# Measurement of MHD turbulence properties by synchrotron radiation techniques

Jian-Fu Zhang 1,2,∗, Ru-Yue Wang 1

###### Abstract

It is well-known that magnetohydrodynamic (MHD) turbulence is ubiquitous in
astrophysical environments. The correct understanding of the fundamental
properties of MHD turbulence is a prerequisite for revealing many key
astrophysical processes. The development of observation-based measurement
techniques has significantly promoted MHD turbulence theory and its
implications in astrophysics. After describing the modern understanding of MHD
turbulence based on theoretical analysis and direct numerical simulations, we
review recent developments related to synchrotron fluctuation techniques.
Specifically, we comment on the validation of synchrotron fluctuation
techniques and the measurement performance of several properties of magnetic
turbulence based on data cubes from MHD turbulence simulations and
observations. Furthermore, we propose to strengthen the studies of the
magnetization and 3D magnetic field structure’s measurements of interstellar
turbulence. At the same time, we also discuss the prospects of new techniques
for measuring magnetic field properties and understanding astrophysical
processes, using a large number of data cubes from the Low-Frequency Array
(LOFAR) and the Square Kilometre Array (SKA).

##  1 Keywords:

Magnetohydrodynamic turbulence, Polarization radiation, Interstellar magnetic
fields, Statistic methods

##  2 Introduction

Turbulence is ubiquitous in our universe. As is perceived in everyday life,
the movement of the fluid leads to the development of hydrodynamic turbulence.
Unlike our terrestrial world, conductive fluids in astrophysics naturally
induce magnetic field fluctuations leading to MHD turbulence due to the
presence of almost all material in an ionization plasma state. The self-
similarity over a large dynamical range of MHD turbulence in astronomical
environments has been well confirmed by observations, as shown by the power-
law spectrum of interstellar electron density fluctuations on the plane of the
sky (Armstrong et al., 1995; Chepurnov & Lazarian, 2010). In addition, the
presence of MHD turbulence has been also pointed out by observations that far
as in galaxy clusters (Zhuravleva et al., 2019) as well as by in-situ direct
measurements in the local solar wind (Bruno & Carbone, 2013).

It is known that MHD turbulence plays an important role in many critical
astrophysical processes, such as star formation (Mac Low & Klessen, 2004;
McKee & Ostriker 2007; Ostriker, 2009; Dib et al., 2010; Crutcher, 2012),
magnetic reconnection (Lazarian & Vishniac, 1999, henceforth LV99; Lazarian et
al., 2020), magnetic field amplification (Federrath, 2016), cosmic rays
acceleration (Gaches et al., 2021; Zhang & Xiang, 2021) and diffusion (Yan &
Lazarian, 2008; Xu & Lazarian, 2018; Lazarian & Xu, 2021; Fornieri et al.,
2021 for a comprehensive numerical study), and heat conduction in galaxy
clusters (Yuan et al., 2015; Bu et al., 2016). Naturally, a thorough
understanding of MHD turbulence is essential in order to better describe a
large number of astrophysical processes that operate over a large number of
physical scales.

MHD turbulence with the nonlinear complexity generally exhibits diverse
properties at a microscopic plasma level, but it shows regularity behind
chaotic phenomena for turbulent fluctuations and allows a statistical
description at a macroscopic level (Biskamp, 2003). Taking statistical
description into account, commonly used methods are the power spectrum, the
correlation and structure functions (see Monin & Yaglom (1975) for an old
book; Boldyrev et al., 2002a; 2002b), by which one mostly considers volume and
time averaging of physical variables to implement the statistical ensemble
measurements of turbulence. In addition, the Delta-variance spectrum (Stutzki
et al., 1998; Dib et al., 2021), as another manifestation of the power
spectrum, has been used to characterize turbulence properties.

In recent years, numerical simulations or experiments combined with analytical
theory has significantly promoted the study of MHD turbulence (Goldreich &
Sridhar, 1995, thereafter GS95; LV99; Cho & Lazarian, 2002; Kowal & Lazarian,
2010; Beresnyak 2017). In particular, the numerical experiments are gradually
alleviating the gap between the realistic MHD turbulence physics and the
analytical theory. However, due to the large-scale characteristics inherent in
astrophysical systems resulting in a high Reynolds number $R_{e}>10^{10}$, it
is challenging to simulate a realistic astrophysical environment. The
currently available 3D MHD simulations are limited by the Reynolds number of
about $R_{\rm e}\simeq 10^{5}$ (e.g., Beresnyak, 2019). Therefore, a new
observation-based research perspective is to break the bottleneck of the
direct numerical simulation.

In addition to the study of electron density information, there are two
additional sources of information regarding MHD turbulence in astrophysics:
turbulent velocity and magnetic fields. The former can be retrieved from
spectrometric observations of emission lines (e.g., Lazarian et al., 2002;
Lazarian & Pogosyan, 2004), while the latter, by radio observations (Gaensler
et al., 2011; Lazarian & Pogosyan, 2012; 2016, hereafter, LP12 and LP16). The
main purpose of this paper is to review the progress made in measuring the
fundamental properties of MHD turbulence using the radio synchrotron radiation
method.

Section 2 describes the fundamentals of MHD turbulence theory necessary to
understand what MHD turbulence is and develop magnetic field measurement
techniques. In Section 3, we provide some basic formulae of synchrotron
radiative processes in MHD turbulence. Section 4 presents various statistical
techniques that are being currently developed, the feasibility of which is
tested using synthetic, realistic simulation and observational data in Section
5. Finally, we give a discussion in Section 6.

##  3 Fundamentals of MHD turbulence theory

###  3.1 Turbulence driving mechanism

The turbulence energy source driving varies with different astrophysical
environments. Here, to simplify considerations, we divide the driving sources
into two categories: external and internal driving mechanisms. The former has
been considered as a common way for turbulence driving in the simulations of
interstellar turbulence, due to the convenience of setting external forces.
The most typical cases are supernova explosions in ISM (Spitzer, 1978;
Ostriker, 2009; Chamandy & Shukurov, 2020; Dib et al., 2006), merger events
and active galactic nuclei (AGN) outflows in the intercluster medium
(Subramanian et al., 2006; Chandran, 2005; Lazarian et al., 2020) and outflow
in molecular clouds (Carroll et al., 2009) and young stellar objects
(Federrath, 2016). In contrast, the latter is also accepted generally, such as
magnetic reconnection driving/mediating turbulence in various regimes (Kowal
et al., 2017; Huang & Bhattacharjee, 2016; Cerri & Califano, 2017; Franci et
al., 2017; Dong et al., 2018), the magnetorotational instability-generated
turbulence (Gardiner, 2005; Fromang, 2013; Kunz et al., 2016; Riols et al.,
2017; Zhdankin et al., 2017; Sellwood & Balbus, 1999). Indeed, for realistic
astrophysical processes such as molecular clouds and supernovae, their
drivings may be involved in internal and external mechanisms simultaneously.

Sometimes the complexity of turbulent driving is manifested by the coexistence
of multiple driving mechanisms in the same environment. One of the most
classic cases is the driving process in the ISM, where various drivers may
occur simultaneously: expanding shells and gravitational instability at about
1000 pc scales arising from galaxy mergers; expanding shells, MRI and cloud
collisions at about 100 pc scales from supernova explosion; and stellar
feedback at 10 pc to sub-pc scales by outflow. In the turbulence research
community, for simplicity, one usually uses external driving with the
solenoidal or compressible approach, which is more effective at limited
numerical resolution. In recent years, internal driving mechanisms have
gradually become the focus of research, but only at the cost of expensive
numerical resources (Beresnyak, 2017; Kowal et al., 2017).

###  3.2 Basic properties of MHD turbulence

In the spirit of understanding the nature of turbulence from first principles,
we here provide the visco-resistive equations that govern the evolution of a
magnetized plasma as follows111 These equations can be applied to describe the
plasma dynamics in the case of the long wavelengths and low frequencies with
respect to plasma characteristic scales and frequencies, and the associated
plasma approximations such as quasi neutrality, no displacement current in
Ampére’s law because of non-relativistic limit, as well as resistive Ohm’s law
(refer to Chapter 3 of Krall & Tivelpiece (1973) for further details).:

$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho{\bf v})=0,$ (1)
$\rho(\frac{\partial}{\partial
t}+{\bf{v}}\cdot\nabla){\bf{v}}={\frac{{\bf{j}}\times{\bf{B}}}{c}}-\nabla{{p_{\rm
g}}}+\rho\nu{\nabla}^{2}{\bf{v}},$ (2) $\frac{\partial{p_{\rm g}}}{\partial
t}+{\bf v}\cdot\nabla{p_{\rm g}}+\Gamma{p_{\rm g}}\nabla\cdot{\bf v}=0,$ (3)
$\frac{\partial{\bf B}}{\partial t}-\nabla\times({\bf{v}}\times{\bf
B})=\eta\nabla^{2}{\bf{B}},$ (4) $\nabla\cdot{\bf B}=0,$ (5)

describing the continuity, Eq. (1), the momentum conservation, Eq.(2), the
adiabatic invariance, Eq. (3), the magnetic induction, Eq. (4), and the
divergence of magnetic field, Eq. (5), respectively. Here, ${p_{\rm g}}=c_{\rm
s}^{2}\rho$ is the gas pressure, $c_{\rm s}=\sqrt{\Gamma p_{\rm g}/\rho}$ the
speed of sound, $t$ the evolution time of the fluid, $\nu$ the kinematic
viscosity, ${\eta}=c^{2}/(4\pi\sigma)$ the magnetic diffusivity, $\sigma$ the
conductivity, and $\Gamma$ the adiabatic index. Other physical quantities have
their usual meanings. The interaction of plasma fluids and magnetic field was
formulized in the above equations, by which a series of physical quantities
characterizing the properties of the MHD turbulence can be better understood.

