# Efficiency of turbulent reacceleration by solenoidal turbulence and its
application to the origin of radio mega halos in cluster outskirts

Kosuke Nishiwaki Institute for Cosmic Ray Research, The University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan  Gianfranco Brunetti INAF
Istituto di Radioastronomia, Via P.Gobetti 101, I-40129 Bologna, Italy  Franco
Vazza Dipartimento di Fisica e Astronomia, Universita di Bologna, Via Gobetti
93/2, I-40129 Bologna, Italy  INAF Istituto di Radioastronomia, Via P.Gobetti
101, I-40129 Bologna, Italy  Hamburger Sternwarte, Universität Hamburg,
Gojenbergsweg 112, 41029 Hamburg, Germany  Claudio Gheller INAF Istituto di
Radioastronomia, Via P.Gobetti 101, I-40129 Bologna, Italy

###### Abstract

Recent radio observations with Low-Frequency Array (LOFAR) discovered diffuse
emission extending beyond the scale of classical radio halos. The presence of
such mega halos indicates that the amplification of the magnetic field and
acceleration of relativistic particles are working in the cluster outskirts,
presumably due to the combination of shocks and turbulence that dissipate
energy in these regions. Cosmological magnetohydrodynamical (MHD) simulations
of galaxy clusters suggest that solenoidal turbulence has a significant energy
budget in the outskirts of galaxy clusters. In this paper, we explore the
possibility that this turbulence contributes to the emission observed in mega
halos through second-order Fermi acceleration of relativistic particles and
the magnetic field amplification by the dynamo. We focus on the case of Abell
2255 and find that this scenario can explain the basic properties of the
diffuse emission component that is observed under assumptions that are used in
previous literature. More specifically, we conduct a numerical follow-up,
solving the Fokker–Planck equation using a snapshot of a MHD simulation and
deducing the synchrotron brightness integrated along the lines of sight. We
find that a volume-filling emission, ranging between 30 and almost 100% of the
projected area depending on our assumptions on the particle diffusion and
transport, can be detected at LOFAR sensitivities. Assuming a magnetic field
$B\sim 0.2\mu$G, as derived from a dynamo model applied to the emitting
region, we find that the observed brightness can be matched when $\sim$1%
level of the solenoidal turbulent energy flux is channeled into particle
acceleration.

Galaxy clusters (584)

††journal: ApJ

##  1 Introduction

Galaxy clusters are filled with a hot intra-cluster medium (ICM), that has a
characteristic temperature similar to the the cluster’s virial temperature.
This suggests that the ICM is heated by the gravitational energy released in
the hierarchical merger and accretion processes of clusters (e.g., Press &
Schechter, 1974; Kravtsov & Borgani, 2012). A fraction of the energy can also
be channeled into non-thermal components, such as relativistic particles and
magnetic fields. Shocks and turbulence could be favorable sites for the
particle acceleration and the amplification of the field (see Brunetti &
Jones, 2014, for review).

Radio observations of galaxy clusters probe those non-thermal components by
studying diffuse synchrotron emission of relativistic (cosmic-ray) electrons
(CRe). Radio halo is a diffuse emission with an extension of $\sim$1 Mpc,
often found in the central region of merging clusters (see van Weeren et al.,
2019, for review). The radiative cooling time of CRe is significantly shorter
than the time required for diffusion or advection over $\sim 1~{}{\rm Mpc}$,
implying that there is an in situ mechanism that produces CRe. Reacceleration
by merger-induced turbulence is the most plausible scenario (e.g., Brunetti et
al., 2001; Petrosian, 2001; Fujita et al., 2003; Cassano & Brunetti, 2005),
although cosmic-ray protons (CRp) in the ICM may be important ingredients in
the physics of those phenomena (e.g., Dennison, 1980; Blasi & Colafrancesco,
1999). For example, they can provide seed CRe to re-accelerate through the
hadronic pp collision with the thermal protons in the ICM (e.g., Brunetti &
Lazarian, 2011a; Brunetti et al., 2017; Pinzke et al., 2017; Nishiwaki &
Asano, 2022). It has been shown that the observed statistical properties of
radio halos are in line with the reacceleration model considering the resonant
interaction between compressible turbulence (Cassano & Brunetti, 2005;
Nishiwaki & Asano, 2022; Cassano et al., 2023). Reacceleration by turbulence
is also proposed to explain radio emission detected on larger scales, such as
radio bridges (Brunetti & Vazza, 2020) that are radio filaments connecting
massive pairs in the early stage of mergers discovered by the Low Frequency
Array (LOFAR) (Govoni et al., 2019; Botteon et al., 2020a).

More recently, Cuciti et al. (2022) reported the existence of radio “mega
halos” in four clusters, using the LOFAR observation. The volume of the mega
halos is almost 30 times larger than that of radio halos, suggesting that the
entire volume of the cluster is filled with CRe and magnetic field. The radio
power of mega halos is comparable to or even larger than that of classical
halos. The detection of synchrotron radiation at the large distance (1 - 2
Mpc) from the cluster center also constrains the magnetic field strength in
this region. Since the pressure of non-thermal components, including magnetic
field and CRe, should be smaller than the thermal one as indicated from the
observations and numerical simulations (e.g., Vazza et al., 2016; Eckert et
al., 2019), the magnetic field should be in the range of $0.1\mu{\rm
G}\lesssim B\lesssim 1.7\mu{\rm G}$ (Botteon et al., 2022). The lower bound
($0.1\mu{\rm G}$) is at least one order of magnitude larger than the value
expected by the compression of primordial fields. One possible mechanism of
this non-linear amplification of the field is dynamo in a turbulent medium.

Cosmological simulations of galaxy clusters suggest that turbulence and shocks
driven by continuous accretion of matter fill the entire volume of the cluster
up to the virial radius (e.g., Vazza et al., 2011; Nelson et al., 2014;
Miniati, 2015; Steinwandel et al., 2023). As in the cluster center, the ICM in
the outskirts is a weakly-collisional plasma, and the perturbations would
cause instabilities that effectively reduce the mean free path (mfp) of
thermal protons (e.g., Schekochihin et al., 2005; Kunz et al., 2011; Brunetti
& Lazarian, 2011b), potentially establishing a well-developed inertial range
(e.g., Schekochihin et al., 2009).

Indeed, the infall of the clumps of mass drives the turbulence with a typical
scale of a few 100 kpc in the cluster outskirts (e.g. Vazza et al., 2017). The
timescale of the turbulent cascade, $t_{\rm cas}\sim 600$ Myr (e.g., Brunetti
& Lazarian, 2007), is much shorter than the Hubble time, which allows the
turbulent dynamo to work in that region.

Nowadays, the best studied case of the cluster hosing diffuse radio emission
on the entire cluster volume is the case of Abell 2255 (hereafter A2255)
(Botteon et al., 2022). A2255 is a nearby ($z=0.0806$) cluster, which shows a
complex dynamical state in optical and X-ray observations (e.g., Burns et al.,
1995; Yuan et al., 2003; Golovich et al., 2019; Feretti et al., 1997; Akamatsu
et al., 2017). The cluster is known to host a radio halo and radio relics, and
they have been studied in wide range of frequencies (e.g., Jaffe & Rudnick,
1979; Feretti et al., 1997; Govoni et al., 2005; Pizzo et al., 2008; Pizzo &
de Bruyn, 2009; Botteon et al., 2020b).

More recently, LOFAR observations found that the cluster hosts diffuse
emission extending in very large scales and enveloping the classical halo and
relics (Botteon et al., 2022). Such emission is complex, showing a number of
relic-like features embedded in a truly diffuse component, and a spectral
index distribution between 40 - 144 MHz ranging from 0.6 to 2.5. The flat
spectrum emission is coincident with the radio relics located in the north and
southwest parts of the cluster, while the steep emission is associated with
the diffuse component.

