Supernova remnants (SNRs) are main energy sources of the dynamical evolutions of the interstellar medium (ISM, McKee and Ostriker, 1977). The dynamics of the ISM is driven by the pressures of the thermal particles, turbulence, magnetic field, and cosmic-rays (CRs), which are comparable with each other in our galaxy (Boulares and Cox, 1990; Ferrière, 2001). The CRs are energetic, charged particles that move through the ISM at nearly the speed of light. The supernovae can provide the CRs and ISM turbulence though the origin of the CRs is still under debate (Gabici et al., 2019). There is another suggestion in which the ISM turbulence is driven by gravitational instabilities (Krumholz and Burkhart, 2016; Krumholz et al., 2018). As a consequence of the ISM dynamics driven by the supernovae, the molecular clouds formation and subsequent star formation occur (Inutsuka et al., 2015; Hennebelle and Inutsuka, 2019). Burkhart (2021) recently reviews diagnostics of turbulence in the neutral and molecular ISM. The formed stars with a mass larger than ∼ 10 M ⊙ finally explode as supernovae and the ISM dynamics continues. Krumholz et al. (2019) recently provided reviews on the star clusters, suggesting a picture of their life cycle. It is also pointed out that the gaseous matter can be blown away from the galactic disk by the supernovae, and the star formation can be self-regulated (e.g., Kim and Ostriker, 2018; Armillotta et al., 2020, and see also Diesing and Caprioli (2018)). The CR pressure also affects the dynamics of the thermal gas and can drive the galactic wind (e.g., Girichidis et al., 2018; Hopkins et al., 2018; Mao and Ostriker, 2018; Holguin et al., 2019). Shimoda and Inutsuka (2021) recently showed that the ISM conditions observed in our galaxy are sufficient to drive the galactic wind over a long time and the mass loss rate of the galactic disk due to the wind is comparable to the star formation rate. Although the processes of the CR production in the SNRs, the CR transport in the ISM, and interactions between the CRs and the ISM have been widely investigated, they remain to be resolved. Thus, our understandings of the interplay between the CRs and the ISM (background plasma), and the galaxy evolution, are far from sufficient.

The interplay between the CRs and background plasma is also important for the CR production in the SNR shocks. The SNR shocks are formed by the interaction between the particles and electromagnetic fields. Such shock is called as “collisionless shock” in which the particle-particle collision is inefficient. The electric current of the CRs affects the magnetic field via plasma instabilities and the field is amplified (e.g., Bell, 2004). Since the CRs are scattered by disturbances in the magnetic field, the amplified field also affects the behavior of the CRs. Thus, the CR production in the SNR shocks are closely related to the magnetic-field disturbances. From the energy spectrum of the CR proton measured around the Earth, the SNR shocks are expected to produce the CR protons with energies up to 10^15.5 eV, which is the so-called the knee energy. To realize the knee energy CR protons in the SNR shocks, the strength of the amplified field should be > 100 μ G (e.g., Lagage and Cesarsky, 1983a; Lagage and Cesarsky, 1983b; Bell, 2004). Although many kinetic simulations studying the collisionless shock and the production of the nonthermal particles are reported on (e.g., Ohira, 2016a; Ohira, 2016b; Matsumoto et al., 2017; Caprioli et al., 2018; Caprioli et al., 2020; Marcowith et al., 2020; Haggerty and Caprioli, 2020, see also Crumley et al. (2019); Haggerty and Caprioli (2019) for the case of relativistic shocks), the CR-production process is still unsettled issue due to the limitation of too short simulation time compared to the actual SNR shocks. Related to the CR-production issues, the electron heating at the collisionless shocks is also widely studied (e.g., Ohira and Takahara, 2007; Ohira and Takahara, 2008; Rakowski et al., 2008; Laming et al., 2014; Guo et al., 2017; Guo et al., 2018) The actual nature of the magnetic-field disturbances is important clue to understand the CR production in the SNRs.

The plan of the paper is as follows. In Section 2, we briefly review a standard scenario of the CR production in the SNR shocks. The role of magnetic field is also described. In Section 3, the magnetic-field structure in the SNRs obtained by the radio synchrotron polarimetry is introduced. The field in the SNRs shows a radial orientation which is non-trivial. We review a possible origin suggested in previous studies. In Section 4, diagnostics of disturbances in the magnetic-filed by spatial two-point correlation analysis of the synchrotron intensity are reviewed. We also introduce the results reported on. Finally, we summarize the review and discuss prospects of the magnetic-filed diagnostics and the CR production in the SNRs.

Here we review the diffusive shock acceleration (DSA, Bell, 1978; Blandford and Ostriker, 1978) which is the most generally accepted mechanism of the CR production in the SNR shocks. We mainly discuss on the maximum energy of the CRs feasible in the SNR shocks. Then, the diffusion coefficient of the CR is briefly introduced by following Kulsrud (2005). It is convenient to consider the problem in the inertial frame of reference moving at the speed of the propagation of the shock front. We consider a one-dimensional shock along with the z-axis for simplicity (see Figure 1). The velocity of the fluid (background plasma) incoming from the far upstream region is defined as u (−∞) = u ₀ and the velocity of the downstream region is defined as u (+∞) = u ₂. We set the position of the shock front at z = 0.