Let us replace the operator $\nabla$ by ${\nabla}/{L}$ and set ${\bf
v}=v{\bf\tilde{v}}=L/t{\bf\tilde{v}}$ in Eq. (2) and ${\bf B}={B\bf\tilde{B}}$
in Eq. (4), where $L$ is the characteristic scale length at the typical
velocity $v$, as well as $\bf\tilde{v}$ and $\bf\tilde{B}$ are dimensionless.
After a simple analytical derivation (including only the viscosity term in Eq.
2), we obtain

$\frac{\partial{\bf\tilde{v}}}{\partial
t}+{\bf\tilde{v}}\cdot{\nabla}{\bf\tilde{v}}=\frac{\nu}{vL}{\nabla}^{2}{\bf\tilde{v}},$
(6)

$\frac{\partial{\bf\tilde{B}}}{\partial
t}-{\nabla}\times({\bf\tilde{v}}\times{\bf\tilde{B}})=\frac{\eta}{Lv}\nabla^{2}{\bf\tilde{B}}.\\\
\ $ (7)

Through the coefficients of the equations, we define two dimensionless
physical quantities

$R_{\rm e}=\frac{Lv}{\nu},\ \ \ R_{\rm m}=\frac{Lv}{\eta},$ (8)

where $R_{\rm e}$ is the kinetic Reynolds number characterizing the ratio of
the viscous timescale to the advection timescale, and $R_{\rm m}$ is the
magnetic Reynolds number denoting the ratio of resistive diffusion timescale
to the advection timescale. Considering the temperature $T\simeq 10^{4}$ K
related to $\eta\simeq 0.42\times 10^{7}T_{\rm eV}^{-3/2}\rm cm^{2}/s$, the
scale length $L\simeq 1$ pc, and the velocity $v\simeq 1$ km/s, one can obtain
a large $R_{\rm m}\simeq 10^{16}$ as an example of turbulence setting (e.g.,
Brandenburg & Subramanian, 2005). The relative dominance between the kinematic
viscosity and magnetic diffusivity is described by the magnetic Prandtl
number:

$P_{\rm r}=\frac{R_{\rm m}}{R_{\rm e}}=\frac{\nu}{\eta}.$ (9)

Similarly, combining the adiabatic invariance, Eq. (3) and momentum
conservation, Eq. (2), one can obtain the sonic Mach number (Biskamp, 2003 for
a detailed derivation)

$M_{\rm s}=v/c_{\rm s},$ (10)

for the high-$\beta$ plasma case. This parameter can be used to describe the
information of compressibility of MHD turbulence. For the low-$\beta$ plasma
case, the momentum conservation Eq. (2) is dominated by the magnetic pressure.
On multiplying Eq. (2) by ${\bf v}=v\bf\tilde{v}$, the quantity characterizing
the magnetization strength is called the Alfvénic Mach number

$M_{\rm A}=v/V_{\rm A},$ (11)

where $V_{\rm A}=B/\sqrt{4\pi\rho}$ is the Alfvénic velocity.

Considering current density ($\nabla\times{\bf B}=\frac{4\pi}{c}{\bf j}$)
related to the magnetic field by Ampere’s law, the Lorentz force can be
written in the following form:

$\frac{{\bf j}\times{\bf B}}{c}=\frac{({\bf B}\cdot\nabla){\bf
B}}{4\pi}-\nabla\frac{{\bf B^{2}}}{8\pi},$ (12)

where the first term on the right-hand side describes the magnetic tension
force and the second term magnetic pressure force. The plasma parameter
$\beta$, denoting the ratio of gas pressure to magnetic one, is defined as

$\beta=\frac{p_{\rm g}}{{B^{2}}/8\pi},$ (13)

which can be rewritten as

$\beta=\frac{2M_{\rm A}^{2}}{M_{\rm s}^{2}},$ (14)

with the sonic and Alfvén Mach numbers.

As for the description of the properties of helicity in MHD turbulence, one
can introduce another two quantities termed the magnetic helicity and cross
helicity, which are respectively written as

$H^{M}=\int_{V}{\bf A}\cdot{\bf B}dV,$ (15) $H^{C}=\int_{V}{\bf v}\cdot{\bf
B}dV.$ (16)

Here, ${\bf A}$ is the vector potential with ${\bf B}=\nabla\times{\bf A}$.
The former can measure the twist and linkage of the magnetic field lines, and
the latter can reflect the overall correlation of magnetic and velocity fields
(Falgarone & Passot, 2003).

###  3.3 Cascade processes of MHD turbulence

####  3.3.1 Power law scaling of turbulence cascade

The pioneering work regarding incompressible hydrodynamic turbulence
(Kolmogorov, 1941, henceforth K41) demonstrated that the energy is injected at
large scale (called the injection scale) and dissipated at small scales (the
dissipation scale). The range between these two scales is called the inertial
range, where the energy cascades from injection scale to dissipation scale
with negligible energy losses, that is, the energy transfer rate in this range
keeps a constant,

$\epsilon\thickapprox u_{l}^{2}/t_{\rm cas}=const.,$ (17)

where $u_{l}$ is a characteristic velocity, and $t_{\rm cas}=l/{u_{l}}$ is the
cascading timescale. This expression indirectly reflects the relationship
between velocity $u_{l}$ of fluid elements and their scales $l$, namely
$u_{l}\propto({\epsilon l_{\perp}})^{1/3}$. Therefore, we have the well-known
Kolmogorov 1D spectrum

$E(k)\sim{\epsilon}^{2/3}k^{-5/3},$ (18)

where $k$ is the wave number. Alternatively, the 3D spectrum is expressed as

$E_{\rm 3D}(k)\sim k^{-11/3}.$ (19)

It is noted that this 3D spectrum can be transformed into the 1D one by the
transformation of the volume element $d^{3}{\bf{k}}\sim k^{2}dk$, i.e.,
$E(k)\sim k^{2}E_{\rm 3D}(k)\sim k^{-5/3}$.

For incompressible magnetic turbulence, two classical works (Iroshnikov, 1963;
Kraichnan, 1965, henceforth IK) introduced nonlinear energy models to study
the power-law scaling from turbulence cascade. Different from the K41 energy
cascade by the interaction of eddies, they considered that the energy was
transferred via the Alfvén waves along the local magnetic field. The relation
between the velocity and scale is expressed as $u_{l}\propto(\epsilon V_{\rm
A}l)^{1/4}$ after considering the influence of the magnetic field. The power
spectrum in the IK theory follows

$E(k)\sim(\epsilon V_{\rm A})^{1/2}k^{-3/2}$ (20)

in the inertial range. Although this model considered the role of the magnetic
field, it ignored a critical issue that turbulence should be anisotropic in
the presence of the magnetic field. This limitation was pointed out by later
studies (Montgomery, 1982; Shebalin et al., 1983).

With the anisotropy due to turbulent magnetic fields, GS95 studied the
nonlinear energy cascade of incompressible and strong MHD turbulence. They
predicted that the motions of eddies perpendicular to the magnetic field have
similar properties as Kolmogorov turbulence, i.e., $v_{\perp}\propto(\epsilon
l_{\perp})^{1/3}$. The resulting Kolmogorov-like energy spectra is given by

$E(k_{\perp})\varpropto\epsilon^{2/3}k_{\perp}^{-5/3},$ (21)

where $k_{\perp}$ represents the wave-vector component perpendicular to the
magnetic field. Notice that the magnetic field in the original paper refers to
the global frame of reference which has been corrected as the local frame of
reference by the later intensive studies (LV99; Cho & Vishniac, 2000; Maron &
Goldreich, 2001; Cho & Lazarian, 2002, henceforth CL02; 2003, henceforth
CL03). In particular, LV99 theory pointed out that the reconnection of
turbulent magnetic fields takes place within the eddy turnover time and the
motions of eddies perpendicular to the magnetic field are not controlled by
the magnetic field tension. The magnetic field here refers to the local
magnetic field around the eddies.

