text:
\section{Introduction}

Fast radio bursts (FRBs) are mysterious radio transients with millisecond durations and extremely high brightness temperatures at cosmological distances. So far, over 600 FRB sources have been detected, dozens of which exhibited a repeating behavior \citep[e.g.,][]{CHIME21}.
However, their physical origin is still not well understood due to complexity and diversity of the observations \citep[e.g.,][]{Cordes19,Zhang20,Xiao21}. For example, a Galactic FRB, FRB 200428, was detected to be associated with the magnetar SGR J1935+2154 \citep{Bochenek20,CHIME20,Mereghetti20,Li20,Ridnaia20,Tavani20}, implying that at least some FRBs originate from magnetars born from the core collapse of massive stars \citep{Katz16,Murase16,Beloborodov17,Kumar17,Yang18,Yang21,Metzger19,Wadiasingh19,Lu20,Margalit20,Zhang22,Wang22b,Qu22}.
However, such a magnetar formation is challenged by the observation of another nearby FRB, FRB 20200120E, which was localized to be in a globular cluster of the nearby galaxy M81 \citep{Bhardwaj21,Kirsten22}. The extremely old age of the globular cluster implies that it is more likely produced by an old object or a system associated with a compact binary merger \citep{zhang20c,Kremer21,Lu22}.
Therefore, multiple physical origins for the FRB population seem increasingly likely.

In addition to the FRB sources themselves, propagating effects, e.g., dispersion, Faraday rotation, temporal scattering, scintillation, depolarization, gravitational/plasma lensing, also play important roles to interpret FRB observations and constrain the properties of the FRB environment \citep[e.g.,][]{Xu16,Cordes17,Yang17,Li18b,Yang20b,Er20,Beniamini22,Yang22,Kumar22}.
Dispersion measure (DM) and Faraday rotation measure (RM) are two of the most important measurable quantities for FRBs. For a repeating FRB source, the variations of its DM and RM would provide clues to study the properties of its near-source plasma. \cite{Yang17} studied various possible origins to cause DM variations of an FRB repeater and concluded that the plasma local to the FRB source is the most likely cause.
Different from DM that usually has a significant contribution from the intergalactic medium, the observed RM of an FRB, especially when it is large, is likely contributed by
a highly magnetized environment near the FRB source, because the RM contribution from the intergalactic medium is very small with a typical value of $|{\rm RM_{IGM}}|\ll 10~{\rm rad~m^{-2}}$ \citep{Akahori16} and because the contribution from the interstellar medium in the Milky Way is usually $|{\rm RM_{MW}}|\lesssim100~{\rm rad~m^{-2}}$ at high latitudes \citep{Hutschenreuter22}.
The first known repeater, FRB 121102, shows the largest RM of $|{\rm RM}|\sim10^5~{\rm rad~m^{-2}}$ in all FRBs \citep{Michilli18}, which it decreased by $\sim30\%$ during one year \citep{Hilmarsson21}. This may be caused by
the expansion of a young supernova remnant (SNR) \citep{Piro18}, a magnetar nebula \citep{Margalit18}, or an ejecta from a compact binary merger \citep{Zhao21} around the FRB source, although other scenarios  (see discussion below) may also be possible. Another active repeater, FRB 190520B, has an extremely large host DM with ${\rm DM_{host}}\sim900~{\rm pc~cm^{-3}}$ \citep{Niu22}, which is nearly an order of magnitude higher than those of other FRBs. Meanwhile, its RM value reaches $\sim10^4~{\rm rad~m^{-2}}$ that is second in line next to FRB 121102, and there appears an RM sign reversal during a few months \citep{Anna-Thomas22,Dai22}. Such a large RM and significant RM reversal directly suggest that the FRB environment is magnetized and dynamically evolving.
It is worth noting that both FRB 121102 and FRB 190520B
are associated with a compact persistent radio source with a wide emission spectrum \citep{Chatterjee17,Niu22}, which implies that a persistent radio source and a large RM are likely physically related to each other \citep{Yang20a,Yang22}.

Other FRB repeaters also show complex and diverse RM variations. FRB 20201124A showed an irregular RM variation over one month. Some bursts appeared to have circular polarization and frequency-dependent oscillating polarization properties \citep{Xu21}. FRB 180916B with a 16.33-day periodic activity \citep{CHIME20b} exhibited RM variation with a stochastic component and a secular component at different periods \citep{Mckinven22}.
In summary, RM variations seem to be a common feature for all FRB repeaters.

Such significant RM variations suggest that repeating FRBs are likely surrounded by a inhomogeneous and dynamically evolving Faraday screen. When a radio burst propagates in the screen,
two important effects are involved: 1) the radio bursts would be depolarized due to different RMs at different paths. Very recently, \citet{Feng22} reported that active FRB repeaters exhibit conspicuous frequency-dependent depolarization that can be well described by the multi-path propagation effect in an inhomogeneous magnetized plasma \citep{Yang22}. 2) a significant circular polarization would be generated due to the superposition of electromagnetic waves with different phases and polarization angles from different paths \citep{Beniamini22}.

In this paper, we investigate the possible physical mechanisms that may cause RM variations from a repeating FRB source, and discuss the physical implications of the observed RM variations. The paper is organized as follows. We discuss the necessary conditions and theoretical implications of Faraday rotation in Section \ref{general}. The physical origins of random and secular RM evolution are generally analyzed in Section \ref{FaradayScreen}, where
the random RM variation is caused by the relative motion between the FRB source and a Faraday screen with an inhomogeneous magnetized medium, and the secular RM evolution may be caused by an expanding shell or the orbital motion in a binary system.
In Section \ref{scenarios}, we discuss different astrophysical scenarios, including SNRs with an inhomogeneous medium in Section \ref{SNR}, stellar winds from a massive/giant star companion in Section \ref{stellarwind}, stellar flares from a low-mass star companion in Section \ref{stellarflare}, pulsar winds, pulsar wind nebulae and magnetar flares in Section \ref{pair}, and magnetized outflows from a massive black hole in Section \ref{BH}). The results are discussed and summarized in Section \ref{discussion}. Some detailed calculations are presented in the Appendices.

\section{Rotation measure: a general discussion}\label{general}

Observationally the Faraday rotation measure, ${\rm RM_{obs}}$, is measured by the frequency(wavelength)-dependent polarization angle of linearly polarized waves
\begin{equation}
\psi={\rm RM_{obs}}\lambda^2,
\end{equation}
where $\psi$ is the polarization angle of the electromagnetic wave of wavelength $\lambda$ with respect to that of infinite frequency.
The necessary conditions to measure ${\rm RM_{obs}}$ of a source include: 1) the electromagnetic waves must contain a significant linear polarization component; 2) the polarization angle must satisfy $\psi\propto\lambda^2$.
For an extragalactic FRB, the observed RM can be decomposed in terms of the contributions of various plasma components along the line of sight, i.e.
\begin{equation}
{\rm RM_{obs}}={\rm RM_{\rm ion}}+{\rm RM_{MW}}+{\rm RM_{IGM}}+\frac{{\rm RM_{host}}}{(1+z)^2}+\frac{{\rm RM_{loc}}}{(1+z)^2},
\end{equation}
where $z$ is the redshift of the host galaxy, ${\rm RM_{\rm ion}}$ is the contribution from the Earth ionosphere, which is of the order of $|{\rm RM_{\rm ion}}|\sim(0.1-1)~{\rm rad~m^{-2}}$  \citep{Mevius18,Mckinven22},
${\rm RM_{MW}}$ is the contribution from the interstellar medium in the Milky Way, which has a typical absolute value $|{\rm RM_{MW}}|\lesssim100~{\rm rad~m^{-2}}$ at high latitudes \citep{Hutschenreuter22},
${\rm RM_{IGM}}$ is the contribution from  the intergalactic medium which has a very small value of $|{\rm RM_{IGM}}|< 10~{\rm rad~m^{-2}}$ \citep{Akahori16}, ${\rm RM_{host}}$ is the contribution from the interstellar medium in the FRB host galaxy, which might be of the same order of magnitude as the Milky Way, and ${\rm RM_{loc}}$ is the contribution from the local plasma near the FRB source. Therefore, an observed RM with a large absolute value of $|{\rm RM_{obs}}|\gtrsim10^3~{\rm rad~m^{-2}}$ is expected to be mainly contributed by the local plasma ${\rm RM_{loc}}$. On the other hand, since both the intergalactic medium and interstellar medium cannot have short-term evolution \citep{Yang17}, the observed significant RM variations can only be attributed to the local plasma. In the following discussion, we are only interested in ${\rm RM_{loc}}$, and hereafter directly use the symbol ${\rm RM}$ to denote ${\rm RM_{loc}}$.

For a non-relativistic magneto-ionic (ions+electrons) cold plasma with magnetic field $B$ and electron density $n_e$, the RM could be calculated by the dispersion relation of the electromagnetic wave, and the classical result is
\begin{align}
{\rm RM}&=\frac{e^3}{2\pi m_e^2c^4}\int n_eB_\parallel ds \nonumber\\
&\sim0.81~{\rm rad~m^{-2}}\fractionz{\left<n_e\right>_L}{1~{\rm cm^{-3}}}\fractionz{\left<B_\parallel\right>_L}{1~{\rm \mu G}}\fractionz{s}{1~{\rm pc}} \nonumber\\
&\sim0.81~{\rm rad~m^{-2}}\fractionz{\left<B_\parallel\right>_L}{1~{\rm \mu G}}\fractionz{{\rm DM}}{1~{\rm pc~cm^{-3}}},\label{rm}
\end{align}
where $\left<...\right>_L$ denotes to the average along the line of sight in the local plasma near an FRB source.
Although there is usually a component of the field that is spatially coherent at the scale of a certain object, the field lines are disordered and there are magnetic fluctuations in the scales covering orders of magnitude due to MHD turbulence. Thus, both a large-scale ordered magnetic field and a turbulent magnetic field are involved in various astrophysical scenarios, as shown in Figure \ref{figfield}.
The average of $B_\parallel$ depends on the geometric configuration of the magnetic fields along the line of sight.
For example, we consider that the electron density and magnetic field strength are approximately uniform but the magnetic geometry might change along the line of sight.
If the magnetic field is ordered in large scale (see panel (a) of Figure \ref{figfield}), the average parallel magnetic field $\left<B_\parallel\right>$ would be approximately of the order of the local parallel magnetic field $B_\parallel$, i.e., $\left<B_\parallel\right>_L\sim B_\parallel$. However, if the magnetic field is totally turbulent with coherent lengthscale of $s_B$ (in a region with lengthscale $s_B$, the field could be treated as approximately uniform with a certain direction, see panel (b) of Figure \ref{figfield}), the average parallel magnetic field would be $\left<B_\parallel\right>_L\sim (s_B/s)^{1/2} B_\parallel$ due to the Poisson r.m.s. fluctuations of the polarization angle \citep[e.g.,][]{Beniamini22,Yang22}. In general, for a local plasma one may write
\begin{align}
\left<B_\parallel\right>_L&\simeq1.23~{\rm \mu G}\fractionz{{\rm RM}}{1~{\rm rad~m^{-2}}}\fraction{{\rm DM}}{1~{\rm pc~cm^{-3}}}{-1}\nonumber\\
&\sim
\left\{
\begin{aligned}
&B_\parallel,&&\text{for a large-scale field},\\
&(s_B/s)^{1/2}B_\parallel,&&\text{for  turbulent fields}.
\end{aligned}
\right.\label{Bfield}
\end{align}
The above equation has been used to estimate the strength of total magnetic field in the emission region using RM and DM according in many papers, if the RM and DM originate from the same region. However, if the field is totally turbulent, the real local field strength would be enhanced by a factor of $(s/s_B)^{1/2}$.