In the DSA mechanism, the injection of energetic particles at the shock front is assumed. The energetic particles (CRs) are assumed to be bouncing back and forth between the upstream and downstream region by scattering particles. The scattering is supposed to result from the interaction between the particles and disturbances in the magnetic field (e.g., Jokipii, 1966, we analyze this later). Then, the CRs can diffuse ahead the shock upstream via the scattering. The CRs gain an energy by the upstream scattering because the interaction between the CRs and the upstream disturbances convected toward the downstream region can be regarded as a head-on collision. On the other hand, the CRs loose their energy by the downstream scattering because the collision can be regarded as a rear-end collision. The convection speed of the upstream disturbances u ₀ is always larger than that of the downstream disturbances u ₂. Therefore, the CRs accelerate per one cycle of going the upstream and downstream regions. When the mean free path of the accelerated particles becomes much larger than the size of the SNR, they escape from the system and the acceleration is finished. The mean free path depends on the intensity of the magnetic-field disturbances (e.g., Jokipii, 1966). Thus, the nature of the magnetic field is important to predict the maximum energy of the accelerated particles.

Since the pressure of the CRs is usually expected to be comparably large with the upstream ram pressure, the fluid incoming from the upstream region decelerates before crossing the shock front (e.g., Drury and Voelk, 1981). Moreover, the electric current of the CRs can induce and amplify magnetic-field disturbances especially in the upstream region (e.g., Wentzel, 1968; Kulsrud and Pearce, 1969; Bell, 2004). The strength of the induced field can reach up to ∼ 100 μ G for the SNR shocks (Bell, 2004). These back reaction effects prevent us from quantifying the behavior of the CRs by the diffusion coefficient for example. Detail investigations of the back reaction effects, nature of the magnetic field, and resultant diffusion coefficient are one of the most important issues in the modern scenario of the CR production. The CR accelerating region (i.e., confined region) is often referred to the CR precursor region like the radiative shock. We define the fluid velocity at the CR precursor region as u = u ₁(z) and suppose the velocity field satisfies the conditions of u ₁ (−l _pre) = u ₀ and u ₁ (0) = u ₂, where l _pre > 0 indicates the length of the precursor region (e.g., Ohira et al., 2010). The value of l _pre may depend on the CR energy density (ultimately the injection rate at the shock front) which is one of the most uncertain parameters in the DSA mechanism. At the region x < − l _pre, the intensity of the magnetic-field disturbances weakens. We also notice that the length of the CR accelerating region depends more directly on the intensity of disturbances in the magnetic field via the scattering particles. Therefore, the value of l _pre may also depend on the intensity of the turbulence pre-existing in the upstream medium.

Here we evaluate the maximum energy of the CRs feasible in the SNR shocks (cf., Ohira et al., 2010). The CRs diffusing toward the far upstream region are simultaneously convected toward the downstream region at the speed of u. This means that if the particles have a large spatial diffusion coefficient compared with the convection speed, they escape from the system. From a dimensional argument, the diffusion length-scale of the CRs with a momentum of p is estimated as l d p , x ∼ D p , x u , (1) where D (p, x) is the spatial diffusion coefficient of the particle with a momentum of p. As we pointed out above, a precise expression of the diffusion coefficient is currently not available. If the intensity of electromagnetic disturbances is weak, the diffusion coefficient can be calculated with the test particle approximation (e.g., Schlickeiser, 2002, we analyze a simple case later). In the modern scenario of the CR acceleration at the shock, generations or amplifications of the electromagnetic disturbances are important. Since the injection process of the CRs is not fully understood, the resultant electromagnetic disturbances and diffusion coefficient are unsettled (see e.g., Caprioli and Spitkovsky, 2014, for numerical studies of the diffusion coefficient). The smallest diffusion coefficient is given by D (p, x) = r _L (p, x)c/3 ≡ D _B, where the r _L is the Larmor radius of the particle with a momentum of p, and c is the speed of light, respectively. The spatial dependence of r _L arises from the strength of the magnetic field. This smallest diffusion coefficient can be valid in very turbulent media and is called as the Bohm diffusion. Note that r _L = γmcv/qB, where γ is the Lorentz factor, m is the particle rest mass, v is the speed of particle, q is the electric charge, and B is the strength of the magnetic field. For a relativistic particle, the energy is given by E ≃ cp ≃ γmc ² and thus r _L ≃ E/qB. Since the coefficient in the Bohm diffusion shows a dependence of D _B ∝ E/B, the particles with a higher energy have a longer diffusion length. The CR precursor length l _pre should be equal to or smaller than the diffusion length of the maximum energy CRs, l _d,max. The situation of l _d,max < l _pre represents that the CRs are well confined around the shock. This can be achieved if the shock inject a large amount of the energetic particles, or if the intensity of turbulence pre-existing in the upstream media is quite large. In this review, we suppose this limit (see Ohira et al., 2010, for the case of l _d,max ∼ l _pre). Then, the diffusion length of the maximum energy CRs should be comparable with the shock propagation length scale u ₀ t _age, where the t _age is the time elapsed from the supernova explosion to the present. This is the so-called age-limited acceleration (Lagage and Cesarsky, 1983a; Lagage and Cesarsky, 1983b). The maximum energy of accelerated protons is obtained from l _d (E _max, − l _pre) ∼ u ₀ t _age as E max ∼ 8.4 × 1 0 15 e V B 100 μ G u 0 3000 kms − 1 2 t age 1 k y r . (2) Thus, to accelerate the knee-energy CRs in the SNR shocks, the strength of the magnetic field around the shock should be ∼ 100 μ G though typical strength of the magnetic field in the ISM is only ∼ 1 μ G (Beck, 2001). It should be emphasised that this estimate is based on the Bohm-diffusion limit which results in the smallest diffusion-length and is valid only in highly turbulent media. Bell (2004) pointed out that the field strength of ∼ 100 μ G can be achieved by the back reaction effects of the accelerated CRs with assuming the injection rate of the CRs at the shock front. However, the nature of disturbances in the magnetic field and the resultant diffusion coefficient are yet unclear in actual SNR shocks.