In this paper, we attempt to explore the possibility that turbulence
contributes to the observed emission via magnetic field amplification and
particle reacceleration. We use cosmological MHD simulation of a cluster to
examine the turbulent and magnetic fields in the cluster outskirts. In Sect.
2, we describe the setup of the MHD simulation. In Sect. 3, we review the
reacceleration model and claim that that is compatible with the observed
spectrum of the mega halo. In Sect. 4, we numerically solve the Fokker–Planck
(FP) equation, considering the distribution of turbulence and the projection
along the line of sight. The limitations of our models are discussed in Sect.
5. Finally, we summarize the results in Sect. 6.

##  2 Cosmological MHD simulation

To examine the property of the ICM and turbulence in cluster outskirts, we use
a snapshot of a high-resolution cosmological ideal MHD simulation of a massive
galaxy cluster, produced with grid code ENZO (Bryan et al., 2014). We use the
same simulated cluster as in Botteon et al. (2022), which has a mass of
$M_{200}=8.65\times 10^{14}~{}\rm M_{\odot}$ at $z=0$. This simulation
includes eight levels of Adaptive Mesh Refinement (AMR) to increase the
spatial and force resolution within the virial volume of the cluster, reaching
a peak spatial resolution of $\Delta x=3.95\rm~{}kpc/cell$ (comoving) (Vazza
et al., 2018).

The simulation was started assuming a uniform ”primordial” seed magnetic field
of $B_{0}=0.1\rm~{}nG$ (comoving) at $z=40$. The low redshift properties of
the magnetic field in the cluster volume are found to be fairly independent of
the exact origin scenario, due to the effect of efficient small-scale dynamo
amplification (Vazza et al., 2018). At the late stage of the cluster
evolution, most of the central volume within $<1$ Mpc is simulated with the
finest resolution $\Delta x=3.95\rm~{}kpc/cell$. The virial volume of clusters
is refined at least up to $\Delta x=15.8\rm~{}kpc/cell$ (Domínguez-Fernández
et al., 2019). At 1-2 Mpc from the center, the spatial resolution is
comparable to, or coarser than, the MHD scale, $l_{\rm A}\sim 1$ kpc, where
the turbulent velocity becomes comparable to the Alfvén velocity, so the field
amplification may be underestimated in the simulation. Thus, we evaluate the
field strength in this region in a post-process (see Sect. 3 for the detail).

The turbulent kinetic flux is calculated with small-scale filtering explained
in Vazza et al. (2017), which allows us to reconstruct (and remove) the
contribution from shock waves to the total turbulent energy budget. For each
cell of the simulation, the dispersion of the velocity field $\sigma_{v}(L)$
is measured within a scale $L$. The outer scale of the turbulence $\Lambda$ is
defined as the scale where the change in $\sigma_{v}(L)$ with increasing $L$
becomes sufficiently small (see Vazza et al., 2017, for the detail). The
turbulent kinetic energy flux in each simulated cell can be calculated as:

$\mathcal{F}_{\rm turb}=\frac{1}{2}\frac{\rho\sigma_{v}^{3}(L)}{L}(\Delta
x)^{3}\>,$ (1)

where $\rho$ is the gas mass density in the cell and $\Delta x$ is the cell
size. Based on previous works, we assume that the turbulent spectrum in the
inertial range roughly follows the Kolmogorov scaling (e.g. Vazza et al.,
2011, 2017), and so the value of $\mathcal{F}_{\rm turb}$ is insensitive to
the specific value of $L$ as long as it is in the inertial range. To study the
quantities in the cluster outskirts, we extract a box of a region of 1.6 Mpc
on a side located at 1.2 Mpc from the center of the simulated cluster. Each
cell has a volume of $16^{3}~{}{\rm kpc}^{3}$, and the box is composed of
$10^{6}$ cells. In this region, the outer scale $\Lambda$ is found to be
typically $\Lambda\approx 200~{}{\rm kpc}$. In the following, we use the
turbulent velocity measured at $L=160~{}{\rm kpc}$.

Figure 1: Maps of the X-ray weighted gas temperature along the line of sight
(left) and of the gas velocity along the line of sight (right) for the
simulation snapshot analyzed in this work. The additional white contours show
the regions which can be approximately detected with X-observations in the
0.5-2 keV energy band, while the green square shows the location of the box
used to extract the turbulent flow properties used in Sec.4. Each image is
made of 1024 $\times$ 1024 pixels.

Fig. 1 gives the visual impression of the line of sight gas velocity and of
the gas temperature in the volume of the simulated clusters, in particular in
the box used to extract the physical parameters used in our Fokker–Planck
calculations. The selected box is at the cluster periphery, in a region only
partially detectable through typical X-ray observations, and along the
direction of a prominent filamentary accretion, but without the presence of
massive substructures, or prominent shock waves.

Figure 2: Left: Histogram of turbulent energy flux per unit volume for
solenoidal (blue) and compressible (red) modes in the cluster peripheral
region shown in Fig. 1. The vertical axis shows the number of cells in the
simulated box. The dotted vertical lines show the mean values for each mode.
The total number of cells in the extracted box is $N=10^{6}$. Right:
Cumulative number of cells with the turbulent kinetic flux larger than $F_{\rm
turb}$. The vertical dashed line shows $F_{\rm turb}=10^{44}~{}{\rm
erg/s/Mpc^{3}}$

Fig. 2 (left panel) shows the histogram of turbulent kinetic flux in the
extracted region. We decompose the turbulent velocity into solenoidal
($\nabla\cdot\bm{v}=0$) and compressible ($\nabla\times\bm{v}=0$) modes, using
the procedure of Vazza et al. (2017). We find that the solenoidal mode
typically has a larger kinetic flux than the compressible mode, in line with
previous simulations (e.g., Miniati, 2015; Porter et al., 2015; Vazza et al.,
2017). The mean value of the solenoidal kinetic flux per unit volume is
$F_{\rm turb}=1.1\times 10^{44}~{}{\rm erg/s/Mpc^{3}}$, which is almost 10
times larger than the compressible component. In the right panel, we show the
cumulative number of cells that have turbulent kinetic flux larger than
$F_{\rm turb}$. More than 30% of the cell has the solenoidal turbulent flux
larger than $10^{44}~{}{\rm erg/s/Mpc^{3}}$, while the fraction decreases to
$\approx 3$% in the case of the compressible mode. Note that we are
considering the sector where the feature of mass accretion from the nearby
cluster can be seen (Fig. 1) and it is more turbulent than other sectors of
cluster outskirts (see also Sect. 5). For comparison, we study other regions
with the same volume and the distance from the cluster center and find a
factor $\sim$5 smaller turbulent flux, whereas the solenoidal mode dominates
the compressible one in every region. In the following sections, we focus on
the solenoidal velocity to calculate the acceleration efficiency and the
dynamo field. Note that $F_{\rm turb}$ denotes only the volumetric turbulent
kinetic flux of the solenoidal component.

##  3 CR reacceleration and field amplification by solenoidal turbulence

Turbulence driven through the formation process of galaxy clusters is
typically sub-sonic ($M_{\rm s}<1$) and super-Alfvénic ($M_{\rm A}>1$)
(Brunetti & Lazarian, 2007). A fraction of the turbulent energy can be
converted into non-thermal components such as cosmic rays and the magnetic
field through stochastic acceleration and turbulent dynamo. Turbulence in
astrophysical environments may accelerate particles through different
mechanisms, including resonant and non-resonant mechanisms (e.g., Ptuskin,
1988; Schlickeiser & Miller, 1998; Cho & Lazarian, 2006; Brunetti & Lazarian,
2007; Lynn et al., 2014; Brunetti & Lazarian, 2016; Bustard & Oh, 2022;
Lemoine, 2021; Lazarian & Xu, 2023). Recent MHD simulations of galaxy clusters
suggested that the turbulence in the ICM is dominated by the solenoidal
(incompressible) mode (e.g., Miniati, 2015; Vazza et al., 2017). One possible
acceleration mechanism working in incompressible turbulence is the
acceleration due to the interaction between magnetic field and particles
diffusing in super-Alfvénic turbulence (Brunetti & Lazarian, 2016; Brunetti &
Vazza, 2020).

