the effect of this drag, we also conducted a lower resolution
simulation ($384^{3}$ resolution elements) where we drive turbulence only in
gas with $T>10^{6}$ $\mathrm{K}$, with all other conditions similar to f0.10HR
run. We show the $\mathrm{VSF}_{2}$ for this run in fig. 8. The cold phase
$\mathrm{VSF}_{2}$ is still parallel to the hot phase $\mathrm{VSF}_{2}$ at
all scales for this run, denoting the importance of hydrodynamic drag at all
scales. The amplitude of the $\mathrm{VSF}_{2}$ of the cold phase gas is
slightly smaller, mainly because we do not impart any velocities to the cold
phase at the driving scale, and cold phase is driven only by drag from the hot
phase.

Figure 8: Hot and cold phase $\mathrm{VSF}_{2}$ where we drive turbulence only
in gas with $T>10^{6}$ $\mathrm{K}$, shown in cyan. We also show the fiducial
f0.10HR run for comparison. The cold phase $\mathrm{VSF}_{2}$ is slightly
smaller since it is not driven directly. The hot phase $\mathrm{VSF}_{2}$ is
larger since we are only driving the hot phase, while maintaining global
thermal balance with 10% heating provided by turbulence. The
$\mathrm{VSF}_{2}$ of the two phases are still roughly parallel to each other.

####  5.1.2 Spatial correlation function between cold and hot phase velocities

Figure 9: The two point cross correlation function
$\mathrm{VCF}_{\mathrm{hc}}(r)$ (see equation 6) between cold and hot phase
velocities for our three standard set of runs, normalised by the rms
velocities of the hot and cold phases. At small scales, the hot and cold phase
velocities are positively correlated. With increasing separation, the strength
of correlation decreases.

Spatial correlation functions are another useful tool to understand the
coupling between two quantities. We define the correlation function
$\mathrm{VCF}_{\mathrm{hc}}(r)$ between the 3D velocity fields of the cold and
hot phases as:

$\mathrm{VCF}_{\mathrm{hc}}(r)=\langle\mathbf{v}_{\mathrm{cold}}(\mathbf{x})\cdot\mathbf{v}_{\mathrm{hot}}(\mathbf{x}+\mathbf{e_{1}}r)\rangle.$
(6)

We choose the pairs $\mathbf{x}$ and $\mathbf{x}+\mathbf{e_{1}}r$ such that
gas at $\mathbf{x}$ is in the cold phase and gas at
$\mathbf{x}+\mathbf{e_{1}}r$ is in the hot phase.

In fig. 9, we show the $\mathrm{VCF}_{\mathrm{hc}}(r)$ normalised by the rms
velocities of the two phases. For all of our runs, the cold and hot phase
velocities are positively correlated at small scales. With increasing
separation, the $\mathrm{VCF}_{\mathrm{hc}}(r)$ first increases and reaches a
peak. This peak scale may correspond to the cool cloud radius (see the third
row of fig. 4), but note that the value of the $\mathrm{VCF}_{\mathrm{hc}}(r)$
at small scales ($r/L<0.04$) may be affected by numerical diffusion.After this
peak, with increasing separation, the correlation becomes weaker. At scales
close to the driving scale ($r=L/2$), the velocities are uncorrelated. The
coupling between the two phases is weaker for larger $\chi$. It is also
stronger in the presence of magnetic fields, for our f0.10magHR run (which
also has the smallest $\chi$). This trend of stronger coupling between the
phases in the presence of magnetic fields is also seen in the cluster scale
simulations of W21.

###  5.2 Effect of magnetic fields

In the first part of the study, we showed that the $\mathrm{VSF}_{2}$ in
idealised MHD turbulence is closer to the IK scaling, which is shallower than
the K41/Goldreich-Sridhar scaling. Grete et al. (2021) argue that magnetic
tension could suppress the kinetic energy cascade and make the power spectra
(and $\mathrm{VSF}_{2}$) shallow, suppressing the bottleneck effect. This is
inline with our results.

In our multiphase MHD turbulence runs, we find that magnetic fields couple the
hot and cold phases even at the driving scale (see fig. 6). This efficient
coupling is also reflected in stronger spatial correlation between the hot and
cold phases in fig. 9. The hot phase gas has slightly shallower
$\mathrm{VSF}_{2}$ compared to the idealised MHD runs without cooling. But at
scales smaller than the driving scale, the $\mathrm{VSF}_{2}$ of the cold
phase gas is much steeper than the hot phase gas. We expect this to be due to
the magnetic flux being frozen into the cold phase regions as they condense
and contract, as we explained in paragraph 4.3.1.

###  5.3 Effect of projection

Figure 10:  The $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ of the projected velocity
field with different weights during projection–emission (emw,
$\propto\rho^{2}$), mass (mw, $\propto\rho$) and volume (vw,
$\propto\rho^{0}$) for our fiducial f0.10HR run. These weights are indicated
in the subscripts of the data and fit legends. We show the effects of these
weights on the $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ for gas at all temperatures
(first panel), only hot phase gas ($T>10^{7}$ $\mathrm{K}$, second panel) and
only cold phase gas ($T<2\times 10^{4}$ $\mathrm{K}$, third panel). We fit a
single power-law to the $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ in the first panel
but broken power-laws to the other two panels. The
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$ shows the maximum cancellation along the LOS
and steepening for the volume weighted projection of gas at all temperatures.

