# Synchrotron Intensity Gradient Revealing Magnetic Fields in Galaxy Clusters

Yue Hu  Department of Physics, University of Wisconsin-Madison, Madison, WI
53706, USA  Department of Astronomy, University of Wisconsin-Madison, Madison,
WI 53706, USA  e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>C.
Stuardi  Dipartimento di Fisica e Astronomia, Università di Bologna, via
Gobetti 93/2, I-40129 Bologna, Italy  INAF - Istituto di Radioastronomia di
Bologna, Via Gobetti 101, I-40129 Bologna, Italy  A. Lazarian  Department of
Astronomy, University of Wisconsin-Madison, Madison, WI 53706, USA  e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>G. Brunetti  INAF - Istituto di
Radioastronomia di Bologna, Via Gobetti 101, I-40129 Bologna, Italy  A.
Bonafede  Dipartimento di Fisica e Astronomia, Università di Bologna, via
Gobetti 93/2, I-40129 Bologna, Italy  INAF - Istituto di Radioastronomia di
Bologna, Via Gobetti 101, I-40129 Bologna, Italy  Ka Wai Ho  Department of
Astronomy, University of Wisconsin-Madison, Madison, WI 53706, USA
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545,
USA

###### Abstract

Magnetic fields and their dynamical interplay with matter in galaxy clusters
contribute to the physical properties and evolution of the intracluster
medium. However, the current understanding of the origin and properties of
cluster magnetic fields is still limited by observational challenges. In this
article, we map the magnetic fields at hundreds-kpc scales of five clusters
RXC J1314.4 -2515, Abell 2345, Abell 3376, MCXC J0352.4 -7401, and El Gordo
using the synchrotron intensity gradient technique in conjunction with high-
resolution radio observations from the Jansky Very Large Array (JVLA) and the
Karoo Array Telescope (MeerKAT). We demonstrate that the magnetic field
orientation of radio relics derived from synchrotron intensity gradient is in
agreement with that obtained with synchrotron polarization. Most importantly,
the synchrotron intensity gradient is not limited by Faraday depolarization in
the cluster central regions and allows us to map magnetic fields in the radio
halos of RXC J1314.4 -2515 and El Gordo. We find that magnetic fields in radio
halos exhibit a preferential direction along the major merger axis and show
turbulent structures at higher angular resolution. The results are consistent
with expectations from numerical simulations, which predict turbulent magnetic
fields in cluster mergers that are stirred and amplified by matter motions.

## Introducion

Magnetic fields are a ubiquitous aspect of the cosmos [1, 2, 3], and the
largest-scale cosmic magnetic fields observed to date are found in galaxy
clusters [4, 5, 6, 7]. These magnetic fields are a fundamental part of cosmic
magnetogenesis, either arising from the turbulent amplification of seed fields
or being injected by active galactic nuclei and galaxies [8, 9]. They are
critical to maintaining energy balance within the intracluster medium (ICM)
through heat conduction [10, 11], coupling cosmic rays (CRs) to the
intracluster gas [12, 13, 14], and generating synchrotron radiation by
gyrating CR electrons [15]. Despite the crucial importance, the origin of
magnetic fields in ICM remains the grand challenge problem. To check the
existing theoretical predictions that the magnetic field can be amplified
during the galaxy mergers [16, 17, 18, 19], a comprehensive understanding of
magnetic field topology in galaxy clusters is imperative.

The study of magnetic fields in galaxy clusters is typically based on
observing polarized radio emissions from background or embedded radio galaxies
and diffuse synchrotron sources. The Faraday Rotation Measure (RM) of a
polarized source, obtained from the rotation of its polarization angle with
wavelength, reveals the line-of-sight (LOS) magnetic field weighted by thermal
electron density. An RM grid constructed using many background polarized
sources provides a valuable probe of the cluster’s magnetic field structure
[20, 21]. This approach established that the magnetic fields of the cluster
were turbulent and tangled on spatial scales between 5 and 500 kpc [22].
However, these findings are limited by the number of detected background
polarized sources, and the uncertainties related to the RM grid interpretation
are still not well defined [23]. The plane-of-the-sky (POS) magnetic field
orientation is revealed by direct detection of linearly polarized emission [1,
24], but, as we discussed further, this way of probing the magnetic field
fails for most of the galactic cluster volume.

Polarized diffuse synchrotron emissions, particularly from radio relics and
halos, allow the study of magnetic fields within selected regions of galaxy
clusters. For instance, strongly polarized radio relics (with polarization
fraction up to 60%) that are typically located at the cluster’s periphery can
be explored this way [6, 25, 19]. However, depolarization effects, such as
Faraday depolarization caused by thermal electrons and turbulent magnetic
fields along the LOS, as well as beam depolarization due to a randomized
magnetic field distribution in the POS, prevent this type of studies for most
extended regions of clusters, i.e., galactic halos [26, 27]. Thus, so far, no
polarization mapping of magnetic fields in radio halos has been carried out.
This is one of the main challenges faced by the next-generation radio
facilities, such as the Square Kilometre Array (SKA) [28, 29].

