11institutetext: Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112,
D-21029 Hamburg, Germany
11email<EMAIL_ADDRESS>22institutetext: Department of Physics
and Astronomy, Michigan State University, East Lansing, MI 48824, USA
22email<EMAIL_ADDRESS>
# Turbulence in the intragroup and circumgalactic medium

W. Schmidt 11 J. P. Schmidt 11 P. Grete 22

(Preprint)

###### Abstract

Context. In massive objects, such as galaxy clusters, the turbulent velocity
dispersion, $\sigma_{\mathrm{turb}}$, is tightly correlated to both the object
mass, $M$, and the thermal energy.

Aims. Here, we investigate whether these scaling laws extend to lower-mass
objects in dark-matter filaments.

Methods. We perform a cosmological zoom-in simulation of a filament using an
adaptive filtering technique for the resolved velocity component and a
subgrid-scale model to account for the unresolved component. We then compute
the mean turbulent and thermal energies for all halos in the zoom-in region
and compare different definitions of halo averages. Averaging constrained by
density and temperature thresholds is favored over averages solely based on
virial spheres.

Results. We find no clear trend for the turbulent velocity dispersion versus
halo mass, but significant correlation and a scaling law with exponent
$\alpha\sim 0.5$ between the turbulent velocity dispersion and thermal energy
that agrees with a nearly constant turbulent Mach number, similar to more
massive objects.

Conclusions. We conclude that the self-similar energetics proposed for galaxy
clusters extends down to the CGM of individual galaxies.

###### Key Words.:

galaxies: groups: general, galaxies: evolution, galaxies: star formation,
hydrodynamics, turbulence, methods: numerical

##  1 Introduction

In contrast to galaxy clusters, groups consist of a small number of galaxies
and their halos have a typical mass $\sim 10^{13}M_{\odot}$. Most importantly,
groups of galaxies along with isolated galaxies are mainly found in the dark-
matter filaments of the cosmic web (Lietzen et al. 2012; Cautun et al. 2014;
Tempel et al. 2014). Even so, they host nearly half of all galaxies and there
are observations indicating that the intragroup medium (IGrM) constitutes a
significant fraction of the baryons in the Universe (Mulchaey 2000; Freeland &
Wilcots 2011). Observations of groups also suggest that they are typically out
of equilibrium (O’Sullivan et al. 2014). In particular, scaling laws for the
X-ray luminosity of groups are observed to deviate from clusters (Bharadwaj et
al. 2015; Vajgel et al. 2014; Lovisari et al. 2015) in two ways. First, groups
exhibit a more pronounced scatter than clusters. Second, there exist some
indications of a change of slope in the luminosity-mass and luminosity-
temperature relations.

In numerical simulations, various processes affecting the IGrM have been
identified. For example, the rate of galaxy mergers is relatively high in
groups (Díaz-Giménez & Mamon 2010). Outflows produced by active galactic
nuclei (AGN) and supernovae, which are enhanced by mergers, are expected to
have a significant impact on thermal and non-thermal properties of IGrM (Mihos
& Hernquist 1996; Planelles et al. 2013; Liang et al. 2016; Martin et al.
2018; Patton et al. 2020). In addition to tides and outflows, galaxies
interact via gas stripping with the surrounding medium (Iapichino et al. 2008;
Roediger et al. 2015). All of these processes produce turbulence and heat the
IGrM.

By computing luminosity relations for simulated groups, Liang et al. (2016)
and Paul et al. (2017) were able to confirm deviations from the scaling that
follows from self-similar structure formation. Objects in the mass range of
groups show a steeper slope and relatively high entropy. This suggests that
the energy budget of the IGrM is not predominately controlled by the depth of
the gravitational potential well. Indeed, the analysis of Paul et al. (2017)
indicates that many groups are far from virial equilibrium. If this is so,
tidal tails of merging galaxies, turbulent wakes produced by galaxies falling
into the potential well of a group, and outflows of galaxies should elevate
the kinetic energy by stirring turbulent motions in the IGrM. Since outflows
are also expected for isolated galaxies, they should produce turbulence in the
gas surrounding the galaxies, i.e. the circumgalactic medium (CGM), see, e.g.,
Tumlinson et al. (2017); Lochhaas et al. (2020).

As a result, the turbulent velocity dispersion is an important indicator of
non-equilibrium conditions. Unfortunately, the computation of the turbulent
velocity dispersion is nontrivial. From a physical point of view, it is
important to distinguish between bulk motions (that are induced, for instance,
by gravity) and fully nonlinear, turbulent flows. For example, the accretion
of gas into the potential well of a halo produces bulk flows on large scales,
while eddies produced by hydrodynamic instabilities in the wake of a moving
galaxy or an outflow propagating into the surrounding medium are turbulent.
Separating these components is not possible with the commonly applied method
to compute standard deviations of the radial velocity component in spherical
shells. To circumvent this issue, Schmidt et al. (2016) computed the three-
dimensional turbulent velocity dispersion $\sigma_{\mathrm{turb}}$ by means of
an adaptive algorithm that filters out bulk flows (e.g., accretion flows). It
was shown that both in the intracluster medium (ICM) and in the warm-hot
intergalactic medium (WHIM), the turbulent velocity dispersion follows power
laws. In particular, a strong correlation of turbulent and thermal energies
was established, indicating what is called second self-similarity (Miniati &
Beresnyak 2015; Schmidt et al. 2017). This relation can be expressed as
$\sigma_{\mathrm{turb}}\propto e^{\,\alpha}$, where $e$ is the specific
thermal energy and $\alpha$ the power-law exponent. For the ICM,
$\alpha\approx 0.5$ was found, implying that the turbulent Mach number in
halos is roughly constant. Moreover, the turbulent Mach number in halos was
generally found to be close to unity. This is physically expected, as
turbulence is driven by accretion shocks, mergers, stellar feedback,and active
galactic nuclei. Supersonic flows generated by these processes rapidly decay
to subsonic turbulence, which in turn decays more slowly.

