Revisiting the Polarization of the Emission of the Internal Shock in the Jet of Blazars

Recent Imaging X-ray Polarimetry Explorer (IXPE) observations of blazars tend to support the shock model for the X-ray emission, but report a low degree of polarization (Π ∼ 10%) in X-rays compared with the previous theoretical expectations in the shock model. In order to reconcile the theoretical expectations with observations, we revisit the polarization of the shock emission by considering different types of directions in the distribution of the shock-generated magnetic fields (sgMFs). Here, w sg ′ ∝ ( sin θ ′ ) ζ sg ?> with θ ′ = 0 ?> along the shock normal direction is used to describe the direction in the distribution of sgMFs in the shock comoving frame. It is found that the polarization in the X-ray and radio emission for a general jet in blazars can be described as &Pgr; ∼ 44.5 [ 1 − exp ( − ζ sg / 2.6 ) ] % ?> and &Pgr; ∼ 20 [ 1 − exp ( − ζ sg / 2.4 ) ] % ?> , respectively. Correspondingly, one can have ζ sg ∼ 1−1.5 according to IXPE observations. Besides the sgMFs, the magnetic fields generated by the Richtmyer–Meshkov instability (RMI) (rmMFs) are supposed to be present in the jets. The direction of the rmMFs is mainly distributed along the shock normal in the simulations and thus w rm ′ ∝ ( cos θ ′ ) ζ rm ?> is adopted to describe the direction of the distribution in rmMFs. We find that rmMFs are likely to significantly affect the polarization properties at low-frequency emission, especially when the sgMFs decay rapidly. Based on contemporaneous radio and X-ray observations, we find that the emission of electrons in rmMFs makes a significant contribution to the low-frequency emission and the ordered background magnetic fields can be neglected.

X-ray emission, but report a low degree of polarization (Π ∼ 10%) in X-rays compared with the previous theoretical expectations in the shock model.In order to reconcile the theoretical expectations with observations, we revisit the polarization of the shock emission by considering different types of directions in the distribution of the shock-generated magnetic fields (sgMFs).Here, q ¢ µ ¢ z ( ) w sin sg sg with q¢ = 0 along the shock normal direction is used to describe the direction in the distribution of sgMFs in the shock comoving frame.It is found that the polarization in the X-ray and radio emission for a general jet in blazars can be described as , respectively.Correspondingly, one can have ζ sg ∼ 1−1.5 according to IXPE observations.Besides the sgMFs, the magnetic fields generated by the Richtmyer-Meshkov instability (RMI) (rmMFs) are supposed to be present in the jets.The direction of the rmMFs is mainly distributed along the shock normal in the simulations and thus q ¢ µ ¢ z ( ) w cos rm rm is adopted to describe the direction of the distribution in rmMFs.We find that rmMFs are likely to significantly affect the polarization properties at low-frequency emission, especially when the sgMFs decay rapidly.Based on contemporaneous radio and X-ray observations, we find that the emission of electrons in rmMFs makes a significant contribution to the low-frequency emission and the ordered background magnetic fields can be neglected.

Introduction
Blazars are part of the radio-loud subclass of active galactic nuclei (Blandford & Rees 1978;Blandford & Königl 1979;Blandford et al. 2019; see Blandford et al. 2019 for a review), which power a relativistic jet's point at a small angle θ obs to the Earth's line of sight (Urry & Padovani 1995;Schlickeiser 1996) and exhibit a broad two-hump structure in their characteristic spectral energy distributions (SEDs; Urry 1999).The lowenergy hump (from the radio-to-X-ray frequency range) is generally considered the synchrotron emission of the relativistic electrons in the jet (Ghisellini et al. 1998), while the highenergy hump (from the X-ray to-γ-ray bands) is usually believed to originate from synchrotron self-Compton or a hadronic process (Tavecchio et al. 2001;Abdo et al. 2011;Böttcher et al. 2013).Blazars usually show extreme variability in their light curves (Hovatta & Lindfors 2019), accompanied by the significant/obviously linearly polarization in multi-wave band observations (Agudo et al. 2018;Blinov et al. 2021;Di Gesu et al. 2022;Liodakis et al. 2022;Middei et al. 2023b;Otero-Santos et al. 2023;Peirson et al. 2023).
Multiband polarimetric measurements have been considered an effective tool in breaking the degeneracies in the modeling of the SEDs between the leptonic model (Maraschi et al. 1992;Sikora et al. 1994) and hadronic model (Böttcher et al. 2013;Cerruti et al. 2015), and to explore the magnetic field configurations of the emission region (Jones 1988;Marscher 2014;Tavecchio et al. 2018Tavecchio et al. , 2020;;Tavecchio 2021).In previous work, it has been reported that the variations in the flux and linear polarization of blazars were largely stochastic in nature and this phenomenon could be explained by the magnetic field turbulence in the emission region (Marscher 2014;Marscher & Jorstad 2021).However, some recent works (Kiehlmann et al. 2017;Blinov et al. 2018) have found that the radio and optical polarization properties, such as the polarization angle (PA) rotations, are not simply consistent with a purely random process.This indicates that the magnetic field is not purely stochastic in the emission region, and parts of the magnetic field could be in order (Marscher et al. 2008;Peirson & Romani 2018;Marscher & Jorstad 2021).The ordered magnetic field component could exist in the shock-compressed large-scale background magnetic field (Lyutikov et al. 2005;Hovatta et al. 2012;Zhang et al. 2016), the shock-generated turbulence (Tavecchio et al. 2018(Tavecchio et al. , 2020;;Gabuzda 2021), or in a kinkinstability-induced magnetic reconnection environment (Bodo et al. 2021;Zhang et al. 2021Zhang et al. , 2023)).For the scenario invoking the large-scale kink instabilities, a high optical polarization value of ∼20% and a degree of smooth polarization/PA modulation are predicted by Bodo et al. (2021), with a relatively low X-ray polarization expected by Zhang et al. (2021).In the weakly magnetized shock scenario, the magnetic field in the high-energy emission region is more orderly than in the low-energy emission region, so the degree of polarization is predicted to increase with the energy band (Marscher & Gear 1985;Angelakis et al. 2016;Tavecchio et al. 2018).
Recently, the Imaging X-ray Polarimetry Explorer (IXPE) observed the blazar Mrk 501 (Liodakis et al. 2022) and found that its X-ray polarization (∼10%) was higher than its optical polarization (∼5%), which supported the shock scenario with an energy-stratified electron population.However, the observed degrees of polarization of Mrk 501 (Liodakis et al. 2022;Hu et al. 2024), together with the other blazar polarization observations (Mrk 421, Di Gesu et al. 2022;PG 1553+113, Middei et al. 2023a; BL Lacertae, Middei et al. 2023b;1ES 1959+650 and PKS 2155-304, Hu et al. 2024), are significantly lower than theoretical predictions (e.g., ∼30%, Tavecchio et al. 2018).In Tavecchio et al. (2018), the decaying behavior of the shockgenerated magnetic fields (sgMFs; e.g., Schureet al. 2012) is considered.However, the sgMFs are set to be completely parallel to the shock front, which is the same by setting a high ζ sg in Equation (7) (also see the discussion in Section 3.1).The difference between the IXPE observations and theoretical predictions could come from the fact that the turbulence in the emission region will reduce the anisotropy of the shock magnetic field.In this paper, we study the polarization properties of sgMFs at different degrees of anisotropy.
The Very Long Baseline Array carried out radio polarization imaging of the nearest and brightest blazar Mrk 421 (de Vaucouleurs et al. 1991), and found that the observed direction of the PA was almost perpendicular to the jet axis on the subparsec scale (Lico et al. 2013(Lico et al. , 2014)), which means that there is a large-scale radial magnetic field.Interestingly, the large-scale radial magnetic field was also found in the supernova remnant (Jun & Norman 1996;Zirakashvili & Ptuskin 2008;Inoue et al. 2013;West et al. 2017); for example, the recent X-ray polarization imaging of Cassiopeia A by IXPE revealed the existence of radial magnetic fields (Vink et al. 2022).Inoue et al. (2013) proposed that the large-scale radial magnetic field of the supernova remnant could be driven by the Richtmyer-Meshkov instability (RMI), in which the RMI grows once the incident shock strikes the corrugated contact discontinuity separating two fluids of different densities (Richtmyer 1960;Meshkov 1969).In blazars, the RMI could grow gradually when the shock sweeps through the emission region with nonuniform density distributions, and then the radial (parallel to shock normal) magnetic field component could be amplified from the background magnetic field (Sano et al. 2012).In this paper, we introduce the large-scale radial magnetic fields generated by the RMI (rmMF) in the shock model.
This paper is structured as follows.In Section 2, we present our model and basic calculation method.In Section 3, we present the polarization behavior of the internal shock emission in the blazar case.In Section 4, the internal shocks in blazars are discussed based on the IXPE observations.Finally, we summarize our conclusions in Section 5.