\begin{figure}
    \centering
	\includegraphics[width = 1.0\linewidth, trim = 0 50 0 50, clip]{field.pdf}
    \caption{Schematic configurations of a magnetized FRB environment. There are two extreme scenarios: (a) the magnetic fields in the environment are large-scale ordered. (2) the magnetic fields in the environment are turbulent. The yellow regions correspond to the FRB environment. The orange regions correspond to the clumps in a turbulent medium. The blue lines correspond to magnetic field lines.}\label{figfield}
\end{figure}

For some FRB repeaters with large RMs, e.g., FRB 121102 and FRB 190520B \citep{Michilli18,Anna-Thomas22,Dai22}, the main contributions of observed RMs and DMs are likely to originate from different regions: a large part of DM is contributed by the intergalactic medium, but a large RM of $\gtrsim10^3~{\rm rad~m^{-2}}$ is more likely contributed from the local plasma in the FRB environment, because the intergalactic magnetic field is extremely low and disordered in large scale  \citep[e.g.,][]{Akahori16}.
Some authors \citep[e.g.,][]{Katz21} estimated the magnetic field strength of the local plasma by assumeing that the variations of RMs and DMs originate from the same region, so that
$\left<B_\parallel\right>_L\simeq1.23~{\rm \mu G}(\delta{\rm RM}/1~{\rm rad~m^{-2}})(\delta{\rm DM}/1~{\rm pc~cm^{-3}})^{-1}$.
This formula can be easily applied from the observational data, because both $\delta {\rm RM}$ and $\delta {\rm DM}$ are measurable quantities for a repeating source and the contributions from the ISM and IGM may be estimated for both DM and RM. However, such a formula is inapplicable of the scenario that the RM variation is caused by magnetic configuration variation, which is likely a cause of RM variations and has been confirmed by the RM reversal observed in FRB 190520B \citep{Anna-Thomas22,Dai22}.
For example, let us consider a case that
the electron density and magnetic field strength is unchanged but the magnetic field configuration is changing. This gives $\delta {\rm DM}\sim 0$ and $|\delta {\rm RM}|>0$.
This would give the unphysical conclusion $\left<B_\parallel\right>_L\rightarrow\infty$ with the formula.
The correct treatment should be to take
differentiation of Eq.(\ref{rm}) and obtain
\begin{equation}
\frac{\delta {\rm RM}}{{\rm RM}}\simeq\frac{\delta {\rm DM}}{{\rm DM}}+\frac{\delta \left<B_\parallel\right>_L}{\left<B_\parallel\right>_L},
\label{eq:deltaRM}
\end{equation}
which suggests that the relative RM variation is the sum of the relative variations of DM and average parallel magnetic field.
For example, the RM of FRB 121102 decreased by $|\delta{\rm RM}|\sim |{\rm RM}|\sim10^{5}~{\rm rad~m^{-2}}$ during a few years \citep{Michilli18,Hilmarsson21}. According to Eq.(\ref{eq:deltaRM}), there are two possible reasons for its large relative RM variation ($|\delta {\rm RM}/{\rm RM}|\sim1$): 1) It is due to a large relative DM variation, $\delta{\rm DM}/{\rm DM}\sim1$.
Since the observed DM variation of FRB 121102 is small, $\delta{\rm DM}\sim 1~{\rm pc~cm^{-3}}$ \citep{Hessels19}, the DM of the region contributing the large RM is required to be also small, i.e. ${\rm DM}\sim\delta{\rm DM}\sim 1~{\rm pc~cm^{-3}}$. Thus, the estimated average magnetic field strength is $\left<B_\parallel\right>_L\sim 0.1~{\rm G}$ according to Eq.(\ref{Bfield}), suggesting that the magnetic field near FRB 121102 is extremely strong, much larger than the observed magnetic field of most SNRs and pulsar wind nebulae in the Milky Way \citep[e.g.,][]{Reynolds12}. 2) It is due to the change of the field configuration, $\left|\delta\left<B_\parallel\right>_L/\left<B_\parallel\right>_L\right|\sim1$, which implies that the magnetic field configuration near the source is dynamically evolving during the past few years, similar to the scenario of FRB 190520B \citep{Anna-Thomas22,Dai22}.

At last, we should note that the condition of $\psi\propto\lambda^2$ for the RM measurement to be relevant requires that $\omega\gg\omega_B,\omega_p$ is satisfied according to the dispersion relation of circularly polarized waves, where $\omega_B=eB/m_ec$ is the cyclotron frequency, and $\omega_p=(4\pi e^2n_e/m_e)^{1/2}$ is the plasma frequency. For a given observed frequency, the necessary condition for $\psi\propto\lambda^2$ can be translated to a constraint on the magnetic field strength, i.e.
\begin{equation}
B\ll B_c=\frac{2\pi m_ec\nu}{e}\simeq360~{\rm G}\fractionz{\nu}{1~{\rm GHz}}.\label{Bc}
\end{equation}
Therefore, although the FRB engine (e.g. a neutron star, black hole, or even a white dwarf) has a strong magnetic field near the engine, the region with $B>B_c$ cannot contribute to the observed RM\footnote{The medium near the engine (e.g. a neutron star) is likely a relativistic pair plasma. The RM contribution by such a pair plasma is not important anyway, see discussion in Section \ref{pair} and Appendix \ref{RMpair}.}.

\section{Physical origins of random and secular RM variations}\label{FaradayScreen}

Recent observations show that many FRB repeaters appear to have complex, diverse RM variation patterns.
For example, FRB 121102 exhibited a non-linear decrease of RM absolute value during a few years with several weeks of fluctuations in a short term
\citep{Chatterjee17,Hilmarsson21}.
In an active cycle, FRB 20201124A showed an irregular RM variation during the first 36 days, which is followed by an almost constant RM during a later 18 days \citep{Xu21}. FRB 180916B showed stochastic, small RM variations
followed by a significant secular increasing component over the nine month period \citep{Mckinven22}. It seems that observations show both random fluctuations and systematic secular evolution. In the following, we present some general discussions on these two scenarios.

\subsection{Random RM variations}
For random RM variations, the most likely scenario is that there is a Faraday screen with inhomogeneous medium near the FRB source, and the relative motion between the FRB source and the screen causes irregular RM variations.
For turbulence induced inhomogeneity, this requires that the timescale of relative motion is shorter than the timescale of turbulence, i.e. eddy turnover time.
The ubiquitous turbulence in astrophysical plasmas naturally induces fluctuations in density and magnetic fields, and hence, RM fluctuations \citep{Minter96,Vogt2005,Xu16}.
We consider a Faraday screen with thickness $\Delta R$ and assume a statistical homogeneity of the medium. The ``RM structure function'' is used to represent the mean-squared RM difference between two paths separated by a transverse distance $l$, i.e.,
\begin{equation}
D_{\rm RM}(\overrightarrow{l})\equiv\left<[{\rm RM}(\overrightarrow{x}+\overrightarrow{l})-{\rm RM}(\overrightarrow{x})]^2\right>,\label{SF}
\end{equation}
where $\left<...\right>$ represents an ensemble average. Physically, the RM structure function depends on the power spectra $P(k)$ of the fluctuations of RM density $(n_e B_\parallel)(\overrightarrow{x})$, where $k=2\pi/l$ is the spatial wavenumber. For simplicity, we assume a power-law distribution of\footnote{In the following discussion, we do not separate the RM density fluctuations into fluctuations arising from electron density and magnetic field. If the fluctuations of electron density and magnetic field have different spectral indexes, the spectral index of RM density $(n_e B_\parallel)(\overrightarrow{x})$ would be dominated by the one with larger relative variation \citep{Xu16}, $\delta X/X$, where $X$ denotes $n_e$ or $B_\parallel$.} $P(k)\propto k^\alpha$ for $2\pi L^{-1}<k<2\pi l_0^{-1}$, where $L$ and $l_0$ correspond to the outer scale and inner scale, respectively.
The power spectrum with the 3D spectral index $\alpha<-3$ is called a ``steep spectrum'' (e.g., $\alpha=-11/3$ for the Kolmogorov scaling), and the fluctuations are dominated by the large scale at $\sim L$, which corresponds to the energy injection scale of turbulence;
The power spectrum with $\alpha>-3$ is called a ``shallow spectrum'', and inhomogeneity structures are dominated by small-scale fluctuations near $\sim l_0$, which is the energy dissipation scale of turbulence \citep{Lazarian04,Lazarian06,Lazarian16}.
Shallow density spectra are commonly seen in cold interstellar phases with supersonic turbulence, where the small-scale density enhancement is caused by turbulence compression \citep{Xu17,Xu20}.
For a shallow spectrum,
the fluctuations on scales larger than $L$ are
model-dependent, and $L$ is most likely the largest scale of the fluid system, $L\sim\Delta R$,
for turbulence driven within the system.

We define the correlation length scale $l_{\rm RM}$ with a correlation $\kappa^2\left<\delta (n_eB_\parallel)(\overrightarrow{x}+\overrightarrow{l_{\rm RM}})\delta (n_eB_\parallel)(\overrightarrow{x})\right>=\sigma_{\rm RM}^2/2$, where $\sigma_{\rm RM}^2=\kappa^2\left<\delta(n_eB_\parallel)^2\right>$ correponds to the variance of RM density fluctuations multiplying by $\kappa=e^3/(2\pi m_e^2c^4)$.
For the steep spectrum ($\alpha<-3$), the correlation scale is $l_{\rm RM}\sim L$;
while, for the shallow spectrum ($\alpha>-3$), the correlation scale is $l_{\rm RM}\sim l_0$ \citep{Lazarian16,Xu16}.
For most astrophysical scenarios, the Faraday screen is considered to be thick\footnote{Here, the definitions of ``thick'' and ``thin'' of a Faraday screen is based on the relation between the correlation length $l_{\rm RM}$ and the screen thickness $\Delta R$ \citep{Lazarian16}. Since the correlation length $l_{\rm RM}$ cannot exceed the largest scale of a system for turbulence driven within it, in most astrophysical scenarios involving a shell with the fluctuations of magnetic field and density as the Faraday screen, the thick screen condition ($l_{\rm RM}<\Delta R$) is usually satisfied.}, i.e., $\Delta R>l_{\rm RM}$.
Therefore, the RM structure function of a thick screen could be written as (\cite{Lazarian16,Xu16}; see Appendix \ref{RMSF} for a detailed derivation)
\begin{align}
D_{\rm RM}(l)\sim
\left\{
\begin{aligned}
&\sigma_{\rm RM}^2\Delta R l\fraction{l}{l_{\rm RM}}{-(\alpha+3)},&&l_0<l<l_{\rm RM}\sim L,\\
&\sigma_{\rm RM}^2\Delta R l_{\rm RM},&&l>l_{\rm RM}\sim L.
\end{aligned}
\right.\label{sf1}
\end{align}
for a steep spectrum ($\alpha<-3$) and $L\sim l_{\rm RM}$, and
\begin{align}
D_{\rm RM}(l)\sim
\left\{
\begin{aligned}
&\sigma_{\rm RM}^2\Delta R l\fraction{l}{l_{\rm RM}}{-(\alpha+3)},&&l_0\sim l_{\rm RM}<l<L,\\
&\sigma_{\rm RM}^2\Delta R^2\fraction{\Delta R}{l_{\rm RM}}{-(\alpha+3)},&&l> L.
\end{aligned}
\right.\label{sf2}
\end{align}
for a shallow spectrum ($-3<\alpha<-2$) and $L\sim\Delta R$. Therefore,
the Kolmogorov scaling
with $\alpha=-11/3$ has the RM structure function $D_{\rm RM}(l)\propto l^{5/3}$ in the inertial range and $D_{\rm RM}(l)\sim\text{constant}$ beyond the inertial range.