The projection of velocities along the LOS has very different effects on the
$\mathrm{VSF}_{2}$ in the without cooling runs, and the hot and cold phase
$\mathrm{VSF}_{2}$ in the with cooling runs. In part 1, for both hydro and MHD
runs, the $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ becomes steeper by a factor of
$r^{0.7}$. This is slightly different from the expectations in ZuHone et al.
(2016), who propose a steepening by $r^{1}$ upon projection. Their result is
based on the steepening of the power spectrum by a factor of $k$ upon
projection, followed by the conversion between the scaling of the power
spectrum to the projected second order structure function. However, we note
that the conversion between the scaling of the second order structure function
(in real space) and the power spectrum (in Fourier space) is non-trivial. We
refer the reader to the appendix of Stutzki et al. (1998) for a derivation of
the relation between the scaling of these two quantities for a homogeneous,
isotropic field following different power-law spectral indices. However, our
turbulence velocity fields do not necessarily follow these assumptions.
Moreover, Esquivel & Lazarian (2005) show that for a Gaussian random field
with a steep spectral index $-11/3$ (K41 turbulence has spectral index of
$-11/3$), the velocity structure functions steepen less than the factor of $r$
upon projection. This is in agreement with our results.

In general, the effect of projection in all our runs depends mainly on the
volume-filling fraction of the gas/phase and the uniformity of its emission.
The more volume-filling phases have more cancellation upon projection, since
larger number of eddies are likely to get cancelled along the LOS during
projection. Similarly if the emission is roughly uniform, different eddies
along the LOS will have a similar projection weight and are thus more likely
to be cancelled upon projection. For the subsonic ($\mathcal{M}\approx 0.2$)
turbulence without cooling runs, the gas is volume-filling and roughly uniform
over the entire volume. So the cancellation effect is strong for the
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$ in the HR and magHR runs.

Among our runs with cooling, the f0.10vwHR run has uniform volume filling hot
phase (see first row of fig. 4), so it shows stronger cancellation of eddies
during projection and the $\mathrm{VSF}_{2}$ steepens by $r^{0.74}$ upon
projection. For the fiducial f0.10HR run, the hot phase distribution is
scattered and clumpy and the $\mathrm{VSF}_{2}$ becomes shallow on projection.
Since most of the cold phase emission occurs from small dense clouds, the
cancellation effect of projection is small for the cold phase.

The distance between two points in the plane of projection
$r_{\mathrm{proj}}\leq r$. Upon projection, this effect can shift large scale
power to smaller projected scales and make the value of
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$ larger (smaller) at small (large)
$r_{\mathrm{proj}}$. It is particularly important for runs with steep 3D
$\mathrm{VSF}_{2}$, such as the cold phase $\mathrm{VSF}_{2}$ in the
f0.10magHR run. Since all 3D $\mathrm{VSF}_{2}$ are steep at scales smaller
than the dissipation scale, this spread of power from large scales to small
projected scales can explain the flattening of the
$\mathrm{VSF}_{2}^{\mathrm{LOS}}$ at $r_{\mathrm{proj}}\lesssim 0.04$, namely
the scales smaller than the dissipation scale.

In fig. 10, we show the different projected $\mathrm{VSF}_{2}^{\mathrm{LOS}}$
for the f0.10HR run, where we compare different weights on the
$v_{\mathrm{LOS}}$ during projection, such as weighted by net emission (emw),
weighted by mass (mw) and weighted by volume (vw). The mw and vw projections
are constructed by replacing $\mathcal{L}$ with $\rho$ and unity, respectively
in eq. 4. We compare these three projection weights for different temperature
filters - (i) where we consider gas at all temperatures in the first panel,
(ii) only consider gas in the hot phase ($T>10^{7}$ $\mathrm{K}$) in the
second panel and (iii) only gas in the cold phase ($T<2\times 10^{4}$
$\mathrm{K}$) in the third panel. We have fit power-laws to the scaling range
of the $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ of all gas and broken power-laws to
the scaling range of hot and cold phase $\mathrm{VSF}_{2}^{\mathrm{LOS}}$. The
break scale for the latter two is denoted by a vertical line, similar to fig.
7.

In the first panel, for projected $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ for gas at
all temperatures, we observe the strongest variation with the different
projection weights. Since the emission and density are larger for the cold
dense regions, these projection weights mainly track the small dense clouds
and not the volume-filling hot phase. As a result, we do not observe much
steepening of the $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ due to cancellation of
eddies. But the volume projection weight mainly tracks the hot and
intermediate phase, which fill up the volume. This vw projection has more
cancellation of eddies along the LOS, especially at smaller scales and shows a
much steeper $\mathrm{VSF}_{2}^{\mathrm{LOS}}$. This steepening due to
projection is similar to the turbulence without cooling runs in fig. 3.

We do not observe very strong effects of changing projection weights for the
hot and cold phase $\mathrm{VSF}_{2}^{\mathrm{LOS}}$. This happens because the
volume distribution of both hot and cold phases is clumpy and not volume-
filling, as seen in fig. 4. Since the gas phases themselves are not volume-
filling, projection along the LOS does not introduce much cancellation. The
slopes of the $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ steepen slightly when we
change the projection weight from emw ($\propto\rho^{2}$) to mw
($\propto\rho$) to vw ($\propto\rho^{0}$), as the weights become less biased
towards the small-sized high density regions and have somewhat more
cancellation. Note that we do observe more cancellation for the f0.10vwHR run
which has a volume-filling hot phase.

