# Parametric decay instability of circularly polarized Alfvén wave in
magnetically dominated plasma

Wataru Ishizaki  0000-0002-7005-7139<EMAIL_ADDRESS>Astronomical Institute, Graduate School of Science, Tohoku University, Sendai
980-8578, Japan  Frontier Research Institute for Interdisciplinary Sciences,
Tohoku University, Sendai 980-8578, Japan  Center for Gravitational Physics,
Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa
Oiwake-cho, Sakyo, Kyoto 606-8502, Japan  Kunihito Ioka  0000-0002-3517-1956
<EMAIL_ADDRESS>Center for Gravitational Physics, Yukawa
Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwake-cho,
Sakyo, Kyoto 606-8502, Japan

###### Abstract

We investigate parametric decay instability (PDI) of circularly polarized
Alfvén wave into daughter acoustic wave and backward Alfvén wave in
magnetically-dominated plasma, in which the magnetization parameter $\sigma$
(energy density ratio of background magnetic field to matter) exceeds unity.
We analyze relativistic magnetohydrodynamics (MHD), focusing on wave
frequencies sufficiently lower than the plasma and cyclotron frequencies. We
derive analytical formulae for the dispersion relation and growth rate of the
instability as a function of the magnetization $\sigma$, wave amplitude
$\eta$, and plasma temperature $\theta$. We find that PDI persists even in
high magnetization $\sigma$, albeit with a decreased growth rate up to
$\sigma\to\infty$. Our formulae are useful for estimating the decay of Alfvén
wave into acoustic wave and heat in high magnetization $\sigma$ plasma, which
is a ubiquitous phenomenon such as in pulsars, magnetars, and fast radio
bursts.

††preprint: APS/123-QED

##  I Introduction

Fast radio bursts (FRBs) are the most luminous radio transients in the
universe with millisecond duration [1, 2, 3, 4, 5, 6]. The identification of
the host galaxies [7, 8, 9, 10, 11, 12, 13, 14, 15] and the excess of the
dispersion measure over the free electron density in the Galaxy [1, 2, 3] have
shown that almost all of the sources are at cosmological distances. FRBs have
very high brightness temperatures, suggesting that such emission is due to a
coherent mechanism, i.e., a nonlinear process in which many particles emit
photons of the same phase simultaneously, rather than a superposition of
individual particle emissions [16, 17, 18, 19]. The mechanism of such high-
intensity emission is still unresolved and remains an important problem in
astrophysics [20].

The mechanism and the source of FRBs are not known [19, 21, 22]. The discovery
of an FRB-like burst, FRB200428, coinciding with an X-ray burst from the
Galactic magnetar SGR 1935+2154 has made highly magnetized neutron stars
promising candidates for the source objects of FRBs [23, 24, 25, 26, 27, 28,
29]. Two models are intensively debated regarding the emission region. One is
that the origin of FRBs is the maser emission in strongly magnetized
relativistic shock waves generated outside the magnetosphere [30, 31, 32]. It
has been pointed out that this model has the difficulty that the model
conflicts with the observation of the polarization [33, 34, 35] and the short
time variability in some bursts [36, 37]. The other model is that the energy
of the magnetic field of the neutron star is released and converted into
emission inside the magnetosphere [38, 39, 40, 41, 42]. Since the energy
density of magnetic fields is strong near the surface of stars, magnetic
reconnection [43, 44, 45, 46, 47, 32, 48, 49] or starquake [50, 43, 42, 44,
51] at the surface of stars are discussed as possible energy sources. Even
though we consider phenomena occurring within the magnetosphere of a neutron
star, the emission site remains uncertain. If FRBs are generated in the
magnetosphere near the stellar surface, the difficulty has been pointed out
that their highly intense photons cannot escape from the magnetosphere [52,
53, 54, 41, 55, 56, 57, 58]. Therefore, the energy released at the stellar
surface needs to be transported to an outer emission region and converted into
radio waves [42, 59, 60, 61].

On the surface of the neutron star, the energy should be released in the form
of magnetic pulse (including Aflvén wave) [42, 55, 59, 60, 62, 49] or
thermalized fireball [63, 64, 65, 66]. This paper specifically examines the
cases where the release occurs in the form of Alfvén wave. Fluctuations in the
magnetic field propagate through the magnetosphere as Alfvén waves (although
fast magnetosonic waves are also conceivable, they can traverse the magnetic
field and, thus, should disperse). In the neutron star magnetosphere, the
energy density of the magnetic field significantly surpasses that of the
plasma, so the Alfvén waves are expected to heat the plasma to relativistic
energy as they dissipate, and then, to generate a bunch of charged particles,
which might lead to coherent emission via some plasma instabilities [42, 59].
This expectation has motivated several FRB models using Alfvén waves. Since
the amplitude of Alfvén waves in the magnetosphere increases as they propagate
outward, nonlinear effects during the propagation are important [55, 42, 59,
60, 61]. It remains a non-trivial question whether these large amplitude waves
can traverse the magnetosphere without significant dissipation before reaching
the FRB emission region.

Finite amplitude Alfvén waves decay through the excitation of daughter
acoustic waves and daughter Alfvén waves [67, 68, 69]. This process is one of
the parametric instability, particularly one of three-wave resonant
interactions. In the context of plasma physics, it is known as induced
Brillouin scattering. In this paper, we call this process parametric decay
instability (PDI). In the non-relativistic regime, this phenomenon has been
extensively explored in the context of the solar coronal heating problem and
the mechanism for launching the solar wind [70]. When Alfvén waves propagate
from the chromosphere into the corona, the PDI generates daughter acoustic
waves, which immediately heat the coronal plasma through some dissipative
process (e.g., shock heating) [71, 72, 73]. In the magnetosphere of a neutron
star, in contrast to solar physics, a magnetic field energy density is much
larger than the matter energy density [74, 75, 76, 77, 78]. Although such
systems are often analyzed using the force-free approximation [79, 80, 81],
this approach is inadequate for discussing Alfvén wave stability because this
formulation lacks acoustic waves (i.e., slow waves). Additionally, within the
force-free approximation, an Alfvén wave alone does not decay into lower-
frequency Alfvén or fast waves through three-wave interaction (except for the
interaction with $\omega=0$ waves [82, 83, 84, 85] ). Plasma heating due to
the decay of the Alfvén wave is also considered important in the context of
magnetar X-ray bursts (e.g., [43, 79, 81], and in particular, [86] for
calculation without force-free approximation111 Their calculation [86] is
mainly focused on the macroscopic view of how much the plasma in the
magnetosphere can be heated, rather than on the fundamental processes of the
decay of the Alfven wave, which is our focus. See Section III.6 for a
comparison of their results with ours. ). Consequently, an investigation of
the stability of Alfvén waves in magnetically-dominated plasmas, without
imposing the force-free approximation, is essential.

The PDI in magnetically-dominated plasmas remains largely unexplored.
Matsukiyo & Hada (2003) [87] investigated the stability of Alfvén waves in
two-fluid pair plasmas, providing analytical expressions for the growth rate
of the instability. However, as we will see later, the expression they
obtained is inaccurate when the sound speed of plasma is sufficiently slow or
fast (see equation (76) and the following text for details). Lopez et al.
(2012) [88] incorporated full relativistic effects, but their work lacks an
analytical discussion of the instability growth rate, impeding a systematic
understanding. Deeply in the neutron star magnetosphere, the plasma frequency
$\omega_{p}$ and cyclotron frequency $\omega_{c}$ significantly exceed the
frequency of interest, even in comparison to the typical frequency of FRB
emission [41, 63, 89]. Thus, the magnetohydrodynamic (MHD) model, rather than
the two-fluid model, is easy to handle for describing Alfvén wave propagation
in the neutron star magnetosphere.

In this paper, we study the stability of Alfvén waves in magnetically-
dominated plasmas based on the MHD model, aiming to achieve a systematic
understanding of the timescale of the instability in the neutron star
magnetosphere. We investigate the stability of Alfvén waves propagating
parallel to the background magnetic field in the fluid’s rest frame. For
simplicity, we analyze the 1-D case (i.e., both parent and daughter waves are
propagating parallel to the background magnetic field) and focus on circularly
polarized plane waves as parent Alfvén waves, which are easy to handle in
analytical treatment due to the existence of exact analytical solutions. We
determine the growth rate of the PDI and elucidate its dependence on system
parameters, particularly the magnetization parameter $\sigma$ (the ratio of
magnetic to matter energy density). The structure of this paper is as follows:
In Section 2, we formulate the governing equations for perturbations added to
the exact solution of Alfvén waves with finite amplitude. Section 3 derives
the dispersion relation from the perturbation equations, leading to the
determination of the growth rate of the PDI. Section 4 summarizes the obtained
results, discusses the interpretation and application of our findings, and
outlines future prospects.

##  II Formulation

###  II.1 Basic equations

In this paper, we analyze the stability of the Alfvén wave in the initially
rest frame of plasma (we call it the reference frame). We assume that the
frequencies of waves considered in this paper are sufficiently smaller than
both the cyclotron frequency and the plasma frequency, and that the plasma is
quasi-neutral. Therefore, we adopt the ideal magnetohydrodynamic (MHD)
equation as the governing equation. Furthermore, for simplicity, in this
paper, we only consider the case where the waves propagate in a direction
parallel to a constant magnetic field oriented in the $z$ direction. When
there is no change of physical quantities in the $x$ and $y$ directions (i.e.,
we consider a 1D problem), the special relativistic MHD equations can be
written as follows:

$\frac{1}{c}\frac{\partial}{\partial
t}\left[(\epsilon+p)\gamma^{2}-p+\frac{1}{8\pi}\left(E^{2}+B^{2}\right)\right]+\frac{\partial}{\partial
z}\left[(\epsilon+p)\gamma^{2}\beta_{z}+\frac{1}{4\pi}(\bm{E}\times\bm{B})_{z}\right]=0,$
(1) $\frac{1}{c}\frac{\partial}{\partial
t}\left[(\epsilon+p)\gamma^{2}\beta_{z}+\frac{1}{4\pi}(\bm{E}\times\bm{B})_{z}\right]+\frac{\partial}{\partial
z}\left[(\epsilon+p)\gamma^{2}\beta_{z}^{2}-\frac{1}{4\pi}\left(E_{z}^{2}+B_{z}^{2}\right)\right]+\frac{\partial}{\partial
z}\left[p+\frac{E^{2}+B^{2}}{8\pi}\right]=0,$ (2)
$\frac{1}{c}\frac{\partial}{\partial
t}\left[(\epsilon+p)\gamma^{2}\bm{\beta}_{\bm{xy}}+\frac{1}{4\pi}(\bm{E}\times\bm{B})_{xy}\right]+\frac{\partial}{\partial
z}\left[(\epsilon+p)\gamma^{2}\bm{\beta}_{\bm{x}\bm{y}}\beta_{z}-\frac{1}{4\pi}\left(\bm{E}_{\bm{xy}}E_{z}+\bm{B}_{\bm{xy}}B_{z}\right)\right]=0,$
(3) $\frac{1}{c}\frac{\partial\bm{B}_{\bm{x}y}}{\partial
t}=[\nabla\times(\bm{\beta}\times\bm{B})]_{xy},$ (4)

where $\epsilon$ is the energy density (including the rest mass energy
density) of matter in the fluid rest frame, $p$ is the pressure, $\beta$ is
the 3-velocity normalized by the speed of light $c$,
$\gamma=(1-\beta^{2})^{-1/2}$ is the Lorentz factor, $E$ and $B$ are the
electric and magnetic fields in the reference frame, respectively. The
subscript $z$ means the component in the $z$ direction and $xy$ means the
projection onto the $xy$ plane. The electric field satisfies the ideal MHD
condition:

$\mbox{\boldmath$E$}=-\mbox{\boldmath$\beta$}\times\mbox{\boldmath$B$}.$ (5)

Note that the magnetic field in the $z$ direction should be constant
independent of $t$ and $z$, as we can see from the $z$ component of the
induction equation and the solenoidal condition
$\nabla\cdot\mbox{\boldmath$B$}=0$. Moreover, note that a stationary and
uniform plasma ($\mbox{\boldmath$B$}=B_{0}\mbox{\boldmath$e$}_{z}$,
$\epsilon={\rm const}$, $p={\rm const}$, and $\mbox{\boldmath$\beta$}=0$) is a
solution to equations (1)–(5). We also assume that the fluid is adiabatic. As
the thermodynamic equation relates between the energy density $\epsilon$ and
the pressure $p$, we adopt the following:

$\beta_{s}^{2}\equiv\left(\frac{\partial p}{\partial\epsilon}\right)_{\rm
ad},$ (6)

where $\beta_{s}$ is the sound speed normalised by $c$ and the subscript ${\rm
ad}$ means the adiabatic change.