Besides, for the weak MHD turbulence, the energy transfer rate is expressed
$\epsilon\propto(V_{\rm L}^{4}l_{\parallel}/V_{\rm A}l_{\perp}^{2})$,
resulting in $v_{l}\propto V_{\rm L}(l_{\perp}/L_{\rm in})^{1/2}$. Thus, the
power spectrum of weak turbulence follows (LV99; Galtier et al., 2000)

$E(k_{\perp})\sim k_{\perp}^{-2}.$ (22)

####  3.3.2 Anisotropy

Except for the prediction of the exponent of $-5/3$ in the perpendicular
direction mentioned above, another important contribution of GS95 theory is
the prediction of the scale-dependent anisotropy. They assumed that the
relative motion perpendicular and parallel to the magnetic field is described
by the critical balance condition, namely $l_{\perp}^{-1}v_{\perp}\sim
l_{\parallel}^{-1}V_{\rm A}$, where $v_{\perp}$ is fluctuation velocity of
turbulence at the scale $l_{\perp}$, $l_{\parallel}$ and $l_{\perp}$ represent
the scales parallel and perpendicular to the magnetic field, respectively.
Based on the critical balance condition and the relation of
$v_{\perp}\propto(\epsilon l_{\perp})^{1/3}$, two scales are related by

$l_{\parallel}\sim V_{\rm A}\epsilon^{{-1}/{3}}l_{\perp}^{{2}/{3}},$ (23)

which is the so-called scale-dependent anisotropy theory. This implies that
the smaller the scale, the larger the anisotropy. Note that this anisotropic
relationship holds in the local frame of reference rather than in the global
frame of reference. As for the latter, the anisotropy is scale-independent and
is determined by the anisotropy of the largest eddies (CL03).

It should be stressed that GS95 focused on trans-Alfvénic turbulence, i.e.,
$M_{\rm A}\approx 1$, with the turbulence kinetic energy equal to the magnetic
energy. Later, the theory was generalized to sub-Alfvénic $M_{\rm A}<1$ and
super-Alfvénic $M_{\rm A}>1$ cases, respectively (LV99; Lazarian, 2006). When
the turbulence is driven with the injection velocity $V_{\rm L}$ less than
$V_{\rm A}$ at the injection scale $L_{\rm in}$, it initially undergoes a weak
turbulence cascade process with $v_{l}\propto l_{\perp}^{1/2}$, where the
parallel scale $l_{\parallel}$ does not change (LV99; Galtier et al., 2000).
With the intensification of the interaction of wave packets, the turbulence
changes to strong turbulence at the transition scale $L_{\rm tr}=L_{\rm
in}M_{\rm A}^{2}$ and continues up to the dissipative scale $L_{\rm dis}$. In
the inertial range, the scales perpendicular and parallel to the magnetic
field have the following relationship

$l_{\parallel}\approx L_{\rm in}^{1/3}l_{\perp}^{2/3}M_{\rm A}^{-4/3}.$ (24)

Compared with the trans-Alfvénic turbulence, this relationship has an
additional factor related to the magnetization $M_{\rm A}^{-4/3}$. In
addition, the turbulent velocity is written as

$v_{l}\approx V_{\rm L}(\frac{l_{\perp}}{L_{\rm in}})^{1/3}M_{\rm A}^{1/3},$
(25)

which still follows the Kolmogorov-type cascade, i.e., $v_{l}\propto
l_{\perp}^{1/3}$.

In the case of the super-Alfvénic turbulence, there is almost no constraint of
the magnetic field at the injection scale $L_{\rm in}$, so the turbulence
follows the law of hydrodynamic turbulence. With the development of
turbulence, it experiences a transition from hydrodynamic turbulence to MHD
one at the transition scale $L_{\rm A}=L_{\rm in}M_{\rm A}^{-3}$. In the
inertial range from $L_{\rm A}$ to $L_{\rm dis}$, the turbulence exhibits
again an anisotropy consistent with GS95 theory, with the scale relation of

$l_{\parallel}\approx L_{\rm in}^{1/3}l_{\perp}^{2/3}M_{\rm A}^{-1},$ (26)

and the velocity relation of

$v_{l}\approx V_{\rm L}(\frac{l_{\perp}}{L_{\rm in}})^{1/3}.$ (27)

In the strongly turbulent region, the scale-dependent anisotropy becomes more
significant as the scale decreases.

###  3.4 Compressibility of MHD turbulence

Compressibility is an important addition to the MHD turbulence theory. In
fact, in a real astrophysical environment such as a molecular cloud, the MHD
turbulence is essentially compressible and is composed of three plasma modes,
i.e., Alfvén, slow and fast modes. The idea of the decomposition of MHD
turbulence into individual modes was originally formulated in Dobrowolny et
al. (1980), where they mainly dealt with small amplitude perturbations in
incompressible MHD turbulence.

The actual decomposition in numerical simulation has been implemented using
Fourier transformation (CL02; CL03). Covering the gas-pressure-dominated and
magnetic-pressure-dominated turbulence regimes, they revealed the spectrum and
anisotropy in the compressible MHD turbulence. Typically, the Alfvén mode in
the high- and low-$\beta$ regimes follows the scaling slope and scale-
dependent anisotropy of GS95

$E^{A}(k)\propto k^{-5/3},\ \ \ k_{\parallel}\propto k_{\perp}^{2/3}.$ (28)

The slow mode, as a passive mode, is dominated by the Alfvén mode, and has
properties similar to those of the Alfvén mode:

$E^{s}(k)\propto k^{-5/3},\ \ \ k_{\parallel}\propto k_{\perp}^{2/3}.$ (29)

The fast mode, different from the Alfvén and slow modes, possesses isotropic
properties similar to acoustic turbulence, with the following scaling
relations

$E^{f}(k)\propto k^{-3/2},\ \ \ k_{\parallel}\propto k_{\perp}.$ (30)

Later, Kowal & Lazarian (2010) has advanced the studies on mode decomposition
using a wavelet method to treat the large amplitude perturbations. This method
enables us to trace the direction of the local magnetic field and has
significant advantages over the Fourier transform done in the frame of
reference of the mean magnetic field.

###  3.5 Controversy

To explain the numerical simulations in Maron & Goldreich (2001), several
studies that have attracted community attention proposed a particular process
termed as dynamical alignment to modify the GS95 spectrum from $k^{-5/3}$ to
$k^{-3/2}$ (Boldyrev, 2005; Boldyrev, 2006; Mason et al., 2006). At present,
there are numerical results that are compatible with a dynamic alignment model
(e.g., Chandran et al., 2015). Moreover, solar-wind observations seem to find
some scale-dependent alignment (e.g., Wicks et al., 2013) or the emergence of
three-dimensional anisotropy (e.g., Chen et al., 2011) which would also point
toward a scale-dependent alignment of some sort (see also Wang et al., 2020b
for similar findings from the end of the MHD cascade to the kinetic scales).

At scales close to the dissipation scale, another modification of GS95 theory
was proposed by Mallet & Schekochihin (2017) for their intermittency model and
Mallet et al. (2017) for their reconnection-mediated turbulence model (see
also Loureiro & Boldyrev, 2017), the testing/confirming of which is difficult
due to current limitations in computational abilities.

However, the anisotropy predicted in Bolryrev’s work is inconsistent with some
recent numerical simulations (Beresnyak & Lazarian, 2019; Beresnyak, 2019).
Many studies claimed that the deviation from the spectral index -5/3 is
transient, namely localized in the vicinity of the injection scale, and does
not extend to the whole inertial range (Beresnyak & Lazarian, 2010; Beresnyak,
2013; Beresnyak, 2014). To the best of my knowledge, this issue is still open.
Anyway, the current attempts still do not change the paradigm of the modern
MHD turbulence theory.

Synchrotron fluctuation techniques reviewed in this paper are developed
closely based on the modern understanding of MHD turbulence theory (e.g.,
GS95; LV99; CL02) without more concern for its controversial appearance.
Fortunately, modifications attempted to fundamental theory are not expected to
seriously affect the measurement of the techniques. This is because the
development of measurement techniques is independent of a specific scaling
slope of turbulence and is constrained in the strongly turbulent range, while
modifications to turbulence theory focus on the driving or dissipation scale.

Figure 1: Schematic diagram of the principle of reconstructing the 3D magnetic
field structure using SPDGs. The spatial gradient of polarization intensity
$\nabla(\triangle P)$ at two neighboring wavelengths $\lambda$ reflects the
mean magnetic field structure between the slice $[L_{2},L_{1}]$ (Lazarian &
Yuen, 2018a).

##  4 Synchrotron Radiative Processes in MHD turbulence

Magnetic field and relativistic electrons, as two key factors, play a decisive
role in emitting synchrotron radiation (Ginzburg & Syrovatskii, 1965;
Ginzburg, 1981). Assume that relativistic electrons display a power-law
spectrum

$N(E)dE=KE^{-p}dE,$ (31)

where $N$ represents the number density of relativistic electrons with the
energy between $E$ and $E+dE$, $K$ the normalization constant, and $p$
spectral index of the electrons. In an environment containing regular and
random magnetic fields, synchrotron emissivity can be expressed as the
combination of homogeneous ($j_{\rm h}$) and random ($j_{\rm r}$) parts
(Crusius & Schlickeiser, 1986)

$j(\nu,\theta)=\chi j_{\rm h}(\nu,\theta)+\frac{1}{2}(1-\chi)j_{\rm r}(\nu),$
(32)

where $\chi$ and $1-\chi$ are the percentage of homogeneous and random parts,
respectively. $\theta$ is the angle between field lines and the direction of
emission, and the factor $\frac{1}{2}$ denotes half of the power $j_{\rm
r}(\nu)$ measured in the each of two polarization directions for turbulent
magnetic fields.

