Radiation Process in Relativistic MHD Waves: The Case of Circularly Polarized Alfvén Waves

Turbulence in highly magnetized plasma can be relativistic and induce an electric field comparable to the background magnetic field. Such a strong electric field can affect the emission process of nonthermal electrons. As the first step toward elucidating the emission process in relativistic turbulence, we study the radiation process of electrons in relativistic circularly polarized Alfvén waves. While the induced electric field boosts the average energy of low-energy electrons with a Larmor radius smaller than the wavelength, the emissivity for such electrons is suppressed because of the elongated gyromotion trajectory. The trajectory of high-energy electrons is shaken by the small-scale electric field, which enhances the emissivity. Since the effective Lorentz factor of E × B drift is ≃2 in the circularly polarized Alfvén waves, the deviation from the standard synchrotron emission is not so prominent. However, a power-law energy injection in the waves can produce a concave photon spectrum, which is similar to the GeV extra component seen in GRB spectra. If the turbulence electric field is responsible for the GeV extra component in GRBs, the estimates of the typical electron energy and magnetic field should be largely altered.


INTRODUCTION
The most promising launching mechanism of relativistic jets is the Blandford-Znajek mechanism (Blandford & Znajek 1977), where the rotation energy of a black hole is extracted via a magnetic field in the ergosphere.In this case, the jet energy is dominated by the magnetic field (McKinney et al. 2012(McKinney et al. , 2014)).Relativistic winds from pulsars or magnetars are also dominated by magnetic fields (Goldreich & Julian 1969).
In high-σ plasma, the Alfvén velocity is almost the speed of light.The turbulence velocity can be relativistic.In such turbulence, the induced electric field is comparable to the magnetic field.The electric field affects the trajectory of non-thermal charged particles so that the emission property can be different from the standard synchrotron emission.In most cases, the magnetic fields in the emission region have been estimated by spectral modeling with the standard synchrotron, inverse Compton, and π 0 -decay processes.If the synchrotron emission is largely modified by the turbulence electric field, the estimate of the magnetic field may be misinterpreted.
In this paper, as the first step toward unveiling the emission property in high-σ turbulence, we investigate the radiation process in relativistic circularly polarized Alfvén waves, which is analytically described.
The structure of this paper is as follows.In Section 2, we review the radiation process in the uniform electric field case.In Section 3, we discuss the emission properties in a relativistic circularly polarized Alfvén wave.In Section 4, our numerical method to calculate radiation from electron trajectories is shown.In Section 5, we show the numerical results of the radiation for monoenergetic and power-law injections of electrons.In Section 7, we summarize our results and discuss their implication for spectral modeling.

UNIFORM ELECTRIC FIELD
In the ideal magnetohydrodynamics (MHD), relativistic turbulece with a turbulent velocity δV ∼ c induces the motional electric field with a strength |E| = | δV c ×B|, which can be comparable to the magnetic field strength |B|.The radiation process of charged particles in such relativistic turbulence can be different from conventional synchrotron radiation.In this section, to clarify the effect of an electric field, we first review the radiation process of electrons in the uniform field case.
When a uniform electric field E is perpendicular to a uniform magnetic field B, an observer moving with the drift velocity v E×B = c E×B B 2 observes zero electric field as long as . (1) In this frame (drifting frame), the motion of electrons is spiral around the magnetic field B ′ = B/Γ E×B , and the emission from electrons is the usual synchrotron radiation in the manetic field B ′ .(Hereafter, the prime ′ denotes quantities in the drifting frame.)Considering electrons isotropically injected with a Lorentz factor γ i in the original frame, the synchrotron power, namely photon energy emitted per unit time is given by (Rybicki & Lightman 1979;Jackson & Fox 1999) where γ ′ ∼ Γ E×B γ i is the typical Lorentz factor of the electrons in the drifting frame, and σ T is the Thomson scattering cross section.
In the original frame, the time-averaged Lorentz factor of the electrons is The average energy is boosted from the injection value by the electric field.The drift motion of a charged particle in fields of E ≃ B is elongated in the direction of the drift velocity.This asymmetric motion stretches the time interval for γ > γ ave .As explained in Appendix, this effect slightly increases γ ave compared to the above estimate.The radiation power is Lorentz invariant (Rybicki & Lightman 1979;Jackson & Fox 1999).Using γ, the radiation power is written as The radiation power is suppressed by the factor 1/Γ 4 E×B compared to the usual synchrotron formula without an electric field.In Appendix, we show more details of analytically calculations of the radiation power.