###  5.4 Comparison with other studies

Figure 11:  The $\mathrm{VSF}_{1}$ of the 3D velocity field of the hot phase
and the $\mathrm{VSF}_{1}^{\mathrm{LOS}}$ of the cold phase gas in our
different runs. We also show the $\mathrm{VSF}_{1}^{\mathrm{LOS,obs}}$ from
the observations of three clusters from Li20. We have scaled them
($v_{\mathrm{scl}}=200$ $\mathrm{km/s}$) such that they align with
$\mathrm{VSF}_{1}^{\mathrm{LOS}}$ of our cold phase data at the driving scale
($L/2$). The trends across different runs in the hot $\mathrm{VSF}_{1}$ and
cold $\mathrm{VSF}_{1}^{\mathrm{LOS}}$ for different runs are similar to the
trends in hot $\mathrm{VSF}_{2}$ and cold $\mathrm{VSF}_{2}^{\mathrm{LOS}}$,
respectively. The $\mathrm{VSF}_{1}^{\mathrm{LOS,obs}}$ are roughly parallel
to the cold $\mathrm{VSF}_{1}^{\mathrm{LOS}}$ at large scales but become
steeper at smaller scales.

In fig. 11, we compare the cold phase projected first-order velocity structure
functions $\mathrm{VSF}_{1}^{\mathrm{LOS}}$
($=(\mathrm{VSF}_{1,x}+\mathrm{VSF}_{1,y}+\mathrm{VSF}_{1,z})/\sqrt{3}$) and
the hot phase 3D $\mathrm{VSF}_{1}$888Note that we have shown first-order
$\mathrm{VSF}$ to compare directly with data from Li20. But we have discussed
the second-order $\mathrm{VSF}$s in rest of the paper. . We also compare the
projected cold phase $\mathrm{VSF}_{1}^{\mathrm{LOS}}$ with the
$\mathrm{VSF}_{1}^{\mathrm{LOS,obs}}$ of H$\alpha$ filaments from Li20. We
have imported data from fig. 4 of Li20 and have scaled the
$\mathrm{VSF}_{1}^{\mathrm{LOS,obs}}$ by $v_{\mathrm{scl}}=200$
$\mathrm{km/s}$ so that it has similar amplitude as our cold phase
$\mathrm{VSF}_{1}^{\mathrm{LOS}}$. We have also scaled the radial separation
($r_{\mathrm{proj}}/\ell_{\eta}$) to align the peaks of the Perseus and Virgo
data. We have used the same scaling constants for all three clusters.

The general trends in the hot phase $\mathrm{VSF}_{1}$ and the cold phase
$\mathrm{VSF}_{1}^{\mathrm{LOS}}$ from our simulations are similar to the
trends in their second-order counterparts, which we have discussed in the
previous sections. Overall, the cold phase $\mathrm{VSF}_{1}^{\mathrm{LOS}}$
has a smaller amplitude than the hot phase $\mathrm{VSF}_{1}$. This can be
explained by two main effects - first due to smaller velocity dispersion of
the cold phase gas compared to the hot phase (mainly in the two hydro runs)
and second due to the shift of power from large scales ($r$) to smaller
projected scales ($r_{\mathrm{proj}}$) during projection. At large scales, the
slope of the cold phase $\mathrm{VSF}_{1}^{\mathrm{LOS}}$ is roughly parallel
to the hot phase $\mathrm{VSF}_{1}$ for the f0.10HR run, but it becomes
steeper for the f0.10vwHR run and much steeper for the f0.10magHR run. We have
discussed the steepening of the cold phase 3D $\mathrm{VSF}_{2}$ for the
f0.10magHR run in paragraph 4.3.1 and section 5.2. Even though the
$\mathrm{VSF}_{1}^{\mathrm{LOS}}$ flattens strongly due to projection, it is
still steeper than the hot phase $\mathrm{VSF}_{1}$ for this run. All the cold
phase projected $\mathrm{VSF}_{1}^{\mathrm{LOS}}$ flatten at small scales. We
have discussed the causes of this flattening in section 5.3.

Comparing with the observational data from Li20, both
$\mathrm{VSF}_{1}^{\mathrm{LOS,obs}}$ and our
$\mathrm{VSF}_{1}^{\mathrm{LOS}}$ show a peak at large scales. This peak
corresponds to the driving scale in our simulations and to the size of the
AGN-inflated bubbles in the observational data. At scales larger than the
break scale, the $\mathrm{VSF}_{1}^{\mathrm{LOS}}$ are parallel/slightly
shallower than the $\mathrm{VSF}_{1}^{\mathrm{LOS,obs}}$.

But at scales smaller than the break scale, $\mathrm{VSF}_{1}^{\mathrm{LOS}}$
become flatter for all our runs, whereas the
$\mathrm{VSF}_{1}^{\mathrm{LOS,obs}}$ become much steeper and quickly approach
zero. However, it is important to note that our turbulence with cooling
simulations are not convergent with resolution, and the spatial extent of the
cold clouds decreases with increasing resolution (see fig. 13 of M21, second
and third rows). We have discussed the resolution convergence criteria in
section 3.6.3 of M21. With increasing resolution, the break-scale and the
flattening of the projected $\mathrm{VSF}_{1}^{\mathrm{LOS}}$ could happen at
smaller scales.