We assume that the relative transverse velocity between the FRB source and the Faraday screen is $v_\perp$.
Based on the definition of $D_{\rm RM}(l)$ in Eq.(\ref{SF}), the r.m.s. variation of RM during a time $t$ is
\begin{equation}
|\delta{\rm RM}(t)|\sim \sqrt{D_{\rm RM}(v_\perp t)}.
\end{equation}
The largest RM amplitude contributed by the Faraday screen can be estimated as $|{\rm RM}|\sim|\delta {\rm RM}(l>L)|$, where $|\delta {\rm RM}(l>L)|\sim\sqrt{D_{\rm RM}(l> L)}$ is given by the last equations of Eq.(\ref{sf1}) and Eq.(\ref{sf2}).
Thus, during time $t$, the relative RM variation can be estimated as
\begin{align}
\left|\frac{\delta {\rm RM}}{{\rm RM}}\right|\sim
\left\{
\begin{aligned}
&\fraction{v_\perp t}{L}{-(\alpha+2)/2},&&v_\perp t< L,\\
&1,&&v_\perp t> L,
\end{aligned}
\right.\label{RMvariation}
\end{align}
for both steep and shallow spectra. We emphasize again that in the above equation $L\sim l_{\rm RM}$ is considered for a steep spectrum, and $L\sim\Delta R$ is considered for a shallow spectrum.

The measurements of the RM structure function of some FRB repeaters revealed that \citep{Mckinven22}
$D_{\rm RM}(t)\propto t^{0.2-0.4}$,
which implies that the power spectrum is
$\alpha\sim-(2.2-2.4)$.
Therefore, the Faraday screens of these FRB repeaters have shallow spectra in the inertial ranges, which means that the variation is dominated by small-scale RM density fluctuations.
Notice that as magnetic fluctuations arising from nonlinear turbulent dynamo \citep{Xu16} and turbulent compression have the magnetic energy spectrum basically following the turbulent energy spectrum,
the magnetic energy spectrum is usually steep.
Thus, the observed result implies that a shallow density spectrum is more likely to dominate the RM fluctuations for these particular FRB sources.
Physically, a shallow density spectrum naturally arises in supersonic turbulence, e.g., in star-forming regions \citep{Hennebelle12}.

On the other hand, many FRB repeaters, e.g, FRB 121102, FRB 190520B, FRB 180916B, show large RM variations $|\delta{\rm RM}/{\rm RM}|\sim 1$ during a few months to a few years \citep{Michilli18,Hilmarsson21,Anna-Thomas22,Dai22,Mckinven22}, implying that the outer scale of the inertial range satisfies
\begin{equation}
L\lesssim v_\perp t\simeq10^{-4}~{\rm pc}\fractionz{v_\perp}{100~{\rm km~s^{-1}}}\fractionz{t}{1~{\rm yr}}.\label{outerscale}
\end{equation}

\subsection{Secular RM evolution}

A secular RM evolution may be attributed to the expansion of a magnetized shell or orbital motion of a binary system.
First, we consider the scenario of an expanding magnetized shell with the magnetic field configuration unchanged during a short-term, which is applicable to young SNRs (see Section \ref{SNR}).
or companion flares in a binary system (see Section \ref{stellarflare}).
In a certain astrophysical environment, the electron density and magnetic field are usually related, e.g.
$B\propto n_e^{\gamma_B}$,
where $\gamma_B=1/2,2/3,1$ corresponds to an energy-equipartition plasma, a magnetic freezing plasma, or a shocked compressed plasma, respectively \citep[e.g.,][]{Yang22}.
Due to the expansion, the electron density might decrease with shell radius
$n_e\propto r^{-\gamma_n}$,
with $\gamma_n=0,2,3$ corresponding to a shock compressed medium (upstream medium is assumed to be uniform), a wind medium, and free expansion, respectively. We assume that the time-dependent shell radius is $r\propto t^{\gamma_r}$ with $\gamma_r=1,2/5$ corresponding to free expansion and the Sedov–Taylor phase, respectively. Therefore, the RM evolution satisfies
\begin{equation}
{\rm RM}\propto n_eBr\propto t^{\gamma_r(1-\gamma_n-\gamma_B\gamma_n)}.
\end{equation}

Next, we are interested in the secular RM evolution caused by orbital motion of a binary system, and the companion could be a stellar object (see Section \ref{stellarwind}) or a massive black hole (see Section \ref{BH}). Since the large-scale magnetic fields are contributed by the magnetized companion (i.e., large-scale dipole field for a companion with weak wind, magnetic field in the disk of Be stars, etc.), the RM variation would be periodic with the same period as the orbital period,
\begin{equation}
P=2\pi\fraction{a^3}{GM_{\rm tot}}{1/2}.\label{period}
\end{equation}
where $a$ is the semi-major separation of the binary system, and $M_{\rm tot}$ is the binary total mass. A large RM variation of $|\delta {\rm RM}/{\rm RM}|\sim1$ should occur in the timescale of $\lesssim P$.
Such a scenario could be tested by long-term monitoring of RM variations for
particular FRB repeaters.

\section{Different astrophysical scenarios generating RM variations}\label{scenarios}

In this section, we will discuss a list of possible astrophysical processes that might cause RM variations of a particular FRB repeater.

\subsection{RM variations contributed by a supernova remnant}\label{SNR}

Radio polarization observations of young SNRs suggest that
magnetic fields in SNRs are largely disordered, with a small radial preponderance \citep[e.g.,][]{Dickel76,Milne87,Reynolds12}. There are two possible explanations to the radial preponderance \citep{Jun96,Blondin01,Zirakashvili08,Inoue13,West17}: 1. The Rayleigh-Taylor instability stretches the field lines preferentially along the radial direction; 2. Turbulence with a radially biased velocity dispersion may be induced. In older, larger SNRs, the field lines are often disordered but sometimes tangential. The tangential fields could be explained by the shock compression of the upstream medium \citep[e.g.,][]{Dickel76,Milne87,Reynolds12}.
In summary, polarization and imaging observations indicate that the magnetic fields in a SNR are turbulent and evolving.

\begin{figure}
    \centering
	\includegraphics[width = 1.0\linewidth, trim = 0 0 0 0, clip]{SNR.pdf}
    \caption{Schematic configuration of an FRB source in an SNR expanding in the interstellar/circumstellar medium with gas clouds. The SNR has radius $R$ and thickness $\Delta R$. The red regions correspond to ejected large-scale clumps in the SNR, which reflects the SNR local anisotropy. The blue regions correspond to the gas clouds with average scale $l_{\rm cloud}$ in the interstellar/circumstellar medium. The grey circle corresponds to the FRB source. The blue curves indicate magnetic field lines. The RM is contributed by the parallel component by the magnetic field fluctuations.}\label{figSNR}
\end{figure}

If magnetic geometry along the line of sight keeps unchanged, the RM contribution from an SNR would show a long-term evolution due to the expansion of the SNR \citep[][see the discussion in Section \ref{FaradayScreen}]{Piro18,Zhao21b}. Although this scenario seems to be consistent with the observation of FRB 121102 \citep{Michilli18,Hilmarsson21}, it cannot explain the non-monotonic irregularity exhibited by some FRB repeaters reported recently \citep{Xu21,Anna-Thomas22,Dai22}. In particular, FRB 190520B appears to have a significant RM reversal \citep{Anna-Thomas22,Dai22}, which directly confirms that the field configuration near the FRB source is changing along the line of sight. In the following discussion, we mainly focus on the RM variations contributed by the turbulence in the SNR.

When an SNR propagates in a highly inhomogeneous interstellar/circumstellar medium, as shown in Figure \ref{figSNR}, it generates a pair of shocks upon interacting with the dense gas clouds, and induce turbulence in the magnetized medium \citep[e.g.,][]{Hu22}.
Therefore, for the SNR scenario, one may take the largest outer scale of the turbulence in the SNR as
\begin{equation}
L=\min(\xi \Delta R,l_{\rm cloud}),\label{largestscale}
\end{equation}
where $\Delta R$ is the SNR thickness,  $\xi\Delta R$ is the transverse scale of the ejected large-scale clumps in the SNR (see Figure \ref{figSNR}), with the parameter $\xi$ describing the local anisotropy of the SNR (the smaller the value of $\xi$, the more significant the SNR local anisotropy), and $l_{\rm cloud}$ is the typical scale of the gas clouds \citep[e.g.,][]{Heiles03,Inoue09}.
The intensity map of the molecular emission shows that the density distribution within molecular clouds appears to have shallow spectra characterized by small-scale, large density. The smallest scale of clouds is
\citep[see Figure 10 of][]{Hennebelle12}
\begin{equation}
l_{\rm cloud}\lesssim 0.1~{\rm pc}.
\end{equation}
Notice that the above upper limit of the characteristic size of density structures is due to the limited resolution of observations.

We generally discuss a wide range of the possible transverse relative velocity as $v_\perp\sim(10-10^4)~{\rm km~s^{-1}}$. The lower limit of $v_\perp\sim10~{\rm km~s^{-1}}$ corresponds to the possible minimum intrinsic velocity of the neutron star FRB source \citep[e.g.,][]{Hansen97}, and the upper limit of $v_\perp\sim10^4~{\rm km~s^{-1}}$ corresponds to the initial velocity of an expanding SNR \citep[e.g.,][]{Yang17}.

First, we consider that the scenario of $\xi\Delta R< l_{\rm cloud}$.
During the observing time $t\sim1~{\rm yr}$, the relative distance is of the order $l\sim v_\perp t\sim(10^{-5}-10^{-2})~{\rm pc}\lesssim l_{\rm cloud}$.
According to Eq.(\ref{RMvariation}) and Eq.(\ref{largestscale}), the large relative RM variation $|\delta{\rm RM}/{\rm RM}|\sim 1$ implies $l\sim v_\perp t\sim\xi\Delta R\sim \xi\eta v_{\rm SNR}t_{\rm SNR}$, where $v_{\rm SNR}$ is the SNR expanding velocity, $t_{\rm SNR}$ is the SNR age, and the SNR thickness is $\Delta R=\eta R\sim\eta v_{\rm SNR}t_{\rm SNR}$, and $R$ is the SNR radius. Thus, the typical SNR age is given by
\begin{equation}
t_{\rm SNR}\sim\fractionz{v_\perp}{\xi\eta v_{\rm SNR}}t.
\end{equation}
For the SNR with a small radius of $R\sim\eta^{-1}\Delta R<(\xi\eta)^{-1} l_{\rm cloud}$ and for $0.01\lesssim\xi\eta\lesssim1$, the SNR might have a large expanding velocity of $v_{\rm SNR}\gtrsim10^3~{\rm km~s^{-1}}$ \citep[e.g.,][]{Yang17}. Considering that the intrinsic velocity of most neutrons stars might not exceed $\sim 10^3~{\rm km~s^{-1}}$ \citep[e.g.,][]{Hansen97}, one may have $v_\perp\lesssim v_{\rm SNR}$ in this scenario.
The large relative RM variation $|\delta{\rm RM}/{\rm RM}|\sim 1$ implies that the SNR is young with an age of $t_{\rm SNR}\lesssim100~{\rm yr}(\xi\eta/0.01)^{-1}(t/1~{\rm yr})$ for significant local anisotropy or a thin thickness.
On the other hand, if the SNR is thick and almost locally isotropic with $\xi\eta\gtrsim 0.1$, the observations of $|\delta{\rm RM}/{\rm RM}|\sim 1$ would require that the SNR is extremely young with an age of $t_{\rm SNR}\lesssim (\xi\eta)^{-1}t\sim 10~{\rm yr}(t/1~{\rm yr})$. Such a young SNR would show significant, observable, secular evolution in both DM and RM \citep{Yang17,Piro18,Zhao21b}, and could even be opaque for FRBs. These seem to be inconsistent with the current observations.