In turbulent reconnection theory (Lazarian & Vishniac, 1999), the Alfvén scale
$l_{\rm A}\equiv LM_{\rm A}^{-3}$ (for the Kolmogorov scaling) is the dominant
scale, where the reconnection speed may reach $v_{\rm rec}\sim v_{\rm A}$. In
the regions where the magnetic field dissipates due to the reconnection, the
particles trapped in contracting islands gain energy through a mechanism that
is similar to the first-order Fermi mechanism (Kowal et al., 2012; del Valle
et al., 2016). On the contrary, the particles are expected to cool in the
regions where the dynamo is efficient and the magnetic field lines diverge.

According to Brunetti & Lazarian (2016) particles diffusing through this
complex pattern experience a second-order acceleration mechanism. In this
mechanism, the diffusion coefficient in the momentum space is

$D_{pp}=3\sqrt{\frac{5}{6}}\frac{c_{s}^{2}}{c}\frac{\sqrt{\beta_{\rm
pl}}}{L}M_{s}^{3}\psi^{-3}p^{2},$ (2)

where $p$ is the momentum of the particle, $\beta_{\rm pl}\equiv 2\gamma_{\rm
ad}^{-1}c_{\rm s}^{2}/v_{\rm A}^{2}$ is the plasma beta with $\gamma_{\rm
ad}=5/3$ is the adiabatic index, and $\psi\equiv\lambda_{\rm mfp}/l_{\rm A}$
with $\lambda_{\rm mfp}$ is the mean free path (mfp) of the particle. The mfp
is an important parameter in the model as it determines the spatial diffusion
coefficient,

$D\sim\frac{1}{3}c\lambda_{\rm mfp}\sim 10^{31}{\rm cm}^{2}{\rm
s}^{-1}\left(\frac{\psi}{0.5}\right)\left(\frac{l_{\rm A}}{1~{}{\rm
kpc}}\right),$ (3)

and consequently the efficiency of particle crossing regions of the size
$l_{\rm A}$. Combining the requirement that the fractional change of momentum
in each scattering should be $\Delta p/p\ll 1$ and the effect of particle
scattering due to mirroring in a super-Alfvénic flow, $\psi$ should satisfy
$0.01\ll\psi\lesssim 0.5$; $\psi\sim 0.5$ is the reference value that has been
motivated and used in the previous studies (Brunetti & Lazarian, 2016;
Brunetti & Vazza, 2020).

We consider that the solenoidal turbulence with super-Alfvénic injection
velocity shows a power spectrum with a well-established inertial range. In
such a situation, a fixed fraction ($\eta_{B}\approx 0.05$) of the turbulent
energy flux is consumed through the amplification of the magnetic field (e.g.,
Cho et al., 2009; Beresnyak, 2012; Xu & Lazarian, 2016). MHD simulations of
galaxy clusters succeed in resolving the small-scale turbulent dynamo in the
central region (e.g., ZuHone et al., 2011; Vazza et al., 2018; Domínguez-
Fernández et al., 2019; Steinwandel et al., 2022), but this effect is quenched
in the cluster periphery due to the limited resolution. Thus, we adopt the
same procedure as Brunetti & Vazza (2020) and estimate the dynamo-amplified
field in post-processing, in the output of the MHD simulation (Sect. 2).
Assuming that a fraction $\eta_{B}$ of the turbulent kinetic energy is
channeled into the field amplification, the field strength can be estimated as

$\frac{B^{2}}{8\pi}\sim\eta_{B}F_{\rm turb}t_{\rm
eddy}\sim\frac{1}{2}\eta_{B}\rho_{\rm ICM}\delta v_{\rm turb}^{2},$ (4)

where $t_{\rm eddy}=L/\delta v_{\rm turb}$ is the eddy turn-over time. In this
case, $M_{\rm A}$ and $\eta_{B}$ are related as $M_{\rm
A}\sim\eta_{B}^{-1/2}$. In the simulations of galaxy clusters, the value of
$\eta_{B}$ is as small as $\eta_{B}\approx 0.05$ (Beresnyak & Miniati, 2016;
Vazza et al., 2018). Using $\eta_{B}=0.05$, we obtain the mean field strength
of $\langle B\rangle\sim 0.24\mu{\rm G}$ in the simulated box, which is almost
2 times larger than the mean value of the field strength directly measured in
the simulation ($\langle B_{\rm sim}\rangle=0.11\mu$G). The plasma beta is
$\beta_{\rm pl}\approx 200-500$, which is slightly larger than that in the
cluster center (e.g., Brunetti & Lazarian, 2007). Adopting the dynamo field
(Eq. (4)), $D_{pp}$ (Eq. (2)) can be expressed as a function of $\eta_{B}$ as
(Brunetti & Vazza, 2020)

$D_{pp}\sim 3\frac{\delta v_{\rm turb}^{2}}{cL}\eta_{B}^{-1/2}\psi^{-3}p^{2}.$
(5)

The spectral index map of A2255 in Botteon et al. (2022) is a mixture of flat
($\alpha_{\rm syn}\sim 1.0$) and steep ($\alpha_{\rm syn}\sim 2.0$) regions.
The flattest index is coincident with the arc-shaped structure, suggesting the
particle energization by shocks. On the other hand, the emission with steep
radio indices comes from the diffuse envelope permeating the cluster volume
(see Fig. S2 of Botteon et al., 2022) . The mean value of the index is around
$\alpha_{\rm syn}\approx 1.6$. If we assume a scenario where particles emit in
a homogeneous field, those steep indices suggest that the observing frequency
is close to the steepening frequency, at which the spectrum starts to decline
rapidly due to the cooled spectrum of CRe.

In the turbulent reacceleration model of CRe, one can define the break energy
$\gamma_{\rm b}$ as the energy where the cooling timescale $t_{\rm cool}$
becomes comparable to the acceleration timescale $t_{\rm acc}$. As shown in
Cassano et al. (2010), the steepening frequency $\nu_{\rm s}$ in the
synchrotron spectrum appears at a factor $\xi\sim 5-7$ times larger than the
break frequency $\nu_{\rm b}$ corresponding to $\gamma_{\rm b}$, i.e.,
$\nu_{\rm s}=\xi\nu_{\rm b}$. The cooling time of ultra-relativistic CRe due
to synchrotron and inverse-Compton radiation can be written as $t_{\rm
cool}=6\pi m_{e}c\gamma^{-1}/(\sigma_{\rm T}(B^{2}+B_{\rm CMB}^{2}))$, where
$\gamma$ is the Lorenz factor of the particle, $\sigma_{\rm T}$ is the Thomson
cross section and $B_{\rm CMB}=3.25(1+z)^{2}\mu{\rm G}$ is the inverse-Compton
(IC) equivalent field. Note that 100 MHz emission is mainly produced by the
synchrotron radiation of CRe with $p=\sqrt{\gamma^{2}-1}\sim 10^{4}$, and the
Coulomb loss is negligible for those CRe. Using the magnetic field estimated
from the dynamo model of Eq. (4), the cooling timescale at $\nu_{\rm b}$ can
be estimated as

$\displaystyle t_{\rm cool}$ $\displaystyle=$ $\displaystyle\frac{\sqrt{27\pi
em_{e}c}}{\sigma_{\rm T}}\frac{B^{1/2}}{B^{2}+B_{\rm
CMB}^{2}}\xi^{\frac{1}{2}}\nu_{\rm s}^{-\frac{1}{2}},$ (6) $\displaystyle\sim$
$\displaystyle 450~{}{\rm
Myr}\left(\frac{\eta_{B}}{0.05}\right)^{\frac{1}{4}}(1+z)^{-\frac{5}{2}}\left(\frac{\mu}{0.59}\right)^{\frac{1}{4}}\left(\frac{n_{\rm
ICM}}{10^{-4}~{}{\rm cm}^{-3}}\right)^{\frac{1}{4}}$
$\displaystyle\;\times\left(\frac{\delta v_{\rm turb}}{300~{}{\rm
km/s}}\right)^{\frac{1}{2}}\left(\frac{\xi}{7}\right)^{\frac{1}{2}}\left(\frac{\nu_{\rm
o}}{100~{}{\rm MHz}}\right)^{-\frac{1}{2}},$

where $\mu$ is mean molecular weight, $\nu_{\rm o}=\nu_{\rm s}/(1+z)$ is the
observing frequency. The cooling is dominated by the radiation through the IC
scattering ($B^{2}\ll B_{\rm CMB}^{2}$) in the cluster outskirts, so we
neglected the $B^{2}$ term in the denominator. The acceleration timescale
should be comparable to this value to sustain the emission observed at the
LOFAR frequency.