Based on a comprehensive understanding of the pervasive MHD turbulence in the
ICM, the refs. [30, 31, 32] introduced synchrotron intensity gradient (SIG) as
a way to map the magnetic fields. The fast turbulent reconnection in MHD
turbulence leads to the mixing of magnetized plasma perpendicular to the
direction of the local magnetic field. This results in anisotropic structures
of turbulent velocity, magnetic field, and synchrotron radiation elongating
along the magnetic field [33, 34], as confirmed by numerical simulations [35,
36, 37, 38] and in situ measurements in the solar wind [39, 40, 41]. As a
result, the direction of SIG determines the magnetic field, as was proven by
comparing the magnetic fields traced by SIG and synchrotron polarization for
the Milky Way magnetic fields [30]. This analysis was further substantiated
through numerical simulations that replicated the turbulent conditions within
the Milky Way [30]. Nevertheless, it is crucial to recognize that the nature
of turbulence within galaxy clusters is distinctly different from the
turbulence observed within the Milky Way [42]. Therefore, so far there has
been only one attempt to apply SIG for magnetic field mapping in the relaxed
cluster Perseus [32]. The application assumed the Alfvén scale, i.e., the
scale at which magnetic fields become dynamically important [43], is
observationally resolved. Based on the current study of turbulence and
magnetic field strength in the ICM, the Alfvén scale is constrained from 1 to
60 kpc (see methods, subsection "SIG: theoretical consideration"). However,
our numerical simulations (see methods, subsection "SIG: numerical testing")
demonstrated that resolving the Alfvén scale is not essential for SIG. We
showed that at scales larger than the Alfvén scale, magnetic fields passively
follow the motion of the large-scale flow, with SIG aligned perpendicular to
the magnetic field, similar to the case when the Alfvén scale is resolved (see
methods, subsection "SIG: theoretical consideration"). This understanding
underlines our confident application of SIG to the ICM.

Here we apply SIG to five disturbed galaxy clusters (RXC J1314.4-2515, Abell
2345, Abell 3376, MXCX J0352.4 - 7401, and El Gordo), which are of particular
interest in terms of magnetic field amplification and the acceleration of
cosmic rays by magnetized turbulence [15, 44], using high-resolution radio
observations obtained with the Jansky Very Large Array (JVLA) [25, 22] and the
MeerKAT array [45]. Our choice is motivated by the fact that magnetic fields
in Abell 3376’s and El Gordo’s relics have already been studied by
polarization [46, 47], while the fields in the relics of RXC J1314.4-2515 and
Abell 2345 was also recently mapped [25, 22]. Thus, these four clusters
provide a valuable test bed for applying SIG. Encouraged by the consistency of
the SIG results with observational and numerical testing, we use SIG to map
the magnetic fields in the MCXC J0352.4-7401 cluster [45] and we present the
magnetic field measurements in RXC J1314.4-2515 and El Gordo radio halos,
revealing the structure of magnetic fields on the largest scales ever
measured.

## Results

Magnetic field morphology in RXC J1314.4 - 2515, Abell 2345, Abell 3376, MCXC
J0352.4 - 7401, and El Gordo: We follow the SIG[32] adapted to clusters as
discussed in methods, subsection "SIG: practical implementation" to map Plane
of Sky (POS) magnetic fields in galaxy clusters RXC J1314.4-2515[25, 45],
Abell 2345[22], Abell 3376 [45], MCXC J0352.4 - 7401 [45], and El Gordo [45].
We define the Alignment Measure (AM) to quantify the alignment of the SIG and
the polarization measurements: ${\rm AM}=2(\cos^{2}\theta_{r}-\frac{1}{2})$,
where $\theta_{r}$ is the relative angle between the POS magnetic field
inferred from two methods. ${\rm AM}=1$ corresponds to a perfect parallel
alignment, while ${\rm AM}=-1$ represents a perpendicular alignment. The
values of the AM are illustrated by chromatically superimposing the values
onto the magnetic field vectors as inferred from polarization measurements and
are shown in Figs. 1 and 2.

The detection of polarized synchrotron emission in radio relics in the
peripheral regions of RXC J1314.4 - 2515 and Abell 2345 has been achieved
through JVLA observation at a frequency range of 1-2 GHz [25, 22]. The
resolution of the polarization signal, represented by the full width and half
maximum (FWHM) of the Gaussian beam, is around $25^{\prime\prime}$ (or 120
kpc) for RXC J1314.4 - 2515 and around $30.5^{\prime\prime}$ (or 110 kpc) for
Abell 2345. For RXC J1314.4 - 2515, SIG is calculated per pixel (beam
resolution about $7.6^{\prime\prime}$ or $30$ kpc) and averaged to FWHM about
$25^{\prime\prime}$, which is similar to that of polarization signal. As shown
in Figs. 1 and 3, the magnetic fields inferred from SIG and polarization are
found to be in agreement (overall AM about 0.70, with a standard deviation of
the mean around 0.01), aligned with the elongated intensity relics along the
south-north direction. The measured AM is a bit lower compared to its AM
obtained for SIG in Milky Way [30]. We attribute this to the higher signal-to-
noise and the smaller Faraday rotation effects in the Milky Way case.