Model
In the internal shock model, the central engine of blazars intermittently ejects shells of relativistic plasma at varying speeds.These shells subsequently collide and a pair of shocks traveling in opposite directions in the frame of the shocked fluid are formed.As the shocks propagate in the unshocked part of shells, they convert the ordered bulk kinetic energy of the plasma into the magnetic field energy and random kinetic energy of the particles.At and near the shock front, the electrons are accelerated and a random magnetic field is formed.While flowing downstream, electrons produce synchrotron emission within the magnetic field in situ and inverse-Compton radiation, where the synchrotron emission and the inverse-Compton radiation are responsible for the lowfrequency (radio-optical/UV) emission and high-frequency (X-ray/γ-ray) emission from blazars, respectively.In this paper, we revisit the polarimetric features of the internal shocks by considering a more general distribution of magnetic fields.
To simplify, we study the situation in which a shock is formed at the collision radius R is = 3 × 10 16 cm and propagates in a cone-shaped jet shell with a thickness of R th = R is /Γ 2 and opening angle Θ j .Hereafter, the superscript prime is used to denote the quantities measured in the comoving frame of the downstream, and ¢ t is used to represent the time measured in the rest frame of the downstream since the collision of two jet shells.Since both the electrons and the magnetic fields are advected in the post-shock region and the corresponding energy decays with time, the subscript " ..., 0 " is used to denote the quantities associated with those generated at time ¢ t 0 .For example, if an electron is injected at the time ¢ t 0 , the Lorentz factor g ¢ e of this electron at time ¢ t is represented by g ¢ ¢ ( ) t e,0 .Here, the distance of this electron relative to the shock front is u ¢ -¢ ¢ ( ) t t 0 sh with u ¢ » c sh being the speed of the shock front in the comoving frame of the jet flow.The radiation at moment ¢ t should be the sum of the radiation from the electrons injected at different ¢ Î ¢ [ ] t t 0, 0 , i.e., an integral of the radiation over the volume of the post-shock region.The 1-axis and 2-axis of the ˆˆk 12 -coordinate are used to describe the observed direction of the polarization of the radiation.In the ˆˆk 12 -coordinate, we introduce a spherical coordinate with polar angle Θ and azimuthal angle Φ to describe the emission region and the PA.Here, Θ = 0 is along the direction of e k and Φ is the azimuthal angle in the ˆ1 2-plane from the 1-axis.By setting the viewing angle of the jet axis as Θ v , one can have

Coordinates Used in the Model
x y -coordinate is used to describe the structure of the magnetic fields in the local jet flow.In this coordinate, the unit vector b¢ e of the b¢-axis is along the direction of the local jet flow and the unit vectors of the ¢ x -axis and ¢ y -axis are as follows: In order to describe the direction in the distribution of the magnetic fields in the b ¢ ¢ ¢ x y -coordinate, we introduce a spherical coordinate with polar angle q¢ and azimuthal angle j¢.The polar axis of this spherical coordinate is along the b¢-axis, and j¢ is the azimuthal angle in the ¢ ¢ x y -plane from the ¢ x -axis.4. In the comoving frame of the jet flow, it is useful to define the ¢ ¢ ¢ ˆˆk 1 2 -coordinate to describe the polarimetric properties of the radiation.The unit vectors of the ¢ 2 -axis and ¢ 1 -axis are, respectively, defined as ( ) e e e e e , . 4 In our calculations, the polarization of the radiation is first decomposed into the direction of ¢ ê 1 and ¢ ê 2 and then converted into the vector in the ˆˆk 12 -coordinate.The details can be found in Lan et al. (2018).In the ¢ ¢ ¢ ˆˆk 1 2 -coordinate, we also introduce a spherical coordinate with polar angle q ¢ B and azimuthal angle j ¢ B to describe the direction of the magnetic field.Here, the q ¢ B would be the pinch angle of the emitting electron in the magnetic field, and j ¢ B is the azimuthal angle of the magnetic field in the ¢ ¢ ˆ1 2 -plane from the ¢ 1 -axis, and thus, is related to the PA.The values of q ¢ B and j ¢ B are used in estimating the synchrotron emission of electrons, e.g., Equations (13)-(15).