Finally, we briefly introduce the diffusion coefficient with a test particle approximation in which the CRs do not affect the background media. In addition, the intensity of the disturbances in the magnetic-field δB is assumed to be small compared with the strength of the mean magnetic-field B ̄ . We analyze the diffusion coefficient for the direction along the mean field (the z-axis) following Kulsrud (2005). Let us suppose a single Alfvén-wave packet of length l and consider the inertial frame of reference such that the wave packet is at rest. In such frame, the electric field vanishes. The Alfvén wave is assumed to be linearly polarized in the x-direction (perpendicular to the mean field) and its perturbed magnetic-field is sinusoidal as δ B x = x ̂ δ B ⁡ sin ( k z − ω t ) , where k = 2π/l, ω = kV _A, and V _A is the Alfvén speed. The CR moving along the z-axis at the speed of v _z enters the wave packet from the minus z-direction (v _z > 0). Then, the Lorentz force at the CR is written as F L = q c v × B = q c v y B ̄ − v x B ̄ 0 + q c 0 v z δ B x − v y δ B x . (3) The first term of the right-hand side of the equation represents the rotation about B ̄ . The second term represents the perturbation in the gyration due to the existence of the Alfvén wave. The z component of the Lorentz force gives the simplest analysis. Let Ω be the cyclotron frequency of the CR, so v y = v ⊥ ⁡ sin Ω t + ϕ , where the ϕ takes into account the random phase between the CR and the Alfvén wave. Substituting the z position of the CR, z = z ₀ + v _z t, we rewrite the z component of the Lorentz force as F L , z = 1 2 q v ⊥ δ B cos k v z − ω + Ω t + k z 0 + ϕ − cos k v z − ω − Ω t + k z 0 − ϕ . (4) For v _z > 0, the first term has a higher frequency of kv _z − ω + Ω and vanishes in a long time average. The second term has a lower frequency of kv _z − ω − Ω and does not average out in a long time average if k v z − ω − Ω ≈ 0 , (5) which is the so-called resonance condition. For the CRs accelerating around the SNR shocks, we can regard as v _z ≫ u ₀ ≫ V _A. Then, the resonance condition is reduced to be kv _z ≈ Ω, or l ≈ 2πΩ/v _z . Thus, the CRs are scattered strongly when they interact with magnetic-field disturbances whose scale-length l = 2π/k is comparable to the Larmor radius of the CRs, r _L = v _z /Ω. If the resonance condition is satisfied and v _z ≫ V _A, the change in the z-component of the momentum can be estimated by averaging the wave period τ = 2 π / ω = 2 π / k v z − k V A ≈ 2 π / k v z as δ p z = ∫ d t F L , z = 1 2 q v ⊥ δ B c τ ⁡ cos k z 0 − ϕ ≈ 1 2 q v ⊥ δ B c 2 π k v z cos ϕ ′ , = π p ⁡ sin ⁡ θ δ B B ̄ cos ϕ ′ (6) where ϕ′ = kz ₀ − ϕ, θ is the pitch angle of the CR, and p is the magnitude of momentum. In the inertial frame of reference moving with the Alfvén wave packet, the electric field is zero. Therefore, the energy of the CR dose not vary during the interaction. Note that in the shock rest frame, a change of the energy due to the scattering should be measured because the electric field dose not vanish in this frame. Thus, in the rest frame of the wave packet, the interaction only results in a change of the pitch angle θ. Let p _z = p cos θ be the momentum component parallel to the mean field. Then, the change of p _z due to the scattering can be written as δp _z = − sin θδθ and we obtain the change of the pitch angle as δ θ = − π δ B B ̄ cos ϕ ′ . (7) The pitch angle increases or decreases according to the relative phase between the CR gyration and the perturbation behavior of the Alfvén wave, ϕ′. Now, let us consider the diffusion of the CR due to a multiple interaction in the random phase approximation on ϕ′. Suppose the CR enters the wave packets one after another. Each wave packet is tightly packed and each interaction occurs individually with the time of τ ≈ 2π/kv _z ≈ 2π/Ω. The CR can be regarded as in a random walk and the number of interaction occurring within a time of t is t/τ. In a random walk process, the square of the total change ⟨ Δ θ 2 ⟩ is equal to the sum of the average of squares of each interaction ∑ ⟨ δ θ 2 ⟩ . Thus, we obtain ⟨ Δ θ 2 ⟩ = ∑ ⟨ δ θ 2 ⟩ = t τ π 2 2 δ B B ̄ 2 , (8) where ⟨ cos ϕ′⟩ = 1/2 is used. The diffusion rate of the pitch angle becomes D θ θ = 〈 Δ θ 2 〉 2 t = π 8 Ω δ B B ̄ 2 . (9) Thus, the scattering particles becomes strong when the disturbances in the magnetic field are intense on the length scale comparable with the Larmor radius of the CRs. In the following, we omit the bracket ⟨…⟩. The large angle scattering (Δθ = π/2) takes a time of τ _sc = (π/2)²/D _θθ . Thus, the spatial diffusion coefficient becomes D z z = 1 3 τ sc v 2 ≈ c 3 r L δ B B ̄ − 2 , (10) where we regard v ≃ c for the CRs. A larger intensity of the disturbances δB results in a smaller spatial diffusion coefficient, indicating that the CRs are well confined by the pitch angle scattering. If the disturbances have a smooth energy spectrum P _B (k) = A _k k ^−a , where A _k is a normalization factor, we obtain δ B / B ̄ 2 ≈ k P B ( k ) ≈ A k r L a − 1 . Here the resonance condition k ≈ 1/r _L is used by omitting a numerical factor. Then, the diffusion coefficient becomes D z z ≈ c A k 3 r L − a + 2 ∝ E B ̄ − a + 2 . (11) The spectral slope of the magnetic energy spectrum is related to the energy dependence of the diffusion coefficient via the resonance condition. It should be noted that our analysis is restricted to the small amplitude Alfvén waves and the CRs are assumed to be the test particles. In magnetohydrodynamics (MHD), there is also fast and slow magnetosonic modes which are compressible modes unlike the Alfvén mode. Yan and Lazarian (2002) pointed out that in turbulent ISM (turbulence described by MHD), the resonant scattering by the fast mode can be more important for the diffusion coefficient than the Alfvén and slow modes. This conclusion is derived from the anisotropy of MHD turbulence: the Alfvén and slow modes propagating perpendicularly to the local magnetic-field cannot have a large amplitude while the fast mode can propagate isotropically. Note that even if the slow and Alfvén modes are likely dominated at a large astrophysical scale, the disturbances at a small gyro motion scale (say, r L ≃ 0.2 A U ( E / 1 G e V ) ( B ̄ / 1 μ G ) − 1 ) is important. Therefore, the anisotropic cascading nature of the MHD turbulence becomes significant to estimate the diffusion coefficient. These arguments are employed to explain some features in CR data by Evoli and Yan (2014), and used for a systematic investigation with global CR transport simulations in Fornieri et al. (2021). Moreover, in turbulent media, the mean magnetic-field would vary spatially and temporally, also affecting the diffusion (e.g., Lazarian and Xu, 2021). When the energy density of the CR is large, the test particle approximation becomes not applicable. The electric current of the CR can induce magnetic-field disturbances and the CR would be scattered by the self-generated disturbances. Caprioli and Spitkovsky (2014) studied the diffusion coefficient in the CR accelerating shocks by using results of hybrid simulations though the hybrid simulations are generally limited to too short a timescale compared to actual SNR shocks. The injection process of the CRs in a well-developed shock is still unclear (i.e., the degree of the back reaction effects is not quantified yet). Diagnosing the nature of the turbulent magnetic-field in the SNR is a necessary step to quantify the diffusion coefficient and to study the CR acceleration.