For the scenario of $\xi\Delta R>l_{\rm cloud}$, according to Eq.(\ref{RMvariation}), Eq.(\ref{outerscale}) and Eq.(\ref{largestscale}), the large relative RM variation $|\delta{\rm RM}/{\rm RM}|\sim 1$ implies that $l_{\rm cloud}\lesssim10^{-4}~{\rm pc}(v_\perp/100~{\rm km~s^{-1}})(t/1~{\rm yr})$. In this case, the SNR could be relatively older, however, it is challenging to resolve such small-scale density structures due to the limited resolution in observations \citep{Hennebelle12}.

\subsection{RM variations contributed by stellar winds from a massive/giant star companion}\label{stellarwind}

There is some evidence suggesting that an FRB source might be in a binary system: 1) FRB 180916B shows a periodic activity with a period of $16.35$ days \citep{CHIME20b}, which might correspond to an orbital period of a binary system as proposed by some authors \citep{Ioka20b,Dai20,Lyutikov20,Zhang20e,Li21d,wada21}; 2) FRB 20200120E was found to be associated with a globular cluster in the M81 galaxy \citep{Bhardwaj21,Kirsten22}, which contains many close binary systems; 3) PSR B1744-24A in a binary system displaying a complex, magnetized environment with Faraday conversion and circularly polarized attenuation, which is similar to some FRB repeaters \citep{Xu21,Li22}; 4) RM variations consisting of a constant component and an irregular component have  been observed both in FRB repeaters \citep{Xu21,Mckinven22} and pulsar binary systems \citep{Johnston96,Johnston05}. The latter requires an elliptical orbit of the pulsar. The similar configuration may apply to FRB repeaters as well \citep[e.g.,][]{Li21d,Wang22}.

\begin{figure}
    \centering
	\includegraphics[width = 1.0\linewidth, trim = 0 0 0 0, clip]{stellarwind.pdf}
    \caption{Schematic configuration of an FRB source in a binary system with the companion generating anisotropic stellar winds. The orange circle corresponds to the companion that might be a massive star or a giant star. The grey circle corresponds to the FRB source moving around the companion. The blue curves denote magnetic field lines.}\label{figstellarwind}
\end{figure}

In this subsection, we consider that the extremely large RM and the significant RM variation of some FRB repeaters are caused by the stellar wind from a companion in a binary system, as shown in Figure \ref{figstellarwind}. In order to provide strong stellar winds, the companion is likely a massive star or a giant star.
We assume that the companion star has a mass $M_c$, a radius $R_c$, and a mass loss rate of $\dot M$. The wind velocity can be calculated as the escape velocity, i.e.
$v_w\sim(2GM_c/R_c)^{1/2}\sim620~{\rm km~s^{-1}}(M_c/M_\odot)^{1/2}(R_c/R_\odot)^{-1/2}$.
The electron density in the stellar wind at a distance $r$ from the star is given by
\begin{align}
n_w(r)&\simeq\frac{\dot M}{4\pi\mu_mm_pv_wr^2}\simeq1.1\times10^6~{\rm cm^{-3}}\nonumber\\
&\times\fraction{r}{1~{\rm AU}}{-2}\fractionz{\dot M}{10^{-8}M_\odot~{\rm yr^{-1}}}\fraction{v_w}{10^3~{\rm km~s^{-1}}}{-1},
\end{align}
where $\mu_m=1.2$ is the mean molecular weight for a solar composition,
and the mass loss rate depends on the stellar type, e.g. $\dot M\sim10^{-7}-10^{-5}M_\odot~{\rm yr^{-1}}$ for O stars \citep{Puls96,Muijres12}; $\dot M\sim10^{-11}-10^{-8}M_\odot~{\rm yr^{-1}}$ for Be stars \citep{Snow81,Poe86}; $\dot M\sim10^{-14}-10^{-10}M_\odot~{\rm yr^{-1}}$ for solar-like stars \citep{Wood02}. We assume that the FRB emission region is close to the FRB source. Since the wind density decrease as $r^{-2}$, most of the local DM would be contributed by the wind at\footnote{Except for the scenario that the companion is in front of the FRB source along the line of sight, leading to a large DM during the eclipsing phase. A similar scenario has  been seen in PSR B1744-24A in the globular cluster Terzan 5 reported by \cite{Li22}). During the eclipsing phase, PSR B1744-24A shows a significant DM variation and depolarization caused by RM variation.} $r\sim a$, where $a$ is the binary separation. Thus, the DM contributed by the stellar wind is estimated as
\begin{align}
{\rm DM}_w&\sim n_wa\simeq5.4~{\rm pc~cm^{-3}}\fraction{a}{1~{\rm AU}}{-1}\nonumber\\
&\times\fractionz{\dot M}{10^{-8}M_\odot~{\rm yr^{-1}}}\fraction{v_w}{10^3~{\rm km~s^{-1}}}{-1}.
\end{align}
In order to estimate the RM contribution of the companion wind, we consider that the magnetic field strength at distance $r$ from the companion center satisfies
\begin{align}
B(r)\sim
\left\{
\begin{aligned}
&B_c\fraction{r}{R_c}{-3},&&R_c<r<R_A,\\
&B_c\fraction{R_A}{R_c}{-3}\fraction{r}{R_A}{-\beta_B},&&R>R_A,
\end{aligned}
\right.
\end{align}
where $1\lesssim \beta_B\lesssim 2$. Typically, one has $\beta_B\simeq1$ for a toroidal field and $\beta_B\simeq2$ for a radial field.
In the above equation, $B\propto r^{-3}$ corresponds to the dipole field near the companion surface and $B\propto r^{-\beta_B}$ corresponds to the magnetic field in the stellar wind outside the Alfv\'en radius $R_A$.
In the inner region $r<R_A$, the magnetic field pressure $P_B$ dominates, and the stellar wind moves along the field lines. In the outer region $r>R_A$, due to the large ram pressure of the stellar wind $P_w$, the magnetic field pressure is sub-dominant and would be carried by the wind. The Alfv\'en radius $R_A$ is defined by
\begin{equation}
P_w=\frac{1}{2}\rho_w(R_A)v_w^2\simeq\frac{\dot Mv_w}{8\pi R_A^2}\sim P_B=\frac{1}{8\pi}B_c(R_A)^2,
\end{equation}
where $\rho_w=\dot M/(4\pi r^2 v_w)$ is the mass density of the stellar wind.
According to the above equation, one has
\begin{align}
R_A&\sim\fraction{B_c^2R_c^6}{\dot Mv_w}{1/4}\simeq0.1R_\odot\fraction{B_c}{1~{\rm G}}{1/2}\fraction{R_c}{1R_\odot}{3/2}\nonumber\\
&\times\fraction{\dot M}{10^{-8}M_\odot~{\rm yr^{-1}}}{-1/4}\fraction{v_w}{10^3~{\rm km~s^{-1}}}{-1/4},\label{alfven}
\end{align}
suggesting that $R_A$ is inside the star for our typical parameters, so that
the magnetic field is dominated by the stellar wind at $r>R_c$.
In other words, the stellar wind would carry the relative strong magnetic field extending to large distances.
The RM contributed by the companion wind is approximately
\begin{align}
{\rm RM}_w&\sim\frac{e^3Bn_wa}{2\pi m_e^2c^4} \simeq2\times10^4~{\rm rad~m^{-2}}\fractionz{B_c}{1~{\rm G}}\fractionz{R_c}{1R_\odot}\nonumber\\
&\times\fractionz{\dot M}{10^{-8}M_\odot~{\rm yr^{-1}}}\fraction{a}{1~{\rm AU}}{-2}\fraction{v_w}{10^3~{\rm km~s^{-1}}}{-1}.\label{RMw1}
\end{align}
for a toroidal field with $\beta_B\simeq1$, and
\begin{align}
{\rm RM}_w&\simeq94~{\rm rad~m^{-2}}\fractionz{B_c}{1~{\rm G}}\fractionz{R_c}{1R_\odot}\nonumber\\
&\times\fractionz{\dot M}{10^{-8}M_\odot~{\rm yr^{-1}}}\fraction{a}{1~{\rm AU}}{-3}\fraction{v_w}{10^3~{\rm km~s^{-1}}}{-1}.\label{RMw2}
\end{align}
for a radial field with $\beta_B\simeq2$.
A large RM value of ${\rm RM}\gtrsim10^4~{\rm rad~m^{-2}}$ is consistent with the observation of FRB 121102 and FRB 190520B \citep{Michilli18,Hilmarsson21,Anna-Thomas22,Dai22}.
Therefore, the environment of FRB repeaters with large RMs might correspond to the stellar wind of a massive star or a giant star with a toroidal magnetic field configuration.
In the following discussion, we will analyze the RM variation within this scenario.

When an FRB repeater is in a binary system, both orbital motion of the binary system and dynamic evolution of the companion wind could cause RM variation.
As discussed in Section \ref{FaradayScreen}, a magnetized companion usually has a large-scale field, and the orbital motion of the radio source in such a large-scale field would cause RM variation \citep{Wang22,Li22}.
Such a scenario has been observed in a binary system. For example, the RM of PSR B1259-63 reaches a few times $10^3~{\rm rad~m^{-2}}$ and significantly reverses sign around the periastron \citep{Johnston96,Johnston05}.
Consider that the total mass of the binary system is $M_{\rm tot}$. The orbital period is given by Eq.(\ref{period}), i.e.
\begin{equation}
P
\simeq0.3~{\rm yr}\fraction{a}{1~{\rm AU}}{3/2}\fraction{M_{\rm tot}}{10~M_\odot}{-1/2}.
\end{equation}
In this scenario, RM variation should have the same period as the orbital period. However, there is no evidence that any FRB repeater shows a periodic RM variation up to the present. In particular, FRB 180916B exhibits a 16.33-day periodicity in its burst activity \citep{CHIME20b}, and the periodic activity might be due to orbital motion as proposed in some models \citep{Ioka20b,Dai20,Lyutikov20,Zhang20e,Li21d,Wang22}. However, the long-term RM evolution and burst-to-burst RM variations appear unrelated to the periodic activity and the activity cycle phase \citep{Mckinven22}. There are two possible reasons: 1. the periodic activity of radio bursts is not due to the orbital motion, and the possible RM variation period corresponding to the orbital period is much larger than the observing time of a few years; 2. the periodic activity of radio bursts is caused by the orbital motion, but the long-term RM variation of FRB 180916B is due to the intrinsic evolution of the stellar wind along the line of sight.

Turbulence in stellar winds could be caused by the anisotropic distribution or the episodic outflow of the stellar wind, which may smear out the apparent RM periodic evolution.
For turbulence of the companion wind at the distance $r\sim a$, the typical outer scale of turbulence might be estimated by
\begin{equation}
L\sim \min(\theta_w a, v_w\Delta t_w),
\end{equation}
where $\theta_w$ is the typical anisotropic distribution angle of the stellar wind, and $\Delta t_w$ is the typical timescale of the wind outflow variation. The episodic wind variation is usually caused by stellar flares, which will be further discussed in the Section \ref{stellarflare}. Here we are mainly interested in the case of persistent wind with $\theta_w a\ll v_w\Delta t_w$, leading to
\begin{equation}
L\sim \theta_w a\simeq1~{\rm AU}\fractionz{\theta_w}{1~{\rm rad}}\fractionz{a}{1~{\rm AU}},
\end{equation}
The Keplerian velocity of the FRB source around the companion is
\begin{equation}
v=\fraction{GM_{\rm tot}}{a}{1/2}\simeq94~{\rm km~s^{-1}}\fraction{M_{\rm tot}}{10M_\odot}{1/2}\fraction{a}{1~{\rm AU}}{-1/2}.
\end{equation}
Therefore, the time of the FRB source crossing the outer scale $L$ is
\begin{equation}
t_L\sim\frac{L}{v}\simeq18~{\rm day}\fractionz{\theta_w}{1~{\rm rad}}\fraction{M_{\rm tot}}{10M_\odot}{-1/2}\fraction{a}{1~{\rm AU}}{3/2}.
\end{equation}
For the adopted typical parameters, this timescale is slightly shorter than the observing time of RM variations of some FRB repeaters.
According to Eq.(\ref{sf1}), Eq.(\ref{sf2}) and Eq.(\ref{RMvariation}), one has $|\delta {\rm RM}/{\rm RM}|\propto t^{-(\alpha+2)/2}$ and $D_{\rm RM}(t)\propto t^{-(\alpha+2)}$ for $t\lesssim t_L$; and $|\delta {\rm RM}/{\rm RM}|\sim 1$ and $D_{\rm RM}(t)\sim\text{constant}$ for $t\gtrsim t_L$.