###  II.2 Circularly polarized finite amplitude Alfvén wave

There is an eigenmode in equations (1)–(4), which is a circularly polarized
Alfvén wave. A circularly polarised Alfvén wave is a solution to the MHD
equation not only at small amplitudes but also at finite amplitudes. The
circularly polarized Alfvén waves propagating in the direction of the
background magnetic field $B_{0}$ are expressed as

$\bm{B}=\bm{B}_{0}+\delta\bm{B},$ (7) $\left\\{\begin{aligned} \delta
B_{x}&=\delta B\cos(kz-\omega t)\\\ \delta B_{y}&=\delta B\sin(kz-\omega t)\\\
\delta B_{z}&=0\end{aligned}\right.,$ (8)
$\bm{\beta}=\delta\bm{\beta}=-\chi_{0}\beta_{A,\eta}\frac{\delta\bm{B}}{B_{0}},$
(9) $\epsilon=\epsilon_{0},\quad p=p_{0},$ (10)

where $\beta_{A,\eta}$ is the phase velocity of the finite amplitude Alfvén
wave, $\delta\mbox{\boldmath$B$}$ and $\delta\mbox{\boldmath$\beta$}$ are the
magnetic field and velocity field fluctuations associated with the Alfvén wave
respectively, and $\chi_{0}$ is the direction of the wave propagation:

$\chi_{0}={\rm sgn}(k).$ (11)

Furthermore, a dimensionless number representing the amplitude of the wave is
introduced as follows:

$\delta B=\eta B_{0}.$ (12)

Substituting these ansatz into equations (1)–(4), we obtain the following from
the $x$ component of the induction equation (4):

$\omega=\chi_{0}\beta_{A,\eta}ck.$ (13)

From the $x$ component of the equation of momentum conservation (3), we obtain

$\left[\left(\epsilon_{0}+p_{0}\right)\delta\gamma^{2}+\frac{B_{0}^{2}}{4\pi}\right]\chi_{0}\beta_{A,\eta}\omega-\frac{B_{0}^{2}}{4\pi}ck=0,$
(14)

where
$\delta\gamma=(1-\delta\beta^{2})^{-1/2}=(1-\eta^{2}\beta_{A,\eta}^{2})^{-1/2}$.
Note that the same equations can be obtained from the $y$ component of the
equations (3) and (4). By eliminating $\omega$ and $k$ from equations (11) and
(12), we obtain the phase velocity $\beta_{A,\eta}$ as follows:

$\beta_{\mathrm{A},\eta}^{2}=\beta_{\mathrm{A}}^{2}\left[\frac{1+\beta_{\mathrm{A}}^{2}\eta^{2}}{2}\left(1+\sqrt{1-\left(\frac{2\beta_{\mathrm{A}}^{2}\eta}{1+\beta_{\mathrm{A}}^{2}\eta^{2}}\right)^{2}}\right)\right]^{-1},$
(15)

where $\beta_{A}$ is a phase velocity of the linear (i.e. infinitesimal
amplitude) Alfvén wave normalized by the light speed, which can be written as

$\beta_{\mathrm{A}}^{2}=\frac{B_{0}^{2}}{4\pi\left(\epsilon_{0}+p_{0}\right)+B_{0}^{2}}.$
(16)

From equation (13), it can be seen that a circularly polarised Alfvén wave
with finite amplitude expressed in the form of equations (7)–(10) is a wave
propagating with a phase velocity expressed in equation (15). The phase speed
of the wave decreases as $\eta$ increases. For $\eta\ll 1$, the phase velocity
can be written as:

$\beta_{\mathrm{A},\eta}=\beta_{\mathrm{A}}\left[1-\frac{1}{2}\beta_{\mathrm{A}}^{2}\left(1-\beta_{\mathrm{A}}^{2}\right)\eta^{2}+\mathcal{O}\left(\eta^{4}\right)\right].$
(17)

As can be seen from this, the phase speed reduction appears on the order of
$\mathcal{O}(\eta^{2})$.

###  II.3 linearization for daughter waves

Figure 1:  Summary of setting perturbations in the governing equations. The
terms enclosed by the red rectangle represent the parent Alfvén wave. Note
that when combined with background $\mathcal{O}(1)$ quantities (i.e.,
background), these $\mathcal{O}(\eta)$ parent waves yield exact solutions to
the MHD equations as represented by equations (7)–(10). The terms enclosed by
the dotted rectangles represent the daughter waves, with the green dotted one
representing the daughter Alfvén wave and the blue dotted one representing the
daughter acoustic waves. In this paper, we investigate whether there are modes
that grow unstable when perturbations represented by dotted lines are added to
finite-amplitude circularly polarized Alfvén wave. In Section II.3,
$\mathcal{O}(\varepsilon)$ is assumed to be small, and equations are derived
within its linear range, with no specific assumption on the magnitude of
$\eta$. In Section II.4 and later, for simplicity, $\eta$ is considered mildly
small, and higher-order terms of $\eta$ are ignored in the perturbation
equations in order $\mathcal{O}(\varepsilon)$.

As seen in the previous section, a circularly polarised Alfvén wave of finite
amplitude is an exact solution of the MHD equation. In this section, we
investigate the stability of Alfvén waves by adding perturbations
$\mbox{\boldmath$b$}_{\perp}$, $\mbox{\boldmath$\beta$}_{\perp}$,
$\epsilon_{\|}$ and $\beta_{\|}$ of magnitude $\mathcal{O}(\varepsilon)$,
expressed in Fig. 1 or in equation (18) below, to the solution of the MHD
equations (7)–(10):

$\left\\{\begin{array}[]{rlrrrr}\bm{B}&=\bm{B}_{0}&+\delta\bm{B}&+\bm{b}_{\perp}&\\\
\bm{\beta}&=&\delta\bm{\beta}&+\bm{\beta}_{\perp}&+\bm{\beta}_{\|}\\\
\epsilon&=\epsilon_{0}&&&+\epsilon_{\|}\end{array}\right.,$ (18)

where $\mbox{\boldmath$b$}_{\perp}$ and $\mbox{\boldmath$\beta$}_{\perp}$ are
the magnetic field and velocity field fluctuations of the component
perpendicular to the $z$ direction (i.e.,
$\mbox{\boldmath$b$}_{\perp}\perp\mbox{\boldmath$B$}_{0}$ and
$\mbox{\boldmath$\beta$}_{\perp}\perp\mbox{\boldmath$B$}_{0}$), respectively,
meaning Alfvén wave-like perturbations. On the other hand, $\epsilon_{\|}$ and
$\beta_{\|}$ are fluctuations of the internal energy density and velocity
parallel to the $z$ direction (i.e.,
$\mbox{\boldmath$\beta$}_{\|}\|\mbox{\boldmath$B$}_{0}$), respectively,
meaning acoustic wave-like perturbations.

Substitute equations (18) into equations (1)–(4) and equate the coefficients
of each order of $\varepsilon$ with $0$. At the $0$-th order of $\varepsilon$,
the non-trivial relations is obtained from equations (3) and (4) as follows:

$\left[\delta\gamma^{2}\left(\epsilon_{0}+p_{0}\right)+\frac{B_{0}^{2}}{4\pi}\right]\frac{1}{c}\frac{\partial}{\partial
t}\left(\delta\mbox{\boldmath$\beta$}\right)-\frac{B_{0}}{4\pi}\frac{\partial}{\partial
z}\left(\delta\mbox{\boldmath$B$}\right)=0,$ (19)
$\frac{1}{c}\frac{\partial}{\partial
t}\left(\delta\mbox{\boldmath$B$}\right)-\frac{\partial}{\partial
z}\left(B_{0}\delta\mbox{\boldmath$\beta$}\right)=0.$ (20)

These equations constitute the wave equation of the parent wave of magnitude
$\mathcal{O}(\eta)$. Note that equations (1) and (2) are trivial at the $0$-th
order. The $1$-st order equations for $\varepsilon$ are as follows (see
Appendix A for derivation):