RADIATION IN ALFV ÉN WAVE
In astrophysical MHD turbulence, there may be a frame where the turbulence is globally isotropic, but locally the fluid velocity and the electromagnetic field are anisotropic.For simplicity, we assume that electrons are isotropically injected in this frame.The turbulence may be injected at a large scale, and cascade into smaller scales, where the wave amplitude of the turbulence is so small (δV ≪ c) that the effect of the electric field is almost negligible.The induced electric field is significant only at the injection scale.To investigate the effect of the turbulence electric field on the radiation, we consider a wave propagating to a certain direction with a finite wave length λ.While various modes of MHD waves as turbulence are possible, the analytical description is possible for circularly polarized Alfvén waves even with non-linear amplitude (δV ∼ c).As a first step, here we focus on this simplest case.

Circularly polarized Alfvén wave
Relativistic perturbation (δV ∼ c) implies non-linear amplitudes of the perturbed fields.Even in linearly polarized Alfvén waves, the induced magnetic pressure leads to compression of fluid.The compressed part of the waves propagates faster and the nonlinear waves steepen (Shikin 1969), so that the analytical description of the non-linear Alfvén waves is difficult.On the other hand, the circularly polarized Alfvén wave can be treated with a constant total pressure in any phase of the wave.
The analytical description of the perturbed electromagnetic fields for a relativistic circularly polarized Alfvén wave propagating along the background magnetic field (z-axis) is given by (Kennel & Pellat 1976) where B 0 is the strength of the background magnetic field, V is the fluid velocity in the Alfvén wave, and k = 2π/λ is the wavenumber.Given the gas energy density ε and the gas pressure p, the phase speed of the Alfvén wave is written as , the phase speed is rewritten as V A = c σ/(σ + 1).As shown in equations ( 5)-( 10), for V ∼ V A ∼ c, the induced electric field is comparable to the background magnetic field.

Long wavelength limit
In this section, we consider the case in which the Larmor radius of electrons is much smaller than the wavelength, In this case, the electric and magnetic fields can be approximated as almost uniform ones.The radiation power of electrons is suppressed as 8π as discussed in section 2, where In the circularly polarized Alfvén wave with B 0 ≃ δB ≃ δE, the E × B drift velocity is which implies Γ E×B ≃ √ 2. Finally, we obtain From the Lorentz transformation of the synchrotron frequency in the E × B rest frame, the typical frequency is where is the typical frequency of isotropic synchrotron radiation.The typical frequency decreases by the factor 4 π /Γ 2 E×B ≃ 2/π compared to the usual synchrotron formula.