Magnetic Fields in the Downstream
The synchrotron emission of the internal shock is related to three types of magnetic fields: ordered background magnetic fields (obMFs) ¢ B ob , shock-generated random magnetic fields (sgMFs) ¢ B sg , and magnetic fields generated by RMI (rmMFs) ¢ B rm .obMFs.The obMFs in the emission region are carried from the central engine of blazars.The geometric configuration of ¢ B ob is usually set to toroidal or radial structures in the emission region.In general, the strength of the obMFs is lower than that of the sgMFs at a very small distance from the shock, but is stronger than that of the sgMFs at a very high distance from the shock.However, the polarization of the emission of the internal shocks with only strong obMFs has been extensively studied (e.g., Zhang et al. 2014Zhang et al. , 2016)).In addition, the strength of the rmMFs is strong compared with that of the obMFs in the postshock region (e.g., Sano et al. 2012;Inoue et al. 2013).This paper is focused on the polarization of the emission of the internal shock with an anisotropic distribution of sgMFs and the magnetic fields generated by RMI.To simplify the problem, the obMFs are set as ob ob sh,0 with f ob = 10 −3 , where ¢ B sh,0 is the initial strength of the sgMF.sgMFs.In the internal shock model, the random magnetic field is generated by the electric current filaments generated from Weibel instability (Weibel 1959).Some numerical simulations show that the filaments only survive within a microscopic scale behind the shock front, and then they interact with each other and force coalescence in the downstream region (Silva et al. 2003;Medvedev et al. 2005;Chang et al. 2008).The simulations reveal that the small-scale turbulent magnetic field generated by the shock decays with time as a power-law form when it flows away from the shock front to the downstream (Chang et al. 2008;Lemoine et al. 2013;Lemoine 2013).Thus, the strength evolution of the sgMFs formed at time ¢ t 0 can be described as where ¢ B sh,0 is the initial strength of the sgMFs generated at time ¢ t 0 , ¢ t is the time since the collision of two jet shells, ¢ t B is the decay timescale of the shock-generated random magnetic field, and α B is a power-law decay index.The effect of the sgMFs' decay behavior on the jet's emission is related to the value of T m sh,0 is the electron synchrotron cooling timescale in the magnetic field of ¢ B sh,0 and g ¢ m is the minimum Lorentz factor of the shock-accelerated electrons (Zhao et al. 2014).Thus, it is useful to associate with τ B as a free parameter.In general, the initial strength of the sgMFs' ¢ B sh,0 is associated with the dissipated energy density e¢ of the shock as where ò B is the fraction of the shock dissipated energy used to form the sgMFs.With the dissipation efficiency ò dis , we set for simplicity, where L jet is the kinetic energy of the jet and Γ = 10 is the Lorentz factor of the emission region.Assuming the fraction of the dissipated energy used to accelerate electrons is ò e , one can have ò dis ò e L k = L obs in the fast cooling case, where L obs is the observed luminosity.To simplify our discussion, the values of ò dis = 5% (Kobayashi et al. 1997), ò e = 0.3, and L jet = 10 46 erg s −1 are used in our calculations.
Simulations reveal that the sgMFs at the shock front are predominantly orthogonal to the shock normal (Tavecchio et al. 2018).In Tavecchio et al. (2018), the sgMFs are set to be completely parallel to the shock front.However, the expected polarization in such type of magnetic field morphology is around ∼30% (e.g., Tavecchio et al. 2018), which is significantly higher than the observations.In order to reconcile the theoretical expectations with the observations, we revisit the polarimetric features of the internal shock model by considering a more general distribution in the direction of the sgMFs.Since the direction of the sgMFs is predominantly orthogonal to the shock normal, the distribution in the direction of the sgMFs is described as where ζ sg is a power-law index and q¢ is the angle between the direction of the sgMF and b¢ e (see Section 2.1).The distribution in the direction of the sgMFs is assumed to remain the same during the decay of the sgMFs and is the same for sgMFs generated at different ¢ t 0 .Equation (7) reveals that the sgMFs with sg sg for the sgMFs generated at time ¢ t 0 , where q q j W¢ = ¢ ¢ ¢ d d d sin .Magnetic fields generated by the RMI.Except for the sgMFs, the magnetic fields can be amplified through the macroscopic turbulence dynamo excited by the so-called RMI.This instability is an inevitable outcome of interactions between shock and ambient density fluctuations (Richtmyer 1960;Meshkov 1969).Simulations show that the magnetic field can grow by at least 2 orders of magnitude compared to the magnetic energy immediately behind the shock (i.e., obMFs), provided the kinetic energy of the turbulence injected by the RMI is greater than the magnetic energy (e.g., Inoue et al. 2011;Sano et al. 2012;Inoue et al. 2013).Thus, we take rm sh,0 with f rm = 2 × 10 −2 , and the intensity of the rmMFs in the post-shock region remains constant.The simulations of the rmMFs are finally dominated by the radial component (e.g., Sano et al. 2012).Then, the distribution in the direction of the rmMFs can be described as where ζ rm is the anisotropy index of the rmMFs.Equation (8) reveals that the rmMF with sin cos , sin sin , cos rm,0 takes a proportion of ò ò for the magnetic fields generated by RMI in the emission region.
The magnetic field in any region.The total magnetic field in an emission region can be expressed as