Our projected $\mathrm{VSF}_{1}^{\mathrm{LOS}}$ do not reproduce the steep
slopes in the observed $\mathrm{VSF}_{1}^{\mathrm{LOS,obs}}$. But in fig. 6,
we showed that the 3D $\mathrm{VSF}_{2}$ of the cold phase could be strongly
magnetised and have a steeper slope than the hot phase $\mathrm{VSF}_{2}$. The
flattening effect of projection could happen at much smaller scales for higher
resolution simulations (this is beyond the scope of the present paper). If the
slope of the $\mathrm{VSF}_{1}^{\mathrm{LOS}}$ remains steep upon projection,
then magnetic fields can explain the steepening of the cold phase
$\mathrm{VSF}_{1}^{\mathrm{LOS,obs}}$.

W21 have studied the $\mathrm{VSF}_{1}$ of hot and cold phases in a $(1$
$\mathrm{Mpc})^{3}$ simulation of an isolated galaxy cluster999The
$\mathrm{VSF}_{1}$ may not be well-converged in a global simulation (see fig.
1 for our convergence test) and the numerical viscous scale would vary
spatially in a refined grid.. Even though our setup is quite different from
that of W21, we observe some similarities between our results. In W21,
turbulence in the hot phase is stirred by the motion of the cold phase under
the influence of gravity. The $\mathrm{VSF}_{1}$s of the cold and hot phases
are parallel for the hydro runs, and the amplitude of the $\mathrm{VSF}_{1}$
of the driving phase (in their case the cold phase) is larger. This is similar
to our f0.10hotdriv run, except our hot phase is the driving phase and and the
cold phase $\mathrm{VSF}_{2}$ follows the same scaling as the hot phase
$\mathrm{VSF}_{2}$, but has a smaller amplitude (see fig. 8). The equivalent
scaling of the cold phase $\mathrm{VSF}_{1}$ with the rest of the ambient gas
$\mathrm{VSF}_{1}$ is also observed in Gronke et al. (2021) (see their fig.
19), although they start with gas already in a cold cloud and drive turbulence
in the ambient hot medium. With the introduction of magnetic fields, in both
W21 and our study, the velocities of the cold and hot phases become coupled
even at large scales.

Hillel & Soker (2020) have analysed the jet-ICM interaction in the central
$(40$ $\mathrm{kpc})^{3}$ region of a cluster. They argue that the steep slope
of the $\mathrm{VSF}_{1}^{\mathrm{LOS,obs}}$ could be due to the interaction
of AGN jets and the ICM over a large range of scales. We observe that the 3D
$\mathrm{VSF}$s follow a similar scaling relation for both hot and cold phases
in our hydrodynamic runs, even when we drive turbulence only in the hot phase
(see fig. 8). Since Hillel & Soker (2020) studied the $\mathrm{VSF}$s only in
the hot phase gas, it is possible that the cold phase $\mathrm{VSF}$s would
have a similar scaling as the hot phase. However, note that we have mainly
studied the effects of steady state turbulence where we inject energy only at
driving scales from $10$–$40$ $\mathrm{kpc}$. So we cannot directly compare
between the results. A possible way to check this using our setup would be to
drive turbulence across a large range of scales and compare the hot and cold
phase $\mathrm{VSF}$s, but this is beyond the scope of this study.

Körtgen et al. (2021) have studied the power spectra of column density and
projected velocity in a simulation of an isolated galaxy with multiple
temperature phases of the interstellar medium. They also identify a break in
their projected velocity power spectra and find that it corresponds to a
transition from 2D disk to 3D isotropic turbulence for the neutral phases. For
the cold phases, the break scale corresponds to the size of the clouds (this
is perhaps also true for the peak in our cold-hot VCFs in fig. 9).

##  6 Caveats and future work

Here we discuss some of the shortcomings of our work and future prospects.

One of the main caveats of our setup is the absence of stratification. The ICM
is stratified due to the gravitational potential of the dark matter halo.
Gravity introduces the free-fall time-scale ($t_{\mathrm{ff}}$), which plays
an important role in the ICM thermodynamics (Choudhury & Sharma, 2016;
Choudhury et al., 2019; Voit, 2021). For strongly stratified turbulence, the
hot phase velocities and their power spectra would be anisotropic, where
motion parallel to the direction of gravity is subdued (Mohapatra et al.,
2020, 2021b) and the eddies form pancake-like structures. On the other hand,
the cold clumps, which are denser than the hot phase in our hydro runs would
have strong motions along the direction of gravity, as seen in W21. The strong
X-ray emission from the denser central regions of galaxy clusters may
introduce an additional bias in the projected hot phase
$\mathrm{VSF}^{\mathrm{LOS}}$. The cancellation effects of projection in real
ICM may be small because of stratification. The brightest regions dominate
emission along any LOS and cancellation may only be partial. We plan to
include stratification and study thermal instability and kinematics of the
multiphase gas in a future study.

We have analysed a simple steady state feedback model, with turbulent driving
only at large scales and thermal feedback distributing energy uniformly across
volume/density. However, the interaction between AGN jets and the ICM could
occur across a large range of scales, as shown in Hillel & Soker (2020).
Temporal variability in energy injection due to cool core cycles is another
effect that can make the idealised steady state simulations differ from
reality (e.g., see fig. 7 of Prasad et al. 2018). These can also affect the
$\mathrm{VSF}^{\mathrm{LOS}}$. Our simple feedback picture is insufficient to
capture the physics of other feedback procedures such as cosmic rays (Ji et
al., 2020; Butsky et al., 2020; Butsky et al., 2021), which are important for
both thermal instability models as well as the kinematics of the multiphase
halo environments.