###  4.1 Radiation in homogeneous Magnetic Field

In a completely homogeneous field, the synchrotron emissivity of relativistic
electrons is expressed by (Ginzburg & Syrovatskii, 1965)

$j_{\rm h}(\nu,\theta)=\int dEN(E)\mathcal{F}(E,\nu,\theta).$ (33)

The integrand $\mathcal{F}$, representing the power per frequency unit emitted
by a single electron of given energy and pitch angle, is given by (see Rybicki
& Lightman, 1979 for more details on the derivation)

$\mathcal{F}(E,\nu,\theta)=\frac{\sqrt{3}{e^{3}}B\sin{\theta}}{m_{\rm
e}c^{2}}[\frac{\nu}{\nu_{\rm c}}\int_{\nu/\nu_{\rm c}}^{\infty}K_{5/3}(t)dt],$
(34)

where $K_{5/3}(t)$ is the modified Bessel function of order $5/3$,
$f(\frac{\nu}{\nu_{c}})=\frac{\nu}{\nu_{c}}\int_{\nu/\nu_{c}}^{\infty}K_{5/3}(t)dt$
is often referred to as a kernel function, and $\nu_{c}=3E^{2}eB{\rm
sin}\theta/4\pi m_{\rm e}^{3}c^{5}$ is a characteristic frequency of
synchrotron emission.

The power per frequency of a single electron, $\mathcal{F}$, is divided into
the perpendicular and parallel components with respect to the magnetic field
$\bf{B}$ (Rybicki & Lightman, 1979)

$\mathcal{F}_{\perp}(E,\nu,\theta)=\frac{\sqrt{3}e^{3}B\sin{\theta}}{2m_{\rm
e}c^{2}}[f(\frac{\nu}{\nu_{c}})+g(\frac{\nu}{\nu_{c}})],$ (35)
$\mathcal{F}_{\parallel}(E,\nu,\theta)=\frac{\sqrt{3}e^{3}B\sin{\theta}}{2m_{\rm
e}c^{2}}[f(\frac{\nu}{\nu_{c}})-g(\frac{\nu}{\nu_{c}})],$ (36)

where $g(\nu/\nu_{c})=\frac{\nu}{\nu_{c}}K_{\frac{2}{3}}(\frac{\nu}{\nu_{c}})$
and $K_{\frac{2}{3}}$ is the modified Bessel function of order $2/3$. As a
result, the degree of linear polarization can be calculated by

$\varPi(\nu)=\frac{\mathcal{F}_{\perp}-\mathcal{F}_{\parallel}}{\mathcal{F}_{\perp}+\mathcal{F}_{\parallel}}=\frac{g(\nu/\nu_{c})}{f(\nu/\nu_{c})}$
(37)

for a single electron radiation. Replacing $\mathcal{F}$ in Eq. (33) with
$\mathcal{F}_{\parallel}$ or $\mathcal{F}_{\perp}$, we can obtain the
perpendicular ($j_{\perp}$) and parallel ($j_{\|}$) components of synchrotron
emissivity for electron polulations. Hence, we have the degree of linear
polarization for the electron populations

$\varPi(\nu)=\frac{j_{\perp}-j_{\parallel}}{j_{\perp}+j_{\parallel}}.$ (38)

After integration, the total emissivity (Eq.33) can be simplified as

$j_{\rm h}(\nu,\theta)\propto(B_{\perp})^{(p+1)/2}\nu^{-(p-1)/2},$ (39)

where the spectral index of photons is $\alpha=\frac{p-1}{2}$ and
$B_{\perp}=B\sin{\theta}$.

###  4.2 Radiation in Random Magnetic Field

In a completely random field, the calculation of synchrotron emissivity will
be more complex compared to a completely homogeneous field, due to the
presence of a stochastic direction. The synchrotron emissivity is formulated
as (Crusius & Schlickeiser, 1986)

$j_{\rm
r}(\nu)=\frac{1}{4\pi}\int_{0}^{2\pi}d\varphi\int_{0}^{\pi}d\theta\sin{\theta}j_{\rm
h}(\nu,\theta),$ (40)

by averaging the total spontaneously emitted power for the homogeneous field
$j_{\rm h}(\nu,\theta)$ over all possible values of the polar ($\theta$) and
azimuthal ($\varphi$) angles. This equation implies that randomness of the
magnetic field is maintained ranging from the Larmor radii of the radiating
electrons to the size of the emitting source.

Defining $x=\nu{\rm sin}\theta/\nu_{\rm c}$, Eq. (40) can be rewritten as

$j_{\rm r}(\nu)\propto B\int_{0}^{\infty}dEN(E)R(x),$ (41)

where $x=\frac{\nu}{(c_{1}BE^{2})}$, $c_{1}=\frac{3e}{4\pi m_{\rm
e}^{3}c^{5}}$ and
$R(x)=\frac{x}{2}\int_{0}^{\pi}d{\theta}\sin{\theta}\int_{x/\sin{\theta}}^{\infty}K_{5/3}(t)dt$
is associated with the Bessel function of order $5/3$. The analytical results
demonstrated that $j_{\rm r}$ remains the same power-law relation for
frequency as that of $j_{\rm h}$ (Crusius & Schlickeiser, 1986). Different
from the situation of homogeneous field, $j_{\rm r}$ is related to the total
magnetic field strength $B$ and not just its perpendicular component
$B_{\perp}$ in $j_{\rm h}$ expression.

###  4.3 Synchrotron self-absorption

When involving in the synchrotron self-absorption, one can adopt Eq. (34) to
obtain the following the self-absorption coefficient

$\kappa(\nu,\theta)=\frac{c^{2}}{8\pi\nu^{2}}\int
E^{2}\frac{d}{dE}(\frac{N(E)}{E^{2}})\mathcal{F}(E,\nu,\theta)dE.$ (42)

We can thus write total radiation intensity by the radiative transfer equation

$I(\nu)=\frac{j}{\kappa}(1-e^{-\tau}),$ (43)

where $\tau=\kappa\ell$ is the absorption optical depth, and $\ell$ is the
size of the emitting region.

For the electrons with a power-law distribution, the absorption coefficient
can be reduced to

$\displaystyle\kappa(\nu,\theta)=\frac{p+10/3}{p+2}\mu(\nu,\theta)=G(p)\frac{e^{3}}{2\pi
m_{\rm e}}$ (44) $\displaystyle(\frac{3e}{2\pi m_{\rm
e}^{3}c^{5}})^{p/2}K(B\sin\theta)^{(p+2)/2}\nu^{-(p+4)/2},$

where $\mu(\nu,\theta)=\frac{p+2}{p+10/3}\kappa(\nu,\theta)$, and the
coefficient
$G(p)=\frac{\sqrt{3}}{4}\Gamma{(\frac{3p+2}{12})}\Gamma{(\frac{3p+22}{12})}$
is only associated with the electron exponent.

To obtain polarization properties, we here introduce the following
coefficients in two mutually perpendicular directions

$\kappa_{\perp}(\nu,\theta)=\kappa(\nu,\theta)+\mu(\nu,\theta),$ (45)
$\kappa_{\parallel}(\nu,\theta)=\kappa(\nu,\theta)-\mu(\nu,\theta).$ (46)

By adapting Eq. (43), we have radiation intensity components

$I_{\perp}(\nu)=\frac{j_{\perp}}{\kappa_{\perp}}(1-e^{-\kappa_{\perp}\ell}),$
(47) $I_{\|}(\nu)=\frac{j_{\|}}{\kappa_{\|}}(1-e^{-\kappa_{\|}\ell}).$ (48)

As for an optically thin medium, i.e., $\kappa_{\perp,\parallel}\ell\ll 1$,
the degree of linear polarization is consistent with Eq. (38). In the case of
optically thick medium, i.e., $\kappa_{\perp,\parallel}\ell\gg 1$,

$\Pi(\nu)={\frac{{j_{\perp}/\kappa_{\perp}-j_{\parallel}/\kappa_{\parallel}}}{j_{\perp}/\kappa_{\perp}+j_{\parallel}/\kappa_{\parallel}}}$
(49)

can be used to calculate the degree of linear polarization.

Figure 2: The influences of telescope angular resolution (left panel) and
Gaussian noise (right panel) on the power spectrum calculated at the
wavelength $\lambda=2.4$ in code units. The symbol $\Delta\sigma$ indicating
standard deviation of Gaussian noise accounts for a fraction of the mean
synchrotron polarization intensity. The symbol $\theta_{\rm FWHM}$ denoting an
effective Gaussian beam is used to convolve 2D maps of Stokes parameters. The
original (ideal) slope without considering the resolution and noise is shown
in the solid line (Zhang et al. 2018).