In particular, if the binary orbit is elliptical, a larger RM variation would occur near the periastron due to a stronger magnetic field and a higher electron density, and would keep almost constant far away from the periastron, as was observed in PSR B1259-63 \citep{Johnston96,Johnston05}. Some FRB repeaters, e.g., FRB 20201124A, also exhibited the similar behaviors \citep{Xu21,Wang22}, although the periodic evolution of RM fluctuations have not been detected so far.
It is worth noting that for this scenario, the periodic evolution of RM variations would be easier to achieve for an elliptical orbit compared with a circular orbit,
because the possible significant fluctuations of electron density and magnetic fields in the latter case might smear out the clean periodic signature in RM variation.

\subsection{RM variations contributed by stellar flares from a low-mass star companion}\label{stellarflare}

Stellar flares are usually defined as catastrophic releases of magnetic energy leading to particle acceleration and electromagnetic radiation accompanied by coronal mass ejections (CMEs) \citep[e.g.,][]{Haisch91}. Frequent flaring occurs on stars with an outer convection zone, and the timescale of energetic fares is longer than that of less energetic flares, regardless of the host star \citep[e.g.,][]{Pettersen89}.
Over short timescales of minutes to a few hours, they emit energy
ranging from $10^{23}~{\rm erg}$ (nanoflares; e.g., \cite{Parnell00}) to
$10^{31}-10^{38}~{\rm erg}$ (superflare; e.g., \cite{Shibayama13,Gunther20}).
In particular, for low-mass stars, due to  strong convection near their surfaces, flares and CMEs are usually frequent. In the following discussion, we will mainly focus on stellar flares/CMEs from low-mass stars.

\begin{figure}
    \centering
	\includegraphics[width = 1.0\linewidth, trim = 0 0 0 0, clip]{stellarflare.pdf}
    \caption{Schematic configuration of an FRB source in a binary system with the companion producing CMEs associated with flares. The orange shell corresponds to the CME where the FRB crosses it at the distance $r$ from the center of the companion star.
    The orange circle corresponds to the companion that might be a low-mass star. The grey circle corresponds to the FRB source moving around the companion. The blue curves denote magnetic field lines.}\label{figstellarflare}
\end{figure}

To estimate the impact of stellar flares/CMEs, we apply the empirical relationship between flare energy in the X-ray band, $E_X$, and the CME mass, $M_{\rm CME}$, found by \cite{Aarnio12}, i.e. $\log M=0.63\log E_X-2.57$. We assume that the ratio between the stellar flare X-ray energy $E_X$ and the CME kinetic energy $E_{\rm CME}$ is $\epsilon$, i.e. $E_X=\epsilon_X E_{\rm CME}$, and adopt a typical value $\epsilon_X\sim0.01$
considering that the energy emitted bolometrically is typically larger than the X-ray energy by a factor of 100 for the same flare strength \citep{Osten15,Gunther20}. Therefore, the CME mass is
\begin{equation}
M_{\rm CME}\simeq 2.1\times10^{16}~{\rm g}\fraction{E_{\rm CME}}{10^{32}~{\rm erg}}{0.63},
\end{equation}
and the CME velocity is estimated as
\begin{equation}
v_{\rm CME}\simeq\fraction{2E_{\rm CME}}{M_{\rm CME}}{1/2}
\simeq980~{\rm km~s^{-1}}\fraction{E_{\rm CME}}{10^{32}~{\rm erg}}{0.185}.
\end{equation}
We define $t_{1/2}$ as the decay time from the peak luminosity to half of that level for a stellar flare. The U-filter flare data show that $t_{1/2}$ depends on the U-band flare energy, i.e. $\log t_{1/2}=0.3\log E_U-7.5$ in cgs unit \citep{Pettersen89}. We assume $E_U=\epsilon_U E_{\rm CME}$ with the efficiency taken as $\epsilon_U\sim0.1$ for U band \citep[e.g.,][]{Osten15}. The flare duration $\Delta t$ is then estimated as
\begin{equation}
\Delta t\sim t_{1/2}\simeq63~{\rm s}\fraction{E_{\rm CME}}{10^{32}~{\rm erg}}{0.3}.
\end{equation}
Therefore, the electron density of the CME at the distance $r$ from the center of the companion star is
\begin{align}
n_{\rm CME}(r)&\simeq\frac{M_{\rm CME}}{4\pi\mu_mm_pv_{\rm CME}r^2\Delta t}\nonumber\\
&\simeq600~{\rm cm^{-3}}\fraction{E_{\rm CME}}{10^{32}~{\rm erg}}{0.145}\fraction{r}{1~{\rm AU}}{-2}.
\end{align}
Different from the scenario of stellar winds, the contributions of DM and RM from the stellar flare depend on the time difference between the stellar flare and FRBs. We consider that an FRB encounter the stellar flare at the distance $r$ from the companion star, as shown in Figure \ref{figstellarflare}. The DM contribution by the flare is approximately
\begin{equation}
{\rm DM}_{\rm CME}\sim n_{\rm CME}r\simeq2.9\times10^{-3}~{\rm pc~cm^{-3}}\fraction{E_{\rm CME}}{10^{32}~{\rm erg}}{0.145}\fraction{r}{1~{\rm AU}}{-1}.
\end{equation}
Similar to the discussion of stellar winds in Section \ref{stellarwind}, the Alfv\'en radius of the stellar flare is given by Eq.(\ref{alfven}), i.e.
\begin{align}
R_A&\sim\fraction{B_c^2R_c^6\Delta t}{M_{\rm CME}v_{\rm CME}}{1/4}\nonumber\\
&\simeq0.6R_\odot\fraction{B_c}{1~{\rm G}}{1/2}\fraction{R_c}{1R_\odot}{3/2}\fraction{E_{\rm CME}}{10^{32}~{\rm erg}}{-0.13}.
\end{align}
Because $r\gg R_A$, the magnetic field in the stellar flare also satisfies $B(r)\propto r^{-\beta_B}$, with $\beta_B$ ranging from 1 to 2. Therefore, the RM contribution by a flare is approximately
\begin{align}
{\rm RM}_{\rm CME}&\sim \frac{e^3}{2\pi m_e^2c^4} Bn_{\rm CME}r\simeq11~{\rm rad~m^{-2}}\nonumber\\
&\times\fractionz{B_c}{1~{\rm G}}\fractionz{R_c}{1R_\odot}\fraction{E_{\rm CME}}{10^{32}~{\rm erg}}{0.145}\fraction{r}{1~{\rm AU}}{-2}.
\end{align}
for a toroidal field with $\beta_B\simeq1$, or
\begin{equation}
{\rm RM}_{\rm CME}\simeq 0.05~{\rm rad~m^{-2}}\fractionz{B_c}{1~{\rm G}}\fractionz{R_c}{1R_\odot}\fraction{E_{\rm CME}}{10^{32}~{\rm erg}}{0.145}\fraction{r}{1~{\rm AU}}{-3}.
\end{equation}
for a radial field with $\beta_B\simeq2$. Notice that the above typical values of RMs are much smaller than those of stellar winds given by Eq.(\ref{RMw1}) and Eq.(\ref{RMw2}). This is because the absolute mass loss rates of stellar flares/CMEs from low-mass stars are much smaller than those of stellar winds from massive/giant stars.

In particular, for a low-mass star with a surface magnetic field $B_c\sim10^3~{\rm G}$ and a radius of $R_c\sim0.1R_\odot$ (due to strong convection near the surface of a low-mass star, its surface magnetic fields are usually stronger than those of a massive star, also see \cite{Kochukhov21}), the RM contributed by its flare at $r\sim1~{\rm AU}$ could reach $\sim10^3~{\rm rad~m^{-2}}$ for a toroidal field.
Therefore, if the large RMs of $\gtrsim10^3~{\rm rad~m^{-2}}$ are contributed by the companion flare, the separation of such a binary system is required to be small, with $a\lesssim1~{\rm AU}$, and each radio burst is required to be emitted just when the CME sweeps the FRB source. These requirements are likely fine turning.
Considering that the burst rate of radio bursts of some FRB repeaters ($\gtrsim 100~\text{bursts}~{\rm day^{-1}source^{-1}}$ for FRB 121102, \cite{Lid21}) is much larger than the rate of stellar flares ($\lesssim10~\text{flares}~{\rm day^{-1}source^{-1}}$ e.g., \cite{Osten15,Davenport16}) and that FRB emission and stellar flaring in a binary system should be physically independent,  the time difference between FRBs and flares should be randomly distributed.
We assume that two radio bursts (Burst A and Burst B) are emitted with a ten-day time delay in a binary system with a separation of $a\sim1~{\rm AU}$ and that the companion star has $B_c\sim10^3~{\rm G}$ and $R_c\sim0.1R_\odot$. Burst A is emitted when the CME just sweeps the FRB source, leading to ${\rm RM}\sim10^3~{\rm rad~m^{-2}}$.
Assuming that the CME velocity is $v\sim1000~{\rm km~s^{-1}}$, the CME would at $r\sim6~{\rm AU}$ when Burst B is crossing the CME. The RM of Burst B is about ${\rm RM}\sim30~{\rm rad~m^{-2}}$. In summary, the RM varies from $10^3~{\rm rad~m^{-2}}$ to $10~{\rm rad~m^{-2}}$ during ten days. Such an extremely RM variation has not been observed in any FRB repeaters, which is inconsistent with the current observations unless flares are more frequent so that Burst B would encounter another newly ejected CME when it is observed.