SIG and polarization are sensitive to variations of magnetic field orientation
with beam size. As shown in Fig. 8, $\rm AM$ decreases for a large beam. In
addition, unlike SIG, polarization is sensitive to Faraday rotation and
Faraday depolarization. As a result, we do expect to see differences between
the polarization and SIs. The systematic difference between the two measures
carries important information that sheds light on the difference in the
physical mechanisms of the processes that reveal magnetic field direction, and
this difference can be explored in future studies to get deeper insight into
the physics of ICM.

Similarly, in Abell 2345, we obtain AM around 0.6 (see Fig. 2). The
misalignment between the SIG and polarization in the south tail of Abell
2345-E can be attributed to the potential uncertainties in both measurements
(as detailed in the Supplementary Information). In particular, we noticed that
the misalignment is associated with a point source. These sources induce
synchrotron intensity gradients that are not associated with magnetic fields.
The removal of the point source increases the correspondence between
polarization and SIG increases. On the other hand, intensity jumps at shock
fronts may induce deviation in the SIG from that of the underlying magnetic
field. However, as we discuss in the Supplementary Information, the
contribution of the shock fronts becomes marginal in the process of the sub-
block averaging method adopted in SIG. For the relics of Abell 2345 the
polarization resolution corresponds to FWHM about $30.5^{\prime\prime}$, which
is higher than that of SIG, which potentially also affects the AM.
Nevertheless, our study confirms the overall consistency between the
directions the polarization and SIG revealed. This finding strengthens the
rationale for using this technique in mapping magnetic fields in galaxy
clusters where polarization has not been reported.

Figs. 4, 5, and 6 present the magnetic field measurement of the merging
clusters Abell 3376, MXCX J0352.4-7401, and El Gordo using the SIG. These
clusters have different redshifts ($z$ approximately 0.87 for El Gordo, 0.127
for MXCX J0352.4-7401, and $0.046$ for Abell 3376). The SIG measurements are
in agreement with earlier partial polarization observations in Abell 3376’s
double relics [46] and El Gordo’s west relic [47]. The magnetic fields
associated with radio galaxy jets and bent by ICM in Abell 3376 have also been
observed in the SIG measurement [48]. These earlier studies only covered
limited portions of the relics with detected polarization signals, but the SIG
measurements cover the entire structure and provide complete magnetic field
maps. In addition, we present the SIG-mapped magnetic field for the detected
relics in MCXC J0352.4-7401. Future high-resolution synchrotron intensity
observations will enable SIG to map magnetic fields at smaller scales.

Measurement of polarized synchrotron emission in radio halos is challenging
due to the strong Faraday depolarization effect. SIG, however, offers a unique
solution to this problem as it is insensitive to depolarization. As a result,
SIG opens an avenue for studying the magnetic field of radio halos. For
example, in RXC J1314.4-2515 and El Gordo, polarization is detected only in
the double relics [25] and the west relic [47], respectively. However, SIG
provides a complete picture of the magnetic fields in the entire cluster,
including halos (see Figs. 1 and 6). Due to the beam and LOS averaging, the
gradient signals caused by small-scale fluctuations are averaged out in the
central radio halo. SIG, however, remains sensitive to the large-scale
component of the magnetic field (see methods, subsection "SIG: theoretical
consideration"). At low resolution, the magnetic field in RXC J1314.4-2515
(see Fig. 1 in the Supplementary Information) is preferentially aligned along
the merger axis, whereas on top of this behavior, vortex-like start appearing
at higher resolution (see Fig. 1), yet limited to effective 120 kpc
resolution. The maps clearly show the transition between radio relics and
halos, which are associated with shock waves where the magnetic field is
compressed in the direction perpendicular to the merger axis (see Figs. 4 and
5 in the Supplementary Information).

## Discussion

The reported magnetic field structure testifies the magnetic field
amplification during galactic mergers. It is in line with previous RM grid
measurements [22] and MHD simulations that predicted magnetic fields evolve
with cluster dynamics [16, 17, 18, 19] (see Fig. 7 for an illustration). The
fields are stretched/stirred and further amplified by large-scale bulk flows
along the merger axis. On the other hand, the presence of large-scale magnetic
fields suggests that the magnetic field amplification through turbulent dynamo
is rather inefficient at its non-linear stage, in which less than 10% of
turbulent kinetic energy is transferred to magnetic field energy [49, 44].
Thus, for stationary turbulence, it would take around ten of the longest eddy
turnover times for the magnetic field to reach equipartition with turbulence.
Since the merging of clusters did not last for such a long time, the more
prominent magnetic field structures are formed through the large-scale
stretching caused by the flows resulting from the merger. We anticipate that
there are more magnetic field stochasticity on the small scale.