$\frac{1+\beta_{s}^{2}\delta\beta^{2}}{1-\delta\beta^{2}}\frac{1}{c}\frac{\partial\epsilon_{\|}}{\partial
t}+\left(\delta\gamma^{2}\left(\epsilon_{0}+p_{0}\right)\right)\frac{\partial\beta_{\|}}{\partial
z}=\\\ -\frac{1}{c}\frac{\partial}{\partial
t}\left[\left(2\delta\gamma^{4}\left(\epsilon_{0}+p_{0}\right)+\frac{B_{0}^{2}}{4\pi}\right)\left(\delta\bm{\beta}\cdot\bm{\beta}_{\perp}\right)+\chi_{0}\frac{\delta\beta\delta
BB_{0}\beta_{\|}}{4\pi}\right]+\frac{B_{0}}{4\pi}\left[\bm{\beta}_{\perp}\cdot\frac{\partial}{\partial
z}(\delta\bm{B})+\delta\bm{\beta}\cdot\frac{\partial\bm{b}_{\perp}}{\partial
z}\right],$ (21)
$\left[\delta\gamma^{2}\left(\epsilon_{0}+p_{0}\right)+\frac{\delta
B^{2}}{4\pi}\right]\frac{1}{c}\frac{\partial\beta_{\|}}{\partial
t}+\beta_{s}^{2}\frac{\partial\epsilon_{\|}}{\partial
z}=-\frac{\partial}{\partial
z}\left(\frac{\delta\bm{B}\cdot\bm{b}_{\perp}}{4\pi}\right)+\frac{B_{0}}{4\pi}\left(\delta\bm{B}\cdot\frac{1}{c}\frac{\partial\bm{\beta}\perp}{\partial
t}+\bm{b}_{\perp}\cdot\frac{1}{c}\frac{\partial}{\partial
t}(\delta\bm{\beta})\right),$ (22)
$\left[\delta\gamma^{2}\left(\epsilon_{0}+p_{0}\right)+\frac{B_{0}^{2}+\delta
B^{2}}{4\pi}\right]\frac{1}{c}\frac{\partial\bm{\beta}_{\perp}}{\partial
t}-\frac{B_{0}}{4\pi}\frac{\partial\bm{b}_{\perp}}{\partial
z}=-\delta\gamma^{2}\left(\epsilon_{0}+p_{0}\right)\beta_{\|}\frac{\partial}{\partial
z}(\delta\bm{\beta})\\\
-\delta\gamma^{2}\left(1+\beta_{s}^{2}\right)\epsilon_{\|}\frac{1}{c}\frac{\partial}{\partial
t}(\delta\bm{\beta})-\delta\bm{\beta}\beta_{s}^{2}\frac{1}{c}\frac{\partial\epsilon_{\|}}{\partial
t}+\frac{B_{0}}{4\pi}\frac{1}{c}\frac{\partial}{\partial
t}\left(\beta_{\|}\delta\bm{B}\right)\\\
-2\delta\gamma^{4}\left(\epsilon_{0}+p_{0}\right)\left(\delta\bm{\beta}\cdot\bm{\beta}_{\perp}\right)\frac{1}{c}\frac{\partial}{\partial
t}(\delta\bm{\beta})+\frac{1}{4\pi}\frac{\partial}{\partial
z}\left[\hat{z}\cdot\left\\{\left(\delta\bm{\beta}\times\bm{b}_{\perp}\right)+\left(\bm{\beta}_{\perp}\times\delta\bm{B}\right)\right\\}\right]\left(\delta\bm{\beta}\times\bm{B}_{0}\right)\\\
\quad-\frac{B_{0}}{4\pi}\delta\bm{\beta}\left[\bm{\beta}_{\perp}\cdot\frac{\partial}{\partial
z}(\delta\bm{B})+\delta\bm{\beta}\cdot\frac{\partial\bm{b}_{\perp}}{\partial
z}-B_{0}\frac{1}{c}\frac{\partial}{\partial
t}\left(\delta\bm{\beta}\cdot\bm{\beta}_{\perp}\right)-\chi_{0}\delta\beta\delta
B\frac{1}{c}\frac{\partial\beta_{\|}}{\partial t}\right]\\\
-\frac{\delta\bm{B}\cdot\bm{b}_{\perp}}{4\pi}\frac{1}{c}\frac{\partial}{\partial
t}(\delta\bm{\beta})-\left(\frac{\delta\bm{B}}{4\pi}\cdot\frac{1}{c}\frac{\partial\bm{b}_{\perp}}{\partial
t}\right)\delta\bm{\beta}+\chi_{0}\frac{\delta
B\delta\beta}{4\pi}\frac{1}{c}\frac{\partial\bm{b}_{\perp}}{\partial
t}+\frac{\delta\bm{B}}{4\pi}\left(\frac{1}{c}\frac{\partial\bm{\beta}_{\perp}}{\partial
t}\cdot\delta\bm{B}\right)+\frac{\delta\bm{B}\cdot\bm{\beta}_{\perp}}{4\pi}\frac{1}{c}\frac{\partial}{\partial
t}(\delta\bm{B}),$ (23) $\frac{1}{c}\frac{\partial\bm{b}_{\perp}}{\partial
t}-B_{0}\frac{\partial\bm{\beta}_{\perp}}{\partial
z}=-\frac{\partial}{\partial z}\left(\beta_{\|}\delta\bm{B}\right).$ (24)

###  II.4 linearlization for parent wave

Hereafter, for simplicity, we assume a situation where the amplitude of the
parent wave is not very large (i.e., $\eta\lesssim 1$), and assume that terms
of magnitude $\mathcal{O}(\varepsilon\eta^{2})$ are ignored. In this case,
from equation (17), $\beta_{A,\eta}\sim\beta_{A}$ for 1-st order equations.
Furthermore, ignoring the term $\mathcal{O}(\varepsilon\eta^{2})$ in equations
(21)-(24), we obtain the following equations:

$\frac{1}{c}\frac{\partial\epsilon_{\|}}{\partial
t}+\left(\epsilon_{0}+p_{0}\right)\frac{\partial\beta_{\|}}{\partial
z}=-\frac{1}{c}\frac{\partial}{\partial
t}\left[\left(2\left(\epsilon_{0}+p_{0}\right)+\frac{B_{0}^{2}}{4\pi}\right)\left(\delta\bm{\beta}\cdot\bm{\beta}_{\perp}\right)\right]+\frac{B_{0}}{4\pi}\left[\bm{\beta}_{\perp}\cdot\frac{\partial}{\partial
z}(\delta\bm{B})+\delta\bm{\beta}\cdot\frac{\partial\bm{b}_{\perp}}{\partial
z}\right],$ (25)
$\left(\epsilon_{0}+p_{0}\right)\frac{1}{c}\frac{\partial\beta_{\|}}{\partial
t}+\beta_{s}^{2}\frac{\partial\epsilon_{\|}}{\partial
z}=-\frac{\partial}{\partial
z}\left(\frac{\delta\bm{B}\cdot\bm{b}_{\perp}}{4\pi}\right)+\frac{B_{0}}{4\pi}\left(\delta\bm{B}\cdot\frac{1}{c}\frac{\partial\bm{\beta}_{\perp}}{\partial
t}+\bm{b}_{\perp}\cdot\frac{1}{c}\frac{\partial}{\partial
t}(\delta\bm{\beta})\right),$ (26)
$\mathcal{E}\frac{1}{c}\frac{\partial\bm{\beta}_{\perp}}{\partial
t}-\frac{B_{0}}{4\pi}\frac{\partial\bm{b}_{\perp}}{\partial
z}=-\left(\epsilon_{0}+p_{0}\right)\beta_{\|}\frac{\partial}{\partial
z}(\delta\bm{\beta})-\left(1+\beta_{s}^{2}\right)\frac{B_{0}\epsilon_{\|}}{4\pi\mathcal{E}}\frac{\partial}{\partial
z}(\delta\bm{B})-\delta\bm{\beta}\beta_{s}^{2}\frac{1}{c}\frac{\partial\epsilon_{\|}}{\partial
t}+\frac{B_{0}}{4\pi}\frac{1}{c}\frac{\partial}{\partial
t}\left(\beta_{\|}\delta\bm{B}\right),$ (27)
$\frac{1}{c}\frac{\partial\bm{b}_{\perp}}{\partial
t}-B_{0}\frac{\partial\bm{\beta}_{\perp}}{\partial
z}=-\frac{\partial}{\partial z}\left(\beta_{\|}\delta\bm{B}\right),$ (28)

where

$\mathcal{E}\equiv\epsilon_{0}+p_{0}+B_{0}^{2}/4\pi.$ (29)

Note that all the terms in the right-hand sides of these equations are
$\mathcal{O}(\varepsilon\eta)$, and this implies that $\varepsilon\ll\eta$ is
assumed because $\mathcal{O}(\varepsilon^{2})$ terms are neglected. On the
right-hand side of these equations, there are some terms that did not appear
in the analysis of parametric decay instabilities based on the non-
relativistic MHD equations [67, 68, 69]. These terms stem from (1) the
velocity squared (i.e., $\mathcal{O}(\beta^{2})$), (2) the internal energy
density that can be comparable to the rest mass energy density, and (3) the
displacement current, which were neglected for the non-relativistic regime.

###  II.5 dimensionless equations

To clarify the dominated terms in the equation, we normalize the equation and
obtain a non-dimensionalized equation. The quantity representing the amplitude
of the parent wave is normalized as follows:

$\delta\bm{u}\equiv\frac{\delta\beta}{\beta_{A}},\quad\delta\bm{e}=\frac{\delta\bm{B}}{B_{0}}.$
(30)

Note that this non-dimensionalisation yields
$|\delta\mbox{\boldmath$u$}|^{2}=|\delta\mbox{\boldmath$e$}|^{2}=\eta^{2}$.
Also, since $\mathcal{O}(\eta^{2})$ is ignored here, the normalization scale
of $\delta\mbox{\boldmath$\beta$}$ is $\beta_{A}$ rather than
$\beta_{A,\eta}$. The amplitude of the daughter Alfvén wave of order
$\mathcal{O}(\varepsilon)$ is normalized as follows:

$\bm{u}_{\perp}\equiv\frac{\beta_{\perp}}{\beta_{A}},\quad\bm{e}_{\perp}=\frac{\bm{b}_{\perp}}{B_{0}}.$
(31)

For the daughter acoustic wave of order $\mathcal{O}(\varepsilon)$:

$u_{\|}\equiv\frac{\beta_{\|}}{\beta_{s}},\quad
e_{\|}\equiv\frac{\epsilon_{\|}}{\epsilon_{0}+p_{0}}.$ (32)

Here, we define two dimensionless parameters that characterize this system.
One is the magnetization parameter $\sigma$, defined as follows:

$\sigma\equiv\frac{B_{0}^{2}}{4\pi\left(\epsilon_{0}+p_{0}\right)}.$ (33)

By using $\sigma$, the Alfvén speed can be written as
$\beta_{A}^{2}=\sigma/(1+\sigma)$. The other is the ratio of the sonic
velocity to the Alfvén velocity:

$\theta\equiv\frac{\beta_{s}}{\beta_{A}}.$ (34)

Note that this corresponds to approximately the 1/2 power of a quantity called
”plasma beta” in the non-relativistic case 222 The plasma beta ($\beta_{\rm
pl}$) is typically defined as the ratio of the plasma pressure to the magnetic
pressure (i.e., $\beta_{\rm pl}\equiv p_{0}/(B_{0}^{2}/8\pi)$). The value
derived from this definition and the square of the ratio of the sound speed to
the Alfvén velocity is nearly equal, in fact for a polytropic gas with a
specific heat ratio $\Gamma$ (i.e., satisfying $\Gamma$-law),
$\theta^{2}=(2/\Gamma)\beta_{\rm pl}$. .