##  5 Statistics of Synchrotron Fluctuations

On the basis of synchrotron emissivity in Section 4, we can obtain observable
synchrotron intensity ($I$) and polarization intensity ($P$) by integrating
along the line of sight (Burn, 1966; Waelkens et al., 2009)

$I({\bf X})=\int_{0}^{L}(j_{\perp}+j_{\parallel})dz,$ (50) ${\bf P}({\bf
X},\lambda^{2})=\int_{0}^{L}(j_{\perp}-j_{\parallel})e^{2i\psi({\bf X},z)}dz,$
(51)

where $L$ is the spatial scale of radiative propagation, $\bf X$ is a 2D
vector on the plane of the sky and $z$ is the distance variable along the line
of sight. The phase (polarization) angle $\psi({\bf
X},z)=\psi_{0}+\lambda^{2}\Phi$ is associated with the intrinsic polarization
angle $\psi_{0}$ and the Faraday rotation measure $\Phi$. This is expressed as

$\Phi({\bf X},z)\propto\int_{0}^{z}n_{\rm e}({\bf X},z^{\prime})B_{\|}({\bf
X},z^{\prime})dz^{\prime},$ (52)

where $n_{\rm e}$ is the thermal electron density, and $B_{\|}$ is the
magnetic field parallel component along the line of sight. Defining the
intrinsic polarization intensity density as ${\bf P}_{\rm i}={\bf Q}_{\rm
i}+i{\bf U}_{\rm i}$ which is independent of wavelengths (this simplification
does not affect the statistical results, as verified in Zhang et al., 2018 ),
we can write the polarization intensity ${\bf P}={\bf Q}+i{\bf U}$ as

${\bf P}({\bf X},\lambda^{2})=\int_{0}^{L}dz{\bf P}_{\rm i}({\bf
X},z)e^{2i\lambda^{2}\Phi({\bf X},z)}.$ (53)

###  5.1 Correlation Analysis of Synchrotron Fluctuations

According to the commonly used method of correlation analysis, the two-point
correlation of the polarized intensities can be expressed as (LP16)

$\displaystyle\langle{\bf P}({\bf X}_{1},\lambda_{1}^{2}){\bf P}^{*}({\bf
X}_{2},\lambda_{2}^{2})\rangle=\int_{0}^{L}dz_{1}\int_{0}^{L}dz_{2}$ (54)
$\displaystyle\langle{\bf P}_{\rm i}({\bf X}_{1},z_{1}){\bf P}_{\rm
i}^{*}({\bf X}_{2},z_{2})e^{2i(\lambda_{1}^{2}\Phi({\bf
X}_{1},z_{1})-\lambda_{2}^{2}\Phi({\bf X}_{2},z_{\rm 2}))}\rangle,$

where $\bf P^{*}$ denotes the conjugate of the polarization intensity, $\bf
X_{1}$ and $\bf X_{2}$ represent two different spatial positions on the plane
of the sky, $\lambda_{1}$ and $\lambda_{2}$ correspond to two wavelengths. Eq.
(54) can not only characterize the correlation of the spatial separation but
also the dispersion of frequencies. Based on this, LP16 proposed two
techniques, namely the polarization spatial analysis (PSA) and the
polarization frequency analysis (PFA).

In analogy with the velocity channel analysis (VCA) technique (Lazarian et
al., 2002), the PSA correlates the polarization signal at different spatial
points of the position-position frequency (PPF) space that has similarities
with the position-position volume (PPV) space. At a fixed wavelength, the PSA
can be expressed as

$\displaystyle\langle{\bf P}({\bf X_{1}}){\bf P}^{*}({\bf
X_{2}})\rangle=\int_{0}^{L}dz_{1}\int_{0}^{L}dz_{2}e^{2i\bar{\phi}\lambda^{2}(z_{1}-z_{2})}$
(55) $\displaystyle\times\langle{\bf P}_{\rm i}({\bf X}_{1},z_{1}){\bf P}_{\rm
i}^{*}({\bf X}_{2},z_{2})\rangle$ $\displaystyle\times
e^{-2\lambda^{4}\langle(\triangle\Phi({\bf X}_{1},z_{1})-\triangle\Phi({\bf
X}_{2},z_{2}))^{2}}\rangle,$

where $\bar{\phi}$ is mean Faraday rotation measure density and $\Delta\Phi$
is the fluctuation of Faraday rotation measure. This represents the
correlation of polarization intensity as a function of spatial separation
${\bf R}={\bf X}_{2}-{\bf X}_{1}$ at the same wavelength.

The PFA that is analogous to the velocity coordinate spectrum (VCS) correlates
the polarization information along a single line of sight at the same spatial
position. At a fixed spatial position, the PFA for different wavelengths is
expressed by

$\displaystyle\langle{\bf P}({\lambda_{1}^{2}}){\bf
P}^{*}(\lambda_{2}^{2})\rangle=\int_{0}^{L}dz_{1}\int_{0}^{L}dz_{2}e^{2i\bar{\phi}(\lambda_{1}^{2}z_{1}-\lambda_{2}^{2}z_{2})}$
(56) $\displaystyle\langle{\bf P}_{\rm i}(z_{1}){\bf P}_{\rm
i}^{*}(z_{2})e^{2i(\lambda_{1}^{2}\triangle\Phi(z_{1})-\lambda_{2}^{2}\triangle\Phi(z_{2}))}\rangle,$

which is complementary to the PSA, so this enables a more comprehensive
acquisition of the properties of MHD turbulence.

Another important aspect of the development of the synchrotron fluctuation
technique is the construction of the statistics of polarization intensity
derivative with respect to the squared wavelength $\lambda^{2}$. The
correlation statistics of $dP/d\lambda^{2}$ is formulated as

$\displaystyle\langle\frac{{\bf dP}({\bf X}_{1})}{d\lambda^{2}}\frac{{\bf
dP}^{*}({\bf
X}_{2})}{d\lambda^{2}}\rangle=\int_{0}^{L}dz_{1}\int_{0}^{L}dz_{2}e^{2i\bar{\phi}\lambda^{2}(z_{1}-z_{2})}$
(57) $\displaystyle\times\langle{\bf P}_{\rm i}({\bf X}_{1},z_{1}){\bf P}_{\rm
i}^{*}({\bf X}_{2},z_{2})\Delta\Phi({\bf X}_{1},z_{1})\Delta\Phi({\bf
X}_{2},z_{2})$ $\displaystyle\times e^{-2{\lambda}^{2}i(\Delta\Phi({\bf
X}_{1},z_{1})-\Delta\Phi({\bf X}_{2},z_{2}))}\rangle,$

which is more sensitive to the fluctuations of Faraday rotation than the
polarization intensity correlation given in Eq. (55). The introduction of this
relationship allows for more information such as the spectrum of Faraday
rotation to be obtained. In addition to considering the correlation function,
one can also use the structure function of polarization intensity to reveal
the anisotropy of MHD turbulence (see LP16 for detailed formulation).

###  5.2 Analysis of Power Spectrum

With the correlation analysis of synchrotron fluctuations, the power spectrum
of synchrotron polarization is described through the Fourier transform of the
two-point correlation tensor as

${E}_{\bf P}({\bf K})=\frac{1}{(2\pi)^{2}}\int d{\bf R}e^{-i{\bf K\cdot
R}}\langle{\bf P}({\bf X}_{1}){\bf P}^{*}({\bf X}_{2})\rangle,$ (58)

where ${\bf R}={\bf X}_{1}-{\bf X}_{2}\propto 1/{\bf K}$ is the spatial lag
vector on the plane of the sky. Similarly, the power spectrum of the
polarization derivative with respect to $\lambda^{2}$ can be written as

${E}_{\bf dP}({\bf K})=\frac{1}{(2\pi)^{2}}\int d{\bf R}e^{-i{\bf K\cdot
R}}\langle\frac{{\bf dP}({\bf X}_{1})}{d\lambda^{2}}\frac{{\bf dP}^{*}({\bf
X}_{2})}{d\lambda^{2}}\rangle$ (59)

for analyzing the multi-frequency observations. Once the power spectrum
distribution of the observed information is obtained through the above two
equations, we can obtain the distribution characteristics of turbulent energy
in space, and also understand several properties related to turbulence, such
as the scaling slope, energy dissipative and injection scales.

###  5.3 Analysis of Anisotropy

Following LP12, we can define the normalized correction function for an
arbitrary observable $\digamma$ as

$\xi_{\rm\digamma}({\bf R})=\frac{{\langle\digamma(\bf
X)\digamma(\bf{X+R})\rangle}-{\langle\digamma(\bf
X)\rangle}^{2}}{\langle\digamma(\bf X)^{2}\rangle-{\langle\digamma(\bf
X)\rangle}^{2}},$ (60)

where $\bf{X}$ and $\bf R$ denote a spatial position and a separation between
any two points on the plane of the sky, respectively. This is related to the
normalized structure function of the observable $\digamma$

$\tilde{D}_{\digamma}=2(1-\xi_{\digamma}),$ (61)

by which we have a multipole expansion

${\tilde{M}}_{n}({\bf
R})=\frac{1}{2}\int_{0}^{2\pi}e^{-in\varphi}{\tilde{D}}_{\digamma}({\bf
R},\varphi)d\varphi,$ (62)

where $n$ represents the order of multipole, and $\varphi$ is the polar angle.
The order of multipole expansion has following meaning: $n=0$ represents the
monopole reflecting the isotropy of statistics, $n=1$ denotes the dipole which
is zero due to the rotation symmetry of the structure function, $n=2$ is
called the quadrupole moment revealing the anisotropy, and $n=4$ is the
octupole moment which has a negligible contribution for the anisotropy. Hence,
we define the ratio of the quadrupole moment to the monopole one (termed
quadrupole ratio, ref. LP16)

$\Re_{\rm Q}=\frac{\tilde{M}_{\rm 2}({\bf R})}{\tilde{M}_{\rm 0}({\bf
R})}=\frac{\int_{0}^{2\pi}e^{-2i\varphi}\tilde{D}_{\digamma}({\bf
R},\varphi)~{}d{\varphi}}{\int_{0}^{2\pi}\tilde{D}_{\digamma}({\bf
R},\varphi)~{}d{\varphi}}$ (63)

to characterize the degree of anisotropy, the absolute value of which is
called the quadrupole ratio modulus.