Short wavelength limit
In the case with r L0 ≫ λ, frequent changes of the field directions should be taken into account.The radiation power of an electron (charge −e) is given by the Liénard formula(Schwinger 1949), where a ⊥ and a ∥ are perpendicular and parallel components of acceleration, respectively.
From the equation of motion and the energy conservation we obtain acceleration of an electron as Then, we obtain the parallel component as and the perpendicular component as For γ ≫ 1, equations ( 21) and ( 22) imply which leads to(Schwinger 1949) From equation ( 19) with δB ≃ δE ≃ B 0 , the fractional change of the Lorentz factor during the wave Similarly, equation ( 18) gives the angle change as Therefore, the electron trajectory is a spiral motion around the background magnetic field with a small oscillation.The velocity components of an electron injected with a pitch angle θ i and an azimuthal angle ϕ i are approximately expressed as where ω B = eB0 γmec is the gyro frequency.As the electron injection is isotropic in this frame, we average over the angles θ i , ϕ i , and a time interval T much longer than the wave crossing time λ/v.Equations ( 5)-( 10), ( 24) and ( 27)-( 30) lead to where B 2 = B 2 0 + δB 2 again.Differently from the case for λ ≫ r L0 in section 3.2, the radiation power is rather enhanced by the perturbed electric field.
We estimate the typical emission frequency.As we consider MHD turbulence, the Larmor radius of nonrelativistic electrons is assumed to be short enough as λ ≫ m e c 2 /eB.In this case, equation (26) implies λ ≫ r L,0 /γ, and ∆θ ≫ 1/γ.This means that the radiation cone with opening angle 1/γ sweeps a certain position before an electron oscillates with a deflection angle ∆θ (Medvedev 2000).With a ≃ a ⊥ , the emission frequency can be estimated (Reville & Kirk 2010) as Averaging with the weight of the radiation power over the angles θ i , ϕ i and time, we can obtain the typical emitted frequency In the case without the turbulence, assuming δE = δB = 0 and using equations ( 5)-( 10), ( 22), ( 24) and ( 27)-( 30), we numerically obtain In the circularly polarized Alfvén wave, a similar calculation assuming δE ≃ δB ≃ B 0 leads to The increase of the peak frequency is because the perturbed electric field δE increases the perpendicular acceleration a ⊥ in equation ( 22).

NUMERICAL METHOD
As we have mentioned, we assume that a monochromatic wave propagates along the z-axis as expressed by equations ( 5)-( 10).The wave locally dominates the turbulence, which is globally isotropic.In the wave rest frame (moving along the z-axis with the velocity V A ), the electric field vanishes.Neglecting the radiative cooling, the electron energy in the wave rest frame is conserved.The electron energy in the original frame periodically oscillates following the change of momentum in the wave rest frame.Therefore, the time averaged Lorentz factor in the original frame does not evolve in this circularly polarized Alfvén wave.
We isotropically inject electrons in the original frame, and solve the equations of motion using the Boris-C solver with second-order accuracy in Zenitani & Umeda (2018) with a time step ∆t = 0.02 min(λ/c, r L0 /c).We isotropically inject 1600 electrons in total.
The average Lorentz factor for electrons injected with a Lorentz factor γ i ≫ 1 is calculated with a time interval T = 200 min(λ/c, r L0 /c) as The radiation power is numerically calculated using the Liénard formula which is also averaged as The radiation power spectrum of an electron is given by the Fourier transformation of the radiation electric field described by the Liénard Wiechart potential (Rybicki & Lightman 1979;Jackson & Fox 1999).For λ ≫ m e c 2 /eB and E ≤ B, the radiation spectrum can be approximately calculated (Reville & Kirk 2010) by where F (x) ≡ x ∞ x K 5 3 (ξ)dξ, and K 5 3 (ξ) is the modified Bessel function of the order 5/3.We resolve frequency range by 20 meshes per logbin.For calculation of F (x), we interpolate the analytical approximate formula in Fouka & Ouichaoui (2013) by the cubic spline in every time step ∆t.
The radiation spectrum is averaged over the injection angles θ i and ϕ i as