Global structures of the magnetic field in SNRs are measured by the radio synchrotron polarimetry (e.g., Dickel and Milne, 1976; Dickel et al., 1991; Reynolds and Gilmore, 1993; DeLaney et al., 2002; Reynoso et al., 2013; Dubner and Giacani, 2015). The synchrotron emission is radiated by relativistic electrons rotating around the magnetic field. The Lorentz force at the electron is always perpendicular to the direction of its motion and the emitted radiation fields is concentrated in the direction about the electron’s velocity due to the relativistic beaming effect. As a consequence, the observed photon is linearly polarized in the direction perpendicular to the direction of the magnetic field projected onto the sky (Westfold, 1959; Rybicki and Lightman, 1979). Hence, the magnetic field orientation is estimated by rotating the observed polarization orientation by 90°. Let N (γ _e) be the number density of electrons with the Lorentz factors between γ _e and γ _e + dγ _e. For a power-law distribution of electrons, N γ e d γ e ∝ γ e − p d γ e , (12) the total intensity per frequency ν (the Stokes I) can be written as i ν = K ν − α B ⊥ 1 + α , (13) where K is a function linearly depending on the total density of the relativistic electrons, α = (p − 1)/2 is the spectral index of the radiated photons, and B _⊥ is the strength of the magnetic-field component perpendicular to the line of sight. The degree of polarization can be derived as (see, Westfold, 1959; Rybicki and Lightman, 1979), Π = α + 1 α + 5 / 3 o r Π = p + 1 p + 7 / 3 . (14) The photon spectra at the radio wavelength band show typically α ≈ 0.6 in the Galactic SNRs (e.g., Green, 2009). Thus, the polarization degree can be up to Π ≈ 0.7 if the magnetic field is completely aligned.

Figure 2 shows the magnetic-field orientation on SNR SN 1006 (Reynoso et al., 2013). The polarization degree is shown in Figure 3. The magnetic fields show the radial orientation with the polarization degree of p ∼ 20–30%. Interestingly, such radial orientation with p ∼ 20–30% is ubiquitously observed at “young” SNRs [Dickel et al. (1991) for SN 1572, DeLaney et al. (2002) for SN 1604, Dubner and Giacani (2015) for SNR Cas A, Zanardo et al. (2018) for SN 1987A, and see also Dickel and Milne (1976)]. In the case of SN 1006, we can find that some regions show transitions of the field orientation: the fields at the shock front are along to the tangential direction of the shell and rotate by 90° at far downstream regions (the southern west region is representative).

Inoue et al. (2013) studied this field orientation by using three-dimensional MHD simulations and showed that the global, radial field-orientation can be explained by the downstream turbulence driven by the effect of rippled shock, namely Richtmyer-Meshkov instability (RMI). The RMI is induced by the interaction between the shock and upstream density fluctuations (see Sano et al., 2012, for details). Figure 4A shows two-dimensional slice of the magnetic-field strength (upper half) and the number density (lower half). The upstream magnetic-field is assumed to be uniformly aligned along the y-direction. The upstream density has fluctuations which follow the observed power spectra in the ISM (the so-called big power-law in the sky Armstrong et al., 1995). The initial density fluctuation is characterized by the mean density ρ ̄ and dispersion Δ ρ ≡ ⟨ ρ 2 ⟩ − ρ ̄ 2 . The power spectrum of the density fluctuation is assumed to be kP _ρ (k) ∝ k ^−2/3 by following the observed spectra (Armstrong et al., 1995) and the fluctuations are given as a superposition of sinusoidal functions with various wave numbers ranging in 2π/L _box ≤ | k | ≤ 256π/L _box, where L _box = 2 pc is the box size. Note that the density fluctuations are assumed to be isotropic.