In light of the observational and numerical results mentioned above, it is
interesting to ask whether the energetics of objects in the mass range of
groups and isolated galaxies reveal fundamental differences compared to more
massive clusters, in other words, whether $\alpha$ changes. In this article,
we use the adaptive mesh refinement (AMR) Enzo to zoom into a filament in a
cosmological volume with a spatial resolution scale of $1\,\mathrm{kpc}$ at
the maximum refinement level. The subgrid physics encompasses a novel subgrid-
scale (SGS) model for numerically unresolved turbulence (Grete et al. 2016) in
combination with standard recipes for star formation and stellar feedback in
galaxies. After describing the numerical methods applied in our simulations in
more detail in the following Section, we present an analysis of the numerical
data in Sect. 3. We begin with a phenomenological discussion and an analysis
of radial profiles for some representative halos. Then we present statistics
for all halos in the selected filament including an analysis of how to
constrain the IGrM or, equivalently, the circumgalactic medium of isolated
galaxies in simulations. Finally, we investigate the correlations of the mean
turbulent velocity dispersion with the halo mass and the thermal energy. In
our conclusions in Sect. 4, we discuss our results with regard to the question
raised above.

##  2 Numerical methods

We use a modified version of the publicly available, open source cosmological
AMR code Enzo (Bryan et al. 2014; Brummel-Smith et al. 2019).111See also
website enzo-project.org. Our modifications are available at
https://github.com/pgrete/enzo-dev. Commit 6300b03 was used in this work. The
code is MPI parallelized, features N-body dynamics based on a second-order
drift-kick-drift algorithm in combination with cloud-in-cell interpolation to
compute the joint gravitational potential of gas and particles, and a variety
of different finite volume solvers for gas dynamics and magnetic fields. In
our simulations, we apply the monotonic upstream-centered scheme for
conservation laws (MUSCL) with a local Lax-Friedrichs (LLF) solver for
sufficient numerical robustness. In addition, we utilize a structural subgrid-
scale (SGS) model that computes effects of numerically unresolved turbulence
on the basis of the instantaneous flow structure at the smallest resolved
scales, see Grete et al. (2016, 2017b) for idealized setups and Grete et al.
(2019) for a cosmological application. This approach avoids the difficulties
and computational cost of solving an additional partial differential equation
(Maier et al. 2009). While the impact on gas properties in halos is typically
negligible compared to the statistical variation among comparable objects
evolving from different initial conditions (Grete et al. 2019), it was found
that the employed model improves higher-order statistics of quantities such as
the vorticity in compressible turbulence simulations (Grete et al. 2017b).
Since turbulence is confined to the cosmic web (Schmidt et al. 2016), we
implemented the Kalman filtering technique for spatially inhomogeneous
turbulence into Enzo. Since no no fixed smoothing scale is applied, this
method is particularly suitable for spatially inhomogeneous turbulence. In the
statistically stationary regime, the filter operates like an exponential low-
pass filter with a characteristic time-scale of $5\,$Gyr and a characteristic
velocity scale of $100\,{\mathrm{km/s}}$ (see statistics discussed in Sect.
3.2). As a result, Kalman filtering allows us to separate the turbulent
fluctuation $\@vec{v}^{\prime}$ from non-turbulent bulk flows such as gas
accretion into the potential wells of halos and filaments:

$\@vec{v}^{\prime}=\@vec{v}-[\@vec{v}],$ (1)

where $\@vec{v}$ and $[\@vec{v}]$ is the unfiltered and filtered velocity,
respectively. In combination with the SGS model for the numerically unresolved
specific kinetic energy $E_{\mathrm{sgs}}$, we define the turbulent velocity
dispersion by

$\sigma_{\rm turb}^{2}=|\@vec{v}^{\prime}|^{2}+2E_{\mathrm{sgs}},$ (2)

A detailed description and numerical tests of the algorithm can be found in
Schmidt et al. (2014).

To achieve sufficient mass resolution for groups of galaxies and individual
galaxies in a filament, the zoom-in technique is applied. With the help of
MUSIC (Hahn & Abel 2011), we generated initial data for a $256^{3}$
cosmological box of co-moving size $50\,\mathrm{Mpc}/h$ and four nested grids
(levels 1-4) in a volume of about $3.9\times 8.6\times
3.9\,(\mathrm{Mpc}/h)^{3}$. As cosmological parameters, we chose $h=0.673$,
$\Omega_{\rm b}=0.0487$, and $\Omega_{\rm m}=0.315$ from the Planck
Collaboration et al. (2014).222Corrections in later releases of the Planck
data are not significant for this work. The particle mass is about $8\times
10^{8}\,M_{\odot}$ at the coarsest (root-grid) resolution and $2\times
10^{5}\,M_{\odot}$ in the zoom-in region. To reach the targeted spatial
resolution of $1\,\mathrm{kpc}$, up to four AMR levels (levels 5 to 8) were
added in the course of the simulation by applying refinement by dark matter
and baryon mass. The total number of cells in our highest-resolution run
culminated above $5\times 10^{8}$. As an illustration, Figure 1 shows a slice
through the nested-grid region around a prominent filament. We use the Python
package yt (Turk et al. 2011) for postprocessing and visualization.

Figure 1: Slice of the gas density in a zoom-in simulation of a filament at
redshift $z=0$. The shown region is a close-up view of the central quarter of
the simulation box ($74.3\,\mathrm{Mpc}$ physical size). Gray lines show edges
of Enzo subgrids (i.e. pieces distributed among processors) up to level 4. AMR
levels 5-8 are hidden.