\subsection{RM variations contributed by pulsar winds, pulsar wind nebulae, or magnetar flares}\label{pair}

In this subsection, we consider the RM contribution from a pair plasma, including a pulsar wind, a pulsar wind nebula, or a magnetar flare. A pulsar wind could be produced by a neutron star as the companion of the FRB source in a binary system (see the panel (a) of Figure \ref{figpulsarwind}) or by the FRB source itself (see the panel (b) of Figure \ref{figpulsarwind}).
In order to place a strong constraint, we first assume that the FRB source is at the center of the pulsar wind in the following discussion (Figure \ref{figpulsarwind}b). If the FRB source significantly deviates from the center of the pulsar wind (Figure \ref{figpulsarwind}a), the corresponding RM would be smaller.
In general, since the pulsar wind is composed of relativistic electron-positron pairs, its RM contribution would be very small as proved below.

\begin{figure}
    \centering
	\includegraphics[width = 1.0\linewidth, trim = 0 150 0 100, clip]{pulsarwind.pdf}
    \caption{Schematic configuration of an FRB source in a pulsar wind. There are two possibilities: (a) the FRB source is in a binary system with a pulsar as its companion. (b) the pulsar wind is produced by the FRB source itslf. The grey circle corresponds to the FRB source, and the blue circle corresponds to the companion pulsar. The blue curves denote magnetic field lines.}\label{figpulsarwind}
\end{figure}

For a neutron star with radius $R$, dipolar magnetic field at the pole $B_p$ and angular velocity $\Omega$, the spin-down power of the neutron star is
\begin{equation}
L_{\rm sd} \simeq \frac{B_p^2R^6\Omega^4}
{6c^3}
\simeq9.6\times10^{36}~{\rm erg~s^{-1}}\fraction{B_{p}}{10^{13}~{\rm G}}{2}\fraction{P}{0.1~{\rm s}}{-4}.
\end{equation}
The magnetic field is nearly dipolar inside the light cylinder $R_{\rm LC}=c/\Omega$ but become toroidal in the pulsar wind. Thus, the field strength at $r>R_{\rm LC}$ is given by
\begin{align}
B(r)&=\frac{B_p}{2}\left(\frac{R}{R_{\rm LC}}\right)^{3}\left(\frac{R_{\rm LC}}{r}\right)\nonumber\\
&=2.2~{\rm G}\fractionz{B_p}{10^{13}~{\rm G}}\fraction{P}{0.1~{\rm s}}{-2}\fraction{r}{10^{13}~{\rm cm}}{-1},\label{B}
\end{align}
where $B_p/2$ is the mean surface magnetic field strength.
The Goldreich-Julian particle ejection rate from the polar cap may be estimated by
\begin{equation}
\dot N_{\rm GJ}=2cA_{\rm cap}n_{\rm GJ}=2.7\times10^{33}~{\rm s^{-1}}\fractionz{B_p}{10^{13}~{\rm G}}\fraction{P}{0.1~{\rm s}}{-2},
\end{equation}
where $n_{\rm GJ}=B_p/Pec$ is the Goldreich-Julian density at the neutron star pole \citep{Goldreich69}, and $A_{\rm cap}\simeq\pi R^3/R_{\rm LC}$ is the area of the polar cap for $R_{\rm LC}\gg R$.
For a pair multiplicity $\mathcal{M}$, the electron/positron number density at distance $r$ is
\begin{align}
n_{e}(r)&=\frac{\mathcal{M}\dot N_{\rm GJ}}{4\pi c r^2}=0.07~{\rm cm^{-3}}\fractionz{\mathcal{M}}{10^3}\nonumber\\
&\times\fractionz{B_p}{10^{13}~{\rm G}}\fraction{P}{0.1~{\rm s}}{-2}\fraction{r}{10^{13}~{\rm cm}}{-2}.\label{n}
\end{align}
We consider that the comoving Lorentz factor of the pulsar wind is $\gamma$, and the pair plasma is hot with thermal Lorentz factor $\gamma_{\rm th}$ in the comoving frame.
For the pair plasma, due to the symmetry of positive and negative charges, its RM would be suppressed by the multiplicity $\mathcal{M}$. In other words, only the net charges contribute to Faraday rotation (see Appendix \ref{RMpair} for details).
Meanwhile, in the comoving frame, the RM contribution from relativistic hot electrons is suppressed by a factor of $\gamma_{\rm th}^2$ due to the relativistic mass $m_e\rightarrow\gamma_{\rm th} m_e$ \citep{Quataert00}.
Therefore, the RM contributed by the pulsar wind is (see Appendix \ref{RMpair} for  details)
\begin{equation}
{\rm RM}=\frac{e^3}{\pi m_e^2c^4}\frac{1}{\gamma_{\rm th}^2\mathcal{M}}\int_{r_c}^dn_eB_\parallel ds,\label{RMw}
\end{equation}
Notice that the above equation does not directly involve the wind Lorentz factor $\gamma$ (see Appendix \ref{RMpair} for detailed reason).
In Eq.(\ref{RMw}), $d$ corresponds to the radius of a pulsar wind nebula, and $r_c$ is the critical radius where the electron cyclotron frequency is equal to the wave frequency in the comoving frame, leading to
\begin{equation}
B(r_c)\sim \frac{2\pi m_ec}{e}\left(\frac{\nu}{\gamma}\right)\simeq3.6~{\rm G}\fraction{\gamma}{100}{-1}\fractionz{\nu}{1~{\rm GHz}}.\label{Brc}
\end{equation}
Different from Eq.(\ref{Bc}), here a factor of $1/\gamma$ is involved due to the Doppler effect of relativistic motion of the pulsar wind and the unchanged parallel field in different frames (see Appendix \ref{RMpair}).
In the region with $r<r_c$, the requirement of $\psi\propto\lambda^2$ for the RM measurement could not be satisfied according to the wave dispersion relation (see Appendix \ref{RMpair}).
According to the equations of $B(r)$ and $n_e(r)$ (Eq.(\ref{B}) and Eq.(\ref{n})), one has
\begin{equation}
\frac{B}{n_e}\simeq30~{\rm G~cm^3}\fraction{\mathcal{M}}{10^3}{-1}\fractionz{r}{10^{13}~{\rm cm}},\label{Bn}
\end{equation}
which is independent of the surface magnetic field and period of the neutron star.
Due to $n_e\propto r^{-2}$ and $B\propto r^{-1}$, most RM is contributed at $r_c$. Thus, according to Eq.(\ref{RMw}), Eq.(\ref{Brc}) and Eq.(\ref{Bn}), the RM contributed by a pulsar wind is estimated by
\begin{equation}
{\rm RM}\simeq2.3\times10^{-3}~{\rm rad~m^{-2}}\gamma_{\rm th}^{-2}\fraction{\gamma}{100}{-2}\fraction{\nu}{1~{\rm GHz}}{2},\label{RMw1}
\end{equation}
which is very small. Note that in the above equation, the frequency-dependent RM is due to the integral lower limit $r_c$. For an FRB with a finite bandwidth between $(\nu_{\min},\nu_{\max})$, the RM measurement based on the condition $\psi={\rm RM}\lambda^2$ implies that $\nu$ in Eq.(\ref{RMw1}) should be replaced by the observed minimum frequency $\nu_{\rm min}$.

Thus, the RM and its variation contributed by a pulsar wind is very small, which is independent of the surface magnetic field, period and multiplicity of the neutron star.
Note that the above discussion assumes that the parallel component of the magnetic field is of the order of the total field, $B_\parallel\sim B$. For the pulsar wind scenario with the field almost perpendicular to the wind velocity \citep[e.g.,][]{Becker09}, the parallel component $B_\parallel$ would be much smaller, leading to an even smaller RM contribution.

We are also interested in the scenarios of pulsar wind nebulae and magnetar flares. A pulsar wind nebula is produced by the interaction between the pulsar wind and the SNR/interstellar medium. In this process, the kinetic energy of the pulsar wind is transferred to thermal energy, i.e., $\gamma\rightarrow\gamma_{\rm th}$. Meanwhile, more pairs could be generated via magnetic reconnection, but the number of the net charges keeps unchanged. Because $n_e\propto r^{-2}$ and $B\propto r^{-1}$, at the pulsar wind nebula radius that is much larger than $r_c$, the RM contribution would be much smaller.

For magnetar flares, since a part of flare energy is transferred to relativistic pairs \citep{Thompson95}, their RM contribution is also expected to be very small for the same reason as in the pulsar winds and pulsar wind nebulae.
The estimated RM contributions from pulsar winds (nebulae) and magnetar flares are also consistent with the observations of most Galactic radio pulsars and radio loud magnetars that have relatively small RMs mainly contributed from the interstellar medium.
In particular, FRB 200428 was produced during the active phase of the magnetar SGR J1935+2152 \citep{Bochenek20,CHIME20}, which was associated with a hard X-ray burst \citep{Mereghetti20,Li20,Ridnaia20,Tavani20}. However, its RM is almost consistent with the value during its radio pulsar phase (Zhu et al. 2022, submitted). This implies that a magnetar flare cannot contribute significantly to RM variations.

\subsection{RM variations contributed by magnetized outflows from a massive black hole}\label{BH}

The extremely large RMs with ${\rm RM}\gtrsim 10^4~{\rm rad~m^{-2}}$ have been observed in the vicinities of massive black holes \citep{Bower03,Marrone07,Eatough13}. For example, the radio-loud magnetar PSR J1745-2900, which resides just $0.12~{\rm pc}$ from Sgr ${\rm A}^\ast$ \citep{Eatough13}, shows a large but relatively stable DM of $1800~{\rm pc~cm^{-3}}$ (consistent with a source located within $<10~{\rm pc}$ of the Galactic center, in the framework of the NE 2001 free electron density model of the Galaxy \citep{Cordes02}) and RM of $10^4-10^5~{\rm rad~m^{-2}}$ with an RM variability of $\sim 3500~{\rm rad~m^{-2}}$ \citep{Desvignes18}. Thus,
it has been suggested that the large RMs observed in some FRB repeaters might be a result that the source is located in the vicinity of a massive black hole \citep{Michilli18,Zhang18,Anna-Thomas22,Dai22}, as shown in Figure \ref{figblackhole}.

\begin{figure}
    \centering
	\includegraphics[width = 1.0\linewidth, trim = 0 0 0 0, clip]{blackhole.pdf}
    \caption{Schematic configuration of an FRB source in the vicinity of a massive black hole generating anisotropic outflow. The central black circle with blue bar corresponds to the mass black hole with accretion disk. The grey circle corresponds to the FRB source moving orbiting the massive black hole. The green regons corresponds to the gas clouds near the massive black hole. The blue curves correspond to the magnetic field lines.}\label{figblackhole}
\end{figure}

Since the wind from a massive black hole is attributed to its accretion, the mass loss rate of the massive black hole can be normalized to the Eddington accretion rate $\dot M_{\rm Edd}$ with a dimensionless parameter $f$,
\begin{align}
\dot M&=f\dot M_{\rm Edd}=\frac{4\pi Gm_p}{\epsilon_{\rm BH}\sigma_Tc}fM_{\rm BH}\nonumber\\
&\simeq2.2\times10^{-3}M_\odot~{\rm yr^{-1}}f\fractionz{M_{\rm BH}}{10^5M_\odot},
\end{align}
where $M_{\rm BH}$ is the black hole mass, and $\epsilon_{\rm BH}\sim0.1$ is the radiative efficiency of a black hole accretion disk. Thus, the electron density at distance $r$ from the massive black hole is
\begin{align}
n_e(r)&\simeq\frac{\dot M}{4\pi\mu_mm_pvr^2}\nonumber\\
&\simeq2\times10^3~{\rm cm^{-3}}f\fractionz{M_{\rm BH}}{10^5M_\odot}\fraction{v}{0.1c}{-1}\fraction{r}{10^{-2}~{\rm pc}}{-2}.
\end{align}
Similar to the discussion on stellar winds (see Section \ref{stellarwind}), the DM contribution is
\begin{equation}
{\rm DM}\sim n_ea
\simeq20~{\rm cm^{-3}}f\fractionz{M_{\rm BH}}{10^5M_\odot}\fraction{v}{0.1c}{-1}\fraction{a}{10^{-2}~{\rm pc}}{-1},
\end{equation}
where $a$ is the separation between the FRB source and the massive black hole,
and the RM contribution is
\begin{align}
{\rm RM}&\sim \frac{e^3}{2\pi m_e^2c^4} B_rn_ea\simeq1.6\times10^4~{\rm rad~m^{-2}}f\nonumber\\
&\times\fractionz{M_{\rm BH}}{10^5M_\odot}\fraction{v}{0.1c}{-1}\fractionz{B_r}{1~{\rm mG}}\fraction{a}{10^{-2}~{\rm pc}}{-1},
\end{align}
where $B_r$ is the field strength at $r\sim a$ from the center of the massive black hole.