Electrons' Prescription and Evolution
The calculation of the evolution of the electron energy spectrum is taken from the work of Zhou et al. (2023).Again we write it as follows.It is generally assumed that the accelerated electrons near the shock front obey a power-law distribution, i.e., for others, where g ¢ ¢ ( ) is the Lorentz factor of the electrons at time ¢ t for these injected at the time ¢ t 0 , the normalization factor ´-7 10 3 cm −3 • s −1 is simply set, p is the power-law index and is taken as 4.7 in our calculation, and tot,0 0 e 3 1 2 is the maximum Lorentz factor of the shock-accelerated electrons, and m p , m e , and q e are the rest mass of the proton, the rest mass of the electron, and the charge of the electron, respectively.In our scenario, both the electrons and the magnetic fields are advected in the postshock region, and the corresponding energy decays with time.During the advection in the post-shock region, the electrons suffer from radiation cooling by synchrotron radiation and inverse-Compton radiation, i.e., where Y is the Compton parameter.Generally, the radiation cooling of the electrons is strong compared with that due to the expansion of the jet shell in the internal shocks.Thus, the adiabatic cooling of electrons is neglected in Equation (10).Since the magnetic field decays with time, the value of Y can be described as where Ỹ denotes the ratio of inverse-Compton radiation to synchrotron radiation power at the shock front and = Ỹ 0.5 is set in this paper (Zhao et al. 2014).With Equations (10) and (11), the Lorentz factor of an electron varies with time as remain the same in Equation (12).That is to say, the effect of Ỹ on the synchrotron radiation spectrum can be reflected by changing the value of ¢ ¢ ( ) B t tot,0 0 .This behavior has been partially shown in Figure 3 and discussed in Section 4 of Zhao et al. (2014).In this paper, we set = Ỹ 0.5 to simplify our discussion and fittings.
In general, the energy distribution of electrons is obtained based on the standard continuity equation (e.g., Chiaberge & Ghisellini 1999;Liu et al. 2021).However, one can obtain the energy distribution of electrons based on the following process.In our scenario, if an electron is injected at time ¢ t 0 with Lorentz factor g ¢ ¢ ( ) t e,0 0 , the Lorentz factor of this electron would decay to g ¢ ¢ ( ) t e,0 at time ¢ t .Correspondingly, the total number of electrons in the g g g ,0 ,0 e,0 0 , which is used in Equations ( 13)-(15).That is to say, the energy distribution of the electrons at time ¢ t for the electrons injected at time ¢ t 0 can be described as e,0 e,0 0 e,0 .

Radiation and Polarization
The detailed process for estimating the polarization of the emission of the synchrotron emission has been presented in previous works (e.g., Del Zanna et al. 2006;Toma et al. 2009;Lan et al. 2018s).The calculations of the synchrotron emission in this paper are the same and presented as follows.
Synchrotron emission for electrons injected at time ¢ t 0 .In our scenario, both the electrons and the magnetic fields are advected in the post-shock region.During the advection, the electrons injected at time ¢ t 0 are cooled to the Lorentz factor of g ¢ ¢ ( ) in situ, the synchrotron emission of the electrons with g ¢ ¢ ( ) in the ¢ ¢ ¢ ˆˆk 1 2 -coordinate can be described as ,0 tot,0 B e,0 e,0 0 where q ¢ B is the pitch angle of the emitting electron in the magnetic field, i.e., is the characteristic frequency of the photon emitted by the electron with the Lorentz factor g ¢ ¢ ( ) , and K 5/3 (x) and K 2/3 (x) are the modified Bessel function of 5/3 and 2/3 order, respectively.One should note that if an electron is injected at time ¢ t 0 , the Lorentz factor of this electron is g ¢ ¢ ( ) t e,0 at time ¢ t .Since only the energy of the electrons decays with time ¢ t , the total number of the electrons with the Lorentz factor in the g g g ,0 ,0 can be described as e,0 0 , which is used in Equations ( 13)-( 15).Since the distribution is in the sgMFs and the rmMFs, the synchrotron emission from a region can be described as ,0 ,0 sg sg sg rm rm rm sg rm ,0 ,0 sg sg sg rm rm rm sg rm ,0 ,0 sg sg sg rm rm rm sg rm Based on the above results, the degree of polarization and PA in the comoving frame can be written as Correspondingly, the Stokes parameters in the the observer frame are where cos tan ,0 , and Φ is the azimuthal angle of the emission region in the ˆˆk 12 -coordinate (see Section 2.1).For the details of the relation of χ ν,0 and c ¢ n,0 , please see Equation (5) in Lan et al. (2019).
Synchrotron emission of electrons in the post-shock region.The total synchrotron emission is from the entire post-shock region.Considering the equal arrival time surface (EATS) effects, the observed synchrotron radiation flux density and the Stokes parameters are described as ò ò ò is the solid angle of the emission region, and "EATS" represents the integration over the EATS.For the jet flow shocked at time ¢ t 0 , the observed time of a photon emitted from are the radius of the jet flow at the collision time and ¢ t , respectively.According to Equations ( 21)-( 23), the observed degree of polarization and angle is estimated as As is known, the polarization of the emission depends not only on the magnetic field structure but also on the observational angle Θ v (e.g., Tavecchio et al. 2018).This paper is focused on the polarization of the emission of the internal shock with the anisotropic distribution of sgMFs and rmMFs.In such types of magnetic fields, Tavecchio et al. (2018) have shown that if a low viewing angle Θ v is set to, e.g., Θ v  Θ j , the degree of polarization of the synchrotron emission is significantly low.In our calculations, we also find that the polarization of the emission from the situation with Θ v = Θ j is significantly lower than that from the situation where Θ v = Θ j + 1/Γ, which is inconsistent with the IXPE observations (∼10%, e.g., Di Gesu et al. 2022; Liodakis et al. 2022).In addition, a large part of the blazars is likely to be off-axis in observations (e.g., Giroletti et al. 2008;Lico et al. 2014;Abe et al. 2023a).Thus, we set Θ v = Θ j + 1/Γ in our calculations.
Here, z = 0.03 is the cosmological redshift, and u ¢ = c sh is set.In order to be consistent with the observed radiation spectrum, the value of p = 4.7 is adopted for the electron injection.