To treat chemical species and radiative cooling, we use the Grackle library.
In our simulations, we applied a 6-species atomic H and He network, metal
cooling using the Cloudy tables, and heating and cooling rates and UV
background rates from Haardt & Madau (2012).333For further details, see
grackle.readthedocs.io/en/latest/Parameters.html In addition to cooling, star
formation and feedback are crucial ingredients to determine the thermodynamic
state of the gas. Unfortunately, star formation is notoriously difficult to
model in cosmological simulations (see review by Naab & Ostriker 2017). For
this reason, we evaluated three models in test runs at lower resolution (two
AMR levels, six levels of refinement in total) in preparation for our fiducial
run. 444Usually, the total star formation rate in a cosmological volume with
periodic boundary conditions is normalized to stellar mass per unit time and
unit volume. However, since stars form only in the nested-grid region in our
simulations, we need to estimate the normalized star formation rate based on
this region. This allows us to compare trends for different star formation
models. First, the model of Kravtsov (2003) (K03) that assumes that star
formation is proportional to the local gas density. This assumption
corresponds to a global Schmidt law.555To reduce fluctuations, we modified the
code to support stochastic star formation also for the K03 model. The
coefficient of proportionality is given by the inverse of a time scale, which
we set to $1\,\mathrm{Gyr}$. With a decline by about one order of magnitude
toward low redshift, see dot-dashed orange line in Fig. 2, the K03 model is
qualitatively in agreement with the observed redshift dependence of the star
formation rate (Behroozi et al. 2013). Nevertheless, the model breaks down
close to $z=0$, as reflected by the sudden burst of star formation. At
intermediate redshifts ($z\sim 1$), the KO3 model roughly agrees with an
expected star formation efficiency $\varepsilon=0.01$. The second model by Cen
& Ostriker (1992) (CO92) directly employs a stochastic star formation recipe
with $\varepsilon$ as free parameter controlling the star formation
efficiency. The two test runs with $\varepsilon=0.01$ and $\varepsilon=0.1$
(dotted blue and turquoise lines in Fig. 2) span the range of efficiencies
that can be reasonably assumed. However, both runs result in unrealistically
high star formation at lower redshifts. The third model we tested is the
dynamical model proposed by Semenov et al. (2016) (S+16). In this case, the
computation of the star formation efficiency is based on the local turbulent
Mach number following from the SGS model. While the model seems to perform
well in simulations of isolated disk galaxies, our results suggest that the
model initially overproduces stars, followed by a too rapid decline, see green
dotted line in Fig. 2. Apart from the underlying assumptions of the model, a
potential problem is that one would need to scale the turbulent energy down to
the scale of star forming clouds, which is much below the spatial resolution
scale of our simulations.

Figure 2: Mean star formation rate per unit volume in zoom-in region for
different models and resolutions: Cen & Ostriker (1992) model (CO92) with
stochastic star formation, where $\varepsilon$ is the star formation
efficiency, a dynamical model with local star formation efficiency (Semenov et
al. 2016, S+16), and the Kravtsov (2003) model (K03) with threshold number
density $n$ and a fixed star formation time scale of $1\,\mathrm{Gyr}$. The
number of levels specified in the legend refers to the number of AMR levels on
top of nested-grid levels. Our fiducial model is shown as solid (online
version: red) line.

Overall, the K03 model follows most closely the observed star formation rates.
Thus, we decided to use the K03 model in our fiducial run. Finally, the
threshold density above which gas can be turned into star particles needs to
be adjusted to the numerical resolution. While we achieved good agreement for
two and three AMR levels (see dashed and dot-dashed lines in Fig. 2), we
increased the threshold by a factor of ten to a number density of
$50\,\mathrm{cm}^{-3}$ in the case of the highest resolution. Otherwise, the
star formation rate for $z\lesssim 2$ would have become too high, however, at
the cost of a systematically lower star formation rate in the high-redshift
regime.

The K03 model employs supernova feedback similar to the prescription of CO92.
Basically, a certain fraction $f_{m\ast}$ of the stellar mass is assumed to be
ejected, resulting in momentum feedback. Thermal feedback is controlled by the
fraction $f_{\mathrm{SN}}$ of a star particle’s rest energy that is deposited
as thermal energy into the gas. The only difference compared to CO92 is that
feedback is applied instantaneously, which is a reasonable approximation for
resolutions in the kpc range, where the numerical time step $\Delta t\gtrsim
10^{6}\,\mathrm{yr}$ is comparable to or greater than the life time of massive
stars. Momentum feedback potentially contributes to the production of
turbulence, although gas motions that are induced locally at the grid scale
will be strongly affected by numerical dissipation. Thermal feedback produces
hot bubbles, particularly during episodes of intense star formation. The
expansion of such bubbles can also give rise to turbulence. For this work, we
adopt the commonly used default parameters $f_{m\ast}=0.25$ and
$f_{\mathrm{SN}}=10^{-5}$. For further details, see Bryan et al. (2014) and
references therein.

Figure 3: Slices of gas density, temperature, and turbulent velocity for the
example compact group halo with a halo mass of $M_{\rm halo}=1.17\times
10^{12}M_{\odot}$ at redshift $z=1$. The panels are scaled to a size of
$4R_{\rm vir}$, where the virial radius of the halo is $R_{\rm vir}=298\,$kpc.

##  3 Results

In this section, we begin with a phenomenological discussion of individual
halos accompanied by an analysis of halo profiles. This is the basis for
determining constraints to distinguish the CGM/IGrM from other gas phases.
Then we analyze mean CGM/IGrM values for all halos in the zoom-in region. We
concentrate on the the cosmological epoch after the observed peak of star
formation and AGN activity. To study evolutionary trends, statistics for $z=0$
and $1$ are compared.

###  3.1 Individual halos

In the following, we analyze data from our high-resolution runs (4 nested grid
levels, 4 AMR levels, global Schmidt law for number densities
$n>50\,\mathrm{{}^{-3}}$). We applied the HOP finder of yt, to identify halos
in the zoom-in region (Eisenstein & Hut 1998). The HOP algorithm defines halos
by searching for density peaks and grouping particles in distinct groups based
on nested density contours (peak, saddle, and outer boundary). The resulting
halos can be either isolated galaxies or groups of galaxies.666For a brief
outline of the algorithm, see https://yt-
project.org/doc/analyzing/analysis_modules/halo_catalogs.html?highlight=hop#hop.
We applied the halo finder with default parameters. At redshift $z=1$, 102
objects with halo masses between $10^{10}\,M_{\odot}$ and $1.32\times
10^{12}\,M_{\odot}$ and virial radii in the range from $28$ to
$298\,\mathrm{kpc}$ were identified.777The halo mass is defined by the total
mass of the particle group identified as halo by the HOP finder. Halos of
lower mass are excluded from our analysis. Apart from relatively poor mass
resolution, turbulent structures smaller than about ten times the grid scale
(i.e., $\lesssim 10\,\mathrm{kpc}$ at the maximum refinement level) are
strongly damped by numerical dissipation (Grete et al. 2017a), making
estimates of turbulent velocities in small, low-mass halos infeasible.

Figure 4: Same as in Fig. 3 for the isolated galaxy example within a lower
mass halo ($M_{\rm halo}=9.90\times 10^{10}M_{\odot}$, $R_{\rm vir}=73\,$kpc).