When an FRB source is moving near a massive black hole with a Keplerian velocity of
\begin{equation}
v\simeq\fraction{GM_{\rm BH}}{a}{1/2}\simeq210~{\rm km~s^{-1}}\fraction{M_{\rm BH}}{10^5M_\odot}{1/2}\fraction{a}{10^{-2}~{\rm pc}}{-1/2},
\end{equation}
the RM variation is accounted for by the changes of magnetic field or electron density due to orbital motion or the inhomogeneous wind medium similar to the scenarios of binary systems.
Different from the stellar wind scenario, the outflow from a massive black hole might interact with the gas clouds in the vicinity of the massive black hole.
For turbulence in the outflow at the distance $r\sim a$, the typical turbulence outer scale may be estimated as
\begin{equation}
L\sim \min(\theta_w a, l_{\rm cloud}),
\end{equation}
where $\theta_{\rm out}$ is the typical anisotropic distribution angle of the outflow from the massive black hole, and $l_{\rm cloud}$ is the typical size of the clouds for the turbulence induced by the interaction of the outflow with the circumnuclear clouds. If anisotropy of the outflow is significant, $\theta_w a<l_{\rm cloud}$, one has
\begin{equation}
L\sim \theta_w a\simeq10^{-2}~{\rm pc}\fractionz{\theta_{\rm out}}{1~{\rm rad}}\fractionz{a}{10^{-2}~{\rm pc}},
\end{equation}
and the timescale of the FRB source crossing the outer scale $L$ is
\begin{equation}
t_L\sim\frac{L}{v}\simeq47~{\rm yr}\fractionz{\theta_{\rm out}}{1~{\rm rad}}\fraction{M_{\rm BH}}{10^5M_\odot}{-1/2}\fraction{a}{10^{-2}~{\rm pc}}{3/2}.
\end{equation}
If the typical cloud size is small with $l_{\rm cloud}<\theta_w a$, one has
\begin{equation}
L\sim l_{\rm cloud}\simeq10^{-2}~{\rm pc}\fractionz{l_{\rm cloud}}{10^{-2}~{\rm pc}},
\end{equation}
and the time of the FRB source crossing the outer scale $L$ is
\begin{equation}
t_L\sim47~{\rm yr}\fractionz{l_{\rm cloud}}{10^{-2}~{\rm pc}}\fraction{M_{\rm BH}}{10^5M_\odot}{-1/2}\fraction{a}{10^{-2}~{\rm pc}}{1/2}.
\end{equation}
According to Eq.(\ref{sf1}), Eq.(\ref{sf2}) and Eq.(\ref{RMvariation}), one has $|\delta {\rm RM}/{\rm RM}|\propto t^{-(\alpha+2)/2}$ and $D_{\rm RM}(t)\propto t^{-(\alpha+2)}$ for $t\lesssim t_L$; and $|\delta {\rm RM}/{\rm RM}|\sim 1$ and $D_{\rm RM}(t)\sim\text{constant}$ for $t\gtrsim t_L$. At last, if the orbit of the FRB source is elliptical, a large RM variation would occur near the periastron and keep almost constant far away from the periastron, and a significant periodic evolution of RM variation could be tested through long-term monitoring of the FRB sources.

In general, the timescale is too long for the supermassive black hole scenario to interpret the observed short-term RM variations, which requires a scaled down $a$ and $l_{\rm cloud}$. The FRB source must be very close to the black hole in order to have the observed rapid RM variability.

\section{Discussions and Conclusions}\label{discussion}

FRBs are mysterious radio transients with the physical origin still unknown. As cosmological radio transients, their propagating effects (including dispersion, Faraday rotation, temporal scattering, scintillation, depolarization, etc.) are important probes to reveal the physical properties of the astrophysical environments the radio waves propagate through, including the near-source plasma, interstellar medium and the intergalactic medium. Different from the DM that is mainly contributed by the intergalactic medium, a large absolute RM value of $\gtrsim(10^2-10^3)~{\rm rad~m^{-2}}$ observed at a high Galactic latitude can be only contributed by the magnetized environment near an FRB source. Furthermore, since the interstellar medium and intergalactic medium are not expected to vary in a short time, an observed RM variation can be only attributed to the dynamical evolution, or the relative motion of the near-source plasma with respect to the FRB source.

Very recently, some FRB repeaters were found to show significant RM variations \citep{Michilli18,Hilmarsson21,Xu21,Anna-Thomas22,Dai22,Mckinven22}, and the relative variation amplitudes of some repeaters reach $|\delta{\rm RM}/{\rm RM}|\sim1$ during a few months to a few years, e.g., FRB 121102 and FRB 190520B \citep{Michilli18,Anna-Thomas22,Dai22}. The RM variations of FRB repeaters reflect that the near-source environments of FRB repeaters are dynamically evolving (e.g., SNR, stellar flare, etc.) or there is a significant relative motion between the FRB source and the environment (e.g., FRB source in a binary system or in the vicinity of a massive black hole). If the magnetized environment is inhomogeneous, when an FRB propagates in the environment, the electromagnetic waves will be depolarized due to the multi-path propagation effects \citep{Beniamini22,Yang22}, which have been recently confirmed by the observations of some FRB repeaters \citep{Feng22}.

In this work, we have investigated some astrophysical processes that may cause RM variations of an FRB repeater, including SNRs, winds and flares from a companion in a binary system, pair plasma (pulsar winds, pulsar wind nebulae and magnetar flares), and outflows from massive black holes, and turbulence induced in these processes.
First, we make a general discussion about the statistical properties of random RM variations. We consider that the power spectrum of the RM density ($n_eB_\parallel$) fluctuations satisfies $P(k)\propto k^{\alpha}$, then the RM structure function is $D_{\rm RM}(t)\propto t^{-(\alpha+2)}$ for $v_\perp t<L$ and $D_{\rm RM}(t)\sim\text{constant}$ for $v_\perp t>L$, where $v_\perp$ is the transverse relative velocity between the FRB source and the environment, and $L$ is the outer scale of the inhomogeneous medium.
During the observing time $t$, the relative RM variation is $|\delta {\rm RM}/{\rm RM}|\sim(v_\perp t/L)^{-(\alpha+2)/2}$ for $v_\perp t<L$ and $|\delta {\rm RM}/{\rm RM}|\sim1$ for $v_\perp t>L$. The measurements of the RM structure function of some FRB repeaters reveal that \citep{Mckinven22} $D_{\rm RM}(t)\propto t^{0.2-0.4}$, leading to $\alpha\sim-(2.2-2.4)$, which implies that the power spectrum of the RM density fluctuations is shallow, and the RM variations are mainly contributed by the electron density fluctuations at small scales. On the other hand, the large relative RM variations of FRB 121102 and FRB 190520B imply that the outer scale of the fluctuations is $L\lesssim v_\perp t\simeq10^{-4}~{\rm pc}(v_\perp/100~{\rm km~s^{-1}})(t/1~{\rm yr})$ for observations of a few years.

On the other hand, secular RM evolution could be attributed to the expansion of a  magnetized shell (e.g., SNR, stellar flares, etc.) or the orbital motion of a binary system (e.g., an FRB source in the stellar winds of a companion in a binary system or in the outflows of a massive black hole). The former scenario predicts that the RM exhibits a long-term monotonic evolution, and the latter scenario suggests that a periodic RM evolution could be detected when the companion has a large-scale strong magnetic field, especially if the orbit is elliptical.

The long-term evolution of the RM contributed by an SNR has been discussed in some previous papers with the assumption that the geometry of magnetic fields along the line of sight keeps unchanged \citep[e.g.,][]{Piro18,Zhao21b}. However, such a model cannot explain the non-monotonic irregular RM evolutions exhibited by some FRB repeaters. In this work, we consider that the medium in an SNR is inhomogeneous due to turbulence and instabilities, and the irregular RM variation is due to the relative motion between the FRB source and the SNR.
We find that
there are two possibilities to contribute a large relative RM variation during a few years: 1) the SNR is young with an age of a few hundreds years, meanwhile, the SNR shell must be thin compared with its radius and appears significantly anisotropic locally. In addition, a young SNR would predict observable secular evolution of RM and DM, which can be tested in future observations. 2) The SNR could be relatively older, and the size of dense clouds around the SNR is extremely small. However, such small-scale clouds have not been detected due to the limited resolution of observations.

The significant RM variations can also be contributed by the medium from the companion in a binary system. Meanwhile, some evidence suggests that the FRB source might be in a binary system (see a detailed discussion in Section \ref{stellarwind}). When an FRB repeater is in a binary system, the RM variation could be caused by the orbital motion of the binary system or the dynamical evolution of the medium from the companion.
For a persistent stellar wind,
we consider that the RM variation is due to the inhomogeneity arising from the turbulence in the anisotropic distribution of the stellar wind. We find that in order to explain the large RM of $\sim(10^4-10^5)~{\rm rad~m^{-2}}$ of some FRB repeaters, the mass loss rate of the companion wind is required to be $\dot M\sim 10^{-8}M_\odot~{\rm yr^{-1}}$ if the FRB source is at $r\sim1~{\rm AU}$. Such a large mass loss rate implies that the companion is a massive main sequence star or a giant star. Meanwhile, for a binary system with a mass $M_{\rm tot}\sim10M_\odot$, a separation $a\sim1~{\rm AU}$, and the companion wind with an anisotropic distribution angle $\theta_w\sim1~{\rm rad}$, the relative RM variation could be $|\delta {\rm RM}/{\rm RM}|\sim1$ and $D_{\rm RM}(t)\sim\text{constant}$ for the observing time longer than a few weeks.
In particular, if the binary orbit is elliptical, a large RM variation would occur near the periastron, and a periodic evolution of RM variation is expected \citep{Wang22}.

Different from the stellar wind case, stellar flares are catastrophic release of magnetic energy and are accompanied by CMEs, which are more frequent in low-mass stars. Based on the observed empirical relations of the stellar flares, we calculated the CME RM contribution in a binary system. We found that the RM is almost independent of the CME energy, but is more related to the companion's surface magnetic field and the position where the FRB crosses it.
Although a large RM value can be generated if the companion is a low-mass star with a strong field and a small separation from the FRB source, a large RM variation is expected during short terms due to the different positions of the CME where the FRB crosses through at different times. The current observation seems not to support such a scenario unless the flares are very frequent.

In the above discussion, the RM is mainly considered to be contributed by the cold non-relativistic magneto-ionic (ion+electron) plasma. In some astrophysical scenarios, including pulsar winds, pulsar wind nebulae and magnetar flares, the plasma is composed of relativistic pairs.
Due to the symmetry of positive and negative charges, the Faraday rotation effect would be canceled, and only the net charges contribute the RM (see Appendix \ref{RMpair} for details). On the other hand, the relativistic motion of electrons would significantly suppress the RM due to the large kinetic mass. Therefore, pulsar winds, pulsar wind nebulae and magnetar flares cannot contribute significantly to RM and RM variations. This is consistent with observations of most Galactic pulsars and magnetars, especially for the observations of FRB 200428 and radio pulses from SGR J1935+2154.

At last, we discussed the RM contribution by the plasma near a massive black hole. The extremely large RMs with ${\rm RM}\gtrsim 10^4~{\rm rad~m^{-2}}$ have been observed in the vicinity of massive black holes \citep{Bower03,Marrone07,Eatough13}, which has been proposed to be the environment of some FRB repeaters with extremely large RMs. In such a scenario, the random RM variation can be due to the turbulence in the anisotropic distribution of the outflow from the massive black hole or by the interaction between the outflow and nearby clouds.
Similar to the stellar wind scenario, if the orbit of the FRB source is elliptical, a large RM variation would occur near the periastron, and a periodic evolution of RM variation is expected for long-term monitoring. It is worth noting that the mass of the massive black hole cannot be too large in this scenario, because many FRB repeaters were localized at positions far from the centers of their host galaxies \citep{Chatterjee17,CHIME20b,Xu21}.