Our simulations are not converged with resolution (this is true for most
simulations of the multiphase ICM), as we discussed in section 3.6.3 of M21.
Higher resolution simulations form more and smaller scale cold gas and have
faster cooling rate. Due to our feedback method, this leads to stronger
turbulence driving and a cooler hot phase (because of turbulent
mixing/heating) upon increasing resolution. Compared to f0.10 run in M21, our
f0.10HR run has a slightly lower hot phase temperature ($\approx 10^{7}$
$\mathrm{K}$) and transonic $\mathcal{M}_{\mathrm{hot}}=0.91$. The ICM is
observed to be subsonic (Hitomi Collaboration, 2016). With increasing
resolution, we also expect the size of the cold clouds to be smaller. These
smaller clouds could shift the break scale and the flattening of our projected
$\mathrm{VSF}^{\mathrm{LOS}}$ to even smaller scales.

##  7 Conclusions

In this work we have studied velocity structure functions in homogeneous
idealised turbulence and multiphase ICM turbulence. Here are some of our main
conclusions:

  * •

The 3D $\mathrm{VSF}_{2}$ of the hot phase gas do not follow the Kolmogorov
turbulence scaling ($r^{2/3}$) for any of our turbulence with cooling runs.
For our hydro runs, it is steeper than K41 slopes for transonic
$\mathcal{M}_{\mathrm{hot}}$ and shallower for subsonic
$\mathcal{M}_{\mathrm{hot}}$. For the MHD run, it follows the Iroshnikov-
Kraichan scaling ($r^{0.5}$).

  * •

The 3D $\mathrm{VSF}_{2}$ of the cold phase gas has a slightly steeper scaling
with $r$ compared to the 3D $\mathrm{VSF}_{2}$ of the hot phase gas for the
hydro runs. It has a smaller amplitude and the ratio of the amplitudes of the
cold and hot phase $\mathrm{VSF}_{2}$ decreases with increasing $\chi$, the
ratio of the densities of the cold and hot phases.

  * •

For the MHD run, the cold and hot phase gas have similar rms velocity, due to
the coupling of the hot and cold phases by the magnetic fields due to magnetic
tension. But the cold phase 3D $\mathrm{VSF}_{2}$ is steeper than the hot
phase 3D $\mathrm{VSF}_{2}$, since the cold phase is magnetically dominated
with a low $\beta$ and small scale motions are constrained and subdued.

  * •

Upon projection, the projected $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ follow a
broken power-law in the scaling range of the $\mathrm{VSF}_{2}$. At large
scales, the slopes of the projected $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ are
steeper than the 3D $\mathrm{VSF}_{2}$ slopes only if the corresponding phase
is volume-filling to have enough cancellation of velocities during projection.
The projected ones are generally shallower than their 3D counterparts at large
scales. All the projected $\mathrm{VSF}_{2}^{\mathrm{LOS}}$ flatten at small
scales.

  * •

The projected cold phase $\mathrm{VSF}_{1}^{\mathrm{LOS}}$ from our
simulations show similar trends as the $\mathrm{VSF}_{1}^{\mathrm{LOS,obs}}$
of H$\alpha$ filaments at large scales. But at small scales, our
$\mathrm{VSF}_{1}^{\mathrm{LOS}}$ flatten due to projection effects, whereas
the observational ones do not. This may be a consequence of our limited
resolution.

## Acknowledgements

This work was carried out during the ongoing COVID-19 pandemic. The authors
would like to acknowledge the health workers all over the world for their role
in fighting in the frontline of this crisis. RM would like to thank Amit Seta
for useful discussions. RM acknowledges Prof. Lisa Kewley and Prof. Matthew
Colless for organising and funding a writing retreat, respectively, where a
part of this work was written. PS acknowledges a Swarnajayanti Fellowship
(DST/SJF/PSA-03/2016-17) and a National Supercomputing Mission (NSM) grant
from the Department of Science and Technology, India. CF acknowledges funding
provided by the Australian Research Council (Future Fellowship FT180100495),
and the Australia-Germany Joint Research Cooperation Scheme (UA-DAAD). We
further acknowledge high-performance computing resources provided by the
Leibniz Rechenzentrum and the Gauss Centre for Supercomputing (grants pr32lo,
pr48pi and GCS Large-scale project 10391), the Australian National
Computational Infrastructure (grant ek9) in the framework of the National
Computational Merit Allocation Scheme and the ANU Merit Allocation Scheme. The
simulation software FLASH was in part developed by the DOE-supported Flash
Center for Computational Science at the University of Chicago.

##  8 Data Availability

All the relevant data associated with this article is available upon request
to the corresponding author.

##  9 Additional Links

Movies of projected density and temperature of different simulations are
available at the following links on youtube:

  1. 1.

Movie of the f0.10HR simulation.

  2. 2.

Movie of the f0.10vwHR simulation.

  3. 3.

Movie of the f0.10magHR simulation.

##  10 Software used

We have used the following software and packages for our work: FLASH (Fryxell
et al., 2000; Dubey et al., 2008), matplotlib (Hunter, 2007), cmasher (van der
Velden, 2020), scipy (Virtanen et al., 2020), NumPy (Harris et al., 2020),
h5py (Collette, 2013), LMfit (Newville et al., 2016) and astropy (Astropy
Collaboration et al., 2018).