Examples are visualized in Figs 3 and 4, respectively, which show slices of
the baryonic gas density $\rho$, the temperature $T$ of the gas, and the
turbulent velocity dispersion $\sigma_{\rm turb}$ defined by equation (2). For
a particular object, the slices are scaled to twice the diameter of the halo.
The object shown in Fig 3 is located outside of the major filament in the
zoom-in region. Its mass of $1.17\times 10^{12}\,M_{\odot}$ is typical for a
galactic halo. Two off-center density peaks in the lower right quarter of the
region shown in Fig 3 are correlated with low temperatures $T\lesssim
10^{4}\,\mathrm{K}$. These structures can be interpreted as satellites of the
galaxy at the center of the halo. Within this halo there are sharp outward
drops in density and temperature are associated with outer accretions shocks.
As discussed in Schmidt et al. (2016), the volume inside of the accretions
shocks is filled with turbulent gas. Compared to the medium in the void,
$\sigma_{\rm turb}$ is at least two orders of magnitude larger in the WHIM and
ICM of clusters. For the halo shown in Fig. 3, the gas outside the accretion
shocks is comparatively quiescent, but the change of $\sigma_{\rm turb}$
across the outer shocks is less pronounced (see right panel). The residuals of
the order of $10\,\mathrm{km/s}$ in low-density regions are caused by slowly
decaying temporal correlations in the filtered flow888See Schmidt et al.
(2014), section 3, for a detailed discussion. Nevertheless, these residuals
are typically a factor of ten smaller than the turbulent velocity dispersion
inside the halo (see radial profiles below).

The object shown in Fig. 4 has a halo mass that is by one order of magnitude
smaller than in the previous example. At the center is a single galaxy with
outflows that can be discerned as hot gas (middle slice) at low densities
(left slice). In this case, a turbulent velocity dispersion of a few
$100\,\mathrm{km/s}$ is found inside the galaxy and in some of the ejected
gas. In most of the CGM, $\sigma_{\rm turb}$ is around $100\,\mathrm{km/s}$.
This is confirmed by the radial profiles shown in Fig. 5, where this halo
(green line) is compared to the halo discussed above (blue line) and two
additional halos with masses of $3.87\times 10^{11}\,M_{\odot}$ (orange line)
and $6.23\times 10^{10}\,M_{\odot}$ (red line). While the gas of the most
massive halo has profiles that resemble those of small galaxy clusters, i.e.
they are relatively flat near the core and fall off steeply at the outskirts
(Schmidt et al. 2016), the lower mass objects have pronounced peaks at the
center and falls off more gradually with radius. As suggested by the profiles
of the two additional halos, the transition is rather gradual and the shapes
of the profiles vary substantially. It is important to keep in mind that
scales $\lesssim 10$ kpc are affected by numerical dissipation and that radial
binning presumes spherical symmetry. Especially for galactic halos, the disk-
like structures can strongly distort the profiles. For example, the central
dip of the temperature profile for the halo with mass $9.9\times
10^{10}\,M_{\odot}$ is not as distinct as one would suppose, given the
temperature slice shown in Fig. 4. This is a consequence of averaging over
low-temperature gas in the ISM and hot gas in the CGM in spherical shells.
Separately, we assess the robustness of the profiles in Fig. 5 by plotting the
mean and median values (solid vs. dashed lines) as well as the interquartile
ranges (IQRs as shaded regions). In addition to the raw IQR, deviations
between the mean and median indicate a large spread in a quantity. Those are
particularly useful for quantities that vary over several orders of magnitude
as the mean values tend to be dominated by the largest values in the sample.
Beyond the central regions ($R\gtrsim 10$ kpc) the profiles exhibit a limited
spread hinting at some first trends. Roughly speaking, more massive halos
typically also have higher temperatures whereas there is no clear trend in the
turbulent velocity dispersion for halo with different masses.

Figure 5: Radial profiles of temperature $T$ and turbulent velocity dispersion
$\sigma_{\rm turb}$ for selected halos at redshift $z=1$. (Mean values per
radial bin are shown as solid lines, medians as dashed lines, and
interquartile ranges as shaded regions.) Figure 6: Resolution dependence of
radial profiles (mean values) for the halo of mass mass $M_{\rm
halo}=3.87\times 10^{11}M_{\odot}$ (see also Figs 4 and 5). The shaded regions
indicate the interquartile range for the highest resolution.

As discussed in Sect. 2, star formation and radiative cooling are sensitive to
numerical resolution. The tuning of star formation and feedback model
parameters to numerical resolution is a common problem in simulations of
galaxy evolution. As an example, the impact of the number of refinement levels
on radial profiles is shown in Fig. 6 for a halo of intermediate mass (i. e.,
$M_{\mathrm{halo}}\sim 10^{11}\,M_{\odot}$). While the turbulent velocity
dispersion profiles are comparable for different spatial resolutions, there
are clearly deviations in the temperature profiles. At lower resolution, star
formation activity is reduced and feedback becomes less efficient, resulting
in a drop of the gas temperature in the core. At least the maximum
temperatures agree for spatial resolutions of $2\,\mathrm{kpc}$ (3 AMR levels)
and $1\,\mathrm{kpc}$ (4 AMR levels). With a radius smaller than a tenth of
the virial radius, the core region fills only a small fraction of the total
volume of halo. As a result, resolution effects can be expected to affect mass
averages more strongly than volume averages.

Figure 7: Same as in Fig. 3 for a halo of mass $M_{\rm halo}=1.58\times
10^{12}M_{\odot}$ and $R_{\rm vir}=373\,$kpc at redshift $z=0$.