Figure 3: Relation between $\eta_{B}$ and $\psi$, required for the observing
frequency of $\nu_{\rm o}=50$ MHz and $\xi=7$. Lines distinguish the results
for the different turbulent velocities at $L=0.16~{}{\rm Mpc}$; $M_{\rm
s}=0.1,0.3,0.4,0.5,0.7$ and 1.0 (from bottom to top). For the ICM properties,
we adopted typical values at the cluster periphery found in MHD simulations,
$n_{\rm ICM}=3\times 10^{-5}~{}{\rm cm}^{-3}$ and $T_{\rm ICM}=4$ keV. The
source at $z=0$ is assumed.

Comparing Eq. (6) with the acceleration timescale $t_{\rm
acc}=p^{2}/(4D_{pp})$ obtained from Eq. (5), one can relate two fundamental
parameters, $\psi$ and $\eta_{B}$. Fig. 3 shows the relation between
$\eta_{B}$ and $\psi$ in the case of $\nu_{o}=50~{}{\rm MHz}$ and $z=0$. We
adopted the typical value for the ICM density and temperature in cluster
outskirts; $n_{\rm ICM}=3\times 10^{-5}~{}{\rm cm}^{-3}$ and $T_{\rm ICM}=4$
keV. Adopting $\eta_{B}\sim 0.05$ and the typical turbulent flux found in MHD
simulation, $F_{\rm turb}\approx 10^{44}~{}{\rm erg/s/Mpc^{3}}$ (Fig. 2), we
find that the observed steep spectrum at LOFAR frequencies can be explained
with the mfp of CRe $\psi\sim 0.5$ of the Alfvén scale. This value is
consistent with that adopted by Brunetti & Lazarian (2016) and Brunetti &
Vazza (2020) to explain radio halos and bridges. The mfp comparable to $l_{\rm
A}$ in turbulence reconnection is supported by the studies of tracers in MHD
simulations (Kowal et al., 2011, 2012).111 more recent theoretical attempts
have also investigated the role of mirroring on particle diffusion and
acceleration (Lazarian & Xu, 2021, 2023).

Figure 4: Cumulative number count of cells as a function of $t_{\rm
acc}/t_{\rm cool}$ in the case of $\psi=0.3$ (red) and 0.5 (blue). We adopted
the same parameters as Fig. 3 ($\nu_{\rm o}$ = 50 MHz, $\xi=7$, and
$\eta_{B}=0.05$). The vertical dotted line shows $t_{\rm acc}=t_{\rm cool}$.

One can calculate the number of cells that are important for the radio
emission by comparing $t_{\rm acc}$ and $t_{\rm cool}$ measured in the
extracted cubic volume. Using Eqs. (5) and (6), the scaling relation for the
fraction of those timescales can be expressed as

$\frac{t_{\rm acc}}{t_{\rm cool}}\propto\rho^{-1/4}\delta v_{\rm
turb}^{-5/2}L\xi^{-1/2}\psi^{3}\eta_{B}^{1/4},$ (7)

In Fig. 4, we show the cumulative fraction of $t_{\rm acc}/t_{\rm cool}$. When
$t_{\rm acc}<t_{\rm cool}$, the acceleration mechanism of Eq.(2) can
efficiently re-accelerate CRe that radiate synchrotron emission in the LOFAR
frequencies.

The fraction of cells is $\sim$ 15% adopting a reference value $\psi=0.5$,
while it increases to $\gtrsim$50 % for $\psi<0.2-0.3$. The dependence on
$\eta_{B}$ is rather weak (Eq. (7)). The fraction ranges form 25% to 10% for
$\eta_{B}=0.01-0.1$, for a fixed $\psi=0.5$. Such a significant fraction
implies a volume-filling emission. One also needs to take care about the
duration of reacceleration, since the reacceleration is important only when it
lasts for several times longer than $t_{\rm acc}$. The filling factor of the
radio-emitting cells will be revisited in Sect. 4.

##  4 Fokker–Planck simulation

Next, we study the synchrotron spectrum in our dynamo reacceleration model
with a numerical calculation. We consider the situation that the pre-existing
population of seed CRe is re-accelerated by incompressible turbulence and
produces the observed radio emission by the synchrotron radiation. As in Sect.
2, we consider the volume of $1.6^{3}~{}{\rm Mpc}^{3}$ located $1.2~{}{\rm
Mpc}$ from the center, and the distribution of the seed CRe is proportional to
the ICM number density. We solve the Fokker-Planck equation of CRe of the
following form (e.g., Cassano & Brunetti, 2005):

$\frac{\partial N_{e}(p,t)}{\partial t}=\frac{\partial}{\partial
p}[\dot{p}N_{e}]+\frac{\partial}{\partial p}\left[D_{pp}\frac{\partial
N_{e}}{\partial p}-\frac{2}{p}N_{e}D_{pp}\right]+Q_{e}(p,t),$ (8)

where $N_{e}(p,t)$ is the spectrum of CRe at time $t$, $\dot{p}$ represents
the momentum loss per unit time, $D_{pp}$ is the momentum diffusion
coefficient, and $Q_{e}$ is the injection spectrum of CRe. The momentum loss
rate $\dot{p}$ includes the effect of radiative(synchrotron and IC) and
Coulomb losses, and $D_{pp}$ is calculated with Eq. (2). We neglect the
injection of the secondary electrons from inelastic $pp$ collisions of cosmic-
ray protons. Due to the low ICM density in the cluster outskirts, the
contribution of the secondary electrons is expected to be smaller than that in
classical radio halos (see Appendix A for further discussion).

We neglect the term related to the spatial transport of CRs in Eq. (8), since
it cannot be properly followed with a snapshot of the MHD simulation. However,
we are considering the case where the mfp of CRe is comparable to the Alfvén
scale ($l_{\rm A}\sim 1~{}{\rm kpc}$), and this leads to a large value of the
spatial diffusion coefficient $D\sim 10^{31}~{}{\rm cm^{2}}/{\rm s}$, as shown
in Eq. (3). The detailed investigation of the effect of spatial transport in
the reacceleration model is left for future works using a Lagrangian tracer
method. In this exploratory work, we adopt following two approaches: a one-
zone approximation (Sect. 4.1), where we adopt the average value of the
physical quantities found in the snapshot, and the calculation considering the
variation of $F_{\rm turb}$ in each cell and the integration along the line of
sight (LOS) (Sect. 4.2).

###  4.1 One zone calculation

We first discuss the energy density of CRe required to reproduce the observed
emission under the one-zone approximation. We adopt the mean values in the
simulation box for the physical quantities; $n_{\rm ICM}=6\times
10^{-5}~{}{\rm cm^{-3}}$, $T_{\rm ICM}=4.5~{}$keV, $v_{\rm sol}=320~{}{\rm
km/s}$ ($M_{\rm s}\approx 0.4$), and $t_{\rm eddy}=0.53~{}{\rm Gyr}$. We adopt
$\eta_{B}=0.05$, which corresponds to $B=0.24\mu{\rm G}$ (Eq. (4)). We
summarize the values of parameters in Tab. 1. As an initial condition, a
cooled spectrum of seed CRe is calculated by integrating Eq.(8) for 2 Gyr with
$D_{pp}=0$ and $Q_{e}(p,t)\propto p^{-\alpha_{\rm inj}}\delta(t)$, where
$\alpha_{\rm inj}=2.2$ and $\delta(t)$ is the Dirac delta function. Due to the
radiative and the Coulomb cooling, the seed CRe has steep cut-offs around
$p_{\rm min}\sim 10$ and $p_{\rm max}\sim 10^{3}$ (Fig. 5 right panel). The
normalization of the initial spectrum is treated as a free parameter.