Polarization in the Situation with Only Anisotropic sgMFs
It has been found earlier that the magnetic fields generated by shocks are not isotropic and are dominated by the component perpendicular to the shock normal (Caprioli & Spitkovsky 2014;Zhang et al. 2016;Tavecchio et al. 2018;Tavecchio 2021).In this paper, we revisit the polarization of the internal shock with such type of sgMFs.Different from previous works (e.g., Tavecchio et al. 2018), Equation ( 7) is introduced to describe the distribution in the direction of the sgMFs in this paper.If a higher value of the anisotropy index ζ sg (> 0) is adopted in Equation (7), more of the sgMFs would be perpendicular to the shock normal.In Figure 1, we show the frequency-dependent Π (upper panel) and χ (bottom panel), where different values of ζ sg , i.e., 0.5 (orange lines), 1 (green lines), 1.5 (blue lines), 2 (red lines), and 3 (purple lines), are adopted.In Figure 1, the solid and dashed lines correspond to different decay behaviors of the sgMFs, i.e., α B = 1.0 and 0.5 are denoted by solid and dashed lines, respectively.By comparing the different colors of the solid or dashed lines, one can find that the anisotropy index ζ sg significantly affects the value of Π for all wave bands.If a higher value of the anisotropy index ζ sg is adopted, the obtained degree of the polarization of Π would be higher for the same wave band.This behavior is not affected by adopting a different decay behavior of the sgMFs, i.e., different α B .Similar to the results found in Tavecchio et al. (2018), α B mainly has an effect on the polarization of the intermediate frequency emission (e.g., the optical band), rather than the high-frequency emission (e.g., the X-ray band) and low-frequency emission (e.g., the radio band).It is worth pointing out that for the case with ζ sg = 2 (purple lines), the polarization of X-rays is around Π ∼ 25% and that of the optical band is around Π ∼ 12%, which is similar to those in Figure 4 in Tavecchio et al. (2018).This reveals that Equation (7) with ζ sg = 2 roughly describes the distribution in the direction of the sgMFs in Tavecchio et al. (2018).
In Figure 2, we show the relations of Π−ζ sg for the X-ray and radio bands, where the "×" and "+" symbols are for the X-ray and radio bands, respectively.The dependence of the polarization on the value of the anisotropy index ζ sg can be easily found.In Figure 2, it can easily be seen that the value of Π increases quickly (black line) are obtained for the X-ray and radio bands, respectively.The above two relations can be understood as follows: (1) In the situation where ζ sg → ∞, most of the sgMFs are along the direction perpendicular to the shock normal in the shock comoving frame.This type of case is the same as that where the electrons are in the ordered magnetic fields.Correspondingly, the polarization in the high-energy emission can be related to the spectral index β as Π = (−2β + 2)/(−2β + 10/3) or the injected index p1 as Π ∼ (p + 1)/(p + 7/3).In Figures 1 and 2, the value of p = 4.7 is adopted for the electron injection, and thus, Π ∼ 81%.For the low-frequency emission, e.g., the radio band, the polarization can be described as Π = (−2β + 2)/ (−2β + 10/3) with β = 1/3, i.e., Π = 50%.One should note that the geometric structure of the emission region affects the polarization of the emission and the asymmetric geometry of the emission region depolarizes the synchrotron emission.In the situation where ζ sg → ∞, the relations of   where the suppression factors of 1.82 and 2.5 in the denominators are related to the effect of the depolarization of the emission region.Since the emission region of the highfrequency emission is small compared with that of the lowfrequency emission, the effect of the depolarization of the emission region is low for the high-frequency emission compared with that for the low-frequency emission.This is consistent with the suppression factor in Equation (26), i.e., 1.82 is lower than that in Equation ( 27), i.e., 2.5.Since the sgMFs are mainly distributed along the direction perpendicular to the shock normal, the polarization of the emission would be parallel to the shock normal for all wave bands.In the bottom panel of Figure 1, the value of χ is indeed 0°for all wave bands.

Polarization in the Situation with Only sgMFs + rmMFs
The RMI appears when an incident shock wave strikes a corrugated contact discontinuity separating two fluids of different densities (Richtmyer 1960;Meshkov 1969).In the blazars, it is very likely that the density in the jets is not uniformly distributed, and thus, RMI may be activated.In this part, we study the polarization of the emission of the internal shock with both sgMF and rmMFs.It is reasonable to believe that the sgMFs would dominate the magnetic field around the shock front.In this situation, the rmMFs have little effect on the polarization of the high-frequency emission, and thus, ζ sg ∼ 1 is required to explain the IXPE observations (i.e., Π ∼ 10% in the X-ray band, Liodakis et al. 2022).Thus, ζ sg ∼ 1 is adopted in the following studies.
In this paper, the sgMFs are assumed to decay with time as Equation (5), and the rmMFs are assumed to not decay with time.If the sgMFs generated at time ¢ t 0 decay quickly, the magnetic field in situ would be dominated by the rmMFs for significantly high ¢ -¢ t t 0 .If not, the sgMFs would dominate the magnetic field entirely.Here, the decay behavior of the sgMFs is reflected by varying the value of α B .In Figure 3, we show the relations of ν −Π and ν−χ in the situation where the rmMFs dominate the magnetic field quickly.Here, the value of α B = 1.0 is adopted.In this figure, the anisotropy indices of the rmMFs ζ rm = 0.25 (green line), 0.5 (blue line), 0.75 (red line), and 1 (black line) are adopted.One can find that there is a transition of χ from 90°to 0°at a transition frequency of ∼10 15 -10 16 Hz based on the bottom panel of Figure 3.This reveals that the magnetic field of the lowfrequency emission is dominated by the rmMFs and that of the high-frequency emission is dominated by the sgMFs.This is owing to the sgMFs and the rmMFs being mainly distributed along the direction perpendicular and parallel to the shock normal, respectively.In this situation, the polarization of the emission in the emission region with the magnetic field dominated by the sgMFs or rmMFs would be parallel (χ = 0°) or perpendicular (χ = 90°) to the shock normal.It is worth pointing out that the Πν relation around the transition frequency is peculiar (also see the green solid line in Figure 6).As ν decreases, the degree of polarization Π decreases from ∼12.5% at the high-frequency emission to ∼0% at around the transition frequency, and subsequently increases to a certain value (e.g., ∼7.5%, the black line) at the low-frequency emission.The value of Π ∼ 0% around the transition frequency reveals that the magnetic field responsible for the transition frequency emission is almost isotropic.
In Figure 4, we show the relations of Π−ζ rm for the optical and radio bands in situations that are the same as those shown in Figure 3 where the suppression factors of 2.8 and 2.2 in the denominators are related to the depolarization effect of the emission region.If the sgMFs do not decay quickly, the sgMFs will dominate the magnetic field in most of the emission region.In Figure 5, we show the polarization of the emission in the situation where α B = 0.5, where the anisotropic indices ζ rm = 0 (black line), 1 (red line), 2 (blue line), 3 (green line), and 4 (orange line) are adopted.It can be easily found that the value of χ remains 0°f or all wave bands, and thus, the polarization of the emission is mainly subject to the distribution of the sgMFs.Compared to the case with only sgMFs, e.g., the green dashed line in Figure 1, the degree of polarization Π in the radio band is low for all the cases shown in Figure 3.In addition, the value of Π decreases as the ζ rm increases according to Figure 5.This behavior reveals that the exit of the rmMFs in the emission region weakens the anisotropy of the magnetic field distribution, even though the rmMFs do not dominate the magnetic field in the main body of the emission region.Taking the case with ζ rm = 0 as an example, the degree of polarization Π ∼ 5.2% is obtained for the radio band, which is lower than that of the case with only sgMFs, i.e., Π ∼ 6.8% (the green dashed line in Figure 1).This indicates that the fraction of emission from the electrons in sgMFs to those in the rmMFs can be expressed as f sg = 5.2%/6.8%= 77% and f rm = 1 − f sg = 23%, respectively.One should note that the main distribution in the direction of the rmMFs is perpendicular to the main distribution in the direction of the sgMFs if ζ rm is not equal to zero.Thus, the depolarization of the radio emission of the jet would be stronger if a higher ζ rm is adopted.We find that the total degree of polarization of the radio band can be roughly described as