MONO-ENERGETIC INJECTION
In this section, we show numerical results for electrons isotropically injected with an initial Lorentz factor γ i for a parameter range of 10 −5 ≤ r L0 /λ ≤ 10 3 .As we mentioned, the MHD approximation implies λ ≫ m e c 2 /eB.So the jitter radiation mechanism (λ < m e c 2 /eB, Medvedev 2000) is not eligible in our case.Our results are shown normalized by the well-known formulae for a uniform magnetic field: the radiation power and the typical frequency ν 0 given by eq. ( 15).The upper panel of Figure 1 shows the average Lorentz factor γ ave as a function of r L0 /λ.In low energy limit (r L0 /λ ≪ 1), the electron energy is boosted by a factor of 2×1.19 ≃ 2.4 as explained in section 2 and Appendix, while γ ave is almost the sama as γ i in high energy limit (r L0 /λ ≪ 1) as explained in §3.3.As shown in the lower panel of Figure 1, the radiation power for r L0 /λ ≪ 1 is suppressed by a factor of 1/Γ 4 E×B ≃ 1/4 as estimated in equation (A10).On the other hand, the power is 1.5-fold enhanced by the electric field for r L0 /λ ≫ 1 as discussed in §3.3.
In Figure 2, we show the radiation spectra νP ν,ave /P 0 .In the case of r L0 /λ = 10 3 (green), the radiation power is enhanced compared to the case without the turbulence (black).The peak frequency is also increased compared to the synchrotron case as discussed in §3.3.
For r L0 /λ = 10 −5 (blue), the radiation power is suppressed compared to the case without the turbulence by a factor of 1/Γ E×B ≃ 1/4.The slight decrease in the typical frequency is consistent with the discussion in equation ( 15).

POWER-LAW INJECTION
In a strong magnetic field, the effect of radiative cooling appears in the high-energy electron energy distribution.Assuming a continuous electron injection with a power-law energy distribution and electron escape, a broken power-law energy distribution is frequently assumed as a steady state.In our case with relativistic turbulence, we need to consider not only cooling but also the energy boost by the electric field, shown in Depending on the cooling break in the electron energy distribution, the photon spectrum shows a variety.We numerically calculate the time evolution of electron energy distribution N (γ, t) in a relativistic Alfvén wave by solving the equation of continuity in the energy space as where γave (γ ave ) is radiative cooling rate of electrons in relativistic Alfvén wave and the injection term Ṅinj (γ ave ) is assumed to be constant with time.By changing the calculation time t, the energy at the cooling break can be adjusted.
As shown in the upper panel of Figure 1, we take into account the boost of Lorentz factor of electrons after injection for the injected electron energy distribution.The initial electron energy distribution at injection is assumed as a power law distribution with an exponential cutoff as where γ i is the initial Lorentz factor at injection and C = 1/ γmax γmin dγ i Ṅinj,0 (γ i )t is the normalization factor.In this paper, we adopt p = 2.Then, we calculate the electron distribution after the energy boost by the wave using γ ave (γ i ) calculated with equation (38) as Ṅinj (γ ave ) = dγ i dγ ave Ṅinj,0 (γ i (γ ave )). ( 46) For the radiative cooling rate γave (γ ave ), we use the average radiation power P ave (γ ave ) calculated by equation (40) as γave (γ ave ) = − P ave (γ ave ) m e c 2 . (47) The radiation spectrum is calculated as where P ν,ave (γ ave ) is the radiation spectrum calculated with eq. ( 42).

Slow cooling
The cooling break should appear at γ ave = γ c , which satisfies t = γ ave / γave .First, we consider the case with γ c ≫ γ cut , where the cooling effect is negligible (slow cooling).As we have discussed, the emission properties change at γ ave = γ λ , where γ λ is the electron Lorentz factor at which the corresponding Larmor radius is comparable to the wavelength, As shown in the upper panel of Figure 3, the electron energy distribution for γ λ ≪ γ min (green) is not affected by the wave, and is almost the same as the case without the wave (dashed black).On the contrary, for γ λ ≫ γ cut (blue), all electrons gain energy by the wave, so that the distribution is shifted to higher energies.In the case for γ min < γ λ < γ cut (red), the distribution shows a concave shape connecting the two cases of γ λ ≪ γ min and γ λ ≫ γ cut around γ ≃ γ λ = 10 8 .
The lower panel of Figure 3 shows the resultant photon spectra.The spectrum and frequency are normalized by and respectively.The boost of electron energies for γ λ ≫ γ cut (blue) leads to a slightly higher peak energy in the photon spectrum.However, as the induced electric field suppresses the emissivity (see Figure 2), the energy boost does not enhance the power in the lowfrequency regime compared to the case without the wave (dashed black).For γ λ ≪ γ min (green), the emissivity is slightly higher compared to the case without the wave owing to the enhanced acceleration as discussed in §5.For γ min < γ λ < γ cut (red), the radiation spectrum is curved slightly around connecting the blue line in low-frequency region and the green line in high-frequency region.However, the modulation is not so prominent.