Our results open perspectives to map magnetic fields in clusters and large-
scale structures and allow for the comparison between numerical expectations
of merging clusters and observations. SIG’s ability to trace the galaxy
clusters’ magnetic fields is confirmed by (a) MHD simulations presented in
Fig. 8, (b) the correspondence of SIG with polarization measurements in radio
relics (see Figs. 1 and 2), as well as (c) the correspondence of the SIG-
traced halo magnetic field structure to those numerically predicted in the
refs. [16, 17, 18, 19]. Being insensitive to the Faraday depolarization, SIG
can be applied to many clusters with diffuse radio emission, which is
especially timely in view of the coming SKA and the Low-Frequency Array
(LOFAR) observations. It opens a unique way of using radio data for the
regions where depolarization masks and distorts the polarized signal. The
prospects of SIG get more exciting in view of recent LOFAR observations that
discovered synchrotron radiation on large scales (several Mpc), on the
outskirts of the cluster or between massive cluster pairs [50, 51, 52]. As
soon as high signal-to-noise images of this very large-scale synchrotron
emission become available, SIG will also enable us to map these largest-scale
magnetic fields and study their statistic properties (see Fig. 6 in the
Supplementary Information for an example of the statistical analysis via the
structure-function). This will provide constraints on the theories of magneto-
genesis of magnetic field and their role within large-scale structures of
Universe evolution.

## Methods

### SIG: theoretical consideration

SIG introduced in ref.[30, 31, 32] is extended in this paper for super-
Alfvénic turbulence present in the ICM. In the super-Alfvénic regime (i.e.,
$M_{\rm A}>1$), turbulent motions at the injection scale are hydrodynamic, and
the kinetic energy of the turbulence follows an isotropic Kolmogorov cascade
($v_{l}\approx l^{1/3}$, where $v_{l}$ is the velocity of the turbulence at
scale $l$). However, at smaller scales, the backreaction of the magnetic field
becomes stronger and the turbulence becomes anisotropic on the Alfvén scale
$l_{\rm A}=LM_{\rm A}^{-3}$, where $L$ is the injection scale [53]. The
typical Alfvén scale in galaxy clusters can be calculated as follows [43]:

$l_{\rm A}\approx 100(\frac{B}{\rm\mu G})^{3}(\frac{L}{\rm
300\leavevmode\nobreak\ kpc})(\frac{v_{L}}{\rm 10^{3}\leavevmode\nobreak\
km\leavevmode\nobreak\ s^{-1}})^{-3}(\frac{n_{e}}{\rm
10^{-3}\leavevmode\nobreak\ cm^{-3}})^{-\frac{3}{2}}{\rm pc},$ (1)

where $B$ is the magnetic field strength, $L$ is the injection scale of
turbulence, and $n_{e}$ is the electron number density. Based on typical
values of $B$ (0.5 - 2.0 $\mu$G), $L$ (300 - 600 kpc), $v_{L}$ (100 - 300 km
s-1), and $n_{e}$ (10-3 cm-3) from literature [54, 55, 56, 57, 58, 59, 25],
the Alfvén scale $l_{\rm A}$ ranges from approximately 1 - 60 kpc. Note that
quoted values of the physical quantities are typically measured for cluster
cores. In the edges of the cluster, these values may be different.

At scales smaller than $l_{\rm A}$, the anisotropic MHD turbulence causes the
turbulent eddies to elongate along the magnetic field, resulting in a gradient
perpendicular to the field. At scales larger than $l_{\rm A}$, the large-scale
gas flows can still regulate the dynamically unimportant magnetic field in the
ICM, causing the field to follow or elongate along large-scale structures, as
shown in Fig. 8. Therefore, the actual number of $l_{A}$ is not important for
using SIG to map the magnetic field, because the gradients are perpendicular
to the magnetic field in both cases.

### SIG: comparison with polarization

SIG and synchrotron polarization are based on different physical effects to
reveal the magnetic field. While synchrotron polarization emerges from the
magnetic fields’ effects on relativistic electrons, SIG is grounded in the
interaction between magnetic fields and conducting fluid. Notably,
measurements obtained from both methods are subject to the effects of LOS
averaging. Given the expected scenario where the LOS integration length for
cluster halos surpasses the scale of magnetic field entanglement, turbulence-
induced fluctuations are not anticipated to manifest a preferential direction,
instead accumulating through a process akin to a random walk. These
turbulence-driven fluctuations play a pivotal role in decreasing the
synchrotron polarization fraction, known as Faraday depolarization. However,
SIG is immune to the depolarization effect, maintaining its reliability in
tracing the magnetic fields in super-Alfénic radio halos. As illustrated in
Fig. 8, the alignment between SIG and magnetic fields remains statistically
stable, even when the LOS integration length increases. On the other side,
polarization is sensitive not only to the value of the magnetic field strength
that aligns parallel to the LOS but also to the density of thermal electrons
within the environment. Given that the orientation of magnetic fields within
galaxy clusters is subject to changes along the LOS, there is a possibility
for differences between contributions to SIG and those to polarization.