## References

  * Astropy Collaboration et al. (2018) Astropy Collaboration et al., 2018, AJ, 156, 123
  * Babyk et al. (2019) Babyk I. V., McNamara B. R., Tamhane P. D., Nulsen P. E. J., Russell H. R., Edge A. C., 2019, ApJ, 887, 149
  * Bambic & Reynolds (2019) Bambic C. J., Reynolds C. S., 2019, ApJ, 886, 78
  * Banerjee & Sharma (2014) Banerjee N., Sharma P., 2014, MNRAS, 443, 687
  * Boldyrev (2005) Boldyrev S., 2005, ApJ, 626, L37
  * Boselli et al. (2019) Boselli A., et al., 2019, A&A, 623, A52
  * Bouchut et al. (2007) Bouchut F., Klingenberg C., Waagan K., 2007, Numerische Mathematik, 108, 7
  * Bouchut et al. (2010) Bouchut F., Klingenberg C., Waagan K., 2010, Numerische Mathematik, 115, 647
  * Burgers (1948) Burgers J., 1948, Elsevier, pp 171–199, doi:https://doi.org/10.1016/S0065-2156(08)70100-5, https://www.sciencedirect.com/science/article/pii/S0065215608701005
  * Butsky et al. (2020) Butsky I. S., Fielding D. B., Hayward C. C., Hummels C. B., Quinn T. R., Werk J. K., 2020, ApJ, 903, 77
  * Butsky et al. (2021) Butsky I. S., et al., 2021, arXiv e-prints, p. arXiv:2106.14889
  * Carilli & Taylor (2002) Carilli C. L., Taylor G. B., 2002, ARA&A, 40, 319
  * Choudhury & Sharma (2016) Choudhury P. P., Sharma P., 2016, MNRAS, 457, 2554
  * Choudhury et al. (2019) Choudhury P. P., Sharma P., Quataert E., 2019, MNRAS, 488, 3195
  * Churazov et al. (2012) Churazov E., et al., 2012, MNRAS, 421, 1123
  * Collette (2013) Collette A., 2013, Python and HDF5. O’Reilly
  * Conselice et al. (2001) Conselice C. J., Gallagher John S. I., Wyse R. F. G., 2001, AJ, 122, 2281
  * Das et al. (2021) Das H. K., Choudhury P. P., Sharma P., 2021, MNRAS, 502, 4935
  * Donzis & Sreenivasan (2010) Donzis D. A., Sreenivasan K. R., 2010, Journal of Fluid Mechanics, 657, 171
  * Dubey et al. (2008) Dubey A., et al., 2008, in Pogorelov N. V., Audit E., Zank G. P., eds, Astronomical Society of the Pacific Conference Series Vol. 385, Numerical Modeling of Space Plasma Flows. p. 145
  * Edge (2001) Edge A. C., 2001, MNRAS, 328, 762
  * Esquivel & Lazarian (2005) Esquivel A., Lazarian A., 2005, ApJ, 631, 320
  * Eswaran & Pope (1988) Eswaran V., Pope S. B., 1988, Computers and Fluids, 16, 257
  * Fabian (1994) Fabian A. C., 1994, Annual Review of Astronomy and Astrophysics, 32, 277
  * Fabian et al. (2003) Fabian A. C., Sanders J. S., Crawford C. S., Conselice C. J., Gallagher J. S., Wyse R. F. G., 2003, MNRAS, 344, L48
  * Federrath (2013) Federrath C., 2013, MNRAS, 436, 1245
  * Federrath et al. (2008) Federrath C., Klessen R. S., Schmidt W., 2008, ApJ, 688, L79
  * Federrath et al. (2009) Federrath C., Klessen R. S., Schmidt W., 2009, ApJ, 692, 364
  * Federrath et al. (2010) Federrath C., Roman-Duval J., Klessen R. S., Schmidt W., Mac Low M. M., 2010, A&A, 512, A81
  * Federrath et al. (2021) Federrath C., Klessen R. S., Iapichino L., Beattie J. R., 2021, Nature Astronomy, 5, 365
  * Fryxell et al. (2000) Fryxell B., et al., 2000, The Astrophysical Journal Supplement Series, 131, 273
  * Fujita et al. (2020) Fujita Y., Cen R., Zhuravleva I., 2020, MNRAS, 494, 5507
  * Gendron-Marsolais et al. (2018) Gendron-Marsolais M., et al., 2018, MNRAS, 479, L28
  * Goldreich & Sridhar (1995) Goldreich P., Sridhar S., 1995, ApJ, 438, 763
  * Govoni & Feretti (2004) Govoni F., Feretti L., 2004, International Journal of Modern Physics D, 13, 1549
  * Grauer et al. (1994) Grauer R., Krug J., Marliani C., 1994, Physics Letters A, 195, 335
  * Grete et al. (2020) Grete P., O’Shea B. W., Beckwith K., 2020, ApJ, 889, 19
  * Grete et al. (2021) Grete P., O’Shea B. W., Beckwith K., 2021, ApJ, 909, 148
  * Gronke et al. (2021) Gronke M., Oh S. P., Ji S., Norman C., 2021, arXiv e-prints, p. arXiv:2107.13012
  * Harris et al. (2020) Harris C. R., et al., 2020, Array programming with NumPy (arXiv:2006.10256), doi:10.1038/s41586-020-2649-2, https://doi.org/10.1038/s41586-020-2649-2
  * Hillel & Soker (2016) Hillel S., Soker N., 2016, MNRAS, 455, 2139
  * Hillel & Soker (2020) Hillel S., Soker N., 2020, ApJ, 896, 104
  * Hitomi Collaboration (2016) Hitomi Collaboration 2016, Nature, 535, 117
  * Hu (1992) Hu E. M., 1992, ApJ, 391, 608
  * Hunter (2007) Hunter J. D., 2007, Computing in Science Engineering, 9, 90
  * Iroshnikov (1964) Iroshnikov P. S., 1964, Soviet Ast., 7, 566
  * Ji et al. (2020) Ji S., et al., 2020, MNRAS, 496, 4221