At redshift zero, the largest halo in the nested-grid region has a mass of
$3.61\times 10^{12}\,M_{\odot}$ and a virial radius $R_{\rm
vir}=467\,\mathrm{kpc}$. In total, we found 62 objects above
$10^{10}\,M_{\odot}$ with virial radii greater than $50\,\mathrm{kpc}$. From
the four halos with masses greater than $10^{12}\,M_{\odot}$, two are massive
galaxies, while the others are composed of multiple objects. An example is
shown in Fig. 7. There are two galaxies close to the center that are
surrounded by relatively dense and hot gas. Quite likely, these galaxies are
interacting. There are also smaller blobs of dense, cool gas that could be
either residuals from an interaction or smaller satellite galaxies. Thus, it
can be interpreted as a compact group at the lower end of the mass range of
groups. In contrast to the halo of comparable mass shown in Fig. 3, the
transition to the intergalactic medium of the filament is rather gradual. This
becomes clear when looking at the temperature profile of this halo, see blue
lines in Fig. 8. The temperature changes only little with radius, except for
the temperature increase in the core region. The profile of the turbulent
velocity dispersion is also rather flat. The halo of mass $1.22\times
10^{12}\,M_{\odot}$ (orange lines) is an isolated, massive galaxy. There is no
drop in temperature in the outskirts either and the temperature profile is
flat throughout the halo. This object might be a fossil group, which is the
end state of former group members merging into a single, dominant galaxy. The
other profiles show two single galaxies (green and red lines) with halo masses
$\sim 10^{11}\,M_{\odot}$. They are dominated by cooler gas close to the
center. Interestingly, they exhibit strong turbulence at radii $\sim
10\,\mathrm{kpc}$, where $\sigma_{\mathrm{turb}}$ is in the range from $100$
to more than $200\,\mathrm{km/s}$. Further outside in the CGM,
$\sigma_{\mathrm{turb}}$ decreases to a significantly lower background level.
This suggests that feedback from the galaxies enhances turbulence.

Figure 8: Radial profiles of temperature $T$ and turbulent velocity dispersion
$\sigma_{\rm turb}$ selected halos at redshift $z=0$. (Mean values per radial
bin are shown as solid lines, medians as dashed lines, and interquartile
ranges as shaded regions.)

###  3.2 Halo statistics

The key element of our analysis is the computation of mean energies for all
halos in the zoom-in region with mass $M_{\rm halo}\leavevmode\nobreak\
>\leavevmode\nobreak\ 10^{10}M_{\odot}$. The standard method is to average
over all cells within the virial sphere, i.e. from the center of the halo to
the radius $R_{\rm vir}$. However, our phenomenological discussion in Sect.
3.1 shows that warm, tenuous gas can be found inside the virial radius, while
hotter gas at higher density can extend beyond the virial radius. For this
reason, we need additional constraints to distinguish the CGM or IGrM from
other gas phases in filaments.

####  3.2.1 Identifying the CGM/IGrM

To remove the warm, tenuous gas we exclude gas below a minimum overdensity
$\delta_{\mathrm{min}}$ relative to the mean density of baryonic and dark
matter and below a minimum temperature $T_{\mathrm{min}}$:

$\delta>\delta_{\mathrm{min}},\quad\mathrm{and}\quad T>T_{\mathrm{min}}.$ (3)

In addition, we only include gas with a neutral hydrogen number density below
$1\,\mathrm{cm^{-3}}$. These constraints are applied to a spherical volume of
radius $R_{\mathrm{max}}=2R_{\mathrm{vir}}$. On the one hand, this includes
gas in the outskirts outside of the virial sphere. On the other hand, dense
gas at low temperatures inside galaxies, i.e., star-forming gas, is excluded.
A maximal radius of $2R_{\mathrm{vir}}$ corresponds to the regions shown Figs
3, 4, and 7. It is sufficiently large to contain structures like the galactic
halo in Fig. 4 while a significant overlap with neighboring halos is avoided.

Figure 9: Volume-weighted (left) and mass-weighted (right) mean values of the
turbulent velocity dispersion and thermal energy of halos for different
constraints at $z=0$. Solid lines show medians of objects in the mass range
$[M/2,2M]$ for each halo of mass $M$.

To quantitatively evaluate different thresholds, we first computed the volume-
and mass-weighted averages of the turbulent velocity dispersion and thermal
energy of all halos at redshift $z=0$ for varying overdensity
$\delta_{\mathrm{min}}$ of 1, 10, and 50 at a fixed temperature limit
$T_{\mathrm{min}}=2.5\times 10^{4}\,\mathrm{K}$, see Fig. 9. While the chosen
thresholds are arbitrary within the range of plausible densities and
temperatures of the CGM/IGrM, they avoid assumptions such as spherical
symmetry or correlations between the properties of a dark-matter halo and its
gas contents (see also the discussion in Schmidt et al. 2016 in the context of
clusters). The turbulent velocity dispersion $\sigma_{\mathrm{turb}}$ exhibits
no clear trend with halo mass, except that the scatter increases toward low-
mass halos. The results are overall not particularly sensitive to the chosen
threshold density. As indicated by the sliding median for a mass window
$[M/2,2M]$ (solid lines in Fig. 9), the typical turbulent velocity dispersion
is of the order of $100\,\mathrm{km/s}$. The mean thermal energies suggest a
drop from halo masses above $M_{\rm halo}\sim 10^{12}M_{\odot}$ to lower
masses. Since the four most massive halos are both galaxies and small groups,
it is unclear whether this drop has any significance or is merely a
statistical fluctuation. Similar to the turbulent velocity dispersion, we find
larger scatter for halos of lower mass. Moreover, some trends with the
thresholds $\delta_{\mathrm{min}}$ and $T_{\mathrm{min}}$ can be discerned:
Toward the low-mass end, thermal energies tend to be higher for lower
$\delta_{\mathrm{min}}$. This can be understood as a consequence of excluding
high-temperature gas at relatively low densities, for example, the shocked gas
that can be seen in Fig. 4. As expected, this trend is reduced for the lowest
density threshold if mass weighing is applied. We also calculated the mean
turbulent velocity dispersion and thermal energy for a higher minimum
temperature of $T_{\mathrm{min}}=5\times 10^{4}\,\mathrm{K}$ at intermediate
overdensity $\delta_{\mathrm{min}}=10$ to assess the effect of
$T_{\mathrm{min}}$. While the overall impact of the temperature threshold is
rather minor, one can see a clustering of the mean thermal energies just above
the chosen temperature threshold. This means that objects with a significant
fraction of gas in the temperature range below the threshold are shifted
upwards, which should be avoided. On the other hand, choosing
$T_{\mathrm{min}}$ significantly lower than $2.5\times 10^{4}\,\mathrm{K}$,
would partially mix up the surroundings of galaxies with warm component of the
ISM.