\section*{Acknowledgements}
We acknowledge helpful discussions with Shi Dai, Yi Feng, Kejia Lee, Di Li, Dongzi Li, Qiao-Chu Li, Fa-Yin Wang, Wei-Yang Wang, Zhao-Yang Xia, and Yong-Kun Zhang.
YPY is supported by National Natural Science Foundation of China grant No. 12003028 and the China Manned Spaced Project (CMS-CSST-2021-B11).

\section*{Data Availability}
This theoretical study did not generate any new data.

\bibliographystyle{mnras}

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Issey Miyake 三宅一生 is a Japanese fashion designer, known for his technology-driven clothing design.
I also like Issey Miyake Spring 2018 in Paris Fashion Week, even the brand is leading by Yoshiyuki Miyamae since 2012. The performance by 3 dancers in Issey Miyake new collections showed the essence of the relationship between body and fabric, make me notice only the texture, shape and forms, but also the space between body and fabric through the movement. I'd love to try to shoot my body movement in my garment at the end of the project.
Among these research, I like these five design and photography particularly, 1st for the silhouettes, 2nd for the different pieces combining to one piece of work, 3rd for grotesque shape.
Besides, what I learned the most from Issey Miyake is about the space between the human body and clothes, to build the relationship between me and the garment which representing me. And Irving Penn's photography also made me start to consider the way how to shoot the relationship between me and the clothes.


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Kevin says he's done Taking crap from anyone He's just tryin to have some fun Playing Rock and Roll But when he sings his song There's notes in there that don't belong Even so it's pretty strong 'Cause he sings with soul It may not go like it should But it's still pretty good We're playing music that nobody knows Songs about dinosaurs, songs about UFOs Playing songs that no-one's ever heard Kevin says it's cool If you make up your own words When his mom comes round She screams "turn that racket down!" Kevin says it proves our sound Is getting pretty close Up and down the block Neighbors they can't help but talk And anyone who wants to rock Will get their daily dose Rehearsal never stops Till someone calls the cops Playing music that nobody knows Songs about dinosaurs, songs about UFOs Playing songs that no-one's ever heard Kevin says it's cool If you make up your own words Playing music that nobody knows Songs about manticores, songs about TV Shows Playing songs that no-one's ever heard Kevin says it's cool If you make up your own words Songs about dinosaurs, songs about UFOs Songs about superstrings, songs about portmanteaus Songs about baby corn, songs about potatoes Kevin says it's cool if you make up your own words
Success 04:11
I've been trying so hard just to open this can I throw it down upon the floor, to go get a bandaid for my hand Then it's open and I'm not prepared for peaches on my feet. And I stumble, and trip on the cat and knock over the bleach. And if that's what success feels like, great. I'll take it. 'Cause I've never had it so good. I've been trying so hard. just to get my son to poop. Then he's jumping on my chest and I finally hear the sound of doom. Then explosion, and I'm wiping, but the wipes are almost through. Then my son says, "I'm sorry papa, but I've got more for you." And if that's what success smells like, great. I'll take it. 'Cause I've never had it so good. I've been trying so hard to be a more professional guy. I wear a button down shirt... with an unbuttoned fly. When I notice, I tell my students I need to go and wash my hands and in the bathroom, I realize that I'm wearing my wife's pants. And if that's what success looks like, great. I'll take it. 'Cause I've never had it so good. Stood up too fast, got dizzy, fell down, and now I'm on the floor. But at least I'm not in that uncomfortable chair no more. So, score. And I can hear the crowds roar: If that's what success feels like, great. I'll take it. 'Cause I've never had it so good. If that's what success feels like, great. I'll take it. 'Cause I've never had it so good.
Growing Up Weird 02:47
All the students and the teachers Sittin up in the bleachers Cheering for every single score And you there in the back Wondering what all this fuss is for There may never come a day when you just blend in You may never come around to where they're at And you may never feel normal But you might feel better Soon as you decide you're okay with that CHORUS Growing up weird, growing up weird Maybe it's not so bad Not as bad as you feared And you're thinking that you're growing up very strangely Every day in school you're Feelin so peculiar Crash-landed from a distant planet Gettin the evil eye Whenever you walk by and who can stand it? Everyone who's anyone stood out like sore thumbs Heroes are weirdos who refuse to quit And you may never feel normal But you might feel alright Soon as you decide just to roll with it CHORUS CHORUS
I Inherited Magic 03:01
I built a cardboard house, wallpapered in gay porn, as big as my mom and dad's garage. I tried to reason with a screaming naked man who chased my friends out of a house I inherited magic. I gave a woman I was dating one of my teeth. And I went to a bar dressed as the Pope. I hosted fencing parties on the roof of my old apartment building I inherited magic. One time I couldn't sleep I couldn't sleep for 2 weeks I felt a dread so deep I was on another plane. The therapist told me it was manic / bipolar And that was grandpa's disorder You might think it was tragic but maybe I inherited.... I inherited magic.
Don't Burn Books 03:07
Don't burn books burn your iPad instead Throw some kindles on the fire 'till it's all just twisted melting wire (x2) It's not uncommon to catch myself having stared at my laptop all day long and I think, Why don't I read a book or go outside Don't burn books burn your iPad instead throw some kindles on the fire 'till it's all just twisted melting wire Don't burn books burn your iPhone instead crush your android with some pliers till it's all just twisted mangled wire Somehow I don't really feel like checking my e-mail or even my facebook page I'd rather write an old fashioned letter or maybe just go out and play Don't burn books burn your iPad instead throw some kindles on the fire 'till it's all just twisted melting wire Don't burn books. Smash your laptop instead You can't punch me in the Facebook when I'm outside riding bikes instead Don't burn books burn your iPhone instead crush your android with some pliers till it's all just twisted mangled wire…. Turn it Turn it off (etc.) Sometimes I think that I can't turn it off anymore (x4) Don't burn books burn your iPad instead Throw some kindles on the fire 'till it's all just twisted melting wire (x2)
Special 02:00
I get it. You want to be special. And not just different from everyone else. No no! You want to be CHOSEN. You want Hagrid to knock on your homely lonely meager Homo sapiens -- ADVENTURE! Maybe put you in DANGER. But then you'd save yourself with that thing that makes you SPECIAL. You wanna know you're SPECIAL. And you can't just hear it from you or your mother. But from some credible external other: A CERTIFICATION OF SPECIALNESS You don't even need to be rich or happy. Your luck doesn't have to be good. You just want a role too important to cut from the play. And if you can't be special like that what then? Resign yourself to the consolation of a contribution to a beautiful world before you are forgotten. ooh ooh ooh ooh... We all want to be, something special We all want, we all want to be-- Because you wanna be --
The Dinosaur 02:14
the dinosaur the dinosaur he didn't know the meteor the meteor would lay them low the burning rock, the aftershock the cloud of dust to block the sun, and everyone could not adjust the end came for the dinosaur upon the Earth. Behold, the noble stegosaur More than just a metaphor Once a real true science fact Before the big rock made impact Into the Gulf of Mexico over 60 million years ago the dinosaur the dinosaur he didn't know the meteor the meteor would lay them low the Big Kaboom, The shroud of Doom the darkened sky The scene went bad And so they had to say goodbye Extinction for the dinosaur Upon the Earth
Go To Sleep 01:35
go to sleep	cause you're driving me crazy close your eyes	cause we're done reading stories I don't care if you're tired because I'm tired If you don't get some sleep You'll be a nightmare go to sleep	cause you're driving me crazy close your eyes	cause we're done reading stories I know this world is too new For you to look away Wait and see It'll be There in the morning go to sleep	cause you're driving me crazy
Finally 03:01
Finally finally, it's all free! Finally finally all for me There's nobody else around and there isn't gonna be ... all for me shop windows full of smart phones and smart tvs finally. no more traffic no more frantic people crowding me ...all for me shop windows full of sneakers and jewelry finally all for free all for me... for only me only me
Let's just say right up front that we really like this album and we want you to have it even if you can't afford to pay for it or just plain don't want to! However, if you DO want to give us money, we also support that choice.
Making these songs was a labor of love, and offered a chance to break new ground; this is our first album to feature drums and bass, for starters.
The songs are about having kids and also about being a kid. And dinosaurs!
Weird doesn't mean bad.
The Half Brothers are:
John Ackermann: Vocals, Mandolin, Bass, Accordion.
Rick Miller: Vocals, Guitars.
David Nixon: Vocals, Banjo, Bass, Drums, Programming.
Ron Carnell played drums on tracks 1,2,3,and 7.
Mike Catts played upright bass on tracks 3 and 9.
David played drums on tracks 4 and 6.
Nathan Mesnard engineered recording for Mr. Catts' bass tracks.
Daniel Spils engineered recording Mr Carnell's drum tracks and David's drums on track 4.
David took the lead in mixing this album, with lots of help from Rick and John.
Digital mastering by Dave Pascal.
Thanks to Pete Remine for production assistance.
Additional vocal on "Kevin Says" by Laura McCabe.
Additional vocal on "Go To Sleep" by Ilsa Ackermann.
Cover photo by David's Mom.
Cover handwriting by Theo Ackermann.
Special thanks to Laura, Jared, Allegra, Sarah, and Jules
Copyright 2018 by the Half Brothers all rights reserved.
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MINSK, 28 October (BelTA) – Belarus is sincerely interested in the establishment of peace and stability in the region, Belarusian Minister of Foreign Affairs Vladimir Makei said in an interview to Russia's Kommersant newspaper, BelTA informs.
According to the minister, the European Union for Belarus is the second main trade and economic partner after Russia. "Our aspiration to have good relations with the EU can be explained by a simple wish to have normal trade and economic relations. Our economy is export-oriented. We have to sell up to 60-70% of the country's output abroad and we are interested in markets. However, we will never do it at the expense of or to the detriment of Russia. It would be immediately evident if we tried to agree with Europe, bypassing Moscow and infridging its interests. We have never made such decisions," the Belarusian Minister of Foreign Affairs said.
"We do not have any rosy expectations that all the doors will open tomorrow and the European Union will welcome Belarus with open arms. However, there is a tendency towards the normalization of relations," Vladimir Makei noted.
Apart from that, the Belarusian Minister of Foreign Affairs stated that the placing of a Russian airbase in the country would not help to lower military and political tensions in the region. "Conversely, the airbase will lead to irritation towards both Minsk and Moscow. Besides, Belarus has enough airbases to deploy hundreds of aircraft in a matter of hours. What is more important for us at present is to discuss the readiness to give a rapid response in case of increasing tensions in the region or a threat to the security of Belarus and Russia," said the minister. Vladimir Makei added that on 27 October the joint board of the Ministries of Foreign Affairs of Belarus and Russia did not discuss the placing of a Russian airbase in the country.
Vladimir Makei was also asked to comment on the situation in Syria. The minister said that Belarus supports Russia's counterterrorism campaign in this Middle Eastern country. "Russia is taking on a really tough challenge there, which is terrorism that poses a threat to the whole humankind. It is a common goal and it is impossible to achieve it without the efforts of all the states. We were surprised that Russia was not invited into the Western coalition when it was set up. Now we see that Russia has made its conclusions from the Libyan history and, let us say, decided to act in Syria, in full compliance with international law," said the minister.
Vladimir Makei added that Belarus had always spoken in favor of normalizing the situation in Syria through a domestic political dialogue with the participation of incumbent President Bashar Assad. "In February 2015 I paid an official visit to Syria and held a meeting with Bashar Assad. Then he stated that he was ready for such a dialogue. At the same time we can state unambiguously that there is foreign interference in Syria. Otherwise, the situation in the country would have been back to normal long time ago," Vladimir Makei added.