Figures 4A,B show the case of Δρ/⟨ρ⟩₀ = 0.3. The Alfvén Mach number estimated from the mean density is about 60 (the mean shock velocity is v ̄ sh ≃ 1800 k m s − 1 ). The shock front is rippled by reacting the upstream density fluctuations. Then, considering at the shock rest frame, the velocity component tangential to the shock front is transmitted to the downstream region without the shock dissipation. As a consequence, shear flows are generated at the downstream region. The sound waves are also generated at the downstream region due to the transmission of the density fluctuations (McKenzie and Westphal, 1968) and the waves modify the local circulation of the shear flow (vorticity mode), leading to a growth of the vorticity (Nishihara et al., 2010). Note that if there is only vorticity modes or sound modes, the vorticity would not grow (it may be understood by Crocco’s theorem in hydrodynamics). Hence, turbulence develops at the downstream region. The growth time of the RMI is estimated as t _RMI ∼ l _Δρ /v _RMI, where l _Δρ is a typical length scale of the density fluctuation, and v RMI ∼ ( Δ ρ / ρ ̄ ) / ( 1 + Δ ρ / ρ ̄ ) v ̄ sh , respectively (Sano et al., 2012; Inoue et al., 2013). The magnetic fields are amplified by the developed turbulence, i.e., by turbulent dynamo processes. From the upper part of Figure 4A, we can find that the fields can be amplified up to ∼ 100 μ G (see also Giacalone and Jokipii, 2007; Sano et al., 2012). Inoue et al. (2013) showed that the RMI-driven turbulence is anisotropic toward the direction of the shock travel. Figure 4B shows results of synthetic observations of the synchrotron polarimetry with assuming a uniform, spatial distribution of the CR electrons in the downstream region. The white bars indicate the magnetic-field directions estimated from the polarization angles and the color indicates the polarization degree. The results show very good agreements with the observations in particular the polarization degree and the transition of the polarization direction. The mean polarization degree is 20–30%. The transition of the magnetic-field directions may result from the turbulent modification of the magnetic field in the RMI-driven turbulence. The characteristic length-scale of the modification can be written as (Sano et al., 2012) l RMI ∼ v ̄ sh r c t RMI ∼ l Δ ρ r c A , (15) where r _c is the shock compression ratio and A ≡ ( Δ ρ / ρ ̄ ) / ( 1 + Δ ρ / ρ ̄ ) is the Atwood number. Since the compression ratio converges quickly to a value of four in the monoatomic gas, the characteristic length and the transition length are insensitive on the Mach number. Inoue et al. (2013) confirmed the insensitivity of the transition length on the Mach number by comparing the cases of v ̄ sh ∼ 1800 k m s − 1 and v ̄ sh ≃ 3600 k m s − 1 (the other parameters are the same as each case). Thus, the radial orientation of the magnetic fields in the young SNRs can be explained by the RMI-driven turbulence. It should be notice that in this scenario, the downstream turbulence is super Alfvénic and the CR back reaction effects are implicitly assumed to be small. If there is significant amplification of the magnetic fields due to the CR back reactions, trans- or sub-Alfvénic turbulence can be driven. If the case that, the “bending” of the magnetic field by turbulent motions is modest, so the RMI scenario should be improved. Further investigations on the nature of the turbulent magnetic-fields are required.

It is known that the Alfvénic, incompressible turbulence is anisotropic with respect to a local magnetic-field direction (Goldreich and Sridhar, 1995; Lazarian and Vishniac, 1999; Cho and Vishniac, 2000; Maron and Goldreich, 2001) though Inoue et al. (2013) assumed an isotropic density fluctuations in the ISM. ¹ The density fluctuations of MHD turbulence are likely associated with the presence of compressive modes (slow and fast magnetosonic modes). The slow modes follow the anisotropy seen in the Alfvénic turbulence with a scaling relation of kP(k) ∝ k ^−2/3 while the fast modes are likely developing a more isotropic cascades with a scaling relation of kP(k) ∝ k ^−1/2. Zhang et al. (2020) recently reported that the large scale turbulence in the multi-phase ISM can be associated to the magnetosonic modes. From the results reported by Armstrong et al. (1995) and Zhang et al. (2020), the slow modes dominated turbulence may be driven in the ISM. The influence of the anisotropy in the pre-existing upstream turbulence on the downstream RMI turbulence may also be interesting subject.

The difference on the associated cascades not only has consequences on the CR diffusion (Yan and Lazarian, 2002; Evoli and Yan, 2014; Fornieri et al., 2021; Lazarian and Xu, 2021; Hu et al., 2022), but potentially also on the differential heating/energization of the ions or electrons (i.e., depending on the main mechanism at play, e.g., Chandran et al., 2010; Howes, 2010; Cranmer, 2014; Comisso and Sironi, 2019; Rowan et al., 2019; Zhdankin et al., 2019; Kawazura et al., 2020; Cerri et al., 2021; Zhdankin, 2021). How much energies left to heat/energize the electrons at smaller scales (e.g., the electron cyclotron scales), after part of it has been dissipated by ion heating/energization at larger scales (e.g., the ion cyclotron scales), is crucial to determine the CR electron population (the density and spectral slope) which is related to the synchrotron emissivity and is assumed to be spatially uniform in the analysis by Inoue et al. (2013) and the synchrotron correlation introduced later. The observational results of the kinetic scale turbulence in the solar wind are summarized by Goldstein et al. (2015).