Table 1: Parameters adopted in the FP calculations | Sect. 4.1 | Sect. 4.2
---|---|---
$\psi$ | 0.3 | 0.6
$\eta_{B}$ | 0.05 | 0.05
$T_{\rm dur}$ [Gyr] | 3 | 3
$L$ [kpc] | 160 | 160
$F_{\rm turb}$ [erg/s/Mpc3] | $1.1\times 10^{44}$ | –aaWe use the values found each simulated cell.
$n_{\rm ICM}$ [cm-3] | $6\times 10^{-5}$ | –aaWe use the values found each simulated cell.
$T_{\rm ICM}$ [keV] | 4.5 | –aaWe use the values found each simulated cell.
$B$ [$\mu$G] | 0.24bbThe magnetic field is calculated with Eq. (4). | –a,ba,bfootnotemark:

As seen from Fig. 3, for $M_{\rm s}\sim 0.4-0.7$, a value $\psi\sim 0.3-0.5$
is compatible with the observed emission. This is confirmed by numerical
calculation in Fig. 5 (left panel), where we plot the synchrotron spectrum in
the one-zone model with a dashed line. While the spectral shape is mainly
determined by the parameters $\eta_{B}$ and $\psi$ (Eq. (7)), the
normalization (brightness) depends on the initial spectrum $N_{e}(p,0)$ and
the duration of reacceleration $T_{\rm dur}$. In this section, we assume that
$T_{\rm dur}$ is sufficiently long and the CRe spectrum reaches a steady state
due to the balance between the reacceleration and the radiative cooling (Eq.
(8)). We find that this steady state is achieved at $T_{\rm dur}\geq 3~{}{\rm
Gyr}$. The duration of $T_{\rm dur}=3~{}{\rm Gyr}$ is comparable to the
dynamical timescale of the ICM in the simulated region, so the mean value of
the turbulent flux $F_{\rm turb}$ can evolve in this timescale. Thus, the
assumption of constant $F_{\rm turb}$ during the FP calculation should be
considered as a rough approximation.

We determine the normalization of the initial spectrum as the synchrotron
brightness at 49MHz matches the observed value. The data points in Fig. 5 show
the range of the brightness of the diffuse envelope measured in 1.5-2 Mpc from
the center of A2255 (Botteon et al., 2022).

We define the efficiency of the reacceleration, $\eta_{\rm acc}$, as the
fraction of the turbulent kinetic flux that turns into the increase of the CRe
energy density and the radiation:

$\eta_{\rm acc}F_{\rm turb}\equiv\frac{d\epsilon_{\rm
CRe}}{dt}+\varepsilon_{\rm rad},$ (9)

where $\epsilon_{\rm CRe}$ is the CRe energy density and $\varepsilon_{\rm
rad}=\varepsilon_{\rm syn}+\varepsilon_{\rm IC}$ is the sum of the frequency-
integrated emissivities of synchrotron and IC radiations. In the steady state
of $N_{e}$, the first term on the right-hand side is negligible. The
synchrotron emissivity $\varepsilon_{\rm syn}$ is calculated in the range of
$10^{4}-10^{10}$ Hz, and $\varepsilon_{\rm IC}$ is calculated with
$\varepsilon_{\rm IC}=(B_{\rm cmb}/B)^{2}\times\varepsilon_{\rm syn}$. Since
$B\ll B_{\rm cmb}$, $\varepsilon_{\rm rad}$ is dominated by $\varepsilon_{\rm
IC}$.

In Fig. 5 (dashed line), we show the CRe energy spectrum in the steady state.
The energy is dominated by 100 MHz-emitting particles and the particles with
$p<10^{3}$ are not energetically important. We find $\epsilon_{\rm rad}\approx
7\times 10^{-30}$ erg/s/cm3 and $\eta_{\rm acc}\approx 2\%$, which is
consistent with the estimate in Botteon et al. (2022). Due to the assumption
of stationary conditions, this result is independent of the initial spectrum.

We find that the spectrum in Fig. 5 is reasonably reproduced by considering a
range of values of $\psi$ around the reference value $\psi\sim 0.5$. On the
other hand, the assumption of a mfp that is significantly reduced, $\psi<0.2$,
generates spectra that are too hard compared to the observation. Note that
this result is based on the assumption that the turbulence and physical
parameters in the simulated ICM are representative of the external regions in
A2255.

Figure 5: Left: The thick solid line shows the synchrotron spectrum in a
single beam, typically seen in our $10^{6}$ cell calculation. For comparison,
the spectrum in the one-zone model is shown with the dashed line. The data
points show the typical value of the brightness measured at 1.5-2 Mpc from the
cluster center (Botteon et al., 2022). The error bars show the ranges of the
brightness in this region including 1$\sigma$ error. We adopt $\psi=0.3$ for
the one-zone model, while $\psi=0.6$ and for the cell-wise calculations. In
both models, we assume $T_{\rm dur}=3$ Gyr. The beam spectrum is decomposed
into the contributions from the cells with $F_{\rm turb}<5\times
10^{44}~{}{\rm erg/s/Mpc^{3}}$ (blue) and $F_{\rm turb}>5\times 10^{44}~{}{\rm
erg/s/Mpc^{3}}$ (red). Right: CRe spectra in the same calculations. Spectra
before and after the reacceleration is shown with thin and thick black lines,
respectively. The dashed lines show the spectra in the one-zone model.

###  4.2 projection along the LOS

Next, we consider the distribution of turbulent energy in the box and study
how the emission from highly turbulent cells affects the spectrum integrated
along the LOS. To consider the CRe spectrum in each simulated cell, we
calculate the FP equation for various $F_{\rm turb}$ found in the simulated
box. We make a histogram of $F_{\rm turb}$ with 160 bins equally spaced in the
logarithmic scale in the range of $10^{38}~{}{\rm erg/s/Mpc^{3}}<F_{\rm
turb}<10^{46}~{}{\rm erg/s/Mpc^{3}}$ (Fig. 2). For simplicity, we adopt the
same values of dynamo $B$ (Eq. (4)) and $D_{pp}$ (Eq. (5)) for the cells in
the same $F_{\rm turb}$ by calculating the mean values of $\rho_{\rm ICM}$ and
$v_{\rm turb}$ in each $F_{\rm turb}$ bin. The synchrotron brightness is
calculated by integrating the emissivity of 100 cells (corresponds to 1.6 Mpc)
along the axis of the simulation. In one projection, there are 100$\times$100
LOS.

We assume the same initial spectrum as in Sect. 4.1, i.e., the spectrum with
cut-offs at $p_{\rm min}\sim 10$ and $p_{\rm max}\sim 10^{3}$. As in the
previous section, we consider a sufficiently long $T_{\rm dur}=3$ Gyr to
ensure that the steady state due to the balance between cooling and
reacceleration is achieved in many of the cells. Note that the turbulent flux
in each cell would change in a few eddy turnover time, $t_{\rm eddy}\sim
0.3-1~{}{\rm Gyr}$. When $T_{\rm dur}$ is short and the steady state is not
achieved, the result may depend on the initial condition. This point is
further discussed in Appendix B, where we show that the observed emission can
be explained with a different combination of $T_{\rm dur}$ and the initial
condition.

The non-dimensional parameters in our model, $\psi$ (Eq. (2)) and $\eta_{B}$
(Eq. (4)), are assumed to be constant over cells. As a test case, we adopt the
same $\eta_{B}=0.05$ as the one zone calculation. We use a snapshot of the
simulation and do not consider the evolution of the background fluid. We
assume that the initial energy density of the CRe in each cell is proportional
to the thermal energy density i.e., $\epsilon_{\rm
CRe}(t=0)\propto\epsilon_{\rm ICM}$. We neglect the particle transport between
cells during the calculation. Those simplification reduces the computational
cost and enables us to calculate the spectrum in every cell in the box for
several Gyrs. The study on the impact of CR transport is left to future works
featuring the Lagrangian tracer approach (Sect. 5).