To gain a qualitative understanding of the thresholds, Fig. 10 shows the gas
temperature inside the regions constrained by the $\delta_{\mathrm{min}}$ and
$T_{\mathrm{min}}$ for the halo with $M_{\rm halo}=1.74\times
10^{11}M_{\odot}$ at $z=0$ (see also Fig. 8). In the case
$\delta_{\mathrm{min}}=1$ and $T_{\mathrm{min}}=2.5\times 10^{4}\,\mathrm{K}$
(top left panel in Fig. 10), only the ISM of the galaxy in the center is
excluded and the region for which the average is computed is simply cut off at
$r=2R_{\rm vir}$. The opposite extreme is $\delta_{\mathrm{min}}=50$
$T_{\mathrm{min}}=2.5\times 10^{4}\,\mathrm{K}$ (bottom left), where only the
most dense part of the CGM is included. The intermediate threshold
($\delta_{\mathrm{min}}=10$, right panels) effectively constrains the CGM to
gas inside $2R_{\rm vir}$. In agreement with the preceding analysis, the
resulting mean values of the thermal energy become higher if gas at lower
densities is included (see Table 1), while the differences between
$T_{\mathrm{min}}=2.5\times 10^{4}\,\mathrm{K}$ (top right) and
$10^{5}\,\mathrm{K}$ (bottom right) are rather small for this halo. The
largest turbulent velocity dispersion is found for $\delta_{\mathrm{min}}=50$.
In this case, turbulent gas in the close vicinity of the galaxy fills a
relatively large volume fraction. As indicated by the radial profile plotted
in Fig. 8, the mean turbulent velocity dispersion tends to decrease if more
gas from regions at radii $\gtrsim 100\,$kpc contributes to the average.

(a) $\delta>1$, $T>2.5\times 10^{4}\;{\mathrm{K}}$

(b) $\delta>10$, $T>2.5\times 10^{4}\;{\mathrm{K}}$

(c) $\delta>50$, $T>2.5\times 10^{4}\;{\mathrm{K}}$

(d) $\delta>10$, $T>10^{5}\;{\mathrm{K}}$

Figure 10: Temperature slices constrained by overdensity $\delta>\delta_{\mathrm{min}}$ and temperature $T>T_{\mathrm{min}}$ for a halo of mass $M_{\rm halo}=1.74\times 10^{11}M_{\odot}$ at redshift $z=0$. The inner circle shows the virial radius $R_{\rm vir}=194\,$kpc of the halo. For the computation of halo averages based on these constraints, a maximal radius of $2R_{\rm vir}$ (outer circle) is applied. Table 1: Mean thermal energy and turbulent velocity dispersion of the halo shown in Fig. 10 for different density and temperature thresholds. $\delta_{\mathrm{min}}$ | $T_{\mathrm{min}}$ |  $\langle\sigma_{\mathrm{turb}}\rangle\,$[km/s] |  $\langle e\rangle\,$[keV]
---|---|---|---
$1$ |  $2.5\times 10^{4}\,$K | 60.5 | 0.0334
$10$ |  $2.5\times 10^{4}\,$K | 53.9 | 0.0190
$10$ |  $10^{5}\,$K | 53.3 | 0.0195
$50$ |  $2.5\times 10^{4}\,$K | 72.6 | 0.0184
Figure 11: Mean values of the turbulent velocity dispersion vs halo mass (top) and mean thermal energy (bottom) for halos at redshift $z=1$ (left) and $z=0$ (right). As indicated in the legend, both volume- and mass-weighted averages are computed for cells within the virial radius or cells with density and temperature above the specified thresholds. While the thick solid lines in the top plots ($\sigma_{\mathrm{turb}}$ vs $M$) are sliding medians as Fig. 9, the lower plots ($\sigma_{\mathrm{turb}}$ vs $e$) show power-law fits with slope $\alpha$. Table 2: Median values of the turbulent velocity dispersion for $M_{\rm halo}\geq 10^{10}M_{\odot}$. $z$ | region (weighing) | median [km/s]
---|---|---
$1$ |  $R_{\mathrm{vir}}$ (vol) | 105.3
$1$ |  $R_{\mathrm{vir}}$ (mass) | 121.3
$1$ |  $\delta>10,T>2.5\times 10^{4}\,$K (vol) | 100.7
$1$ |  $\delta>10,T>2.5\times 10^{4}\,$K (mass) | 110.2
$0$ |  $R_{\mathrm{vir}}$ (vol) | 83.3
$0$ |  $R_{\mathrm{vir}}$ (mass) | 120.2
$0$ |  $\delta>10,T>2.5\times 10^{4}\,$K (vol) | 80.5
$0$ |  $\delta>10,T>2.5\times 10^{4}\,$K (mass) | 84.2

####  3.2.2 Scaling laws

Based on the quantitative and qualitative analysis, we we choose
$\delta_{\mathrm{min}}=10$ and $T_{\mathrm{min}}=2.5\times 10^{4}\,\mathrm{K}$
for the following analysis. Again, to evaluate the robustness of this choice
we also compare the results to the ones obtained from directly averaging over
the virial spheres. The resulting mean values of $\sigma_{\mathrm{turb}}$ at
redshifts $z=1$ and $0$ are plotted in Fig. 11. Both definitions of averages
result in similar distributions of $\sigma_{\mathrm{turb}}$ vs. halo mass,
which is confirmed by the sliding medians. For the complete sample, however,
we find that mass-weighted averages over virial spheres tend to be larger
compared to other averages (see medians of $\sigma_{\mathrm{turb}}$ listed in
Table 2). Since the radial profiles (see Figs 5 and 8) show that
$\sigma_{\mathrm{turb}}$ is larger in the cores than in the outskirts, mass-
weighing emphasizes the peak values of $\sigma_{\mathrm{turb}}$. As shown in
the preceding section, low-mass halos exhibit a stronger scatter. Moreover, at
lower redshift this scatter becomes even more pronounced, suggesting that
evolutionary effects beyond the self-similar gravitational collapse enhance
the diversity of low-mass halos. The overall median values confirm a trend of
decreasing $\sigma_{\mathrm{turb}}$ from $z=1$ to $0$, suggesting that energy
injection is reduced toward lower redshift. This is in agreement with the
decline of stellar feedback, which contributes to the production of turbulence
in the surroundings of galaxies.

The strong scatter of the energy-mass relations is confirmed by the
computation of correlation coefficients. Since the data cannot be assumed to
follow normal distributions,999This is easily confirmed by plotting histograms
of the data. we use Spearman’s nonparametric measure of correlation. The
results are listed in Table 3. While moderate correlations are found for
$z=1$, the halo masses become weakly correlated or uncorrelated with thermal
energy and turbulent velocity dispersion at $z=0$.