Alternatively, the ratio of structure functions in the perpendicular direction
to each other can also depict the level of anisotropy, and is given by

$\Re_{\rm
S}=\frac{\langle|\digamma_{\parallel}(x+R_{\parallel})-\digamma_{\parallel}(x)|^{2}\rangle}{\langle|\digamma_{\perp}(y+R_{\perp})-\digamma_{\perp}(y)|^{2}\rangle},$
(64)

where $R=(R_{\parallel}^{2}+R_{\perp}^{2})^{1/2}$ is the spatial separation on
the plane of the sky. When $\Re_{\rm S}=1$, statistics represents an isotropy
while any deviation from $\Re_{\rm S}=1$ implies an appearance of anisotropy.

###  5.4 Statistics of Spatial Gradients

Combining the local alignment theory of MHD turbulence (GS95; LV99) with the
synchrotron polarization fluctuation predictions (LP16), the gradient
techniques were proposed to measure the properties of MHD turbulence (Lazarian
et al., 2017; Lazarian & Yuen, 2018a). This is analogous to velocity gradient
techniques (González-Casanova & Lazarian, 2017; Yuen & Lazarian, 2017). The
feasibility lies in that magnetic field and velocity fluctuations enter
symmetrically, according to the theory of Alfvénic turbulence. In the
framework of GS95 MHD turbulence theory, it is expected that the alignment of
the magnetic field gradient direction is perpendicular to the local magnetic
field direction with the eddies (for other different anisotropy models, the
sensitivity to this alignment effect may significantly depend on the scales).
Since the synchrotron fluctuations are elongated in the direction of the mean
magnetic fields, the magnetic fields can be traced by the direction of the
spatial gradient of the observed signal.

When involved in Faraday depolarization of synchrotron radiation fluctuations,
synchrotron polarization gradients (SPGs) and synchrotron polarization
derivative gradients (SPDGs) can reveal more detailed information on the
turbulent magnetic field. In particular, the latter can sample the spatial
regions close to the observer with the sampling depth controlled by the
radiation wavelength (see Figure 1), which provides a new approach to map the
3D distribution of Galactic magnetic fields.

Specifically, the decorrelation of the Faraday rotation measure is introduced
as (LP16; Zhang et al., 2020)

$\lambda^{2}\varPhi=0.81\lambda^{2}\int_{0}^{L_{\rm eff}}dzn_{\rm
e}B_{\|}=2\pi,$ (65)

where $L_{\rm eff}$ is an effective spatial depth of the Faraday rotation
sampling. We can thus write the ratio of the depth sampled by the Faraday
rotation to the emitting region size as

$\frac{L_{\rm eff}}{L}\approx\frac{2\pi}{\lambda^{2}L}\frac{1}{\phi},$ (66)

with $\phi={\rm max}(\sqrt{2}\sigma_{\phi},\bar{\phi})$. Here, $\sigma_{\phi}$
denotes the root mean square and $\bar{\phi}$ Faraday rotation density.
According to Eq. (66), the strong and weak depolarization can be characterized
by $L_{\rm eff}/L<1$ and $L_{\rm eff}/L>1$, respectively. In the case of the
strong depolarization, the entire space is divided to two different parts
shown in Figure 1, from which we can know that only the part of $z<L_{\rm
eff}$ experiences the depolarization while the part of $z>L_{\rm eff}$ cannot
contribute to the polarization correlation. When observing frequency interval
$\triangle\nu=c/\lambda^{2}\triangle\lambda$ is small enough, the difference
of ${\bf P}(\lambda_{1})$ and ${\bf P}(\lambda_{2})$ in the range of
$[L_{2},L_{1}]$ is written as

$\triangle{\bf P}\approx\int_{L_{2}}^{L_{1}}dz{\bf P}_{\rm i}({\bf
X},z)e^{2i\lambda^{2}\Phi({\bf X},z)},$ (67)

from which the expression of the SPDGs is given by

$\nabla\frac{d|{\bf P}|}{d\lambda^{2}}\sim\lambda^{-2}\nabla|{\bf
P}(\lambda_{1})-{\bf P}(\lambda_{2})|.$ (68)

From another perspective, namely, based on the spatial gradient analysis of
Stokes parameters $Q$ and $U$ on the complex plane, Gaensler et al. (2011)
proposed the synchrotron polarization gradients to constrain sonic Mach number
in the warm or ionized ISM. After a simple mathematical derivation, the
spatial gradient of polarization intensity can be expressed as

$|\nabla{P}|=\sqrt{(\frac{\partial Q}{\partial x})^{2}+(\frac{\partial
U}{\partial x})^{2}+(\frac{\partial Q}{\partial y})^{2}+(\frac{\partial
U}{\partial y})^{2}}.$ (69)

Subsequently, considering rotational and translational invariance of
coordinates, various diagnostic quantities were developed in Herron et al.
(2018).

Figure 3: The transition of polarization variance from the stochastic Faraday
rotation fluctuation dominant regime for $\eta=\infty$, to the uniform
fluctuation dominant one for $\eta=0.080$ in 3D synthetic simulation with
$4096^{3}$ resolution. The underlying MHD turbulence corresponds to Komolgorov
spectrum of the index $11/3$ (Zhang et al., 2016).

##  6 Application of Statistical Techniques

###  6.1 Measurement of Scaling Slope

LP12 predicted that the power spectrum, correlation, and structure functions
of synchrotron radiation fluctuations, termed two-point statistics, can
recover the scaling slope of turbulent magnetic fields. Numerical simulations
show that the correlation function does not work in revealing scaling slopes,
but it can be used to explore the correlation scale and the effect of the
electron spectral index on turbulence (Herron et al., 2016; Lee et al., 2016;
Zhang et al., 2016). The main reason we think is that LP16 performed
theoretical studies under the condition of the power-law correlation model,
from which they predicted the possibility of obtaining the scaling slope for
the corresponding model. The correlation function itself, however, has a
divergent behavior for numerical simulation or observational data, because
Stokes parameters $Q$ and $U$ contain the information of negative values
arising from the change in the direction of the turbulent magnetic field.

Numerical results demonstrated that the scaling slope of the underlying MHD
turbulence can be recovered approximately by the two-point structure function
of synchrotron radiation fluctuations (Lee et al., 2016; Zhang et al., 2022).
From the perspective of numerical simulations, the direct calculation of the
spatial structure function is more time-consuming, compared with the
correlation function or power spectrum. Alternatively, one can first get the
statistics of the correlation function, provided that it maintains
convergence, and then calculate the structure function using the relation of
${\rm SF}(\bf{R})=2[{\rm CF}(0)-{\rm CF}(\bf{R})]$. In general, the two-point
structure functions are unable to correctly reproduce the scaling of a
spectrum with a slope $\sim-3$ or steeper (sometimes problems may arise
already at slopes slightly shallower than $-3$, if the underlying spectrum is
not a perfect power law). For instance, Lee et al. (2016) measured the scaling
slope of 5/3 with the change in a numerical resolution. It was pointed out
that the scaling slope of the two-point structure function, calculated
directly from 2D synthetic data, is still less than 5/3 expected for the
Kolmogorov spectrum, even with $8192^{2}$ resolution. Thus, using a two-point
structure function to extract the scaling slope of turbulence requires more
precise statistics with higher numerical resolution. Alternatively, the multi-
point structure functions are recommended to achieve better convergence of the
results (see Cho & Lazarian, 2009; Cho, 2019).

Obtaining the scaling slope through the power spectrum of synchrotron
fluctuations should be one of the most efficient ways, as the calculation of
the power spectrum invokes the fast Fourier transform in the wave-number space
to effectively accelerate the process. Adopting both synthetic and MHD
turbulence simulation data, Lee et al. (2016) obtained the power spectrum and
its derivative with respect to the wavelength of synchrotron polarization
arising from synchrotron polarization radiation together with Faraday rotation
fluctuations. This study succeeded in obtaining the scaling slope and
demonstrated that simulations are in agreement with the theoretical prediction
in LP16. In addition, using simulated interferometric observations on how to
recover the scaling slope was also attempted. Notice that the study mentioned
above is based on the spatially coincident synchrotron emission and Faraday
rotation regions. Furthermore, considering spatially separated and compounded
synchrotron radiation and Faraday rotation fluctuations, Zhang et al. (2018)
studied how to extract the scaling slope of synchrotron polarization
intensities. In the short wavelength range, the power spectra reveal
fluctuation statistics of the perpendicular component of turbulent magnetic
fields, and the spectra reflect the fluctuation of the Faraday rotation in the
long-wavelength range. As shown in Figure 2, the effects of telescope angular
resolution and noise structure do not hinder the use of power spectrum methods
to obtain scaling slopes of MHD turbulence in the large-scale inertial range.