We calculate the distribution of the spectral index and compare it with the
observation by Botteon et al. (2022). The spectral index and its statistical
properties would depend on the beam size and sensitivity of the observation,
so we consider those of LOFAR. The beam size in Botteon et al. (2022) is 35”,
which corresponds to almost 3 times the cell size at the redshift of A2255, so
we calculate the spectrum of one beam by summing up the intensities of
$3\times 3$ neighboring LOS. Each simulated beam consists of the emission from
900 cells. Following Botteon et al. (2022), we introduce a cut in our
simulated data, neglecting the beams that has 145 MHz brightness below
$2\sigma_{\rm HBA}$, where $\sigma_{\rm HBA}=200\mu{\rm Jy}$ is the r.m.s.
noise per beam of the LOFAR HBA band.

Figure 6: Fractional contribution from $F_{\rm turb}>5\times 10^{44}~{}{\rm
erg/s/Mpc^{3}}$ (orange) and $F_{\rm turb}<5\times 10^{44}~{}{\rm
erg/s/Mpc^{3}}$ (blue) cells to the flux shown in Fig. 5. The vertical dashed
lines show the LBA and HBA frequencies of LOFAR.

In Fig. 5, we show the typical synchrotron spectrum of beam detectable with
the LOFAR sensitivity (black); here we specifically use $\psi=0.6$. Since more
turbulent cells contribute more than less turbulent cells the results are not
exactly consistent with those in the one-zone model (Sect. 4.1) or with Fig 3.
In fact, the model considering the LOS integration of $\sim 10^{3}$ cells (in
each beam) requires a slightly larger value of $\psi$ and consequently a
slightly less efficient acceleration mechanism.

We decompose the spectrum into the contributions from $F_{\rm turb}>5\times
10^{44}~{}{\rm erg/s/Mpc^{3}}$ (orange) and $F_{\rm turb}<5\times
10^{44}~{}{\rm erg/s/Mpc^{3}}$ (blue) cells. We confirm the trend seen in
Brunetti & Vazza (2020); at higher frequencies ($\nu>100$MHz), the highly
turbulent cells dominate the emission, which occupies only a small fraction
(4.5%) of the volume, while the emission is more volume-filling at lower
frequencies. In Fig. 6, we show the contribution from the turbulent cells as a
function of frequency. CRe spectra are compared in Fig. 5. Unlike one-zone
calculation, the overall spectrum has a bump around $p\sim 100$. CRe in the
cells with smaller turbulent energy dominates the CRe energy, although they do
not significantly contribute to the emission at $\sim 100$ MHz (Fig. 6). The
typical momentum of CRe that corresponds to 100 MHz emission shifts to $p\sim
2-5\times 10^{3}$, since the magnetic field in $F_{\rm turb}\approx 5\times
10^{44}$ erg/s/Mpc3 cells is a factor $\sim$2 stronger than $B=0.24~{}\mu$G
adopted in the one-zone model. As discussed in Appendix B, the contribution
from low $F_{\rm turb}$ can be larger and the emission becomes more volume-
filling under different assumption on the initial condition.

In reality, the CR distribution would be smoothed by diffusion and/or
streaming, and CRs would experience multiple reacceleration within the
dynamical time. In such a situation, the gap between cells with large $F_{\rm
turb}$ and small $F_{\rm turb}$ would be reduced and the emission becomes more
volume-filling. This point would be further studied in future studies with a
Lagrangian tracer method (see also Sect. 5).

We find that 375 beams out of 1089 satisfy the criterion of detection, i.e.,
$S_{\rm HBA}>2\sigma_{\rm HBA}$, so almost 34% of the area of emission region
can be covered by the LOFAR sensitivity. This fraction increases with the
efficiency of reacceleration (smaller $\psi$), as the contribution by the less
turbulent cells ($F_{\rm turb}<5\times 10^{44}~{}{\rm erg/s/Mpc^{3}}$) becomes
more significant. However, we find that, for $\psi\leq 0.3$, the typical
spectrum starts to become flatter than the observed one.

The value of $\eta_{\rm acc}$ (Eq. (9)) differs in cell to cell, and it
becomes $\eta_{\rm acc}\approx 0$ when $F_{\rm turb}$ is very small and the
effect of the reacceleration is negligible. To obtain a typical value of
$\eta_{\rm acc}$ in this model, we calculate

$\eta_{\rm acc}=\frac{\sum(\frac{d\epsilon_{\rm CRe}}{dt}+\varepsilon_{\rm
rad})}{\sum F_{\rm turb}},$ (10)

where the sum is taken for the cells in each beam ($\sim 10^{3}$ cells). We
calculate the mean value of $\eta_{\rm acc}$ for the 375 beams and find
$\langle\eta_{\rm acc}\rangle\approx 1\%$. Although there is a plenty of
turbulent energy in $F_{\rm turb}>5\times 10^{44}~{}{\rm erg/s/Mpc^{3}}$
cells, the occurrence of such cells is small (4.5%). The combination of those
two result in $\eta_{\rm acc}\approx 1\%$, which is slightly smaller than that
found in the one zone model (Sect. 4.1).

Figure 7: Distribution of spectral indices of the beams observable with the
LOFAR sensitivity. The vertical axis shows the fraction of beams with each
$\alpha_{\rm syn}$. The black dotted line shows the observation of Botteon et
al. (2022). In our calculation, the total number of observable beams is
$\approx 300$ (30% of the area), while the observational data shows the
distribution of 836 pixels. For the model parameters, we adopted
$\eta_{B}=0.05$ and $\psi=0.6$. The reacceleration is calculated for $T_{\rm
dur}=3$ Gyr, and the beam size is assumed to be 35”. We tested the projection
along three different axes ($x,y,$ and $z$, distinguished by colors).

Fig. 7 shows the distribution of spectral index in each beam calculated with
$\psi=0.6$ and $\eta_{B}=0.05$. The black dashed line shows the result
reported in Botteon et al. (2022). The mean value of the index in our
calculation is $\langle\alpha\rangle\approx 1.7$, and the difference in the
results of three different projections is marginal. Our results are in line
with the observation of diffuse radio emission enveloping A2255.

##  5 limitations

Clearly, the complex morphology observed by LOFAR suggests that at least
shocks and turbulence contribute to the emission. A full modeling of A2255 is
however beyond the aim of the present work. We have explored the possibility
of turbulent reacceleration in the context of the specific model Brunetti &
Lazarian (2016). We find that the steep-spectrum diffuse emission observed at
1-2 Mpc distance from the center of A2255 can be explained using parameters
($\eta_{B}$ and $\psi$) that are in line with previous literature (Brunetti &
Lazarian, 2016; Brunetti & Vazza, 2020). In this case, the spatial diffusion
coefficient of the CRe becomes $D\sim 10^{31}{\rm cm}^{2}{\rm s}^{-1}$ (Eq.
(3)), i.e., CRe can diffuse over 450 kpc within 1 Gyr. Within the acceleration
time of radio-emitting CRe $t_{\rm acc}\sim 500~{}{\rm Myr}$ (Eq. (6)), the
diffusion length is $\sim 300$ kpc. In addition, CRe can be spread and mixed
by turbulent motion. This implies that CRe can travel $\sim$30 cells during
reacceleration, making the distribution smoother. Since the cells that
efficiently accelerate CRs appear in every $\sim 3^{3}$ cell (i.e., $\sim
4\%$), the diffusion is fast enough to fill the space between turbulent cells.