Effects of the ObMFs or Viewing Angle on the Polarization
In general, the strength of the obMFs is lower than that of the sgMFs at a very small distance from the shock, but is stronger than that of the sgMFs at a much greater distance from the shock.In addition, the strength of the rmMFs is strong compared with that of the obMFs in the post-shock region (e.g., Sano et al. 2012;Inoue et al. 2013).Since the polarization of the emission of the internal shocks with only strong obMFs has been extensively studied (e.g., Zhang et al. 2014Zhang et al. , 2016)), we study the polarization of the emission of the internal shocks with only sgMFs + obMFs, of which the results are shown in Figure 6.Here, the situation with fast decaying sgMFs, i.e., α B = 2, is adopted, and ζ sg = 1 is set.In this figure, different types of obMF morphologies, i.e., toroidal obMFs (black solid line) and radial obMFs (blue solid line) are discussed, and the polarization of the emission of the internal shocks with only sgMFs is also plotted with the green solid line for comparison.From this figure, one can find that the polarization at the low-frequency emission is very different from that of the situation with only sgMFs.It reveals that the obMFs become dominant in the much larger post-shock region, where the lowfrequency emission is produced.Then, it is easy to see that the degree of polarization at the low-frequency emission is very high,  e.g., 20%-40%, which is similar to that with only obMFs (e.g., Zhang et al. 2014Zhang et al. , 2016;;Tavecchio et al. 2018).Interestingly, there is a dip in the Π-ν relation and a transition in the χ-ν relation at ν ∼ 10 15 -10 16 Hz for the situation with radial obMFs.This is similar to those found in Figure 3 (see the discussion in Section 3.2).In addition, a bump appears below the dip in the Πν relation at ν ∼ 10 14 Hz.This is because (1) for the internal shock emission in the situation with only obMFs, the Π remains almost constant in ν < 10 12 Hz and increases with rising ν in ν  10 12 Hz; (2) the high-frequency emission is mainly from the electrons in the sgMF-dominated region, i.e., around the shock front.Moreover, the value of Π in this situation is significantly lower than that in the situation with only obMFs.Thus, the combination of (1) and (2) leads to the formation of a bump at ν ∼ 10 14 Hz.
We also study the situation with slow-decaying sgMFs, e.g., α B = 0.5, of which the polarization of the internal shock emission is similar to that shown in Figure 1, and thus, is not shown in this paper.In addition, the strength of the rmMFs would be strong compared with that of the obMFs in the postshock region (e.g., Sano et al. 2012;Inoue et al. 2013).However, if the strength of the rmMFs is weaker than that of the obMFs, the polarization of the internal shock emission is similar to those shown in Figure 6 or Figure 1.Thus, the situation with sgMFs + rmMFs + obMFs is not shown in this paper.
The polarization of the emission depends not only on the magnetic field structure but also on the viewing angle Θ v (e.g., Tavecchio et al. 2018).For the internal shock with anisotropic distribution sgMFs and rmMFs, Tavecchio et al. (2018) have shown that if a low viewing angle Θ v is set, the degree of polarization of the synchrotron emission would be significantly low.In Figure 6, we also show the polarization of the internal shock emission in the situation withΘ v = Θ j (dashed lines).A very low degree of polarization is indeed found in this figure.This implies that if the internal shock scenario is applied to explain the IXPE observations (∼10%, e.g., Di Gesu et al. 2022;Liodakis et al. 2022), a high viewing angle is required.Indeed, a large part of the blazars are likely to be off-axis observations (e.g., Giroletti et al. 2008;Lico et al. 2014;Abe et al. 2023a).