A factor that can induce misalignment in observational applications is the
beam size. This effect can be illustrated by considering a particular example
of a very large beam, which covers the full field of view of the observation.
Such a beam results in a constant intensity map and the corresponding gradient
vanishes, but polarization still can give one magnetic field orientation for
this large beam. This beam effect, therefore, reduces the alignment as shown
in the right panel of Fig. 8.

### SIG: practical implementation

In this study, SIG serves as the primary tool for analysis. The intensity
gradient calculated from the synchrotron intensity map ($I(x,y)$) allows for
mapping the orientation of magnetic fields. This is achieved through a
pixelized gradient map $\psi(x,y)$ as follows:

$\displaystyle\bigtriangledown_{x}I(x,y)$ $\displaystyle=I(x+1,y)-I(x,y),$ (2)
$\displaystyle\bigtriangledown_{y}I(x,y)$ $\displaystyle=I(x,y+1)-I(x,y),$
$\displaystyle\psi(x,y)$
$\displaystyle=\tan^{-1}\left(\frac{\bigtriangledown_{y}I(x,y)}{\bigtriangledown_{x}I(x,y)}\right),$

here, $\bigtriangledown_{x}I(x,y)$ and $\bigtriangledown_{y}I(x,y)$ represent
the $x$ and $y$ components of the gradient, respectively. Gradients are
blanked out if their corresponding intensity value is less than $3\sigma$
noise level.

The gradient map $\psi(x,y)$ is further processed through the sub-block
averaging method [60]. This method involves taking all gradient orientations
within a sub-block of interest and applying Gaussian fitting to the
corresponding histogram. The peak value of the Gaussian distribution
represents the statistically most probable gradient orientation within that
sub-block. The averaging step ensures that the resulting gradient direction
incorporates turbulence’s statistical properties. The processed gradient map
is denoted as $\psi_{\rm s}(x,y)$, and its uncertainty is related to the sub-
block size. A larger sub-block size guarantees a sufficient amount of data for
statistical fitting, leading to lower uncertainty. Typically, gradients are
averaged over 20 $\times$ 20-pixel sub-blocks, as this size has been
determined through previous numerical and observational studies to guarantee
sufficient statistics for extracting turbulence’s properties [61]. To address
the boundary effect in cases where the number of data points at the edge of an
intensity structure may be less than 20 $\times$ 20 pixels, a minimum of 10
$\times$ 10 pixels is established for averaging.

The averaging procedure for each sub-block is independent. However, this is
not the case for actual magnetic field lines, necessitating a correlation of
the averaged gradient with that of its neighboring. This can be mathematically
handled by performing smoothing on the pseudo-Stokes parameters ($Q_{\rm g}$
and $U_{\rm g}$), which are defined as:

$\displaystyle Q_{\rm g}(x,y)=I(x,y)\cos(2\psi_{\rm s}(x,y)),$ (3)
$\displaystyle U_{\rm g}(x,y)=I(x,y)\sin(2\psi_{\rm s}(x,y)),$
$\displaystyle\psi_{\rm g}(x,y)=\frac{1}{2}\tan^{-1}(\frac{U_{\rm g}}{Q_{\rm
g}}),$

where $\psi_{\rm g}$ is the pseudo polarization angle. Similar to the Planck
polarization, $\psi_{\rm B}=\psi_{\rm g}+\frac{\pi}{2}$ gives the POS magnetic
field orientation. The weighted intensity ensures that (i) $Q_{\rm g}$ and
$U_{\rm g}$ follow a Gaussian distribution, which facilitates the smoothing of
the pseudo-Stokes parameters using a Gaussian filter. The FWHM of the Gaussian
filter is equal to the sub-block size. (ii) The magnetic fields mapped by SIG
are intensity-weighted, which is also the case for the magnetic field inferred
from synchrotron polarization weighted by the polarized intensity.

### SIG: uncertainty

The uncertainty in SIG is mainly due to systematic errors in radio images and
the SIG algorithm itself. We calculated SIG’s uncertainty by considering error
propagation and blanked out the pixels in which the uncertainty is larger than
30 degrees. We presented the uncertainty maps in Supplementary Figs. 2 and 3
and listed the median value of the uncertainty for each cluster in Tab. 2.