Fast cooling
For the fast cooling (γ c ≪ γ min ) case, the electron energy distribution becomes softer than the injection spectrum, and a low-energy component below γ min appears.The upper panel of Figure 4 shows these properties.The low-energy sharp breaks correspond to γ min .The energy distribution is shifted to higher energies by the induced  electric field for γ λ ≫ γ cut (blue).Note that, unlike the slow cooling case, the energy shift for this steeper spectrum leads to a much larger increase of N (γ) for a given γ.The low-energy cut-off at ∼ γ c is relatively high because of the cooling suppression.The distributions for the other cases (green and red) present similar behaviors to those in the slow-cooling cases, though the differences in the cooling efficiency appear in the different low-energy cut-offs.
The normalization for the lower panel of Figure 4 is similar to the slow cooling case with The spectral power for γ λ ≫ γ cut is high compared to the case without the wave, even though the radiation power is relatively suppressed by the induced electric field.This is due to the large increase of N (γ) by the energy shift for a steep energy distribution as we mentioned.Even in the slow cooling case, this enhancement of emissivity can be expected for a steep injection spectrum with e.g.p = 3.In the case of γ λ ≪ γ c shown with the green line, the radiation spectrum is shifted to a slightly higher frequency in the high-frequency region compared to the case without the wave.
In the case of γ min < γ λ < γ cut (red), the spectrum shows a characteristic bump structure due to the distorted electron spectrum.The steeper spectrum above ∼ 10 2 ν min extends to two orders of magnitude in ν.The flat component above ∼ 10 6 ν min can be misunderstood as an extra synchrotron self-Compton component.In this misinterpretation, spectral modeling would lead to a much lower magnetic field than the actual value.