Discussion Based on the IXPE Observations
Recently, the IXPE, combined with other telescopes, performed multiwavelength observations of the blazars Mrk 501, Mrk 421, and PG 1553+113.The blazars Mrk 501, Mrk 421, and PG 1553 +113 are classified as high-synchrotron-peaked BL Lacertae objects (ν peak > 10 15 Hz, Di Gesu et al. 2022;Liodakis et al. 2022;Middei et al. 2023a), of which the X-ray emission is dominated by the synchrotron radiation of the electrons.The IXPE observations may reveal that the particle acceleration in shocks may operate in the blazar jet (Di Gesu et al. 2022;Liodakis et al. 2022).In this section, we apply the IXPE observations associated with the radio/optical observations to estimate the properties of the sgMFs and rmMFs.
1. Liodakis et al. (2022) performed two observations on Mrk 501 with IXPE, accompanied by observations across the electromagnetic spectrum from multiple observatories.The results from these observations (Liodakis et al. 2022) are summarized as follows: (1) The degree of X-ray linear polarization Π X is around 10%, the host galaxy corrected intrinsic degree of optical polarization Π O is ∼5%, and the degree of radio polarization Π R is ∼1.5%.
(2) The polarization of the radio-to-X-ray bands is aligned with the jet axis within uncertainties.
(3) There is no evidence of polarization variability during either IXPE observation.In Section 3, we found that the polarization of the jet emission in the internal shock scenario is along the jet axis if the sgMFs dominate the magnetic field in most of the emission region.Thus, the sgMFs dominate the magnetic field in most of the emission region during the IXPE observations of Liodakis et al. (2022).In the internal shock scenario for the blazar jet's emission, the X-rays are mainly from the shock front, in which the magnetic field is dominated by sgMFs.Thus, based on Figure 2, we can have ζ sg ∼ 0.7 for the shock responsible for the X-rays in Mrk 501.
Correspondingly, the degree of polarization of the radio (optical) band would be around 5% (5%) based on Figure 2 if the rmMFs can be neglected compared with sgMFs.However, the observation of Π R ∼ 1.5% is very low compared with the expectation shown in Figure 2.This may reveal that the emission of the electrons in the rmMFs could make a significant contribution to the radio band and the obMFs can be neglected.The contribution fraction f rm of the emission for the electrons in the rmMFs and the anisotropy index ζ rm have the following relation:  degree of polarization of the radio band in the situation with only sgMFs is Π R ∼ 9%, which is significantly different from the observations.This also reveals that the emission of the electrons in the rmMFs makes a significant contribution to the radio band and the obMFs can be neglected.2023b;Di Gesu et al. 2023).This may imply that the X-ray emission region is different from the optical/radio emission region in these two observations.4. Middei et al. (2023a) reveal an orphan optical polarization swing of PG 1553+113 during the IXPE observation.IXPE performed the observations of PG 1553+113 on 2023 February 1-2 and 2023 February 7-8.The time average degree of X-ray polarization at both observed periods is Π X ∼ 10.1% with χ X ∼ 86°.During the observation period of 2023 February 1-2, the degree of optical polarization Π O ∼ 2.2% with a significant rotation of the PA is found.However, there is no significant variety in the X-ray PA.This implies that the emission region of the optical band may be different from that of the X-ray band during this period.Thus, we do not discuss the observation results during this period.During the observation period of 2023 February 7-8, the optical polarization Π O ∼ 4.2% and radio polarization Π R ∼ 3% are found, and the PA of the radio-X-ray bands is roughly consistent.The observed direction of the X-ray polarization is found to be oblique in the direction of the parsecscale jet with a difference of ∼45°, which is similar to that observed for Mrk 421 during its first IXPE observation in 2022 May.Since the local direction of the jet may be different from the direction of the parsec-scale jet, Di Gesu et al. (2022) proposed that the direction of the polarization of the X-ray band may be in the same direction as the local jet around the emission region.With Equation (26), one can have ζ sg = 0.7 for the sgMFs based on the observed degree of X-ray polarization Π X ∼ 10.1%.Correspondingly, the degree of polarization of the radio band is estimated to be Π R ∼ 5% for the case with only sgMFs.This also reveals that the emission of the electrons in the rmMFs makes a significant contribution to the radio band and the obMFs can be neglected.By comparison with the observed Π R ∼ 3%, the contribution fraction of the emission from the electrons in the rmMFs is f rm ∼ 10%-40%, where the ζ rm -dependent f rm is shown in Figure 7) by the blue line.
In summary, the anisotropy index of the sgMFs is around ζ sg ∼ 0.7−1.5 based on the degree of polarization of the X-ray emission in blazars.Correspondingly, one can estimate the degree of polarization of the radio band expected in the situation with only sgMFs, which is found to be generally higher than the observed degree of radio polarization in the radio band.This reveals that the emission of the electrons in the rmMFs makes a significant contribution to the low-frequency emission, and the contribution fraction is around 40%-70% in the situation where ζ rm = 0.The ζ rm -dependent f rm is shown in Figure 7 for the observations discussed above.If most of the rmMFs are along the shock normal (i.e., ζ rm  3), the contribution fraction f rm is around 10%-23%.We would like to point out that most of the rmMFs are along the direction of the shock normal for the situation where ζ rm  3.

Conclusions
The morphology of the magnetic field in the jet of blazars is of great significance for the study of the radiation mechanism and particle acceleration process in situ.The operation of the IXPE polarization detector provides an unprecedented opportunity to directly study the magnetic field morphology in the X-ray emission region.Recent IXPE observations reveal that the X-ray polarization is higher than the optical and radio polarization and parallel to the shock normal.Thus, the X-ray emission of the blazars is suggested to support the shock scenario with an energy-stratified electron population (Di Gesu et al. 2022;Liodakis et al. 2022).In order to reconcile the theoretical expectations with observations, we revisit the polarization of the emission of the internal shock by considering different types of directions in the distribution of the sgMFs and rmMFs, while also considering the effect of the obMFs.
The sgMFs are mainly along the direction perpendicular to the shock normal.Thus, we introduce q ¢ µ ¢ z ( ) w sin sg sg with q¢ = 0 being along the shock normal to describe the distribution in the direction of the sgMFs in the shock comoving frame.Here, most of the sgMFs are distributed along the shock front in the situation where ζ sg  3.In the cases with only sgMFs, it is found that the polarization in the X-ray and radio emission for a general jet in blazars can be described )] 2.4 % sg , respectively.The degree of polarization of the optical band is between that of the X-ray band and that of the radio band.The directions in the polarization of radio-X-ray bands are the same.In addition, the above results are not affected by the decay behavior of the sgMFs, i.e., different α B is adopted to describe the decay of the sgMFs.In the internal shock scenario for the blazar jet's emission, the X-rays are mainly from the shock front, in which the magnetic field is dominated by the sgMFs.Based on the results of the IXPE observations, the value of ζ sg ∼ 1−1.5 is required in order to reconcile the X-ray observations.
The magnetic fields generated by the RMI, which are triggered once the incident shock strikes the corrugated contact discontinuity separating two fluids of different densities (Richtmyer 1960;Meshkov 1969), are supposed to be present in the jets.Simulations reveal that the rmMFs are mainly distributed along the direction of the shock normal (Sano et al. 2012;Inoue et al. 2013), and thus, q ¢ µ ¢ z ( ) w cos rm rm is introduced to describe the direction of the distribution of rmMFs.In the situation where ζ rm  3, most of the rmMFs are distributed along the shock normal.In the situation with sgMFs and rmMFs, the presence of rmMFs is likely to reduce the degree of polarization in the lowfrequency emission if the emission of the electrons in the sgMFs dominates the emission.If the emission of the electrons in the rmMFs dominates the low-frequency emission, the direction of the polarization of the low-frequency emission would be perpendicular to that of the high-frequency emission (i.e., X-rays).For the situation where the low-frequency emission is dominated by the electrons in the rmMFs, we find that the degree of polarization of the radio emission can be described as .For the situation where the directions of the polarization of the radio-X-ray bands are the same, the degree of polarization of the low-frequency (e.g., radio) emission can be described as Π sg (ζ sg )f sg −Π rm (ζ rm ) × f rm , where f sg and f rm are the contribution fraction of the electrons in the sgMFs and that in the rmMFs for the radio emission, respectively.Based on contemporaneous radio and X-ray observations, we find that the emission of the electrons in the rmMFs makes a significant contribution to the lowfrequency emission.If the rmMFs can be described with ζ rm = 0, the contribution fraction f rm is around 40%-70%.If most of the rmMFs are along the shock normal (i.e., ζ rm  3), the contribution fraction f rm is around 10%−23%.
The obMFs may significantly affect the low-frequency emission polarization.If the sgMFs decay quickly (α B = 2.0) and the obMFs are stronger than the rmMFs, the degree of polarization of the radio and optical bands would be significantly high, e.g., 20%-40%.For the recent IXPE observations on highsynchrotron peaked BL Lacertae objects, the observational degree of polarization in the radio band is significantly lower than that expected from the situation with only sgMFs.This reveals that the obMFs can be neglected in the post-shock region.Correspondingly, if the internal shock scenario is applied to explain IXPE observations, a high viewing angle is required.