West et al. (2017) argued another explanation on the radial magnetic-field orientation assuming the injection and diffusion of the CR electrons. In their scenario, the electrons are selectively accelerated at “locally” quasi-parallel shocks. They assumed that the CR-electron density is concentrated at the region adjacent to the shock front and the density is scaled by cos²2ϕ _B , where ϕ _B is the angle between the shock normal and the post-shock magnetic field. Then, the observed orientation is biased toward the radial direction even if the intrinsic distribution of the magnetic-field directions is isotropic. They also argued the case of quasi-perpendicular shock (the density of the CR electrons is scaled by sin²2ϕ _B ) and pointed out that the polarization properties can be similar to the case of SN 1006 if there is radially aligned magnetic fields. Recent investigations on the injection of CR electrons suggest that the quasi-perpendicular shock is favored to accelerate the electrons efficiently (Katou and Amano, 2019; Amano et al., 2020; Amano and Hoshino, 2022). Thus, the combination of the efficient acceleration at quasi-perpendicular shocks and the existence of the radially aligned magnetic fields may be favored though the assumption about the CR diffusion remains to be resolved.

The spectral slopes of the magnetic energy in SNRs are recently examined by spatial two-point correlation functions of synchrotron intensities (Roy et al., 2009; Shimoda et al., 2018a; Saha et al., 2019; Vishwakarma and Kumar, 2020; Saha et al., 2021). Lazarian and Pogosyan (2012, 2016) provided mathematical formalisms describing how the correlation functions of synchrotron intensities are related to the energy spectrum of magnetic disturbances. Here we review simplify analysis (see also appendix of Shimoda et al., 2018a).

Defining position vectors of the two-point r and r ′, the second-order correlation function of the local synchrotron intensity is written as C i ν , α 2 = ⟨ i ν r i ν r + l ⟩ r , (16) where i _ν is the local synchrotron intensity given by Eq. 13 and l ≡ r ′ − r . The bracket ⟨…⟩ _r means the statistical average which is given by a volume average. The local intensity of the synchrotron radiation depends on B ⊥ 1 + α . Thus, with assuming spatially uniform distribution of CR electrons, the second-order correlation function of the synchrotron intensity is related to the correlation of the magnetic-field strength as C i ν , α 2 ∝ ⟨ B ⊥ r 1 + α B ⊥ r + l 1 + α ⟩ r . (17) When α = 0 (i.e. p = 0), the function reproduces the correlation of the magnetic-field strength. The case of α = 1 (p = 3) is also simple and can be representative for the case of arbitrary α. In the following, we omit the notations as B _⊥( r ) → B _⊥ and B ⊥ ( r + l ) → B ⊥ ′ . Then, the correlation function can be written as C i ν , α = 2 2 ∝ ⟨ B ⊥ 2 B ⊥ ′ 2 ⟩ r = ⟨ B ⊥ 4 + B ⊥ ′ 4 ⟩ r 2 − 1 2 B ⊥ + B ⊥ ′ 2 B ⊥ − B ⊥ ′ 2 r . (18) Decomposing the magnetic-field strength into the mean strength B ̄ ⊥ = ⟨ B ⊥ ⟩ r and the fluctuating component Δ B ⊥ = B ⊥ − B ̄ ⊥ , we can rewrite the above equation as C i ν , α = 2 2 ∝ ⟨ B ⊥ 2 B ⊥ ′ 2 ⟩ r = ⟨ B ⊥ 4 + B ⊥ ′ 4 ⟩ r 2 − 2 B ̄ ⊥ 1 + Δ B ⊥ + Δ B ⊥ ′ B ̄ ⊥ 2 Δ B ⊥ − Δ B ⊥ ′ B ̄ ⊥ 2 r . (19) For small standard deviations of the fluctuating strength | Δ B ⊥ + Δ B ⊥ ′ / B ̄ ⊥ | < 1 , we obtain C i ν , α = 2 2 ∝ 〈 B ⊥ 2 B ⊥ ′ 2 〉 r ≈ 〈 B ⊥ 4 + B ⊥ ′ 4 〉 r 2 − 2 B ̄ ⊥ Δ B ⊥ − Δ B ⊥ ′ B ̄ ⊥ 2 r = 4 B ̄ ⊥ 2 〈 Δ B ⊥ Δ B ⊥ ′ 〉 r + c o n s t . , (20) where we have assumed statistically homogenous turbulent fields as ⟨ B ⊥ 4 ⟩ r = ⟨ B ⊥ ′ 4 ⟩ r and ⟨ B ⊥ 2 ⟩ r = ⟨ B ⊥ ′ 2 ⟩ r . Hence, the correlation function C i ν , α = 2 ( 2 ) reproduces the second-order correlation of the magnetic-field strength. Note that even if the turbulent fields are completely random without the mean component, the approximation of the small standard deviation is valid for small scales. This is because the disturbances on the larger scales act as a guide field for the field disturbances on the smaller scales (Lazarian and Vishniac, 1999; Cho and Vishniac, 2000; Cho and Lazarian, 2003). Lazarian and Pogosyan (2012) showed relations of 〈 B ⊥ 2 B ⊥ ′ 2 〉 r 〈 B ⊥ 4 〉 r − 〈 B ⊥ 2 〉 r 2 ≈ 〈 B ⊥ 1 + α B ⊥ ′ 1 + α 〉 r 〈 B ⊥ 2 1 + α 〉 r − 〈 B ⊥ 1 + α 〉 r 2 (21) for several α for a power-law correlation function of B _⊥. In the range of 0.2 ≤ α ≤ 2, they reported that the maximum difference of ⟨ B ⊥ 1 + α B ⊥ ′ 1 + α ⟩ r from ⟨ B ⊥ 2 B ⊥ ′ 2 ⟩ r is only three per cent. Thus, the correlation functions with 0.2 ≤ α ≤ 2 can be written as C i ν , α 2 ∝ 〈 B ⊥ 1 + α B ⊥ ′ 1 + α 〉 r ≈ D α 〈 B ⊥ 2 B ⊥ ′ 2 〉 r . (22) where D(α) is a function of α. This argument is numerically confirmed by Lee et al. (2016). They performed synthetic observations of synchrotron radiations from simulated magnetic fields and derived the Fourier power spectrum from the correlation of the observed synchrotron polarization intensity for the parameter range of 0.5 ≤ α ≤ 3. They found that the power spectrum successfully reproduced the spectral index of the given magnetic fields. In the case of SNRs, the index of photon spectrum is typically α ≈ 0.6 (e.g., Green, 2009). Thus, the correlation function C i ν , α ( 2 ) can reproduce the second-order correlation of the magnetic-field strength in the SNRs.