CONCLUSIONS & DISCUSSION
Magnetically dominated outflow has been considered for pulsar winds, blazar jets, and gamma-ray bursts.Relativistic turbulence in highly magnetized plasma should induce electric fields with a large amplitude.While the standard synchrotron emission has been adopted for the model of emission from such objects, the large electric field induced in the outflows can affect the emission process from relativistic electrons.
As the first step to investigate the emission processes in relativistic turbulence, we have considered circularly polarized Alfvén waves in this paper.We have calculated the radiation spectrum by numerically following the electron trajectories in the waves, which can be analytically expressed.We have shown the energy dependence of the emission power and spectrum from a single electron.For electrons whose Larmor radius is significantly smaller than the wavelength of the turbulence r L0 ≪ λ, the motional electric field suppresses the emission power by a factor of 1/Γ 4 E×B , where Γ E×B is the Lorentz factor for the E × B drift motion.The value of Γ E×B can be significantly large in relativistic turbulence.However, in the case of the circularly polarized Alfvén wave, Γ E×B is limited to √ 2. Note also that the average energy of electrons injected in the relativistic wave is boosted for r L0 ≪ λ.For r L0 ≫ λ, the emission power and spectral peak frequency are slightly increased by the wave.
We have also demonstrated the emission spectra from electrons injected with a power-law energy distribution.In the slow cooling case, though the complicated effects mentioned above are entangled, the resultant photon spectrum is not drastically modified.However, for the fast cooling case, the photon spectrum can be concave around ν λ , which is the typical photon energy emitted by electrons of r L0 ≃ λ.
The resultant spectrum is similar to GRB photon spectra detected with Fermi-LAT (Abdo et al. 2009(Abdo et al. , 2010;;Ackermann et al. 2010Ackermann et al. , 2011;;Zhang et al. 2011;Ackermann et al. 2014;Yassine et al. 2017).The extra component detected in the GeV energy range has been interpreted as an SSC (Corsi et al. 2010a,b;Asano et al. 2011;Pe'er et al. 2012) or hadronic cascade component (Asano et al. 2009(Asano et al. , 2011;;Asano & Mészáros 2011).If this extra component is due to relativistic waves in highly magnetized plasma, the estimate of the magnetic field and typical electron energy can be largely altered.In our demonstration, the caveat is that electrons are assumed to be isotropic and energetic as r L0 ≫ λ at injection.If a wave comes just after the isotropic injection, our setup for the calculation is justified.
Though the existence of electrons with r L0 ≫ λ is also non-trivial, there are several setups favorable for generating such high-energy electrons.For example, electrons are accelerated by the electric field due to the vacuum gap in the black hole magnetosphere (e.g.Neronov & Aharonian 2007), then such electrons can be injected into a magnetically driven jet outside.In this case, the injection process and turbulence property in the emission site are independent.Another possible injection mechanism is magnetic reconnection and succeeding re-acceleration.In this case, the injection scale of turbulence λ is comparable to the scale of magnetic islands l MI .The induced electric field at the reconnection site is roughly comparable to the background magnetic field.The typical initial energy of accelerated electrons is E typ ∼ eEl MI ∼ eBλ.The Larmor radii of electrons accelerated directly with the electric field in magnetic islands can be the same scale as the magnetic islands as r L0 = E typ /(eB) ∼ λ (Bessho & Bhattacharjee 2012;Sironi & Spitkovsky 2014;Comisso & Sironi 2019).These electrons can be further accelerated via stochastic non-gyro-resonant scattering off the turbulent fluctuations (e.g.see Hoshino 2012;Comisso & Sironi 2019).Such a process may cause an effective injection of electrons with r L0 ≫ λ.
While we have focused on circularly polarized Alfvén waves in this paper, our next step will be the investigation of emission in more general turbulence.Especially for relativistic compressible waves, Γ E×B can be significantly large.The induced large electric field may greatly modify the radiation spectrum.
We first appreciate the advice from the anonymous referee.We also thank Kosuke Nishiwaki, Yo Kusafuka, Tomoki Wada, Takumi Ohmura, Tomohisa Kawashima, Tomoya Kinugawa, Kyohei Kawaguchi for useful comments and discussion.R.G. acknowledges the support by the Forefront Physics and Mathematics Program to Drive Transformation (FoPM).This work is supported by the joint research program of the Institute for Cosmic Ray Research (ICRR), the University of Tokyo, and KAKENHI grant Nos.22K03684, and 23H04899 (K.A.).Numerical computations were in part carried out on FU-JITSU Server PRIMERGY CX2550 M5 at ICRR.

Figure 1 .
Figure 1.Upper: the average Lorentz factor γave of electrons injected with Lorentz factor γi. Lower: the average radiation power Pave normalized by P0.

Figure 2 .
Figure 2. Radiation spectra for different rL0/λ.The synchrotron spectrum without the turbulence is shown with black line as a reference.

Figure 3 .
Figure3.The electron energy distributions (upper) and the radiation spectra (lower) with a power-law (p = 2) distribution for injection in the slow cooling case.The colored solid lines show the results for different γ λ .The case without the wave is shown with the dashed black line.

Figure 4 .
Figure 4. Same as Figure 3 but for the fast cooling case.

Figure 6 .
Figure 6.The average radiation power of electrons normalized by P0.