Figure 4: Quadrupole ratio of synchrotron polarization intensities for Alfvén,
slow and fast modes as a function of radial separation of maps in sub-Alfvénic
regime (Wang et al., 2020a).

Motivated by one-point statistics in LP16 (termed PFA), which is characterized
by the variance of polarized emission as a function of the squared wavelength
along a single line of sight, Zhang et al. (2016) studied how to extract the
scaling slope of underlying MHD turbulence. To depict the level of turbulence,
they defined a ratio $\eta$ of the standard deviation of the line-of-sight
turbulent magnetic field to the line-of-sight mean-magnetic field. As shown in
Figure 3, a large ratio ($\eta\gg 1$) characterizing a turbulent field
dominated regime reflects the polarization variance
$\left<P^{2}\right>\propto\lambda^{-2}$, and a small ratio ($\eta\lesssim
0.2$) characterizing a mean-field dominated regime reveals the polarization
variance $\left<P^{2}\right>\propto\lambda^{-2-2m}=\lambda^{-10/3}$ with
$m=2/3$ for the Kolmogorov scaling. As a result, the turbulent spectral index
was successfully recovered by the polarization variance in the case of the
mean-field dominated regime, i.e., a small $\eta$. At the same time, it was
pointed out that the change in cosmic ray electron spectral indices cannot
affect the scaling slope measurement of magnetic turbulence. This provides a
new method for recovering the scaling index of MHD turbulence and agrees with
the theoretical prediction of LP16. Interestingly, this technique has been
successfully applied to explain the depolarization in the optical wavebands
from the active galactic nucleus (Guo et al., 2017).

###  6.2 Measurement of Anisotropy

As described in Section 3, the prediction of the anisotropy (eddy’s elongation
along the local magnetic field direction) in turbulent cascade is an important
contribution of GS95. It is challenging to test the local anisotropy of MHD
turbulence from an observational point of view because the observational
signal is inevitably involved in the integration along the line of sight in
the observer’s frame of reference. Note that integrated synchrotron
fluctuations along the line of sight were predicted by LP12 to be anisotropic,
with the eddy’s major axis aligned with the direction of the mean magnetic
field instead of the local field within the MHD turbulence volume.

The spatial second-order structure function of radiation fluctuations is a
very direct way to reveal the existence of anisotropy in MHD turbulence,
although the anisotropy here does not directly correspond to the local
anisotropy favored by the modern MHD turbulence theory. If visualizing a
contour map for 2D structure function of observations, one can easily see the
anisotropic distribution, from which the major axis direction of the elongated
structure will qualitatively characterize the mean magnetic field orientation
(see Figure 1 in Wang et al. (2020a) for an example). Considering the ratio of
two measurement directions perpendicular to each other from observed
polarization intensities, called an anisotropic coefficient, we can
quantitatively determine the degree of the anisotropy of the MHD turbulence in
the global frame of the reference; refer to Lee et al. (2019) and Wang et al.
(2020a) for more details.

A new more-precise method to measure the anisotropy is termed as the
quadrupole ratio described in Section 5. This method was proposed in LP12 that
derives many analytical formulae including the predictions from the Alfvén,
slow and fast modes in the high or low $\beta$ plasma regime. A numerical test
of theoretical predictions on the quadrupole ratio was attempted by Herron et
al. (2016) based on MHD turbulence simulation. We have to point out that their
testing used a simplified version rather than a general formula. In
particular, they adopted the compressible MHD turbulence data to study the
isotropic version of formulae, which fails to provide proof of the usability
of the analytical formula. Considering the angle change in between the mean
magnetic field direction and the line of sight, Wang et al. (2022) recently
tested the theoretical prediction of LP12 in detail and confirmed the
correctness of the analytical formulae, providing convenience to study the
anisotropy of MHD turbulence by using the analytical expressions.

How the anisotropic structure changes with observation frequency were studied
in Lee et al. (2019) by the quadrupole ratio. Their studies showed that in the
high-frequency range, the observed polarization exhibits the averaged
structures of both foreground and background regions, while in the low-
frequency range, the large-scale structures are wiped out by a strong Faraday
depolarization and the small-scale anisotropy reflects magnetic field
structure from the foreground region.

By decomposing the compressible MHD turbulence into three plasma modes, i.e.,
Alfvén, slow and fast modes, Wang et al. (2020a) demonstrated that the
quadrupole ratio statistics of the polarization intensity from the Alfvén and
slow modes present a significant anisotropy, while the fast mode statistics
show isotropic structures (see Figure 4). Those findings are in agreement with
the earlier direct numerical simulations provided in CL02. As a result, the
quadrupole ratio method can be applied to the measurement of the anisotropy of
MHD turbulence in various astrophysical environments.

###  6.3 Measurement of Magnetic Field Orientations

The traditional method of measuring the magnetic field direction is called the
synchrotron polarization vector, which is based on the fact that the observed
polarization vectors are perpendicular to the magnetic field directions within
the emitting source. Note that the limitation of this method cannot be applied
to observations with Faraday rotation.

Using synchrotron intensity gradients (SIGs) together with both MHD turbulence
simulation and PLANCK archive data, Lazarian et al. (2017) demonstrated the
practical applicability of the gradient technique described in Section 5,
comparing the measured directions of magnetic fields arising from gradient
technique with that of the traditional polarization vector method.
Furthermore, SIGs were advanced in Lazarian & Yuen (2018a) to SPGs and SPDGs.
Subsequently, to explore the magnetic field measurement capabilities of
synchrotron gradient techniques, SPGs and SPDGs were generalized and applied
to a variety of possible astrophysical environments in a series of studies
including different turbulence regimes, the gradient of various diagnostic
quantities, realistic observations of Galactic diffuse media, as well as
spatially separated synchrotron radiation and Faraday depolarization regions.

####  6.3.1 Gradient Measurement in Various Turbulence Regimes

Based on synthetic observations from MHD turbulence simulations, Zhang et al.
(2019a) extended both SPGs and SPDGs to super-Alfvénic regimes and studied how
to trace the directions of turbulent magnetic fields. Focusing on
multifrequency measurements in the presence of strong Faraday rotation, they
provided the procedures of how to recover the projected mean magnetic fields
on the plane of the sky and the local magnetic fields within a tomographic
slice. The results demonstrate that in the low-frequency strong Faraday
rotation regime, the SPGs have a significant advantage over the traditional
polarization vector method in tracing projected mean magnetic fields and the
SPDGs are applicable to tracing the local magnetic field directions in the
local frame of reference of MHD turbulence. At the same time, their parameter
research provides a testing ground for the application of the new techniques
to a large number of data cubes, such as those from LOFAR and SKA.

####  6.3.2 Gradient Measurement of Various Diagnostic Quantities

The various diagnostic quantities, including the polarization directional
curvature, polarization wavelength derivative, and polarization wavelength
curvature, are derived in Herron et al. (2018) considering the translational
and rotational invariance of the observed quantities. It is noticed that the
authors were not aware of the relation between the magnetic field direction
and gradients of synchrotron radiation gradients, and did not explore whether
the new diagnostics they introduced can trace magnetic field in observations.

The spatial gradients of various diagnostics provided in Herron et al., (2018)
have been proposed to trace the directions of the mean magnetic field in Zhang
et al. (2019b), who found that the gradients of various diagnostics can
effectively determine the direction of the projected magnetic fields on the
plane of the sky. One of these techniques, namely the maximum of the radial
component of the polarization directional derivative, is the best robust
method for tracing the mean magnetic field directions in the Galactic ISM.

The synergy of various diagnostic gradients is complementary in tracing the
actual direction of interstellar magnetic fields, especially in the low-
frequency Faraday rotation regime where the traditional polarization vector
method cannot work. Applying the synergies of synchrotron diagnostic gradients
to the archive data from the Canadian Galactic Plane Survey (Taylor et al.,
2003), Zhang et al. (2019b) showed that various diagnostic techniques make
consistent predictions for the Galactic mean magnetic field directions.

####  6.3.3 Gradient Measurement of Spatially Separated Regions

When confronted with realistic observations, the synchrotron polarization
radiation obtained by the observers could originate from complex turbulence
regions rather than a single spatially coincident radiation and Faraday
rotation region. Can the synchrotron polarization gradient still measure the
direction of the magnetic field for more complex astrophysical settings?

The measurement capabilities of the SPGs were studied in Wang et al. (2021)
through constructing spatially separated models, focusing on exploring how the
Faraday rotation density in the foreground region affects the measurement of
the projected magnetic field. Their numerical results showed that the use of
the SPG technique for a complex astrophysical environment can successfully
trace the projected mean magnetic field direction within the emitting-source
region independent of radiation frequency.

####  6.3.4 Measurement of the Local Magnetic Field Direction

Thanks to the inevitable accumulation of the observed signal along the line of
sight, i.e., less locality, it is extremely challenging to measure the local
magnetic field in MHD turbulence. However, the measurement of the local
magnetic field will be direct evidence to verify the modern MHD turbulence
anisotropy theory since this theory is based on the local magnetic field
reference system (see Section 3.3.1). At the same time, the acquisition of the
local magnetic field information is a prerequisite for reconstructing the
Galactic 3D magnetic field as well.