First
, we define three right-handed Cartesian coordinates, i.e., ˆˆk 12 -coordinate, b ¢ ¢ ¢ x y -coordinate, and ¢ ¢ ¢ ˆˆk 1 2 -coordinate.The related axes are defined as follows: 1.The unit vector e k ( ¢ e k ) of the k-axis ( ¢ k -axis) is along the line of sight and points to the observer in the rest frame of the blazar (comoving frame of the local jet flow).Then, e k and ¢ e k have the following relation: βc = βce β is the velocity of the local jet flow in the rest frame of the blazar, c is the speed of light, bulk Lorentz factor of the local jet flow.2. Taking e jet-axis as the unit vector of the jet axis moving toward us, the unit vectors of the 1-axis and 2

Figure 2 .
Figure 2. Relations of Π−ζ sg for the X-ray (red line) and radio (black line) bands in the case with only sgMFs, where the value of α B = 1.0 is adopted.
but with different ζ rm , where the "×" and "+" symbols represent the numerical data for the optical and radio bands, respectively.By fitting the Π−ζ rm relations with a function of z line) are obtained for the optical and radio bands, respectively.Considering the effect of the geometric structure of

Figure 3 .
Figure3.The frequency-dependent Π (upper panel) and χ (bottom panel) from the case with both sgMFs and rmMFs, where the α B =1.0 and ζ sg = 1 are adopted to describe the sgMFs.The green, blue, red, and black lines correspond to the cases of ζ rm = 0.25, 0.5, 0.75, and 1, respectively.

Figure 4 .
Figure 4. Relations of ζ rm −Π for the optical (red line) and radio (black line) bands in the case with both sgMFs and rmMFs, where the α B = 1.0 and ζ sg = 1 are adopted to describe the sgMFs.
is the contribution fraction of the electrons in the sgMFs (rmMFs) for the radio emission and can be estimated based on the case with ζ rm = 0, Π sg (ζ sg ) estimated based on Equation (27) for a given ζ sg , and Π rm (ζ rm ) estimated based on Equation (29) for a given ζ rm .

Figure 5 .
Figure 5. Same as Figure 3 but with the sgMFs' decay index of α B = 0.5.The black, red, blue, green, and orange lines correspond to the cases of ζ rm = 0, 1, 2, 3, and 4, respectively.Since the value of χ remains zero for all wave bands and all studied cases in this figure, we only show the case with ζ rm = 0 as an example.
and Π rm (ζ rm ) is estimated based on Equation (29) for a given ζ rm .Taking ζ rm = 0 as an example, one has Π rm (ζ rm ) = 0, f sg = 1.5%/5% = 30%, and f rm = 70%.The ζ rm -dependent f rm is shown in Figure 7 with the black line, where the value of f rm varies from 70% to 16% for different ζ rm .It is worth pointing out that most of the rmMFs are along the direction of the shock normal for the case with ζ rm  3. 2. The results from the first observation of Mrk 421 are similar to the observation of Mrk 501, i.e., Π X ∼ 15%, Π O ∼ 2.7%, and Π R ∼ 3% (Di Gesu et al. 2022).Importantly, the PA of the radio-X-ray bands is greatly deviated from the direction of the jet axis.However, the Mrk 421 jet has a wide opening angle of 60°, and the jet may bend tens of degrees on a small scale(Di Gesu et al. 2022).Thus, Di Gesu et al. (2022)  proposed that the PA of radio-X-ray bands is roughly along the direction of the jet.In this scenario, one can have ζ sg = 1.5 for the sgMFs based on Figure2.Correspondingly, the theoretical

Figure 7 .
Figure 7.The ζ rm -dependent f rm for the different blazars discussed in Section 4.
with ζ sg .We fit the Π−ζ sg relations with a function of Based on the relation of Π sg (ζ sg )f sg −Π rm (ζ rm ) × f rm = 3% and Π sg (ζ sg ) = 9%, one can have f sg = 3%/9% = 30% and f rm = 70% for the case with ζ rm = 0.For other values of ζ rm > 0, the value of f rm ∼ 20%-70% can be found in Figure7denoted by the red line.It should be noted that the optical polarization is lower than the radio polarization, which seems to contradict the predictions of the shock-accelerated energy-stratified electron model.It is more likely that the host galaxy makes a significant contribution in the optical band, which has been found in the observation of Mrk 501 (Figure3in Liodakis et al. 2022).3. It should be noted that IXPE conducted three observations of Mrk 421 (Di Gesu et al. 2022; Abe et al. 2023b; Di Gesu et al. 2023) and the latter two observations reported a large angle rotation of the PA only in the X-ray band rather than in the optical and radio bands (Abe et al.