Using equations (30)–(34), we obtain following normalized equations:

$\frac{1}{c}\frac{\partial e_{\|}}{\partial t}+\beta_{s}\frac{\partial
u_{\|}}{\partial z}=-\beta_{A}^{2}\frac{1}{c}\frac{\partial}{\partial
t}\left(\delta\bm{u}\cdot\bm{u}_{\perp}\right)-\sigma\delta\bm{u}\cdot\left(\frac{1}{c}\frac{\partial\bm{u}_{\perp}}{\partial
t}-\beta_{A}\frac{\partial\bm{e}_{\perp}}{\partial z}\right),$ (35)
$\frac{1}{c}\frac{\partial u_{\|}}{\partial t}+\beta_{s}\frac{\partial
e_{\|}}{\partial z}=-\theta^{-1}\beta_{A}\frac{\partial}{\partial
z}\left(\delta\bm{e}\cdot\bm{e}_{\perp}\right)+\sigma\theta^{-1}\delta\bm{e}\cdot\left(\frac{1}{c}\frac{\partial\bm{u}_{\perp}}{\partial
t}-\beta_{A}\frac{\partial\bm{e}_{\perp}}{\partial z}\right),$ (36)
$\frac{1}{c}\frac{\partial\bm{u}_{\perp}}{\partial
t}-\beta_{A}\frac{\partial\bm{e}_{\perp}}{\partial
z}=\theta\beta_{A}^{2}\frac{1}{c}\frac{\partial}{\partial
t}\left(u_{\|}\delta\bm{e}\right)-\frac{1}{1+\sigma}\left[\beta_{s}u_{\|}\frac{\partial}{\partial
z}(\delta\bm{u})+\beta_{A}e_{\|}\frac{\partial}{\partial
z}(\delta\bm{e})+\beta_{s}^{2}\frac{1}{c}\frac{\partial}{\partial
t}\left(e_{\|}\delta\bm{u}\right)\right],$ (37)
$\frac{1}{c}\frac{\partial\bm{e}_{\perp}}{\partial
t}-\beta_{A}\frac{\partial\bm{u}_{\perp}}{\partial
z}=-\theta\beta_{A}\frac{\partial}{\partial
z}\left(u_{\|}\delta\bm{e}\right).$ (38)

Here we used the following normalized equation of the order
$\mathcal{O}(\varepsilon^{0})$:

$\frac{1}{c}\frac{\partial}{\partial
t}\left(\delta\bm{u}\right)=\beta_{A}\frac{\partial}{\partial
z}\left(\delta\bm{e}\right),$ (39) $\frac{1}{c}\frac{\partial}{\partial
t}\left(\delta\bm{e}\right)=\beta_{A}\frac{\partial}{\partial
z}\left(\delta\bm{u}\right).$ (40)

The second term on the right side of equation (35) is found to be a small
quantity of $\mathcal{O}(\varepsilon\eta^{2})$ using equation (37). Therefore,
the equation (35) becomes

$\frac{1}{c}\frac{\partial e_{\|}}{\partial t}+\beta_{s}\frac{\partial
u_{\|}}{\partial
z}=-\frac{\sigma}{1+\sigma}\frac{1}{c}\frac{\partial}{\partial
t}\left(\delta\bm{u}\cdot\bm{u}_{\perp}\right).$ (41)

Similarly for equation (36), we have

$\frac{1}{c}\frac{\partial u_{\|}}{\partial t}+\beta_{s}\frac{\partial
e_{\|}}{\partial z}=-\theta^{-1}\beta_{A}\frac{\partial}{\partial
z}\left(\delta\bm{e}\cdot\bm{e}_{\perp}\right).$ (42)

##  III Dispersion relation

As seen in the previous section, the $\mathcal{O}\left(\varepsilon\right)$
perturbation quantities $\mbox{\boldmath$u$}_{\perp}$,
$\mbox{\boldmath$e$}_{\perp}$, $u_{\|}$, $e_{\|}$ follow the linear
differential equations (37), (38), (41), and (42). In this section, we examine
the stability of the wave solutions in these equations by finding the
dispersion relation.

###  III.1 mode expansion

In order to find the dispersion relation, each physical quantity of the wave
is divided into Fourier components. Since the parent Alfvén wave is a
circularly polarized wave with a given wavenumber $k_{0}$ and frequency
$\omega_{0}$, it can be written as

$\delta\bm{e}=\frac{1}{\sqrt{2}}\left(\delta
e_{0}\exp\left(i\phi_{0}\right)\mbox{\boldmath$e$}_{\rm
R}+\text{c.c.}\right),$ (43) $\delta\bm{u}=-\delta\bm{e},$ (44)

where $\phi_{0}=k_{0}z-\omega_{0}t$ is the phase of the parent Alfvén wave,
$\bm{e}_{\rm R}=\left(\bm{e}_{x}+i\bm{e}_{y}\right)/\sqrt{2}$ is the unit
polarization vector for right-handed polarization, $\mbox{\boldmath$e$}_{j}$
($j=x,y,z$) is the unit vector, and c.c. denotes the complex conjugate. Note
that $|\delta e_{0}|^{2}=\eta^{2}$. Here, although equation (43) seems to
represent only right-handed polarization waves, equation (43) simultaneously
represents not only right-handed polarized waves with positive $\omega_{0}$
and positive $k_{0}$ but also negative $\omega_{0}$ and negative $k_{0}$ at
the same time [90]. Furthermore, in the MHD regime (i.e., where the wave
frequency is much smaller than the cyclotron frequency and plasma frequency),
right- and left-handed polarizations are symmetric, so that what happens with
one polarization also occurs with the other polarization. Therefore, it is
sufficient to discuss wave stability for the parent wave expressed by the
equation (43).

For the acoustic wave perturbation, we write the frequency as $\omega$ and the
wavenumber as $k$ as follows:

$e_{\|}=\frac{1}{2}\left(e_{k}\exp(i\phi)+\text{c.c.}\right),$ (45)
$u_{\|}=\frac{1}{2}\left(u_{k}\exp(i\phi)+\text{c.c.}\right),$ (46)

where $\phi=kz-\omega t$ is the phase of the acoustic wave. Since there are
two daughter Alfvén waves produced by the interaction of the acoustic wave and
the parent Alfvén wave, corresponding to Stokes and anti-Stokes waves, they
are written as follows:

$\bm{e}_{\perp}=\frac{1}{\sqrt{2}}\left(e_{+}\exp\left(i\phi_{+}\right)\mbox{\boldmath$e$}_{\rm
R}+\text{c.c.}\right)+\frac{1}{\sqrt{2}}\left(e_{-}\exp\left(i\phi_{-}\right)\mbox{\boldmath$e$}_{\rm
R}+\text{c.c.}\right),$ (47)
$\bm{u}_{\perp}=\frac{1}{\sqrt{2}}\left(u_{+}\exp\left(i\phi_{+}\right)\mbox{\boldmath$e$}_{\rm
R}+\text{c.c.}\right)+\frac{1}{\sqrt{2}}\left(u_{-}\exp\left(i\phi_{-}\right)\mbox{\boldmath$e$}_{\rm
R}+\text{c.c.}\right),$ (48)

where $\phi_{\pm}=\phi_{0}\pm\phi=k_{\pm}z-\omega_{\pm}t$ is the phase of the
daughter Alfvén waves ($-$ for the Stokes wave, and $+$ for the anti-Stokes
wave), and $k_{\pm}=k_{0}\pm k$ and $\omega_{\pm}=\omega_{0}\pm\omega$ are
wavenumbers and frequencies of daughter Alfvén waves, respectively.

Substituting equations (43)–(48) into equation (37), the following equation
for the Fourier component is obtained from the term proportional to
$\exp(i\phi)$:

$\omega e_{k}-c\beta_{s}ku_{k}=\beta_{A}^{2}\omega\left(u_{+}\delta
e_{0}^{*}+u_{-}^{*}\delta e_{0}\right).$ (49)

From equation (38), the following equation is obtained from the $\exp(i\phi)$
component:

$\omega u_{k}-c\beta_{s}ke_{k}=\theta^{-1}c\beta_{A}k\left(e_{+}\delta
e_{0}^{*}+e_{-}^{*}\delta e_{0}\right).$ (50)

Similarly from the $\exp(i\phi_{+})\mbox{\boldmath$e$}_{\rm R}$ component of
equation (41), we obtain the following:

$\omega_{+}u_{+}+c\beta_{A}k_{+}e_{+}=\frac{1}{2}\delta
e_{0}\left[\left(\theta\beta_{A}^{2}\omega_{+}-\frac{1}{1+\sigma}\theta
c\beta_{A}k_{0}\right)u_{k}+\frac{1}{1+\sigma}\left(c\beta_{A}k_{0}+\theta^{2}\beta_{A}^{2}\omega_{+}\right)e_{k}\right],$
(51)

and from the $\exp(-i\phi_{-})\mbox{\boldmath$e$}^{*}_{\rm R}$ component:

$\omega_{-}u_{-}+c\beta_{A}k_{-}e_{-}=\frac{1}{2}\delta
e_{0}^{*}\left[\left(\theta\beta_{A}^{2}\omega_{-}-\frac{1}{1+\sigma}\theta
c\beta_{A}k_{0}\right)u_{k}+\frac{1}{1+\sigma}\left(c\beta_{A}k_{0}+\theta^{2}\beta_{A}^{2}\omega_{-}\right)e_{k}\right].$
(52)

Similarly, from the $\exp(i\phi_{+})\mbox{\boldmath$e$}_{\rm R}$ component of
equation (42), we obtain

$\omega_{+}e_{+}+c\beta_{A}k_{+}u_{+}=\frac{1}{2}\theta
c\beta_{A}k_{+}u_{k}\delta e_{0},$ (53)

and from the $\exp(-i\phi_{-})\mbox{\boldmath$e$}^{*}_{\rm R}$ component,

$\omega_{-}e_{-}^{*}+c\beta_{A}k_{-}u_{-}^{*}=\frac{1}{2}\theta
c\beta_{A}k_{-}u_{k}\delta e_{0}^{*}.$ (54)

Equations (49)–(54) are six equations for six Fourier components
$e_{k},u_{k},e_{+},u_{+},e_{-}^{*},u_{-}^{*}$.

###  III.2 dispersion relation

Hereafter, we adopt the following unit system:

$\omega_{0}=1,\quad k_{0}=1.$ (55)

From the conditions for the existence of nontrivial solutions to
$e_{k},u_{k},e_{+},u_{+},e_{-}^{*},u_{-}^{*}$ in equations (49)–(54), the
dispersion relation is obtained. By equating the determinant of the
coefficient matrix of equations (49)–(54) with $0$, the dispersion relation,
which is the equation relating $\omega$ and $k$, is obtained as follows:

$\displaystyle(\omega-k)(\omega^{2}-\theta^{2}k^{2})\left\\{(\omega+k)^{2}-4\right\\}$
$\displaystyle=$
$\displaystyle\frac{\eta^{2}}{(1+\sigma)^{3}}\left(S_{0}+S_{1}\sigma+S_{2}\sigma^{2}+S_{3}\sigma^{3}\right)$
(56) $\displaystyle+$
$\displaystyle\frac{\eta^{4}k\omega\sigma}{(1+\sigma)^{4}}\left(T_{0}+T_{1}\sigma+T_{2}\sigma^{2}\right),$

where

$\displaystyle S_{0}$ $\displaystyle=$ $\displaystyle
k^{2}(\omega^{3}+k\omega^{2}-3\omega+k),$ (57) $\displaystyle S_{1}$
$\displaystyle=$ $\displaystyle
2S_{0}-\omega^{3}+k\omega^{2}-\theta^{2}k^{2}\left[2k\omega^{2}+(2k^{2}-3)\omega-k\right],$
(58) $\displaystyle S_{2}$ $\displaystyle=$ $\displaystyle
S_{0}-\omega^{3}+k\omega^{2}-\theta^{2}\left[\omega^{5}+2k\omega^{4}+(k^{2}-3)\omega^{3}-3k(k^{2}+1)\omega^{2}-k^{2}(3k^{2}-7)\omega+k^{3}\right],$
(59) $\displaystyle S_{3}$ $\displaystyle=$
$\displaystyle\theta^{2}k\omega(\omega-k)\left[(\omega+k)^{2}-4\right],$ (60)
$\displaystyle T_{0}$ $\displaystyle=$ $\displaystyle k,$ (61) $\displaystyle
T_{1}$ $\displaystyle=$
$\displaystyle-\omega+2k-\theta^{2}(k\omega^{2}+k^{2}\omega-k),$ (62)
$\displaystyle T_{2}$ $\displaystyle=$
$\displaystyle-\omega+k-\theta^{2}(k\omega-1)(\omega+k).$ (63)

Note that since we have only calculated correctly up to the lowest order of
$\eta$ in this paper, expressions of $T_{j}$ ($\mathcal{O}(\eta^{2})$ times
higher order terms than $S_{j}~{}(j=0,1,2,3)$) may change when the calculation
correctly incorporates higher order terms of $\eta$. Note also that although
we used the dispersion relation $\omega_{0}=c\beta_{A}k_{0}$ for the parent
Alfvén wave propagating in the positive $z$ axis in deriving equation (56),
the above expression is identical for a wave propagating in the opposite
direction (i.e. the negative $z$ direction). Equation (56) is an algebraic
equation for $\omega$ and $k$, and its solution represents the wave that can
propagate through the plasma in the presence of parent Alfvén wave.