### SIG: numerical testing

A numerical test is presented for illustration purposes. In accordance with
the method described in the ref. [30], we utilize several 3D MHD simulations
of turbulence to synthesize synchrotron emission maps, including the
synchrotron intensity $I_{s}(x,y)$, Stokes parameters $Q_{s}(x,y)$ and
$U_{s}(x,y)$, and polarization angle $\psi(x,y)$. Each simulation box is
divided into 5123 cells, with a turbulence injection scale of approximately
256 cells. The calculation of these maps is based on the following equations:

$\displaystyle I_{s}$ $\displaystyle=\int
n_{e,r}(B_{x}^{2}+B_{y}^{2})B_{\bot}^{\gamma}dz,$ (4) $\displaystyle Q_{s}$
$\displaystyle=\int-n_{e,r}(B_{x}^{2}-B_{y}^{2})B_{\bot}^{\gamma}dz,$
$\displaystyle U_{s}$ $\displaystyle=\int-
n_{e,r}(2B_{x}B_{y})B_{\bot}^{\gamma}dz,$ $\displaystyle\psi$
$\displaystyle=\frac{1}{2}\tan^{-1}(\frac{U}{Q}),$

where $B_{\bot}=\sqrt{B_{x}^{2}+B_{y}^{2}}$ is the magnetic field component
perpendicular to the LOS , with $B_{x}$ and $B_{y}$ being the $x$ and $y$
components, respectively, and $n_{e,r}$ is the density of relativistic
electrons. In consideration of the fact that the anisotropy of synchrotron
emission is insensitive to the spectral index of the electron energy
distribution [62], a homogeneous and isotropic distribution with spectral
index $\alpha=3$ is adopted, yielding an index of $\gamma=\frac{\alpha-3}{4}$.

An example is shown in the left panel of Fig. 8. The super-Alfvénic condition
of $M_{\rm A}\approx 3.2$ leads to a turbulence length scale of $l_{\rm
A}\approx 8$ cells. The magnetic field orientation inferred from SIG globally
agrees with that derived from polarization (with an agreement measure of AM
$\approx 0.81$). The AM values for other simulations are listed in Tab. 1,
which demonstrates correspondence of the magnetic field orientation mapped by
polarization and SIG.

### Observational data: intensity of synchrotron emission

The synchrotron emission images used in this work are produced with JVLA and
MeerKAT observations. The summary of the data set is presented in Tab. 2.

RXC J1314.4-2515: polarization images with similar resolution thanks to the
images released by the MeerKAT Galaxy Cluster Legacy Survey Data Release 1
(https://archive-
gw-1.kat.ac.za/public/repository/10.48479/7epd-w356/index.html). The high
resolution 1.28 GHz image from which we derived the SIG measurements (see Fig.
1) has a beam size of $7.3^{\prime\prime}\times 7.6^{\prime\prime}$
corresponding to a spatial resolution of about 30 kpc at $z=0.247$. The RMS
noise of this image is 5 $\mu$Jy/beam. Instead, a low-resolution SIG image was
obtained from the same JVLA dataset from which we derived polarization (see
Fig. 1 in the Supplementary Information). This total intensity image at 3GHz
has a resolution of $17^{\prime\prime}\times 17^{\prime\prime}$, corresponding
to 66 kpc around, and an RMS noise of 13 $\mu$Jy/beam. This image is the same
as the one presented by the ref. [25] but smoothed with a circular beam size.

Abell 2345: The resolution beam of the 1.5 GHz JVLA total intensity map in
Fig. 2 is $19^{\prime\prime}\times 19^{\prime\prime}$, corresponding to a
spatial resolution of 58 kpc at $z=0.179$. With respect to the image presented
by ref.[22] this has been smoothed to obtain a circular beam size. The RMS
noise is 80 $\mu$Jy/beam.

Abell 3376: The resolution beam of the 1.28 GHz MeerKAT total intensity images
of this cluster shown in Fig. 4 is $7.4^{\prime\prime}\times
7.6^{\prime\prime}$, corresponding to a spatial resolution of approximately 7
kpc at $z=0.046$. The RMS noise is 3.1 $\mu$Jy/beam.

MCXC J0352.4 - 7401 (Abell 3186): The total intensity 1.28 GHz MeerKAT images
of this cluster shown in Fig. 5 have a resolution beam of
$6.9^{\prime\prime}\times 7.1^{\prime\prime}$. This corresponds to a spatial
resolution of approximately 16 kpc at $z=0.127$. The RMS noise is 2.6
$\mu$Jy/beam.

El Gordo: The resolution beam of the total intensity 1.28 GHz MeerKAT image of
El Gordo shown in Fig. 6 is $7.1"\times 7.1"$, corresponding to a physical
resolution of approximately 55 kpc at $z=0.87$. The RMS noise is 1.5
$\mu$Jy/beam.