Table 3: Spearman’s correlations (correlation coefficient $r$ and $p$-value) between halo mass $M$, mean thermal energy $e$, and turbulent velocity dispersion $\sigma_{\mathrm{turb}}$ for the data shown in Fig. 11. |  |  $e$–$M$ |  $\sigma_{\mathrm{turb}}$–$M$ |  $\sigma_{\mathrm{turb}}$–$e$
---|---|---|---|---
$z$ | region (weighing) | $r$ | $p$ | $r$ | $p$ | $r$ | $p$
$1$ |  $R_{\mathrm{vir}}$ (vol) | 0.482 | $2.9\times 10^{-7}$ | 0.450 | $2.1\times 10^{-6}$ | 0.638 | $5.3\times 10^{-13}$
$1$ |  $R_{\mathrm{vir}}$ (mass) | 0.390 | $5.0\times 10^{-5}$ | 0.591 | $6.0\times 10^{-11}$ | 0.538 | $5.7\times 10^{-9}$
$1$ |  $\delta>10,T>2.5\times 10^{4}\,$K (vol) | 0.424 | $9.0\times 10^{-6}$ | 0.329 | $7.3\times 10^{-4}$ | 0.654 | $8.8\times 10^{-14}$
$1$ |  $\delta>10,T>2.5\times 10^{4}\,$K (mass) | 0.482 | $3.0\times 10^{-7}$ | 0.395 | $3.9\times 10^{-5}$ | 0.641 | $3.9\times 10^{-13}$
$0$ |  $R_{\mathrm{vir}}$ (vol) | 0.151 | 0.24 | 0.074 | 0.57 | 0.702 | $2.0\times 10^{-10}$
$0$ |  $R_{\mathrm{vir}}$ (mass) | 0.101 | 0.44 | 0.262 | 0.040 | 0.418 | $7.3\times 10^{-4}$
$0$ |  $\delta>10,T>2.5\times 10^{4}\,$K (vol) | 0.305 | 0.016 | -0.028 | 0.83 | 0.540 | $5.9\times 10^{-6}$
$0$ |  $\delta>10,T>2.5\times 10^{4}\,$K (mass) | 0.412 | $8.7\times 10^{-4}$ | 0.006 | 0.96 | 0.442 | $3.2\times 10^{-4}$

A different picture emerges when plotting $\sigma_{\mathrm{turb}}$ vs. the
thermal energy $e$. Similar to the analysis of clusters in Schmidt et al.
(2016), we find a correlation between the turbulent and thermal energies (with
$\sigma_{\mathrm{turb}}$ as proxy of the turbulent kinetic energy. The
correlation coefficients listed in Table 3 are in the range between 0.5 and
0.7 for volume-weighted averages and somewhat lower of mass-weighted averages.
The rather high correlations are confirmed by $p$-values: For $p<0.05$, it can
be excluded at $95\,\%$ confidence level that data sets appear correlated by
chance. However, compared to clusters (halo mass above $10^{13}\,M_{\odot}$),
the scatter is stronger. Power-law fits are shown as straight lines in Fig. 11
and their slopes $\alpha$ are listed in the lower legends. Although there are
small deviations between the slopes following from constraints (3) and volume-
averaged virial spheres, the data basically agree within the scatter. Compared
to mass-weighted averages over virial spheres, however, the discrepancy is
large. By inspecting individual halos, it clearly follows that the mean
thermal energy is shifted to systematically lower values if they are computed
for all gas inside the virial radius. The explanation is quite simple: Mass-
weighing emphasizes dense gas, including the cold gas in the interstellar
medium. This results in a significant bias that is avoided by excluding gas at
densities and temperatures that are characteristic for the ISM. These
contributions are reduced if volume-weighing is applied. Apart from that,
volume-averaging moves the focus away from the close vicinity of galaxies to
the outskirts of the CGM/IGrM. For this reason, we calculated the slopes for
the volume-averaged mean values based on the density and temperature
thresholds at different redshift, $z=\\{1,0.5,0.25,0\\}$, see Table 4. The
resulting $\alpha\sim 0.47$ varies only little with redshift and is consistent
with the scaling $\sigma_{\mathrm{turb}}\propto e^{\,0.5}$, i.e. a roughly
constant turbulent Mach number at given redshift (see also Schmidt et al.
2016). However, the power-law coefficient $\sigma_{0}$, i.e. the turbulent
velocity dispersion at an energy of $1\,\mathrm{keV}$, decreases with
redshift.

Table 4: Fit parameters for the power-law model $\langle\sigma_{\mathrm{turb}}\,[\mathrm{km/s}]\rangle=\sigma_{0}\langle e\,[\mathrm{keV}]\rangle^{\alpha}$, where volume-weighted mean values are computed for overdensities $\delta>10$ and $T>2.5\times 10^{4}\,$K. $z$ | $\alpha$ | $\sigma_{0}\,[\mathrm{km/s}]$
---|---|---
$1$ | $0.477\pm 0.052$ | $793_{-166}^{+210}$
$0.5$ | $0.479\pm 0.063$ | $638_{-151}^{+198}$
$0.25$ | $0.473\pm 0.042$ | $522_{-81}^{+96}$
$0$ | $0.468\pm 0.079$ | $507_{-151}^{+214}$

##  4 Conclusions

We performed nested-grid simulations of a filament, applying AMR to increase
the spatial resolution in halos of masses below $10^{13}\,M_{\odot}$. For
these objects, which can be interpreted as galaxies and groups of galaxies, we
analyzed the thermal and turbulent energy contents of the circumgalactic (CGM)
and, in a few cases, the intragroup (IGrM) medium at the lower-mass end of
groups. We applied a standard recipe for star formation (constant star
formation time scale above a number density threshold) and supernova feedback.

Our study was motivated by the question whether the energy of the CGM/IGrM
scales differently in the mass regime of groups or individual galaxies
compared to halos of higher mass, i.e. in the range of clusters. Since we
approached this question from a physical point of view, we chose
hydrodynamical variables as metrics, namely, the thermal energy of the gas and
the kinetic energy of turbulent gas flows. For the latter, we use the
turbulent velocity dispersion as associated quantity. A meaningful definition
of the turbulent velocity dispersion must be based on an integral quantity,
encompassing velocity fluctuations on all scales. For this reason, we applied
a Kalman filtering algorithm to estimate the numerically resolved component
(Schmidt et al. 2014) and the subgrid-scale model of Grete et al. (2016) for
the unresolved component.