Figure 2:  Dispersion relation (56) with $\eta=0$, and $\theta=0.2$. The
horizontal axis is the wavenumber normalized by that of the parent Alfvén wave
$k_{0}$ and the vertical axis is the frequency normalized by $\omega_{0}$.
Each line corresponds to the $\mathcal{O}(\varepsilon)$ wave propagating in a
uniform plasma in a uniform magnetic field. The backward propagating Alfvén
wave in the first quadrant represents the daughter Alfvén wave (Stokes wave)
with frequency $\omega_{-}=\omega_{0}-\omega$ and wavenumber $k_{-}=k_{0}-k$.
The backward Alfvén wave in the third quadrant is the daughter Alfvén wave
(anti-Stokes wave) with frequency $\omega_{+}=\omega_{0}+\omega$ and
wavenumber $k_{+}=k_{0}+k$.

For $\eta=0$, equation (56) recovers the dispersion relation for waves
propagating parallel to the magnetic field in a uniform plasma. Figure 2 shows
the solution of equation (56) for $\eta=0$ and $\theta=0.2$. Since the parent
Alfvén wave is propagating in the $z$-axis positive direction, $\omega-k=0$
can be written as $\omega_{\pm}=c\beta_{A}k_{\pm}$ (in the physical unit),
which represents the daughter Alfvén waves (Stokes and anti-Stokes waves)
propagating in the same direction as the parent wave. From $\omega+k-2=0$,
this can be written in terms of the frequency and wavenumber of the daughter
Alfvén wave as $\omega_{-}=-c\beta_{A}k_{-}$, indicating that it is the
daughter Alfvén wave (Stokes wave) propagating in the opposite direction of
the parent wave. Similarly, $\omega+k+2=0$ becomes
$\omega_{+}=-c\beta_{A}k_{+}$, which represents the daughter Alfvén wave
(anti-Stokes wave) propagating in the opposite direction to the parent wave.
Furthermore, $\omega^{2}-\theta^{2}k^{2}=0$ represents acoustic waves
propagating backward or forward in the $z$ direction, as shown by the fact
that $\omega^{2}=\beta_{s}^{2}k^{2}$ in the physical unit.

Figure 3:  Dispersion relation (56) for $\eta=0.2$, $\theta=0.2$, and
$\sigma=10$. Same as Figure 2, the horizontal axis is the wavenumber
normalized by $k_{0}$ and the vertical axis is the real part of the frequency
normalized by $\omega_{0}$. The solid blue curves represent the solutions for
stable waves, and the dotted red lines represent the solutions for unstable
growing waves. It can be seen that the instability occurs at the intersection
point ($k\sim\pm 1.7k_{0}$, $\omega\sim\pm 0.3\omega_{0}$) for the forward
acoustic wave and the backward daughter Alfvén wave (Stokes waves).  Figure 4:
Dispersion relation (56) with $\eta=0.2$, $\theta=0.2$, and $\sigma=10$. The
upper panel shows the real part of the frequency $\omega$ (the same figure as
the first quadrant in Figure 3) and the lower panel shows the imaginary part
of the frequency $\omega$. The horizontal axis is the wavenumber normalized by
$k_{0}$. The horizontal line in the lower panel, indicated by the black thick
dotted line, is the maximum growth rate $\Gamma_{\rm max}$, obtained by the
equation (76), which is in good agreement with the maximum value of the growth
rate obtained by numerical calculation. The vertical dashed-dotted lines in
the upper and lower panels represent the upper and lower limits of the range
of wavenumbers $k$ where instability occurs, as expressed in equation (75).
Figure 5:  Dispersion relation (56) calculated with $\sigma=100$, while $\eta$
and $\theta$ are the same as in Figure 4.

Figure 3 shows the dispersion relation obtained by solving the equation (56)
under the parameters $\sigma=10$, $\eta=0.2$, and $\theta=0.2$. The curves
shown in blue in the figure represent solutions with stable waves, while the
curves shown in red dotted lines represent solutions that have non-zero
imaginary parts (i.e., solutions with unstable growth). In the figure, the
instability occurs at the intersection point of the dispersion relations for
the forward acoustic wave and the backward daughter Alfvén wave (Stokes
waves). Figure 4 shows the real (upper panel) and imaginary (lower panel)
parts of the frequency when the dispersion relation (56) is solved with the
same parameters as in Figure 3. As can be seen from these, even for plasma
with relativistic magnetization (in this case $\sigma=10$), the three-wave
interaction which satisfies the resonance conditions
$\omega_{0}=\omega+\omega_{-}$ and $k_{0}=k+k_{-}$ is unstable.

###  III.3 Non-relativistic limit

In order to check the validity of the obtained dispersion relation, we take
the non-relativistic limit and compare it with the known results. The non-
relativistic limit here means the limit where the velocity scale of the system
is much slower than the speed of light and the rest mass energy density is
much larger than the internal energy density. This can be achieved by taking
the limit of Alfvén velocity $\beta^{2}_{A}\ll 1$, or since
$\sigma=\beta_{A}^{2}/(1-\beta_{A}^{2})$, the limit of $\sigma\to 0$. By
taking the limit in equation (56), we obtain

$(\omega-k)\left(\omega^{2}-\theta^{2}k^{2}\right)\left\\{(\omega+k)^{2}-4\right\\}=\eta^{2}k^{2}\left(\omega^{3}+k\omega^{2}-3\omega+k\right).$
(64)

This is the same dispersion relation as previous results obtained for non-
relativistic MHD equation system [68, 69]. Therefore, in this regime
($\sigma\ll 1$), based on this dispersion relation, the daughter Alfvén and
the acoustic waves grow exponentially.

###  III.4 $\sigma\to\infty$ limit

As another extreme case, let us consider the case where $\sigma\to\infty$. The
dispersion relation (56) leaving only the highest order of $\sigma$ is as
follows:

$(\omega-k)\left\\{(\omega+k)^{2}-4\right\\}\left(\omega^{2}-\eta^{2}\theta^{2}k\omega-\theta^{2}k^{2}\right)=0.$
(65)

Since there are only real solutions of $\omega$ to this equation for any $k$,
$\theta$, and $\eta$, it can be seen that instability does not occur in the
limit of $\sigma\to\infty$.

###  III.5 $1/\sigma$ expansion

As seen in the previous two sections, in the limit of small $\sigma$ (i.e.,
non-relativistic limit), the PDI occurs, but in the limit of
$\sigma\to\infty$, no instability occurs. This fact implies that the
parametric decay instability (PDI) is suppressed as $\sigma$ increases. In
order to investigate this quantitatively, we expand the dispersion relation
with $\sigma^{-1}$ as a small quantity and investigate how the instability is
suppressed up to the first order of $\sigma^{-1}$. Hereafter, the term
$\eta^{4}$ in the dispersion relation is ignored as small, and $\theta\ll 1$
is assumed for simplicity. The dispersion relation (56) expanded to the first
order of $\sigma^{-1}$ is as follows:

$(\omega-k)\left(\omega^{2}-\theta^{2}k^{2}\right)\left[(\omega+k)^{2}-4\right]=\eta^{2}\left[S_{3}+\frac{1}{\sigma}\left(S_{2}-3S_{3}\right)\right].$
(66)

Since this equation is a $6$-th-order algebraic equation, it cannot be solved
analytically in general. Therefore, in order to derive the conditions for the
existence of unstable solutions, we set $\eta$ to be small and expand the
dispersion relation around the following resonance frequency $\omega_{1}$ and
wavenumber $k_{1}$, which satisfies both the dispersion relation
$\omega=\theta k$ for forward acoustic waves and $\omega+k-2=0$ for backward
Alfvén waves (Stokes waves). The resonance frequency $\omega_{1}$ and
wavenumber $k_{1}$ can be written as

$\omega_{1}=\frac{2\theta}{1+\theta},~{}~{}k_{1}=\frac{2}{1+\theta}.$ (67)

We take the frequency $\omega$ and the wavenumber $k$ as follows:

$\omega=\omega_{1}+\delta\omega,~{}~{}k=k_{1}+\delta k.$ (68)

We assume that $\delta\omega$ and $\delta k$ are small quantities of magnitude
about $\mathcal{O}(\eta)$. This will be confirmed later to be correct in the
case where the resonance condition is satisfied. Expanding the left-hand side
of equation (66) to the lowest order of $\eta$, we obtain

$(\omega-k)\left(\omega^{2}-\theta^{2}k^{2}\right)\left\\{(\omega+k)^{2}-4\right\\}\sim-32\theta\delta\omega\left(\delta\omega+\delta
k\right).$ (69)

Note that here we used $\theta\ll 1$ and keep only the lowest order of
$\theta$, but assuming that $\theta$ is greater than $\mathcal{O}(\eta)$.
Similarly, the following equation can be obtained for the right-hand side by
expanding to the lowest order of $\eta$:

$\eta^{2}\left[S_{3}+\frac{1}{\sigma}\left(S_{2}-3S_{3}\right)\right]\sim\frac{8\eta^{2}}{\sigma}.$
(70)

Equating the above results, we thus obtain:

$\delta\omega^{2}+\delta k\delta\omega+\frac{\eta^{2}}{4\sigma\theta}=0.$ (71)

If equation (71) has a solution with an imaginary part, equation (66) has an
unstable growing solution. Regarding equation (71) as a quadratic equation of
$\delta\omega$, the condition that its discriminant is negative gives the
following condition that $\delta\omega$ has an imaginary part, namely:

$-\eta\sigma^{-1/2}\theta^{-1/2}<\delta k<\eta\sigma^{-1/2}\theta^{-1/2}.$
(72)

The imaginary part $\Gamma\equiv{\rm Im}(\delta\omega)$ takes its maximum
value $\Gamma_{\rm max}$ when the resonance condition is satisfied (i.e.
$\delta k=0$) and is as follows:

$\Gamma_{\rm max}=\frac{1}{2}\eta\sigma^{-1/2}\theta^{-1/2}.$ (73)

Note that the validity of the assumption that $\delta\omega$ and $\delta k$
are $\mathcal{O}(\eta)$ can be confirmed from equations (72) and (73). As can
be seen from equations (72) and (73), the instability ceases in the limit of
$\sigma\to\infty$.