We tested two cases: the one-zone model in Sect. 4.1 and the cell-wise
calculation in Sect. 4.2. In the former case, we adopt the average values of
the physical quantities found in the simulated box to calculate the FP
equation. In the latter case, we neglected the transport within a short
duration of the reacceleration, and calculated the reacceleration using the
local value of the turbulent flux in each cell. A future study with a
Lagrangian tracer method will be important to discuss the evolution of the CR
spectrum and spatial distribution due to the combination of the
reacceleration, the spatial diffusion, and advection. An observation with
higher angular resolution would also be important to study the correlation
between the flat spectrum and large turbulent kinetic flux predicted in Sect.
4.2, or the gradient of the spectral index around the turbulent region due to
the diffusion.

Concerning the initial condition, we have assumed that there is a cooling
phase with $D_{pp}=0$ before the reacceleration starts, as in many models of
the giant radio halo (e.g., Brunetti et al., 2001; Nishiwaki & Asano, 2022).
However, CRs can be gently re-accelerated for several Gyrs by the modest level
of turbulence continuously driven by mass accretion. This would imply that
there is no clear onset of the mega halo emission, unlike the classical halos
which are supposed to be driven by the mergers of clusters (e.g., Cassano et
al., 2016). In Appendix B, we assume a different initial condition, assuming
that the reacceleration was working before the epoch of the snapshot. In this
case, the observed emission can be explained by a an efficiency that is
slightly reduced, $\eta_{\rm acc}\approx 1.7\%$.

The tracer approach is also important for that issue. Following the energy
gain and loss of CRe with a tracer method in a simulated galaxy cluster,
Beduzzi et al. (2023) demonstrated that $\approx 22-57\%$ of the mega halo
region ($0.4R_{500}<r<R_{500}$) is filled with radio-emitting CRe. Although
the details of the simulation setup and the definition of the volume filling
factor are different from our study, their result suggests that the cluster-
scale diffuse emission is produced through the multiple episodes of turbulent
reacceleration.

When extracting the peripheral region of the simulated cluster in Sect. 2, we
choose the particular sector where a large-scale accretion can be seen (Fig.
1), motivated by the fact that the most turbulent region dominates the
emission in our model (Sect. 4). We note that the typical turbulent energy can
be smaller in other sectors without such an accretion feature. Considering
that the properties of the gas and turbulence seen in the MHD simulation can
be different from those in A2255, the parameters reported in this study are
basically indicative values.

##  6 Conclusions

Recent LOFAR observations reported the presence of diffuse radio emission
permeating the volume of galaxy clusters up to the virial radius. This
emission is termed a mega halo, and its mechanism is still unclear.

The diffuse radio emission in A2255 has a very large extension, enveloping the
example of classical halo and relics reported in previous observations (e.g.,
Botteon et al., 2022). The complex radio morphology is likely generated
through a mix of different processes, e.g., particle acceleration by shocks
and turbulence. The diffuse emission permeating the cluster volume on large
scales is a candidate for turbulent reacceleration. In this paper, we have
explored this hypothesis.

Recent MHD simulations observe that the turbulent energy is dominated by the
solenoidal component in the cluster outskirts, so we focus on the acceleration
mechanism presented by Brunetti & Lazarian (2016), where particles are
accelerated by the interaction between particles and magnetic field lines
diffusing in a super-Alfvénic solenoidal turbulent flow. The acceleration
efficiency $D_{pp}$ depends on the mfp of the CR particle
$\psi\equiv\lambda_{\rm mfp}/l_{\rm A}$, which is treated as a free parameter
(Eq. (2)).

We use a snapshot of a simulated cluster mimicking A2255 and study the
property of turbulence and magnetic field in the cluster outskirts. We extract
a cubic volume of a region in the cluster periphery, where the prominent
feature of mass accretion can be seen (Fig. 1). Since the spatial resolution
of the simulation is not sufficient to resolve the small-scale dynamo at the
Alfvén scale, the magnetic field is estimated in a post-process. We assume
that $\eta_{B}\approx 0.05$ of the turbulent flux is consumed as the dynamo
and obtain $B\approx 0.2~{}\mu{\rm G}$, which is roughly two times larger than
the field found in the simulation. Comparing the efficiency of reacceleration
and radiative cooling, we find that the diffuse emission in A2255 can be
explained with a fiducial value of $\psi$($\sim 0.5$) that was derived from
considerations based on mirroring of electrons in a super-Alfvénic flow
(Brunetti & Lazarian, 2016; Brunetti & Vazza, 2020).

The emission spectrum is calculated by numerically integrating the FP equation
(Eq. 8) with the quantities found in the MHD simulation. Before the
reacceleration, CRe is accumulated around $p\sim 10^{2}$ due to the radiative
and Coulomb cooling before the reacceleration. In Sect. 4.1, we adopted the
one-zone approximation and used the average quantities in the simulation box.
We find that the reacceleration by the solenoidal turbulence is efficient
enough to produce a CRe spectrum peaked at $p\sim 10^{4}$, corresponding to
the synchrotron frequency of $\sim$100 MHz in the $\sim 0.1\mu{\rm G}$ field.
The acceleration efficiency is $\eta_{\rm acc}\approx 0.02$, in line with the
estimate in Botteon et al. (2022).

We also calculate the FP equation in $10^{6}$ cells in the simulation box,
assuming that the seed CRe is uniformly distributed. We find that 100 MHz
emission is dominated by the cells with large turbulent kinetic flux ($F_{\rm
turb}>5\times 10^{44}~{}{\rm erg/s/Mpc^{3}}$), which fill only a few % of the
volume. Considering the LOS integration within the beam size of LOFAR, we find
that a large fraction of the beams ($\gtrsim 30\%$) can be detected with the
LOFAR sensitivity. In this model, only 1% of the turbulent energy needs to be
consumed for the particle acceleration in the turbulent cells. Our model
predicts that the emission will be more volume-filling when observed with
higher sensitivity at lower frequencies.

The reported parameters and the derived efficiencies are indicative values, as
the simulations do not necessarily reproduce the turbulent properties of
A2255. For this reason, our study simply suggests that there is room for
turbulent reacceleration to contribute to the observed emission.

In reality, CRe in each cell would be mixed by the streaming and/or diffusion,
although we neglect those effects in the current study. As discussed in Sect.
5, the mfp comparable to the Alfvén scale leads to a strong diffusion with
$D\sim 10^{31}~{}{\rm cm^{2}/s}$, and CRe can diffuse over a few 100 kpc
within $\sim$1 Gyr. A method that can incorporate those effects, such as the
application of a Lagrangian tracer method (e.g. Vazza et al., 2023), will be
important for more accurate modeling.

## Acknowledgements

The authors thank the anonymous referee for the useful comments. We
acknowledge Andrea Botteon, Katsuaki Asano, and Takumi Ohmura for the fruitful
discussions. This research was supported by FoPM, WINGS Program, the
University of Tokyo. K. N. acknowledges the support by JSPS KAKENHI grants No.
JP23KJ0486. Numerical computations were carried out on PLEIADI at IRA, INAF
(http://www.pleiadi.inaf.it). F. V. acknowledges financial support from the
Horizon 2020 program under the ERC Starting Grant MAGCOW, no. 714196, from the
Cariplo ”BREAKTHRU” funds (Rif: 2022-2088 CUP J33C22004310003), and the usage
of computing time through the John von Neumann Institute for Computing (NIC)
on the GCS Supercomputer JUWELS at Jülich Supercomputing center (JSC), under
project ”radgalicm2”. In this work, we used the enzo code (http://enzo-
project.org), the product of a collaborative effort of scientists at many
universities and national laboratories. The authors gratefully acknowledge the
Gauss center for Supercomputing e.V. (www.gauss-center.eu) for supporting this
project by providing computing time through the John von Neumann Institute for
Computing (NIC) on the GCS Supercomputer JUWELS at Jülich Supercomputing
center (JSC), under project ”radgalicm2” (P.I. F. Vazza).