To infer scaling relations from the simulation data, we need to compute mean
energies. The standard procedure is to apply a halo finder and to average over
the virial sphere of each halo. However, the virial radius is only a crude way
of specifying the boundaries of the gas belonging to a group of galaxies or
the gas surrounding an isolated galaxy. For this reason, we investigated
various criteria for defining the CGM based on density and temperature
thresholds. We find a reasonable constraint when averaging over moderately
overdense (a factor of 10 higher than the mean cosmological density) and warm-
hot gas (above $2.5\times 10^{4}$ K) within a maximum radius of two times the
virial radius. The former excludes the cool-warm ISM inside galaxies whereas
the latter avoids an overlap between halos. We find no clear trend of the
turbulent velocity dispersion with halo mass, except for a larger scatter
toward lower masses (top plots in Fig. 11). The scatter also tends to be more
pronounced at lower redshift. The median for all halos in the nested-grid
region is about $100\,\mathrm{km/s}$ at $z=1$ and $80\,\mathrm{km/s}$ at
redshift zero (see Table 2). It turns out that ISM contributions cause
significant deviations of the statistics inferred from mass-weighted averages
using virial spheres in the mass range of galactic halos. In contrast, we find
only minor differences between volume averages over virial spheres and regions
constrained by density and temperature, which supports the robustness of our
results. The power-law relation between turbulent velocity dispersion and
thermal energy (bottom plots in Fig. 11) with a scaling exponent around
$\alpha\sim 0.47$ at $z=0$ is similar to the relation found in simulations of
galaxy clusters ($\alpha\sim 0.5$, see Schmidt et al. 2016 and references
therein). In other words, halos filled with hot gas also tend to be turbulent,
regardless of the halo mass. This appears to apply all the way from the CGM to
the IGrM to the ICM of massive clusters, although the scatter becomes larger
with decreasing halo mass. Recently, Lochhaas et al. (2021) confirmed for an
individual halo that the turbulent energy is a nearly constant fraction of the
thermal energy over time.

Radial profiles of individual halos suggest that outflows from galaxies may
produce high levels of turbulence in the close vicinity of star-forming
galaxies, while the turbulent velocity dispersion decreases steeply in the
outer regions of the CGM (see, for example, the sigma-shaped profile in Fig.
8). Turbulence in the IGrM can be driven by mergers. In particular, fossil
groups at low redshifts tend to be highly turbulent, as illustrated in Fig. 7.
These effects result in substantially varying turbulence among different
objects, obscuring the scaling relations found for halos of higher mass. Our
results indicate that gravitational potential energy reservoir, for which the
halo mass can be considered as proxy, becomes increasingly modulated by
additional sources of energy in halos of lower mass. Tidal interactions
between galaxies and supernova feedback during episodes of intense star
formation both heat the surrounding gas and stir up turbulence, resulting in
what is sometimes called second self-similarity (Miniati & Beresnyak 2015;
Schmidt et al. 2017). Owing to their transient nature, interactions and
feedback introduce stronger variations and, as a result, weaker correlation.

Although AGN feedback is not decisive for understanding the properties of the
IGrM (Liang et al. 2016), it also has an impact on the galactic environment,
particularly at earlier epochs ($z>1$). For this reason, incorporating AGNs
into the subgrid physics will be an important component in improving the
simulations discussed in this work. Similarly, the CGM and IGrM are weakly
magnetized (Han 2017). Like the ICM , the weakly collisional nature of the
plasma make it prone to fast growing kinetic instabilities that (may) alter
the magnetic field structure and saturation strength in a turbulent
environment(Schekochihin et al. 2005). Thus, a treatment of magnetic fields in
the simulations would further increase their fidelity. In addition, it was
recently shown that the standard adaptive refinement condition based on dark
matter and baryon overdensity, which we also used, (naturally) misses some
details in the structure of the low density outskirts (Peeples et al. 2019;
Hummels et al. 2019). In turn, this may alter the turbulent velocity
dispersion at larger radii. Given the increased amount resources required to
resolve a larger fraction of the filament at very high resolution, we leave
this analysis to a future simulation campaign.

From an observational point of view, the situation is even more complicated
because the X-ray luminosity is the main indicator of the dynamical state of
the IGrM. Indeed, observed group luminosities point toward stronger scatter,
but the question whether the scaling of X-ray luminosity vs temperature breaks
at halo masses characteristic for groups is not settled (Bharadwaj et al.
2015; Vajgel et al. 2014; Lovisari et al. 2015; Liang et al. 2016; Paul et al.
2017). Although the emission of bremsstrahlung depends on $T^{1/2}$ and we
found that $T$ scales with $\sigma_{\mathrm{turb}}$, the dependence on the
squared number density of the gas might substantially weaken the relation
between X-ray luminosity and turbulent energy. In future work, it will be
important to analyze the imprint of the thermal and turbulent energy contents
of the IGrM on X-ray luminosity quantitatively. Moreover, a larger sample will
help to obtain better statistics, including halos in the typical mass range of
groups (i.e. $M\sim 10^{13}M_{\odot}$). This can be achieved by running
nested-grid simulations of a number of filaments in different regions. To
detect breaks in scaling relations under comparable conditions, many halos in
the mass range of clusters have to be computed with a resolution that is
sufficiently high for the sensible application of star formation and feedback
recipes. Although we were not able to extend our analysis to objects outside
of the zoom-in region because of too coarse resolution, our simulations point
at a break down of self-similarity with respect to halo mass, while second
self-similarity prevails in group and galaxy halos.

###### Acknowledgements.

We thank Surajit Paul and Luigi Iapichino for discussions that initiated the
work presented this paper. Moreover, comments by Brian O’Shea helped us to
improve our manuscript. PG acknowledges funding by NASA Astrophysics Theory
Program grant #NNX15AP39G. The simulations presented in this article were
performed on SuperMUC(-NG) at the Leibniz Supercomputing Centre (project
pr62ze). We also acknowledge the yt toolkit by Turk et al. (2011) that was
used for our analysis of numerical data.

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