The correlation functions derived from the “observed” synchrotron emissions suffer from the uncertainty of the geometry of the emission regions via the projection effects. We define the observed intensity of the synchrotron emission per frequency at the two-dimensional sky position X = (x, y) as I ν X = ∫ 0 L X K ν − α B ⊥ X , z 1 + α d z , (23) where the z represents the coordinate along the line of sight and L( X ) is the extent of the emission region. The correlation function for I _ν is written as C I ν 2 λ = ∫ I ν X I ν X ′ ∫ d 2 X ≡ ⟨ I ν X I ν X + λ ⟩ X , (24) where λ = X′ −X is the position vector of two separated positions in the sky X and X ′. Then, the correlation function can be represented as C I ν 2 λ = K ν − α ∫ 0 L X d z ∫ 0 L X ′ d z ′ B ⊥ X , z 1 + α B ⊥ X + λ , z ′ 1 + α X . (25) For the constant L( X ) = L ₀, Lazarian and Pogosyan (2012) and Lee et al. (2016) demonstrated that C I ν ( 2 ) reproduces the scaling relation of a given magnetic-field correlation. If L varies spatially, C I ν ( 2 ) is affected by the geometrical structure of the emission region which is usually unknown. However, the emission regions of some young SNRs are known to be spherical. If we select the two points X and X ′ on the concentric circle of the SNR image, the condition of L( X ) = constant is satisfied (Shimoda et al., 2018a). Figure 5 shows a schematic of the SNR shell and projected image. Hence, the correlation can be analyzed in SNRs with shell like geometry.Shimoda et al. (2018a) demonstrated the spatial two-point correlation analysis of the synchrotron intensity for SN 1572. Figure 6 shows regions where they analyze the two-point correlations. The original image data are published in Williams et al. (2016). The analyzed correlation functions are shown in Figure 7. The intensity correlation functions follow a power-law and the spectral slopes are close to the Kolmogorov scaling λ ^2/3. When the Aflvén Mach number of turbulence is approximately unity, i.e., M _A,turb ≡ u _inj/V _A, where u _inj is the turbulent velocity at the injection scale, the magnetic energy can have such single power-law spectrum (Goldreich and Sridhar, 1995, and see also Brandenburg and Lazarian, 2013 for reviews). The shock velocity of the SN 1572 is estiamted as V _sh ∼ 5000 km s⁻¹ (Williams et al., 2016). If we consider the typical magnetic-field strength in the ISM (∼ 3 μG Beck, 2001) and number density of ∼ 1 cm − 3 , we obtain M _A,turb ∼ 500 with assuming u _inj ∼ V _sh. Thus, the correlations at the outer circles (R ≳ 0.9R _SNR) imply smaller turbulent velocity as u _inj ≪ V _sh and/or larger magnetic-field strength at R ≳ 0.9R _SNR. As we dicussed in Section 3, turbulence can be driven in the downstream region due to the existence of upstream density flucuations with the significant field amplification. Williams et al. (2013) examined the ambient density structure of SN 1572 from infrared dust emissions and found ourder-of-magnitude variations in the density at the length scale of l _Δρ ∼ R _SNR (along the azimuthal angle direction of the shell). Therefore, u _inj ∼ V _sh may be satisfied. Then, the characteristic length of the field amplification due to the RMI-driven turbulence is estimated as l RMI ∼ 0.2 R SNR ( l Δ ρ ) ( r c / 4 ) − 1 , where the Atwood number A ≡ ( Δ ρ / ρ ̄ ) / ( 1 + Δ ρ / ρ ̄ ) is approxmated as unity. Thus, the condition of M _A,turb ≃ 1 can be satisfied at the outer region R ≳ 0.9R _SNR due to the field-amplification by the RMI-driven turbulence. In this scneario, in the dowsntream region just behind the shock front, the magnetic-field strength is not large yet and thus the turbulence should be super-Alfvénic (M _A,turb > 1). In the super-Alfvénic turbulence, the Kolmogorov-like scaling λ ^2/3 is seen on a length scale smaller than l _A, at which turbulent velcoity is equal to the Alfvén velocity. With progress of the turbulent dynamo, the l _A evoleves toward a larger length-scale with increasing the distance from the shock front. On the other hand, if the fields are significantly amplified by the CR back reaction effects, which occur at upstream, such evolution would not be observed. Therefore, we are able to examine the real field-amplification scenario once the most outer circle (R = 1.00R _SNR) is resolved. For such observation, we need a higher sensitivity with sub-arcsecond resolution at GHz band. Future interferometric observations (e.g., by the Squre Kilometer Array; SKA) will provide such data and may distinguish these scenario. The inner circes (R ≲ 0.9R _SNR) show flatter spectra than the outer ones (R ≳ 0.9R _SNR) at larger scalse λ/R ≳ 0.2. In SN 1572, the contact discontinuity, at which Rayleigh-Taylor instability (RTI) works, is located at R ≈ 0.9R _SNR (Warren et al., 2005). The correlation function of the synchrotron intensity can reflect the correlation of the magnetic-field only if the fluctuating components of the field are weak as | ( Δ B ⊥ + Δ B ⊥ ′ ) / B ̄ ⊥ | < 1 . Lee et al. (2016) analyzed synthetically the intensity correlations for magnetic disturbances, which have the Kolmogrov-like power spectrum, without the mean field. They obtained such intensity correlation flattend at nearby the injection scale of turbulence. Note that in such case, fluctuating componets at a larger scale act as a guide field for fluctuations at a smaller scale, so the intensity correlation functions can reporoduce given the magnetic energy spectrum at the smaller scales. Thus, if the RTI behaves as an additional, large energy injection mechanism of turbulence at R ≈ 0.9R _SNR, the intensity correlations could be flattened. This arugumet may require relatively large growth rate of the RTI. The rate depends on the ambitent density sturucture, the initial ejecta profile including the anisotropies, and the effective addiabatic index for instance. These effects on the groeth rate can be studied via the relative distance between the shock front and fronts of the “fingers” of the RTI (e.g., Warren et al., 2005; Orlando et al., 2012; Bao et al., 2021; Orlando et al., 2022). The investigations of the RTI and RMI are on-going (Zhou et al., 2021, for recent reviews).