###  III.6 Decay rate for general $\sigma$

By using the same method used in section III.5 to equation (56), we can obtain
the growth rate and the conditions under which instability occurs for general
$\sigma$. In this section, we continue to use $\mathcal{O}(\eta)$ as a small
quantity and ignore the term of $T_{j}$ on the right-hand side of the equation
(56). Substituting equation (68) into equation (56) and deriving the lowest
order equation for $\eta$, we obtain:

$\delta\omega^{2}+\left(1-\theta\right)\delta k\delta\omega-\theta\delta
k^{2}+\frac{\eta^{2}\left(1-\theta\right)}{4\theta\left(1+\theta\right)^{2}}\frac{\left[1+\sigma\left(1+\theta^{2}\right)\right]^{2}}{\left(1+\sigma\right)^{3}}=0.$
(74)

The condition for $\delta\omega$ to have a solution with an imaginary part is
as follows:

$-\eta\theta^{-1/2}\frac{\sqrt{1-\theta}}{\left(1+\theta\right)^{2}}\frac{1+\sigma\left(1+\theta^{2}\right)}{\left(1+\sigma\right)^{3/2}}<\delta
k<\eta\theta^{-1/2}\frac{\sqrt{1-\theta}}{\left(1+\theta\right)^{2}}\frac{1+\sigma\left(1+\theta^{2}\right)}{\left(1+\sigma\right)^{3/2}}.$
(75)

The growth rate is at maximum $\Gamma_{\rm max}$ when $\delta k=0$, and it can
be written as:

$\Gamma_{\rm
max}=\frac{1}{2}\eta\theta^{-1/2}\frac{\sqrt{1-\theta}}{1+\theta}\frac{1+\sigma\left(1+\theta^{2}\right)}{\left(1+\sigma\right)^{3/2}}.$
(76)

In the non-relativistic limit (i.e. $\sigma\ll 1$) and the low-beta limit
(i.e. $\theta\ll 1$), this expression becomes $\Gamma_{\rm
max}=\eta\theta^{-1/2}/2$ and reproduces the well-known expression of the
growth rate of the PDI for non-relativistic plasma.

This formula (76) for the growth rate agrees with Matsukiyo & Hada (2003) [87]
within the range of $\mathcal{O}(\theta)$ 333Although Matsukiyo & Hada (2003)
assumed $\sigma\lesssim 1$ when deriving the growth rate, the
$\sigma$-dependence of their formula is consistent with our result for general
$\sigma$. They justified neglecting some terms when expanding the dispersion
relation around the resonance point by assuming that $\sigma$ is not too
large. However, considering only points well close to the resonance point is
sufficient to neglect these terms, making the assumption about $\sigma$
unnecessary. . The discrepancy beyond $\mathcal{O}(\theta)$ arises because
their formulation implicitly assumes that the internal energy is significantly
smaller than the rest mass energy. Furthermore, the above result is also
approximately derived from the growth rate of the induced Brillouin scattering
for unmagnetized pair plasma, equation (A3) in Iwamoto et al. (2023) [91], by
multiplying $(\omega/\omega_{\rm c})^{2}$, where $\omega_{\rm c}$ is the
electron cyclotron frequency. This is because the oscillation amplitude of
electrons perpendicular to $\mbox{\boldmath$B$}_{0}$ is suppressed by
$\omega/\omega_{\rm c}$ in the presence of a background magnetic field
compared to the scattering of electromagnetic waves without a magnetic field.

In the bottom panel of Figure 4, the maximum growth rate obtained from
equation (76) is shown by a horizontal thick black dotted line. The upper and
lower limits of the range of wavenumber $k$ in which instability occurs,
represented by the equation (75), are also indicated by the vertical dashed-
dotted lines in the upper and lower panels. As is clear from the figure, the
obtained analytical expressions reproduce the numerical results well. Figure 5
shows the dispersion relation when $\sigma=10^{2}$ and $\eta$ and $\theta$ are
the same as in figure 4. As $\sigma$ increases, not only the maximum growth
rate becomes smaller, but also the range of instability becomes narrower.
Takamoto et al. (2014) [86] performed simulations of Alfvén wave propagation
based on the relativistic MHD equation and obtained a qualitative trend that
Alfvén waves tend to be more stable as $\sigma$ increases. Our result,
equation (76), is consistent with their results and gives a quantitative
interpretation of their nature.

Figure 6:  $\theta$-dependence of the maximum growth rate $\Gamma_{\rm max}$
for $\sigma=10^{3}$ and $\eta=0.2$. Under these parameters, $\theta_{c}\sim
0.01$ (see equation (77)). The red curve is the numerical solution of the
dispersion relation (56) and the blue dotted curve is the growth rate obtained
from the equation (76). For $\theta\gtrsim\theta_{c}$, these are in good
agreement. The black dotted line is the growth rate in the limit where
$\theta\lesssim\theta_{c}$, calculated by equation (80). When $\theta$ is
small, the growth rate takes a constant value independent of $\theta$.

Figure 6 shows the dependence of the maximum growth rate $\Gamma_{\rm max}$ on
$\theta$ when $\sigma=10^{3}$ and $\eta=0.2$. The red curve is the result of
numerically and directly solving the dispersion relation (56), and the blue
dotted curve is the growth rate calculated from equation (76). It can be seen
that the analytical formula (76) is in good agreement with the numerical
solution in $\theta\gtrsim 10^{-2}$. However, for $\theta\lesssim 10^{-2}$,
equation (76) fails to reproduce the numerical result. This is because the
approximation of $|\delta\omega|\ll\omega_{1}$ is broken. The correct growth
rate in such a small limit of $\theta$ will be obtained in the next section.

###  III.7 Strong coupling limit: decay rate for small $\theta$

As seen in Figure 6, when $\theta$ is small, equation (76) fails to reproduce
the numerical solution. This is due to the fact that the condition
$|\delta\omega|\ll\omega_{1}$, which was assumed in obtaining equation (76),
is violated. In the context of plasma physics, such a situation is called
strong coupling limit [92, 93]. The typical $\theta=\theta_{c}$ where the
approximation begins to break down is estimated from
$\omega_{1}\sim|\delta\omega|$ using equations (67) and (76) as follows:

$\theta_{c}\sim 2^{-4/3}\eta^{2/3}\left(1+\sigma\right)^{-1/3}.$ (77)

Since $\Gamma_{\rm max}\sim\Gamma_{\rm max}(\theta=\theta_{c})$ at
$\theta\ll\theta_{c}$ (see Figure 6), we can presume that the growth rate at
$\theta\ll\theta_{c}$ is proportional to $\eta^{2/3}$ from equation (76).
Therefore, we assume that $\delta\omega$ and $\delta k$ are
$\mathcal{O}(\eta^{2/3})$ and expand the dispersion relation again. Note that
$\theta$ in this regime is smaller than $\mathcal{O}(\eta^{2/3})$ because
$\theta_{c}=\mathcal{O}(\eta^{2/3})$. Expanding the equation (56), we obtain
the following equation from the lowest order $\mathcal{O}(\eta^{2})$ terms:

$\delta\omega^{2}\left(\delta\omega+\delta
k\right)+\frac{\eta^{2}}{1+\sigma}=0.$ (78)

The condition that the solution $\delta\omega$ of equation (78) has an
imaginary part is that the wavenumber $k$ is greater than the following value
$k_{\rm min}$:

$k>k_{\rm
min}=2-\sqrt[3]{\frac{27}{4}}\eta^{2/3}\left(1+\sigma\right)^{-1/3}.$ (79)

When $\delta k=0$, the growth rate takes the maximum value $\Gamma_{\rm max}$
as follows:

$\Gamma_{\rm max}=\frac{\sqrt{3}}{2}\eta^{2/3}\left(1+\sigma\right)^{-1/3}.$
(80)

The black dotted line in Figure 6 represents the growth rate calculated by
equation (80). In the parameters of Figure 6, $\theta_{c}\sim 0.01$, it can be
seen that equation (80) reproduces well the growth rate obtained by the
numerical calculation for $\theta\ll\theta_{c}$. Note that, in the non-
relativistic limit (i.e. $\sigma\to 0$), it reproduces the growth rate of PDI
in non-relativistic plasmas at $\theta\sim 0$ [94].

##  IV Discussion

The obtained expression for the growth rate of the instability reveals that
the maximum growth rate $\Gamma_{\rm max}$ is a decreasing function of
$\sigma$, and in the limit of $\sigma\to\infty$, the instability ceases. Since
$\sigma$ is the ratio of the energy densities of the matter and
electromagnetic fields, the limit of large $\sigma$ corresponds to a situation
close to the force-free limit. Our findings are consistent with the fact that
an Alfvén wave is stable to resonant three-wave interactions in the force-free
limit. While it is not evident whether the methods using force-free equations
and the MHD equations employed in our study are mathematically equivalent, we
can demonstrate that they are equivalent within the scope of this study. By
eliminating $\mbox{\boldmath$u$}_{\perp}$ from equations (37) and (38), and
taking the limit of $\sigma\to\infty$, we obtain

$\frac{1}{c^{2}}\frac{\partial^{2}\bm{b}_{\perp}}{\partial
t^{2}}-\frac{\partial^{2}\bm{b}_{\perp}}{\partial z^{2}}=0.$ (81)

This is the equation satisfied by the magnetic field of the free
electromagnetic wave. Similarly, by eliminating $\mbox{\boldmath$e$}_{\perp}$
from equations (37) and (38), and taking the limit of $\sigma\to\infty$, we
obtain

$\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\left(\mbox{\boldmath$u$}_{\perp}-\theta
u_{\|}\delta\mbox{\boldmath$e$}\right)-\frac{\partial^{2}}{\partial
z^{2}}\left(\mbox{\boldmath$u$}_{\perp}-\theta
u_{\|}\delta\mbox{\boldmath$e$}\right)=0.$ (82)

The quantity $\mbox{\boldmath$u$}_{\perp}-\theta
u_{\|}\delta\mbox{\boldmath$e$}$ that satisfies the wave equation in vacuum
can be interpreted as the electric field $\mbox{\boldmath$E$}_{\perp,xy}$
associated with the daughter Alfvén wave perpendicular to the background
magnetic field $\mbox{\boldmath$B$}_{0}$. By multiplying both sides of
equation (82) by $\beta_{A}\mbox{\boldmath$B$}_{0}$ as an outer product from
the left side, we obtain

$\frac{1}{c^{2}}\frac{\partial^{2}\mbox{\boldmath$E$}_{\perp,xy}}{\partial
t^{2}}-\frac{\partial^{2}\mbox{\boldmath$E$}_{\perp,xy}}{\partial z^{2}}=0,$
(83)

where

$\mbox{\boldmath$E$}_{\perp,xy}=-\left(\mbox{\boldmath$\beta_{\perp}$}\times\mbox{\boldmath$B_{0}$}+\mbox{\boldmath$\beta_{\|}$}\times\delta\mbox{\boldmath$B$}\right).$
(84)

This corresponds to the electric field in the $xy$-plane, except for the one
associated with the parent Alfvén wave (see equation (96)). In this way,
equations (37) and (38) describing daughter Alfvén waves reduce to the wave
equation of free electromagnetic waves in the limit $\sigma\to\infty$. On the
other hand, the wave equation for acoustic waves still has a non-zero source
term in the limit of $\sigma\to\infty$:

$\frac{1}{c^{2}}\frac{\partial^{2}\epsilon_{\|}}{\partial
t^{2}}-\beta_{s}^{2}\frac{\partial^{2}\epsilon_{\|}}{\partial
z^{2}}=-w_{0}\left[\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\left(\delta\mbox{\boldmath$\beta$}\cdot\mbox{\boldmath$\beta$}_{\perp}\right)-\frac{\partial^{2}}{\partial
z^{2}}\left(\frac{\delta\mbox{\boldmath$B$}\cdot\mbox{\boldmath$b$}_{\perp}}{B_{0}^{2}}\right)\right],$
(85) $\frac{1}{c^{2}}\frac{\partial^{2}\beta_{\|}}{\partial
t^{2}}-\beta_{s}^{2}\frac{\partial^{2}\beta_{\|}}{\partial
z^{2}}=-\frac{1}{c}\frac{\partial^{2}}{\partial t\partial
z}\left(\frac{\delta\mbox{\boldmath$B$}\cdot\mbox{\boldmath$b$}_{\perp}}{B_{0}^{2}}-\beta_{s}^{2}\delta\mbox{\boldmath$\beta$}\cdot\mbox{\boldmath$\beta$}_{\perp}\right).$
(86)

Although the acoustic waves are generated by the interaction between parent
and daughter Alfvén waves even for $\sigma\to\infty$, the PDI is considered to
be suppressed because the acoustic wave is not possible to excite daughter
Alfvén waves.