### Observational data: polarization measurement

We use the synchrotron polarization measurements obtained from JVLA
observations[25, 22]. We only report here the main characteristics of these
observations while we refer to the original works for a detailed explanation
of the data analysis. The magnetic field orientation is defined as
$\chi_{B}=\chi_{0}+\frac{\pi}{2}$, inferred from the intrinsic polarization
angle $\chi_{0}$ at the source obtained with the Rotation Measure synthesis
technique [63]. This is intended to correct the measured Faraday rotation to
represent the magnetic-field orientation at the relic.

Nevertheless, radio images can be affected by polarization leakage between
Stokes parameters due to instrumental effects. The leakage from Stokes $I$ to
$Q$ and $U$ was estimated at less than 2 % [25, 22]. Since relics are highly
polarized, the leakage should have a marginal effect on the polarization angle
estimates. The uncertainty of mapping magnetic fields with polarization can
also arise from the adopted model of the distribution of thermal and
relativistic electrons within the relics [64]. The uncertainty of the order
$10^{\circ}$ can serve as an estimate for the misalignment of polarization and
SIG vectors, which corresponds to the reported AM variation of $\pm$0.2.

The magnetic field orientation images of the RXC J1314.4-2515 galaxy cluster
were obtained from a 1-2 GHz JVLA observation (with a central frequency of 1.5
GHz and resolution beam of $25^{\prime\prime}$) while a 2-4 GHz JVLA
observation (central frequency of 3 GHz and resolution beam of
$30.5^{\prime\prime}$) was used for Abell 2345. The polarization images are
not smoothed further but re-gridded to match that of the SIG-measured magnetic
field orientation spatially. Polarization images are already masked to show
only pixels detected with a corresponding Gaussian significance level greater
than 5$\sigma$, as explained in refs. [25, 22].

## Tables

$M_{\rm A}$ | 2.4 | 2.9 | 3.2 | 5.2 | 7.8
---|---|---|---|---|---
AM | 0.92 | 0.93 | 0.81 | 0.88 | 0.92
Table 1: Summary of global mean AM values in different Alfvén Mach number $M_{\rm A}$ conditions. The uncertainty of AM calculated from the standard deviation of the mean is around 0.01 - 0.02.  Cluster | Beam resolution (SIG) | $\langle\sigma_{\psi_{\rm g}}\rangle$ | Frequency | Polarization data (resolution)
---|---|---|---|---
RXC J1314.4-2515[45] | 30 kpc (120 kpc) | 7.80∘ | 1.28 GHz | JVLA 3 GHz (120 kpc)[25]
Abell 2345[22] | 58 kpc (180 kpc) | E: 5.79∘, W: 3.83∘ | 1.5 GHz | JVLA 1.5 GHz (110 kpc)[22]
Abell 3376 [45] | 7 kpc (22 kpc) | E: 6.60∘, W: 7.82∘ | 1.28 GHz | -
MCXC J0352.4 - 7401 [45] | 16 kpc (62 kpc) | E: 7.44∘, W: 7.52∘ | 1.28 GHz | -
El Gordo [45] | 55 kpc (400 kpc) | 7.41∘ | 1.28 GHz | -
Table 2: Summary of data sets used in this work. $\langle\sigma_{\psi_{\rm
g}}\rangle$ is the median value of the SIG’s uncertainty over the region of
interest.

Figure 1: The magnetic field orientation of the RXC J1314.4 - 2515 galaxy
cluster. Top: The differences and similarities of the magnetic fields measured
by the two techniques (SIG and polarization) are presented. On the top left,
the morphology of the magnetic fields is revealed through the SIG (FWHM around
$25^{\prime\prime}$ or 120 kpc). On the top right, the magnetic field
morphology is revealed through JVLA synchrotron polarization at 3 GHz (FWHM
approximately 25”). Each magnetic field segment represents the SIG (or
polarization) averaged for $6\times 6$ pixels for visualization purposes. The
colors of the polarization segments represent the AM of the SIG and
polarization. The magnetic field is overlaid on the higher resolution
synchrotron emission image from the MeerKAT survey [45] at 1.28 GHz (FWHM
around $7.6^{\prime\prime}$ or 30 kpc). The pink contours represent X-ray
emission measured by the XMM-Newton and the dotted line indicates the expected
merger axis determined by radio images. The merger axis defined by X-ray
emission’s elongation is studied in the Supplementary Information. Bottom: A
zoom-in view of the magnetic field in RXC J1314.4 - 2515’s halo, indicated by
the green box in the top panel, is provided.  Figure 2: The magnetic field
orientation of the Abell 2345 galaxy cluster. Top: Synchrotron emission image
of the Abell 2345 cluster observed with the JVLA at 1.5 GHz. The pink contours
represent X-ray emission measured by the XMM-Newton and the dotted line
indicates the expected merger axis. Bottom: same as Fig. 1, but for the Abell
2345 cluster’s subregion E (bottom left) and W (bottom right), indicated by
the green boxes in the top panel. The magnetic field inferred from the
synchrotron polarization and background emission image is from the JVLA
observation at 1.5 GHz (FWHM approximately $30.5^{\prime\prime}$ or 110 kpc).
The resolution of the SIG is approximately $50^{\prime\prime}$ (or 180 kpc).