  * Kazantsev (1968) Kazantsev A. P., 1968, Soviet Journal of Experimental and Theoretical Physics, 26, 1031
  * Khatri & Gaspari (2016) Khatri R., Gaspari M., 2016, MNRAS, 463, 655
  * Kolmogorov (1941) Kolmogorov A. N., 1941, Akademiia Nauk SSSR Doklady, 32, 16
  * Körtgen et al. (2021) Körtgen B., Pingel N., Killerby-Smith N., 2021, MNRAS, 505, 1972
  * Kraichnan (1965) Kraichnan R. H., 1965, Physics of Fluids, 8, 1385
  * Lakhchaura et al. (2018) Lakhchaura K., et al., 2018, MNRAS, 481, 4472
  * Li et al. (2020) Li Y., et al., 2020, ApJ, 889, L1
  * McDonald et al. (2012) McDonald M., Veilleux S., Rupke D. S. N., 2012, ApJ, 746, 153
  * Mohapatra & Sharma (2019) Mohapatra R., Sharma P., 2019, MNRAS, 484, 4881
  * Mohapatra et al. (2020) Mohapatra R., Federrath C., Sharma P., 2020, MNRAS, 493, 5838
  * Mohapatra et al. (2021a) Mohapatra R., Jetti M., Sharma P., Federrath C., 2021a, arXiv e-prints, p. arXiv:2107.07722
  * Mohapatra et al. (2021b) Mohapatra R., Federrath C., Sharma P., 2021b, MNRAS, 500, 5072
  * Müller & Biskamp (2000) Müller W.-C., Biskamp D., 2000, Phys. Rev. Lett., 84, 475
  * Nelson et al. (2020) Nelson D., et al., 2020, MNRAS, 498, 2391
  * Newville et al. (2016) Newville M., Stensitzki T., Allen D. B., Rawlik M., Ingargiola A., Nelson A., 2016, Lmfit: Non-Linear Least-Square Minimization and Curve-Fitting for Python (ascl:1606.014)
  * Olivares et al. (2019) Olivares V., et al., 2019, A&A, 631, A22
  * Pfrommer (2013) Pfrommer C., 2013, ApJ, 779, 10
  * Politano & Pouquet (1995) Politano H., Pouquet A., 1995, Phys. Rev. E, 52, 636
  * Pope (2000) Pope S. B., 2000, Turbulent Flows
  * Prasad et al. (2018) Prasad D., Sharma P., Babul A., 2018, ApJ, 863, 62
  * Pulido et al. (2018) Pulido F. A., et al., 2018, ApJ, 853, 177
  * Rose et al. (2019) Rose T., et al., 2019, MNRAS, 489, 349
  * Ruszkowski et al. (2018) Ruszkowski M., Yang H. Y. K., Reynolds C. S., 2018, ApJ, 858, 64
  * Sarzi et al. (2018) Sarzi M., Spiniello C., La Barbera F., Krajnović D., van den Bosch R., 2018, MNRAS, 478, 4084
  * Schmidt et al. (2006) Schmidt W., Hillebrandt W., Niemeyer J. C., 2006, Computers & Fluids, 35, 353
  * Schmidt et al. (2008) Schmidt W., Federrath C., Klessen R., 2008, Phys. Rev. Lett., 101, 194505
  * Sharma et al. (2010) Sharma P., Parrish I. J., Quataert E., 2010, ApJ, 720, 652
  * She & Leveque (1994) She Z.-S., Leveque E., 1994, Phys. Rev. Lett., 72, 336
  * Simionescu et al. (2019) Simionescu A., et al., 2019, Space Sci. Rev., 215, 24
  * Strickland et al. (2004) Strickland D. K., Heckman T. M., Colbert E. J. M., Hoopes C. G., Weaver K. A., 2004, ApJS, 151, 193
  * Stutzki et al. (1998) Stutzki J., Bensch F., Heithausen A., Ossenkopf V., Zielinsky M., 1998, A&A, 336, 697
  * Sun et al. (2021) Sun M., et al., 2021, arXiv e-prints, p. arXiv:2103.09205
  * Sunyaev & Zeldovich (1970) Sunyaev R. A., Zeldovich Y. B., 1970, Ap&SS, 7, 3
  * Sutherland & Dopita (1993) Sutherland R. S., Dopita M. A., 1993, The Astrophysical Journal Supplement Series, 88, 253
  * Tang & Churazov (2018) Tang X., Churazov E., 2018, MNRAS, 477, 3672
  * Tremblay et al. (2018) Tremblay G. R., et al., 2018, ApJ, 865, 13
  * Vantyghem et al. (2019) Vantyghem A. N., et al., 2019, ApJ, 870, 57
  * Vantyghem et al. (2021) Vantyghem A. N., et al., 2021, ApJ, 910, 53
  * Virtanen et al. (2020) Virtanen P., et al., 2020, Nature Methods, 17, 261
  * Voit (2021) Voit G. M., 2021, ApJ, 908, L16
  * Waagan et al. (2011) Waagan K., Federrath C., Klingenberg C., 2011, Journal of Computational Physics, 230, 3331
  * Wang et al. (2021) Wang C., Ruszkowski M., Pfrommer C., Oh S. P., Yang H. Y. K., 2021, MNRAS, 504, 898
  * Werner et al. (2013) Werner N., et al., 2013, ApJ, 767, 153
  * Werner et al. (2014) Werner N., et al., 2014, MNRAS, 439, 2291
  * Yang & Reynolds (2016) Yang H. Y. K., Reynolds C. S., 2016, ApJ, 829, 90
  * Zeldovich & Sunyaev (1969) Zeldovich Y. B., Sunyaev R. A., 1969, Ap&SS, 4, 301
  * Zhang et al. (2018) Zhang C., Churazov E., Schekochihin A. A., 2018, MNRAS, 478, 4785
  * Zhuravleva et al. (2014) Zhuravleva I., et al., 2014, Nature, 515, 85
  * Zhuravleva et al. (2018) Zhuravleva I., Allen S. W., Mantz A., Werner N., 2018, ApJ, 865, 53
  * ZuHone et al. (2016) ZuHone J., Markevitch M., Zhuravleva I., 2016, ApJ, 817, 18
  * van der Velden (2020) van der Velden E., 2020, The Journal of Open Source Software, 5, 2004