Vishwakarma and Kumar (2020) reported similar results of the intensity correlations in SNR Cas A by using the same method as Shimoda et al. (2018a). ² The correlation functions in Cas A show also the Kolmogrov-like scaling and are flattened at inner regoins. These results imply that the magnetic energy spectra in young SNRs can be understood in similar ways to the case of SN 1572. Systematic investigations of the synchrotron intensity in young SNRs with shell-like geometries would bring useful insights about the nature of turbulence in future.

This review introduce recent investigations of the magnetic field structures in young SNRs. The correlation analysis of the synchrotron intensities are significant progress for studying the magnetic energy spectra which are closely related to the diffusion coefficient of the CRs. Future observations with high sensitivity and angular resolution at GHz band could examine whether the CR back reaction effects are significant in the SNR shocks or not; this would be a science case of the SKA.

The relations between the nature of magnetic disturbances and the CR injection at the shock front are also interesting subject, however, the CR injection processes are one of the most uncertain issues in the SNR shocks. Helder et al. (2009) pointed out that if the shock produce the CRs with consuming a significant fraction of its kinetic energy, the downstream temperature becomes lower than the case of adiabatic shocks without the CRs (see also Morlino et al., 2013; Shimoda et al., 2015; Hovey et al., 2018; Shimoda et al., 2018b; Shimoda et al., 2022). Thus, once the shock velocity and downstream temperature are independently measured, we can estimate the CR injection efficiency. The shock velocity is measured from the expansion rate of the SNR shell. The downstream ion temperature is measured from widths of the atomic line emissions. The line width is also affected by the turbulent Doppler broadening. Shimoda et al. (2022) constructed a novel CR injection model in SNR shocks and performed synthetic observations of the X-ray line emissions in SNR RCW 86 considering the turbulent Doppler broadening in the downstream region. They found that the observed line width in the CR accelerating shock becomes sufficiently smaller than the case without the CRs even if the turbulence exists. They also found that future X-ray spectroscopic observations by XRISM (Tashiro et al., 2020) and Athena (Barret et al., 2018) can resolve the line width to distinguish whether the shock accelerates the CRs or not. Thus, by a combination of the future observations of the X-ray atomic line observations and correlation analysis of the radio synchrotron intensities, the interplays between the CRs and background plasma can be observationally studied.

In this review, we introduce studies at relatively “young” SNRs at which the shock speed are large. A larger shock speed is preferred for the confinement of the diffusing CRs around the shock front by the advection. For the study of SNRs as energy sources of the ISM turbulence, investigating an ‘old’ SNR is also important. The blast wave of an SNR expands with accumulating the ISM. When the accumulated mass is comparable to the ejecta mass of the supernova, the blast wave begins to decelerate (Sedov, 1959), then the radiative cooling of the shocked ISM and ejecta becomes important, and finally merges into the ISM. Raymond et al. (2020a,b) recently reported observational results at radiative shocks of the SNR Cygnus Loop, provided various shock parameters in the cooling layer (shock speed, ram pressure, density, compression ratio, dust destruction efficiency, magnetic field strength, and vorticity), and discussed the radio synchrotron emissivity by CR electrons, pion decay emission by CR nuclei, and thermal/thin-shell instabilities of thermal gas. The future (soft) X-ray observations by XRISM and Athena may also contribute or improve such study by providing more accurate plasma diagnostics.