Figure 3: Histograms of AM (between the SIG and polarization) towards RXC
J1314.4-2515 (left) and Abell 2345 (right). The maxima at AM = 1 suggests an
excellent alignment of magnetic fields revealed by the two methods. Figure 4:
The magnetic field orientation of the Abell 3376 galaxy cluster. Top: overall
synchrotron emission intensity map from MeerKAT observation at 1.28 GHz (FWHM
approximately $7^{\prime\prime}$) of the Abell 3376 cluster. The cyan contours
represent X-ray emission obtained from the XMM-Newton archival observations
[65] and the dotted line indicates the expected merger axis. Bottom: The
magnetic field orientation (FWHM approximately $24^{\prime\prime}$
corresponding to a physical scale of 22 kpc), represented by white segments,
in Abell 3376’s relics (indicated by the green boxes in the top panel).
Figure 5: The magnetic field orientation of the MCXC J0352.4 - 7401 cluster
[66]. The magnetic field orientation (white segments), superimposed on the
synchrotron emission intensity map from MeerKAT observation at 1.28 GHz (FWHM
approximately $7^{\prime\prime}$), has resolution FWHM approximately
$24^{\prime\prime}$ corresponding to physical scales of 62 kpc respectively).
Figure 6: The magnetic field orientation of the El Gordo cluster. The
background emission image is from MeerKAT observation at 1.28 GHz (FWHM
approximately $7^{\prime\prime}$). The SIG measurement has a resolution of
FWHM approximately $24^{\prime\prime}$ (or 400 kpc). Each white segment
represents the SIG averaged for $6\times 6$ pixels for visualization purposes.
The blue contours represent the X-ray emission obtained from the Chandra
archival observations [67] and the dotted line indicates the expected merger
axis. Figure 7: Cartoon illustration of magnetic field evolution in cluster
merger. In the merging of two turbulent clusters (panel A): cluster-1 and
cluster-2, the magnetic field is draped and amplified at the merger
(advancing) shock in the first phase (panel B), then the field is stretched
along the merger axis (panel C), and finally it is further amplified by
turbulence generated in the cluster merger (panel D). Figure 8: Numerical test
of SIG. Left: comparison of the magnetic field orientation inferred from SIG
(red segment) and polarization (blue segment). Middle and right: AM of
magnetic fields inferred from SIG and polarization as a function of the
cluster’s LOS thickness (middle) and the beam size (right). Polarization is
smoothed to match the resolution of SIG after averaging.

## Data availability

Source data supporting the SIG plots and other findings presented in this
paper are accessible from the MeerKAT Galaxy Cluster Legacy Survey Data
Release 1: https://archive-
gw-1.kat.ac.za/public/repository/10.48479/7epd-w356/index.html and JVLA data
archive: https://data.nrao.edu/portal/#/. The datasets generated and analyzed
during the current study are available from the corresponding author upon
request.

## Code availability

The simulations of MHD turbulence were conducted with the code MHDFlows, which
can be downloaded from: https://github.com/MHDFlows/MHDFlows.jl. The analysis
of the simulation data can be done with any science program language (e.g.,
Julia) according to the formulas provided explicitly in the manuscript. The
SIG code supporting the findings of this study is available from the
corresponding authors upon request. We have chosen not to publicly share the
code in order to protect proprietary algorithms and maintain compliance with
licensing agreements that restrict open distribution. We are committed to
fostering scientific collaboration and are happy to provide the code to
interested researchers for the purpose of replicating or building upon our
findings.

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## Acknowledgements

Y.H. and A.L. acknowledge the support of the NASA ATP AAH7546 and NSF grants
AST 2307840. K.W.H. acknowledges the support of NASA ATP AAH7546 and the LDRD
program of LANL with project # 20220107DR. NASA provided financial support for
this work through award 09_0231 issued by the Universities Space Research
Association, Inc. (USRA). This work used SDSC Expanse CPU at SDSC through
allocations PHY230032, PHY230033, PHY230091, and PHY230105 from the Advanced
Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS)
program, which is supported by National Science Foundation grants #2138259,
#2138286, #2138307, #2137603, and #2138296. C.S. and A.B. acknowledge support
from the MIUR grant FARE SMS and from the ERC-StG DRANOEL, n. 714245

## Author contributions

All authors discussed the results, commented on the manuscript, and
contributed to its writing. Y.H. and C.S. analyzed the observational data for
radio emission and polarization, while Y.H. and K.W.H. conducted the numerical
analysis. Y.H. and A.L. wrote the original manuscript, and A.B. and C.S.
provided the observational data. G.B. provided crucial comments and
suggestions on the application of SIG to ICM and on physical interpretations
of the results.

## Competing interests

The authors declare no competing interests.

See pages 1,2,3,4,5,6,7,8 of supplementary.pdf