In this work, we have treated the amplitude $\eta$ of the parent Alfvén wave
as constant and calculated the growth rate of infinitesimal amplitude daughter
waves. Consequently, the decay rate of the parent wave cannot be determined
from this analysis. A formulation that considers the case $\epsilon\sim\eta$
is necessary for a more precise derivation. However, according to the general
consideration of three-wave interactions, there is an implication that the
decay rate of the parent wave differs only by a logarithmic factor from the
growth rate of the daughter wave, namely the decay time scale of the parent
wave can be written as

$t_{\rm decay}\sim\Gamma_{\rm max}^{-1}\ln\left(\eta/\epsilon_{{\rm
d},0}\right),$ (87)

where $\epsilon_{{\rm d},0}$ is the normalized initial amplitude of the
daughter waves (e.g., for the Alfvén wave, $\epsilon_{{\rm
d},0}=b_{\perp}/B_{0}$ at the initial state) [67]. Therefore, the growth rate
obtained in this study is effective for estimating the decay rate of the
parent wave.

We can apply our results to relativistically magnetized plasmas, such as the
plasma of the magnetosphere of a neutron star. Let us estimate the timescale
for the decay of Alfvén waves generated on the surface of a neutron star,
assuming they serve as the energy source for the X-ray flare of SGR1935+2154
and FRB 20200428A [23, 24, 25, 26, 27, 28, 29]. The estimated amplitude of the
parent wave, based on the luminosity $L_{\rm X}\sim 10^{41}~{}{\rm
erg~{}s}^{-1}$ of the X-ray burst (i.e., $L_{\rm X}\sim 4\pi R_{\rm
X}^{2}c\eta^{2}B_{\rm NS}^{2}/8\pi$ where $R_{\rm X}\sim 10^{4}~{}{\rm cm}$ is
the size of the energy release region444 The energy release might occur in one
part of the stellar surface rather than the entire surface. Here we adopt a
radius of $R_{\rm X}\sim 10^{4}~{}{\rm cm}$ which is estimated by using the
luminosity $L_{\rm X}\sim 10^{41}~{}{\rm erg~{}s}^{-1}$ and effective
temperature $T_{\rm eff}\sim 80~{}{\rm keV}$ of the X-ray burst [63].  [63],
and $B_{\rm NS}\sim 10^{15}~{}{\rm G}$ is the surface magnetic field), is
$\eta\sim 3\times 10^{-4}$ [42, 60]. Although the magnetization parameter
$\sigma$ in the magnetosphere is not precisely known and depends on the
efficiency of particle production in the magnetosphere, an estimation of
$\sigma\sim 10^{4}$ has been given by previous studies [74, 75, 76, 77, 78,
48] 555 If the $e^{\pm}$ fireball is formed in the X-ray burst, $\sigma$
should be much smaller [63] . In the magnetosphere, where there are a large
number of photons from the X-ray burst, we can expect the temperature of the
magnetosphere to be $T\sim\mathcal{O}({\rm keV})$ due to the Compton heating
[95], so that we evaluate the sound speed of magnetospheric plasma as
$\beta_{\rm s}\sim(1~{}{\rm keV}/511~{}{\rm keV})^{1/2}\sim 0.05$. According
to equation (77), since $\theta_{c}\sim 8\times 10^{-5}$, we find that
$\theta\gg\theta_{c}$ and the growth rate is appropriate to be estimated by
equation (76). This suggests that the Alfvén waves propagate approximately
$2\times 10^{4}$ wavelengths before decaying. Assuming a wave period
comparable to the rotation period of the central neutron star, $\omega_{0}\sim
1$ Hz, $\lambda\sim 10^{10}$ cm, it implies that the waves do not decay
significantly within the light cylinder. However, if we consider waves excited
at a frequency similar to the observed FRB frequency (around $10^{9}$ Hz), the
waves would decay at a distance of approximately $7\times 10^{5}$ cm,
indicating limited escape from the magnetosphere. While these estimates are
rough approximations, our results may be crucial for discussing the radiation
mechanisms of FRBs.

In this study, we analyze the idealized setting of an infinitely extended
uniform plane wave in a dissipationless system. However, when applying these
results to real systems, it is essential to evaluate the impacts of the
following effects. The first is the finite bandwidth effect of the parent
wave. Although stability analysis is conducted for monochromatic waves in this
study, it is more natural to consider waves excited in general systems as wave
packets with a finite bandwidth $\Delta\omega$. In the context of the
parametric instability, it is known that when the bandwidth of the parent wave
is larger than the growth rate of daughter waves in the monochromatic case
($\Delta\omega\gg\Gamma_{\rm max}$), the effective growth rate of the daughter
waves becomes $\Gamma_{\rm eff}\sim\Gamma_{\rm max}^{2}/\Delta\omega$,
suppressed by $\Gamma_{\rm max}/\Delta\omega$ [96]. On the other hand, if
$\Delta\omega\ll\Gamma_{\rm max}$, the phase of the parent wave and the
daughter waves can be considered synchronized during the time scale of
daughter wave growth, allowing us to use the analysis of the monochromatic
parent wave. The second is the effect of dissipation in the plasma. The
dominant dissipation process in the plasma can vary depending on the system
under consideration. Here, we assume an appropriate timescale for the
dissipation process as $\Gamma_{\rm diss}^{-1}$. For instability to occur in
the wave, the timescale of instability without considering dissipation should
be shorter than the dissipation timescale (i.e., $\Gamma_{\rm max}>\Gamma_{\rm
diss}$) [96]. In this case, the growth rate of instability can be estimated as
$\Gamma_{\rm eff}\sim\Gamma_{\rm max}-\Gamma_{\rm diss}$. When the effect of
dissipation is weak, so that the dissipation proceeds over a much longer
timescale compared to the growth timescale of the instability, it is justified
to use the growth rate obtained from the analysis of the monochromatic parent
wave.

In this study, we have only investigated the PDI within the linear stage
(i.e., $\mathcal{O}(\varepsilon)\ll 1$), so that further research is necessary
to understand the development in the nonlinear stage. In systems where
$\sigma\gg 1$, meaning the electromagnetic energy density greatly exceeds the
matter energy density, non-thermal high-energy phenomena such as non-thermal
particle acceleration may occur due to the transfer of electromagnetic energy
to low-density regions. Additionally, for the generated acoustic waves,
damping effects including the Landau damping are expected to occur, thus
requiring an analysis incorporating kinetic effects to further quantitatively
understand the PDI. Magnetohydrodynamic simulation encounters difficulty in
dealing with large $\sigma$ and cannot analyze kinetic phenomena such as
particle acceleration, making Particle-in-Cell (PIC) simulations more suitable
for such studies. While this study also applied linearization to $\eta$,
further extensions are possible. Equations (19)-(22) are formulated for
general $\eta$, and we are currently conducting ongoing research to directly
analyze these equations. When $\eta$ reaches unity, higher-order wave-wave
interactions (e.g., 4-wave interaction) may become significant, although this
phenomenon lies beyond the scope of this study. Exploring higher-order wave-
wave interactions through simulations, particularly PIC simulations, will
provide valuable insights into understanding the stability of Alfvén waves.

##  V Summary

We have investigated the parametric decay instability (PDI) of circularly
polarized Alfvén waves in a plasma with relativistic magnetization, i.e.,
$\sigma\gg 1$, based on the special relativistic magnetohydrodynamic (MHD)
equations. Our results indicate that even in plasmas with relativistic
magnetization, the Alfvén wave is unstable to three-wave interactions where
the parent wave generates two daughter waves - a forward propagating acoustic
wave and a backward propagating Alfvén wave. We have obtained the analytical
expression of the maximum growth rate (see equation (73)) as follows:

$\Gamma_{\rm
max}/\omega_{0}=\frac{1}{2}\eta\theta^{-1/2}\frac{\sqrt{1-\theta}}{1+\theta}\frac{1+\sigma\left(1+\theta^{2}\right)}{\left(1+\sigma\right)^{3/2}},$

where $\eta=\delta B/B_{0}$ is the amplitude of the parent Alfvén wave,
$\sigma=B_{0}^{2}/(4\pi(\epsilon_{0}+p_{0}))$ is the magnetization,
$\theta=\beta_{\rm A}/\beta_{\rm s}$ is the ratio of the sound speed to the
Alfvén velocity, and $\omega_{0}$ is the frequency of the parent Alfvén wave.
This is only valid when the ratio $\theta$ of the sound speed to Alfvén
velocity is greater than $\theta_{c}$, which is expressed by equation (77).
For $\theta\ll\theta_{c}$, the growth rate can be written as follows:

$\Gamma_{\rm
max}/\omega_{0}=\frac{\sqrt{3}}{2}\eta^{2/3}\left(1+\sigma\right)^{-1/3}.$

As can be seen from these equations, we have observed that the instability is
suppressed with increasing magnetization $\sigma$. Our results for the
instability growth rates are consistent with the known non-relativistic
limits. Such a process may provide a clue to the energy conversion mechanism
from electromagnetic field to plasma in environments around compact objects,
where strongly magnetized plasmas are expected to exist, e.g., in the
magnetospheres of neutron stars or black holes.

###### Acknowledgements.

We are grateful to Toshio Terasawa, Kenji Toma, Takayoshi Sano, Yohei
Kawazura, Shuichi Matsukiyo, Masanori Iwamoto, Shoma Kamijima, Rei Nshiura,
Koutarou Kyutoku, Tomoki Wada, Shota Kisaka, Kazumi Kashiyama, and Shuta
Tanaka for fruitful discussion and valuable comments. This work is supported
by Grants-in-Aid for Scientific Research No. 23H01172, 23H05430, 23H04900,
22H00130, 20H01901, 20H01904, 20H00158 (KI), 21J01450 (WI) from the Ministry
of Education, Culture, Sports, Science and Technology (MEXT) of Japan.