Using gradient statistics of synchrotron polarization derivative with respect
to the squared wavelength $dP/d\lambda^{2}$, the measurement of the local
magnetic field direction has been implemented in Zhang et al. (2020) based on
data cubes obtained with MHD turbulence simulations. They demonstrated that in
the low-frequency strong Faraday rotation regime, the statistic analysis of
the spatial gradient of $dP/d\lambda^{2}$ can indeed reveal the local magnetic
field direction and the local anisotropy of the underlying MHD turbulence
which increases with increasing the radiation frequency.

As illustrated in Figure 1, this technique can successfully extract the local
information of MHD turbulence in the tomography space along the line of sight
by using multi-frequency observations at adjacent frequencies. Unlike the
traditional Faraday rotation synthesis (e.g., Burn, 1966), the polarization
gradient techniques are not affected by the Faraday rotation and can improve
the 3D model of the Galactic magnetic field effectively.

###  6.4 Measurement of Magnetization Strength

Several statistical methods, such as Genus, Tsallis, skewness, and kurtosis
statistics, are sensitive to the magnetization level $M_{\rm A}$ of MHD
turbulence (e.g., Burkhart et al., 2012; Herron et al., 2016). In addition,
the correlation function (e.g., Lazarian et al., 2002; Burkhart et al., 2014;
Esquivel et al., 2015) and structure function (Hu et al., 2021; Xu & Hu, 2021)
analysis are also suggested as alternative techniques of measuring $M_{\rm
A}$. At present, these studies are based on velocity information analysis from
MHD turbulence simulations. Similarly, the adoption of synchrotron
fluctuations should be on the agenda.

In analogy with velocity gradient techniques, Lazarian et al. (2018b) first
discussed that synchrotron gradient techniques should be able to measure the
magnetization level. The feasibility of the two methods was explored in Carmo
et al. (2020) to obtain the level of magnetization from synchrotron
polarization fluctuation, namely the top-base and the circular standard
deviation. They claimed that the signal-to-noise ratio is more severe for the
top-base method, but still reliable with the standard deviation method. In
short, the finding on the power-law relation between $M_{\rm A}$ and
synchrotron gradient statistics is important for determining $M_{\rm A}$ from
the current or future available data cubes.

##  7 Discussion

The progress in understanding the role of magnetic fields for processes in a
specific astrophysical environment depends on our ability to obtain their
properties from MHD turbulence. Unfortunately, it is notoriously difficult for
studying the properties of magnetic fields. In practice, almost every
technique presents its limits. For example, the difficulty with dust
polarization measurement lies in the inability to measure the magnetic field
strength along the line of sight and insufficient dust grain alignment
efficiencies at low Galactic latitudes, preventing observers from deriving the
information of magnetic fields in the ISM. For another example, the difficulty
with Zeeman splitting measurement is that it can only measure the line of
sight component strength of the magnetic field, access the high magnitude end
of the interstellar magnetic fields in a mildly turbulent environment, and
provide limited morphology information (Crutcher, 2009). Last but not least, a
serious limitation of the well-known Faraday tomography is that the line-of-
sight component of the magnetic field in turbulence is expected to change its
sign, resulting in the Faraday depth to become ambiguous (see Ho et al. 2019
for a comparison of measure performance between Faraday tomography and SPGs;
refer also to Akahori et al. 2018 for the recent review on other more methods
and applications in specific astrophysics).

The further development of related new techniques can potentially extract
other properties of MHD turbulence. In the framework of synchrotron
fluctuations, many topics deserve to be further studied. As an example,
visualization of the structure or correlation functions of synchrotron
diagnostics can present an anisotropic feature, by which one could trace the
mean magnetic field orientations and measure magnetization strength. It should
be encouraged to explore the feasibility of this technique for measuring
magnetic fields, the fundamental spirit of which is similar to the correlation
function anisotropy technique on velocity information (Lazarian et al., 2002;
Esquivel & Lazarian, 2005; Hu et al., 2021; Xu & Hu, 2021). We would like to
emphasize that the synchrotron dispersion techniques discussed currently are
also complementary to other techniques of magnetic field studies, such as
velocity channel analysis, velocity coordinate analysis, and velocity gradient
techniques. The synergistic use of multiple techniques is necessary because
one will be able to improve the reliability of magnetic field measurements, by
fully utilizing the advantages of each technique.

None of the relevant papers covered in this review consider synchrotron self-
absorption effects (see Section 4 for self-absorption processes). However,
when the brightness temperature of synchrotron radiation approaches the
‘thermal’ temperature of emitting relativistic electrons, the synchrotron
self-absorption effect is expected to be important in the case of low
frequencies. In the framework of new techniques developed, we expect that the
effects of self-absorption can provide yet another method of testing the 3D
structure of the magnetic field. Indeed, only the regions closer to the
observer and less affected by self-absorption are expected to be probed under
strong self-absorption. This effect is similar to the effect of the dust self-
absorption in spectral line statistics that was explored analytically in
Kandel et al. (2017b) and numerically in Yang et al. (2021).

In the recent studies related to synchrotron radiation techniques, one simply
assumes a homogeneous, isotropic non-thermal relativistic electron
distribution with a power-law energy spectrum. In fact, the specific
properties of the relativistic electrons would depend on the cascade process
of MHD turbulence, that is, it is also necessary to solve the so-called
turbulent heating problem between MHD scales and electron plasma scales. Since
part of the initial cascading turbulent energy will be removed by the ion
heating/energizing at ion scales before it reaches the electron scales, only a
fraction of the energy cascading from the large MHD scales will end up in
heating/energizing the electrons which will determine a specific spectral
energy distribution $N(E)$ to produce the observed synchrotron radiation. What
is this fraction is currently a big open question and answering it would
require understanding what happens at turbulent fluctuations during the
transition from MHD scales to kinetic scales, as well as the role of various
collisionless processes leading to particle heating that could be at work
within different plasma regimes (e.g., Quataert, 1998; Schekochihin et al.,
2009; Chandran et al., 2010; Cranmer, 2014; Arzamasskiy et al., 2019; Cerri et
al., 2019; Zhdankin et al., 2019; Vasquez et al., 2020; Cerri et al., 2021).
As a result, different anisotropy models (i.e., different turbulence theories
at MHD scales; see Section 3.5) would determine different properties of
fluctuations at kinetic scales which is essential to determine the
differential heating of ions and electrons.

Compared to the Faraday rotation synthesis to obtain the 3D magnetic fields in
the galaxies (e.g., Burn, 1966; Beck, 2015), where the Faraday depth is
changing its sign along the line of sight, making the determination of the
magnetic field component $B_{\parallel}$ ambiguous (see Ferriere, 2016), the
advantage of the polarization gradient technique is that it is not affected by
Faraday rotation and will effectively improve the measurement of our current
Galactic 3D magnetic field model. The correct understanding of this model, as
a fundamental ingredient, is indeed very important, e.g., for modeling the
anisotropic cosmic-ray propagation in the Galaxy (e.g., Cerri et al., 2017;
Reichherzer et al., 2022) which can explain some anomalies inferred from
gamma-ray emission (Acero et al., 2016; Yang et al., 2016), and is critical to
obtain realistic Galactic evolution (e.g., Beck, 2015).

Furthermore, an accurate understanding of Galactic magnetic fields is
indispensable for searching the B modes of the Cosmic Microwave Background
(CMB, Zucca et al., 2017), which is essential for understanding the evolution
of the early universe. The CMB B mode studies require more accurate removal of
the Galactic foreground contamination, which is achievable provided that the
detailed information of Galactic magnetic fields can be obtained (e.g., Cho &
Lazarian, 2010). Therefore, it is promising that using the new synchrotron
gradients performs foreground removal for studying the CMB B modes on the
plane of the sky. We expect that there are different properties between the
Galactic foreground SPGs and cosmological perturbation one, that is, the
direction traced by the former is perpendicular to the foreground
polarization, while the latter does not follow this relation. In addition, the
CMB E and B mode ratio has been demonstrated to be sensitive to the fraction
of the fast modes in MHD turbulence and the magnetization strength (Kandel et
al., 2017a). Furthermore, one can obtain the local properties of the CMB E and
B modes using the SPDGs, which provides an alternative way of studying the
distribution of compressible MHD turbulence modes over the sky and of
determining the magnetization level of the Galactic ISM.

Most of the work that has been carried out is based on numerical simulations,
which on the one hand helps to test the feasibility of the technique, and on
the other hand, provides the test grounds for studying magnetic fields using a
large number of data cubes from the LOFAR and SKA. Hence, the next step is
necessary to advance the application of the technique using observational data
to test the performance of the technique in a specific astrophysical setting

## Author Contributions

All authors listed have made a substantial contribution to the work and
approved it for publication.

## Funding

This work was partially supported by the National Natural Science Foundation
of China (Grant Nos. 11973035 and 11703020) and by the Hunan Province
Innovation Platform and Talent Plan- HuXiang Youth Talent Project (No.
2020RC3045).

## Acknowledgments

We would like to thank two anonymous referees for valuable comments and Alex
Lazarian for many constructive discussions.

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