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##  Appendix A Cosmic-ray protons and secondary electrons

In addition to the turbulent reacceleration, the injection of secondary
leptons from the the decay of pions produced by $pp$ collision between CRp and
thermal protons is often invoked for the mechanism for the diffuse radio
emission in galaxy clusters (e.g., Dennison, 1980; Blasi & Colafrancesco,
1999; Keshet & Loeb, 2010; Enßlin et al., 2011). The limits on the gamma ray
emission from ICM (e.g., Jeltema & Profumo, 2011; Ackermann et al., 2014,
2016) suggest that the contribution of secondary particle production through
$pp$ collision to the observed emission is sub-dominant (Brunetti et al.,
2017; Adam et al., 2021). However, the secondary leptons can contribute to the
diffuse radio emission by providing the seed population for the reacceleration
(e.g., Brunetti et al., 2017; Pinzke et al., 2017; Nishiwaki et al., 2021).
Also, the mini halos in the dense core of cool-core clusters can originate
from the injection of secondaries (e.g., Pfrommer & Enßlin, 2004; Fujita et
al., 2007; ZuHone et al., 2015; Ignesti et al., 2020). In this section, we
explore the contribution of the secondary electrons to the diffuse emission
found in cluster outskirts.

We calculate the injection of secondary CRe for a given distribution of CRp,
using the procedure of Nishiwaki et al. (2021). We assume a single power-law
spectrum of CRp, $N_{p}\propto p^{-\alpha_{p}}$, with $\alpha_{p}=3.2$ to fit
the radio synchrotron spectrum with $\alpha_{\rm syn}\approx 1.6$. We set
$p/(m_{\rm p}c)=1$ for the minimum momentum of CRp. The energy density of CRp
$\epsilon_{\rm CRp}$ is treated as a free parameter. For simplicity, we
neglect the re-acceleration of both CRe and CRp. We adopt the one-zone
approximation explained in Sect. 4.1.

Figure 8:  Energy densities of CRp and magnetic field required in the pure
hadronic model without reacceleration as a function of the magnetic field. The
red and blue lines show $\epsilon_{\rm CRp}$ and $\epsilon_{\rm B}$,
respectively. The thick black line shows the sum of those two. The energy
densities are normalized by that of the ICM in the simulated box
($\epsilon_{\rm ICM}=1.3\times 10^{-13}~{}{\rm erg/cm^{3}}$). The spectral
index of CRp is assumed to be $\alpha_{p}=3.2$. The vertical dotted line shows
the magnetic field calculated with $\eta_{B}=0.05$ (Sect. 3).

In Fig. 8, we show the energy densities of CRp and the magnetic field required
to explain the observed brightness at 49 MHz in the pure hadronic model
without reacceleration. We find that $\epsilon_{\rm CRp}$ should be as large
as $\epsilon_{\rm CRp}\approx 10^{3}\epsilon_{\rm ICM}$ to explain the
observed brightness when we adopt the same magnetic field as Sect. 4.1, i.e.,
$B=0.24~{}\mu{\rm G}$. The energy density of secondary CRe is negligible
compared to that of CRp. The synchrotron brightness scales as $S_{\rm
syn}\propto K_{e}B^{(\delta+1)/2}$, where $K_{e}\propto\epsilon_{\rm
CRp}/(B^{2}+B^{2}_{\rm CMB}(z))$ and $\delta\approx\alpha_{p}+1$ are the
normalization and the power-law index of secondary CRe spectrum, respectively
(e.g., Brunetti & Jones, 2014). We find that the minimum value of the non-
thermal energy density $(\epsilon_{\rm CRp}+\epsilon_{B})$ is 3 times larger
than $\epsilon_{\rm ICM}$, and it appears when the magnetic field is as large
as $B=6~{}\mu{\rm G}$. Thus, the classical “pure-hadronic” model, which does
not include any reacceleration, is incompatible with the observed mega halo.

Note that the above discussion does not exclude the possibility that the
diffuse emission is produced by the reacceleration of the secondaries injected
through the $pp$ collision. A comprehensive study of this “secondary re-
acceleration” model should incorporate the re-acceleration of CRp, which is
out of the scope of this work. It is worth noting that turbulence might be
significantly damped by the back reaction of the reacceleration of CRp when
$\epsilon_{\rm CRp}\gtrsim\epsilon_{\rm turb}$ (e.g., Brunetti & Lazarian,
2007). Such back reaction is not considered in the derivation of Eq. (2).

##  Appendix B dependence on the initial condition

In Sect. 4, we assumed that the initial spectrum before the onset of the
reacceleration, $N_{e}(p,0)$, is determined by the radiative and the Coulomb
cooling. The spectrum has a peak around $p_{\rm min}\sim 10$ as shown in Fig.
5. However, it is also possible that the “initial spectrum” is affected by the
turbulent reacceleration working before the time of the snapshot. Recent
simulation by Beduzzi et al. (2023) observed that the radio-emitting CRe in
the ICM experience multiple episodes of reacceleration. In such a case, the
peak of the seed CRe spectrum should appear at a larger momentum $p_{\rm
min}\sim 10^{2}-10^{3}$. In this section, we discuss how the results in Sect.
4.2 are modified when we adopt a different initial spectrum. We adopt the same
method as Sect. 4.2, taking into account the LOS integration. We fix
$\eta_{B}=0.05$ also in this section.

To consider the reacceleration before the epoch of the snapshot, we assume the
initial spectrum for the calculation with $p_{\rm min}=10^{3}$. The shape of
the initial spectrum is assumed to be the same in all cells. The initial
energy density of the CRe follows $\epsilon_{\rm CRe}(t=0)\propto\epsilon_{\rm
ICM}$, as in Sect. 4.2.

The difference of the initial spectrum is not important when $T_{\rm dur}$
(calculation time of the FP equation) is very long and the steady state due to
the balance between the cooling and reacceleration is achieved in most of the
cells. In Sect. 4, we find that the steady state is achieved at $T_{\rm
dur}\geq 3$ Gyr when $t_{\rm acc}\approx 0.5$ Gyr.

In this Section, we limit $T_{\rm dur}$ by $\approx 2t_{\rm eddy}$, where
$t_{\rm eddy}$ is the eddy turnover time measured in each cell. For the cells
with $2t_{\rm eddy}\gtrsim 1~{}{\rm Gyr}$, we terminate the calculation at 1
Gyr. The mean value of $t_{\rm eddy}$ in the simulated box is $\approx 0.5$
Gyr, so $T_{\rm dur}=2t_{\rm eddy}$ is shorter than 3 Gyr in most of the
cells.

Figure 9: Same as Fig. 5, but the minimum momentum of the initial spectrum is
assumed to be $p_{\rm min}=10^{3}$ due to the reacceleration before the epoch
of the snapshot. The duration of the FP calculation is limited by $2t_{\rm
eddy}$ and $\psi=0.5$ is adopted.

With the above conditions, we find that the typical spectral
index,$\alpha_{\rm syn}\approx 1.6$, can be reproduced with $\psi=0.5$. Fig. 9
(right) shows the CRe spectra before and after the calculation. Summing up the
CRe spectra over cells included in one beam ($\sim 10^{3}$ cells), we find
that $p_{\rm min}$ after the calculation does not change much from the initial
value, $p_{\rm min}=10^{3}$. This is consistent with the assumption that
$p_{\rm min}=10^{3}$ is caused by the turbulent acceleration prior to the
epoch of the snapshot. We calculate the acceleration efficiency using Eq. (10)
and find a typical value of $\langle\eta_{\rm acc}\rangle\approx 1.7\%$.

Although we assumed a shorter $T_{\rm dur}$ than that in Sect. 4.2, the
contribution of the cells with smaller turbulent energy ($F_{\rm turb}<5\times
10^{44}$ erg/s/Mpc3) increases, and almost 60% of the area of emission region
can be covered at the LOFAR sensitivity.

If we adopt the initial spectrum with $p_{\rm min}\sim 10$ and calculate
reacceleration for $T_{\rm dur}=2t_{\rm eddy}$, the number of CRe with $p\sim
10^{4}$ at the final state becomes smaller than the $p_{\rm min}\sim 10^{3}$
case and that model cannot explain the observed brightness unless $\eta_{\rm
acc}\gtrsim 10\%$.