##  Appendix A Derivation of linearlized equations

###  A.1 Fluid equations with electromagnetic field as external field

In order to derive equations (21)-(24), we first simplify equations (1)-(4).
Using the Maxwell equations, the energy-momentum conservation law can be
rewritten into the fluid equations in the external electromagnetic field as
follows:

$\frac{1}{c}\frac{\partial}{\partial
t}\left[(\epsilon+p)\gamma^{2}-p\right]+\frac{\partial}{\partial
z}\left[(\epsilon+p)\gamma^{2}\beta_{z}\right]=\frac{1}{c}\mbox{\boldmath$j$}\cdot\mbox{\boldmath$E$},$
(88) $\frac{1}{c}\frac{\partial}{\partial
t}[(\epsilon+p)\left.\gamma^{2}\bm{\beta}\right]+\frac{\partial}{\partial
z}\left[(\epsilon+p)\gamma^{2}\beta_{z}\bm{\beta}\right]+\frac{\partial
p}{\partial
z}=\rho_{e}\mbox{\boldmath$E$}+\frac{1}{c}\mbox{\boldmath$j$}\times\mbox{\boldmath$B$},$
(89)

where $\rho_{e}$ is the charge density and $j$ is the current density. The
electromagnetic field is calculated from the Maxwell equations and the ideal
MHD conditions. Since the electric field satisfies the ideal MHD condition
$\mbox{\boldmath$E$}=-\mbox{\boldmath$\beta$}\times\mbox{\boldmath$B$}$, the
evolution of the magnetic field from the Maxwell-Faraday equation is described
by the following induction equation:

$\frac{1}{c}\frac{\partial\mbox{\boldmath$B$}}{\partial
t}=\nabla\times\left(\mbox{\boldmath$\beta$}\times\mbox{\boldmath$B$}\right).$
(90)

The charge density $\rho_{e}$ and the current density $j$ in the right-hand
side of equations (88) and (89), respectively, can be written as follows by
using Maxwell’s equations:

$\rho_{e}=\frac{1}{4\pi}\nabla\cdot\mbox{\boldmath$E$},$ (91)
$\mbox{\boldmath$j$}=\frac{c}{4\pi}\left(\nabla\times\mbox{\boldmath$B$}-\frac{1}{c}\frac{\partial\mbox{\boldmath$E$}}{\partial
t}\right).$ (92)

Note that since the electric field $E$ satisfies the ideal MHD conditions, the
charge density $\rho_{e}$ and the current density $j$ can be obtained from the
magnetic field $B$ and velocity $\beta$.

###  A.2 Derivation of linearized equations

The quantities required in the subsequent calculations up to the order of
$\mathcal{O}(\varepsilon)$ are calculated as follows. Square of the Lorentz
factor of the fluid:

$\gamma^{2}=\frac{1}{1-\mbox{\boldmath$\beta$}\cdot\mbox{\boldmath$\beta$}}\sim\delta\gamma^{2}\left(1+2\delta\gamma^{2}\delta\mbox{\boldmath$\beta$}\cdot\mbox{\boldmath$\beta_{\perp}$}\right),$
(93)

where
$\delta\gamma=(1-\delta\beta^{2})^{-1/2}=(1-\eta^{2}\beta_{A,\eta}^{2})^{-1/2}$.
Enthalpy density:

$\epsilon+p\sim\left(\epsilon_{0}+p_{0}\right)+\epsilon_{\|}\left(1+\beta_{s}^{2}\right).$
(94)

$z$ component of the electric field:

$E_{z}\mbox{\boldmath$e$}_{z}\sim-\left(\delta\mbox{\boldmath$\beta$}\times\mbox{\boldmath$b_{\perp}$}+\mbox{\boldmath$\beta_{\perp}$}\times\delta\mbox{\boldmath$B$}\right).$
(95)

Note that $E_{z}$ is a quantity of $\mathcal{O}(\varepsilon)$. In-plane $xy$
component of the electric field:

$\mbox{\boldmath$E_{xy}$}\sim-\left(\delta\mbox{\boldmath$\beta$}\times\mbox{\boldmath$B_{0}$}+\mbox{\boldmath$\beta_{\perp}$}\times\mbox{\boldmath$B_{0}$}+\mbox{\boldmath$\beta_{\|}$}\times\delta\mbox{\boldmath$B$}\right).$
(96)

Square of the absolute value of the electric field:

$E^{2}\sim\delta\beta^{2}B_{0}^{2}+2\left(\delta\mbox{\boldmath$\beta$}\cdot\mbox{\boldmath$\beta_{\perp}$}\right)B_{0}^{2}+2\chi_{0}\delta\beta\delta
BB_{0}\beta_{\|}.$ (97)

Square of the absolute value of the magnetic field:

$B^{2}\sim B_{0}^{2}+\delta
B^{2}+2\delta\mbox{\boldmath$B$}\cdot\mbox{\boldmath$b_{\perp}$}.$ (98)

Using equations (95)–(98), calculate the right-hand side of equations (88) and
(89). First, the work rate of the electromagnetic field
$\mbox{\boldmath$j$}\cdot\mbox{\boldmath$E$}$ is as follows:

$\mbox{\boldmath$j$}\cdot\mbox{\boldmath$E$}\sim\frac{c}{4\pi}\left[B_{0}\mbox{\boldmath$\beta_{\perp}$}\cdot\frac{\partial}{\partial
z}\left(\delta\mbox{\boldmath$B$}\right)+B_{0}\delta\mbox{\boldmath$\beta$}\cdot\frac{\partial\mbox{\boldmath$b_{\perp}$}}{\partial
z}-B_{0}^{2}\frac{1}{c}\frac{\partial}{\partial
t}\left(\delta\mbox{\boldmath$\beta$}\cdot\mbox{\boldmath$\beta_{\perp}$}\right)-\chi_{0}\delta\beta\delta
BB_{0}\frac{1}{c}\frac{\partial\beta_{\|}}{\partial t}.\right]$ (99)

Here we used the following relations, which hold for circularly polarised
Alfvén waves:

$\delta\mbox{\boldmath$B$}\cdot\frac{\partial}{\partial
z}\left(\delta\mbox{\boldmath$B$}\right)=0,~{}~{}\delta\mbox{\boldmath$\beta$}\cdot\frac{\partial}{\partial
z}\left(\delta\mbox{\boldmath$B$}\right)=0.$ (100)

Next, let us calculate the electric force $\rho_{e}\mbox{\boldmath$E$}$. From
the equation (91), the charge density$\rho_{e}$ is as follows.

$\rho_{e}=\frac{1}{4\pi}\frac{\partial E_{z}}{\partial
z}\mbox{\boldmath$e$}_{z}\sim-\frac{1}{4\pi}\frac{\partial}{\partial
z}\left[\frac{\mbox{\boldmath$B_{0}$}}{B_{0}}\cdot\left(\delta\mbox{\boldmath$\beta$}\times\mbox{\boldmath$b_{\perp}$}+\mbox{\boldmath$\beta_{\perp}$}\times\delta\mbox{\boldmath$B$}\right)\right].$
(101)

As can be seen from this equation, $\rho_{e}=\mathcal{O}(\varepsilon)$. Since
$E_{z}$ is also a quantity of $\mathcal{O}(\varepsilon)$, the electric force
in the $z$ direction $\rho_{e}E_{z}$ is a second order of $\varepsilon$:

$\rho_{e}E_{z}\sim 0.$ (102)

The $xy$ in-plane component of the electric force is as follows:

$\rho_{e}\mbox{\boldmath$E_{xy}$}\sim\frac{1}{4\pi}\left(\delta\mbox{\boldmath$\beta$}\times\mbox{\boldmath$B_{0}$}\right)\frac{\partial}{\partial
z}\left[\frac{\mbox{\boldmath$B_{0}$}}{B_{0}}\cdot\left(\delta\mbox{\boldmath$\beta$}\times\mbox{\boldmath$b_{\perp}$}+\mbox{\boldmath$\beta_{\perp}$}\times\delta\mbox{\boldmath$B$}\right)\right].$
(103)

From equation (92), the $z$ component of the magnetic force
$\mbox{\boldmath$j$}\times\mbox{\boldmath$B$}$ can be calculated as follows:

$\left[\frac{1}{c}\mbox{\boldmath$j$}\times\mbox{\boldmath$B$}\right]_{z}=-\frac{\partial}{\partial
z}\left(\frac{\delta\mbox{\boldmath$B$}\cdot\mbox{\boldmath$b_{\perp}$}}{4\pi}\right)+\frac{B_{0}}{4\pi}\left[\mbox{\boldmath$\delta
B$}\cdot\frac{1}{c}\frac{\partial\mbox{\boldmath$\beta_{\perp}$}}{\partial
t}+\mbox{\boldmath$b_{\perp}$}\cdot\frac{1}{c}\frac{\partial}{\partial
t}(\delta\mbox{\boldmath$\beta$})\right]-\frac{\delta
B^{2}}{4\pi}\frac{1}{c}\frac{\partial{\beta}_{\|}}{\partial t}.$ (104)

The $xy$ in-plane component of the magnetic force is as follows:

$\left[\frac{1}{c}\mbox{\boldmath$j$}\times\mbox{\boldmath$B$}\right]_{xy}=\frac{B_{0}}{4\pi}\frac{\partial}{\partial
z}\left(\delta\mbox{\boldmath$B$}+\mbox{\boldmath$b_{\perp}$}\right)-\frac{B_{0}^{2}}{4\pi}\frac{1}{c}\frac{\partial}{\partial
t}(\delta\mbox{\boldmath$\beta$})-\frac{B_{0}^{2}}{4\pi}\frac{1}{c}\frac{\partial\mbox{\boldmath$\beta_{\perp}$}}{\partial
t}+\frac{B_{0}}{4\pi}\frac{1}{c}\frac{\partial}{\partial
t}\left(\beta_{\|}\delta\mbox{\boldmath$B$}\right)-\frac{\delta\mbox{\boldmath$B$}\cdot\mbox{\boldmath$b_{\perp}$}}{4\pi}\frac{1}{c}\frac{\partial}{\partial
t}(\delta\mbox{\boldmath$\beta$})\\\
-\delta\mbox{\boldmath$\beta$}\left(\frac{\delta\mbox{\boldmath$B$}}{4\pi}\cdot\frac{1}{c}\frac{\partial\mbox{\boldmath$b_{\perp}$}}{\partial
t}\right)-\chi_{0}\frac{\delta
B}{4\pi}\delta\beta\frac{1}{c}\frac{\partial\mbox{\boldmath$b_{\perp}$}}{\partial
t}-\frac{\delta
B^{2}}{4\pi}\frac{1}{c}\frac{\partial\mbox{\boldmath$\beta_{\perp}$}}{\partial
t}+\frac{\delta\mbox{\boldmath$B$}}{4\pi}\left(\frac{1}{c}\frac{\partial\mbox{\boldmath$\beta_{\perp}$}}{\partial
t}\cdot\delta\mbox{\boldmath$B$}\right)+\frac{\delta\mbox{\boldmath$B$}\cdot\mbox{\boldmath$\beta_{\perp}$}}{4\pi}\frac{1}{c}\frac{\partial}{\partial
t}(\delta\mbox{\boldmath$B$}).$ (105)

By substituting the above equations (93)–(105) into equations (88), (89) and
(90), equations (21)–(24) can be obtained by straightforward calculation.

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