## Appendix A: Density distribution function

In fig. A1, we show the volume PDF of normalised density for our three
standard runs and the hot phase driving run. There is a strong contrast in the
density of the two phases for the f0.10vwHR run, less for the f0.10HR run, and
the least for the f0.10magHR and the hot phase driving runs.

Figure A1: The volume PDF of normalised density for our three standard runs
and the hot phase driving run. All the PDFs have two peaks, one at small
$\bar{\rho}$ corresponding to the hot phase and another at large $\bar{\rho}$
corresponding to the cold phase. The hot and cold phase peaks are the most
separated for the f0.10vwHR run, less separated for the f0.10HR run, and
closest for the f0.10magHR and f0.10hotdriv runs.

## Appendix B: structure function slopes and parameters

Table B1: $\mathrm{VSF}_{2}$ fit parameters

Label | $v_{\mathrm{rms}}$ | 3D $\mathrm{VSF}_{2}$ | Projected $\mathrm{VSF}_{2}^{\mathrm{LOS}}$
---|---|---|---
| $(\mathrm{km/s})$ | $m$ | $m_{l}$ | $m_{s}$ | $r_{b}$ | $\delta$
(1) | (2) | (3) | (4) | (5) | (6) | (7)
f0.10HR h | $221\pm 15$ | $0.77\pm 0.003$ | $0.66\pm 0.005$ | $0.53\pm 0.003$ | $0.07\pm 0.001$ | $0.15\pm 0.03$
f0.10HR c | $205\pm 18$ | $0.86\pm 0.001$ | $0.71\pm 0.002$ | $0.54\pm 0.003$ | $0.06\pm 0.001$ | $0.15\pm 0.01$
f0.10vwHR h | $218\pm 12$ | $0.43\pm 0.003$ | $1.17\pm 0.004$ | $0.82\pm 0.01$ | $0.06\pm 0.001$ | $0.15\pm 0.01$
f0.10vwHR c | $167\pm 11$ | $0.51\pm 0.001$ | $0.53\pm 0.01$ | $0.26\pm 0.01$ | $0.07\pm 0.001$ | $0.15\pm 0.03$
f0.10magHR h | $185\pm 14$ | $0.50\pm 0.001$ | $0.70\pm 0.002$ | $0.65\pm 0.002$ | $0.07\pm 0.001$ | $0.13\pm 0.02$
f0.10magHR c | $183\pm 20$ | $1.05\pm 0.001$ | $0.73\pm 0.007$ | $0.62\pm 0.003$ | $0.08\pm 0.001$ | $0.15\pm 0.02$

In table B1, we show the fitting parameters for the turbulence with cooling
runs. We fit a smoothly broken power-law function ‘$\mathrm{SBP}$’ to the
scaling range of the $\mathrm{VSF}_{2}^{\mathrm{LOS}}$. This function is given
by:

$\mathrm{SBP}(r,A,r_{b},m_{s},m_{l},\delta)=A\left(\frac{r}{r_{b}}\right)^{m_{s}}\left[0.5\left(1+\frac{r}{r_{b}}\right)^{1/\delta}\right]^{(m_{l}-m_{s})\delta},$
(B7)

where $A$, $r_{b}$, $m_{s}$, $m_{l}$ and $\delta$ are fitting parameters. For
$r\ll r_{b}$, ‘$\mathrm{SBP}$’ scales as $r^{m_{s}}$ and for $r\gg r_{b}$, it
scales as $r^{m_{l}}$. The parameter $\delta$ denotes the smoothness of the
transition between the two power-laws and has a maximum value of $0.15$. This
function is also used in the broken-power law fit in eq. 11 of Körtgen et al.
(2021).

## Appendix C: VSF convergence

In fig. C1, we show the convergence of the compensated $\mathrm{VSF}_{2}$ with
number of sample pairs of points used to calculate the $\mathrm{VSF}_{2}$ for
a single snapshot at $t=0.911$ $\mathrm{Gyr}$. We use $10^{9}$ sample pairs
for all the 3D $\mathrm{VSF}_{2}$ computations in the main text. Our algorithm
has also been implemented and tested for convergence in Federrath et al.
(2008, 2009, 2021).

Figure C1: The compensated $\mathrm{VSF}_{2}$ of the HR runs ($768^{3}$
resolution without cooling run) for three different sample number of pairs of
points ($n_{\mathrm{pairs}}$). We obtain convergence for
$n_{\mathrm{pairs}}=10^{9}$, and choose this to evaluate the
$\mathrm{VSF}_{2}$ in this study.