Magnetic Field Amplification in Supernova Remnants

Supernova remnants (SNRs) are the most plausible sources of Galactic cosmic rays (CRs; Blandford & Eichler 1987). Magnetic field amplification is expected in SNRs to ensure an efficient diffusive shock acceleration (DSA; Axford et al. 1977; Bell 1978), and is also supported by observational evidence, e.g., the magnetic fields on the order of 100 μG near the shock front inferred from narrow X-ray synchrotron rims (Bamba et al. 2003, 2005a, 2005b; Vink 2012), and the milligauss magnetic fields suggested by the rapid X-ray variability from compact sources in the downstream region (Patnaude & Fesen 2007; Uchiyama et al. 2007; Uchiyama & Aharonian 2008). The milligauss fields are also indicated from radio observations (Longair 1994). Clearly, the amplified magnetic fields in SNRs cannot be accounted for by the shock compression of the interstellar field strength of a few microgauss.

There are extensive studies on the origin of the magnetic fluctuations for confining CRs near the shock, e.g., the non-resonant streaming instability for driving magnetic fields at length scales smaller than the CR Larmor radius (Bell 2004), an inverse cascade of Alfvén waves excited at the Larmor radius to larger wavelengths (Diamond & Malkov 2007). More generally, turbulence is believed to be an efficient agent to amplify magnetic fields via the turbulent dynamo (Kazantsev 1968; Kulsrud & Anderson 1992), which has also been invoked for explaining both the preshock (Beresnyak et al. 2009, hereafter BJL09; Drury & Downes 2012; del Valle et al. 2016) and postshock (Balsara et al. 2001; Giacalone & Jokipii 2007; Inoue et al. 2009; Guo et al. 2012; Fraschetti 2013; Ji et al. 2016) magnetic fields. As turbulence is induced by the interaction between SNR shocks and interstellar turbulent density fluctuations, the turbulent dynamo is inevitable in SNRs with the turbulent kinetic energy dominating over the pre-existing magnetic energy.

The theoretical advances in magnetohydrodynamic (MHD) turbulence have been made since Goldreich & Sridhar (1995) and later works (Lazarian & Vishniac 1999; Cho & Vishniac 2000; Maron & Goldreich 2001) with the conceptual improvement of introducing the local system of reference. These works bring new physical insights into the turbulent dynamo problem. Within the framework of the Goldreich & Sridhar (1995) model of MHD turbulence, a detailed analytical study of the turbulent dynamo process in plasmas with arbitrary conducing and ionization degrees was carried out by Xu & Lazarian (2016, hereafter XL16). A remarkable finding is that the kinematic dynamo in a weakly ionized medium leads to a linearly growing field strength with time,² and the resulting characteristic scale of the magnetic field can significantly exceed the viscous scale of turbulence. This new dynamo regime is referred to as the "damping regime." Since the interstellar media (ISM) that SNR shocks sweep through are frequently partially ionized (Draine 2011), the significant modifications to the kinematic dynamo in the presence of neutrals should be incorporated when studying the magnetic field amplification in the preshock region and its implications on CR acceleration.

For CR acceleration at shocks, strong magnetic fields in both the preshock and postshock regions are necessary to trap and mirror CR particles in order to facilitate multiple shock crossings. The amplification of the preshock magnetic field is crucial. It has been modeled in earlier studies (e.g., BJL09; Drury & Downes 2012; del Valle et al. 2016) in an ideal situation with a fully ionized upstream plasma, and the amplification time is rather limited given a relatively thin precursor (BJL09). To reach a more realistic and generalized description, here we consider the partial ionization of the ISM and examine its influence on the CR diffusion in the amplified magnetic field in the CR precursor.

In the highly ionized postshock medium, the turbulent energy can cascade down to quite small scales. Very likely, the turbulent dynamo starts with an equipartition between the magnetic and turbulent energies at an intermediate scale and fall in the nonlinear regime. Different from the kinematic dynamo with a strong dependence on the microscopic magnetic diffusion, the nonlinear dynamo, which is initiated by the equipartition between the magnetic energy and the local turbulent energy, is mainly subject to the turbulent magnetic diffusion, and consequently the magnetic energy has a linear-in-time growth with a low dynamo efficiency. This theoretical result in XL16 quantitatively agrees with earlier numerical studies, e.g., Cho et al. (2009), Beresnyak (2012).

Motivated by the enhanced field strength in SNRs indicated by observations, in this work we investigate the magnetic field amplification in both the preshock and postshock regions of a SNR by applying the general turbulent dynamo theory developed by XL16 and discuss its implications on the CR acceleration. In Section 2, we analyze the magnetic field amplification in the weakly ionized preshock medium and its implication on the CR diffusion upstream. In Section 3, we study the magnetic field amplification and CR diffusion in the fully ionized postshock medium. Discussions on an alternative acceleration mechanism of CRs in the postshock MHD turbulence are in Section 4. In Section 5, we summarize the main results.

The turbulence in the preshock region can result from the CR pressure gradient in the shock precursor interacting with the density inhomogeneities in the upstream ISM (BJL09). The pre-existing interstellar field of a few μG is expected to be amplified by the resulting turbulence via the small-scale turbulent dynamo process. The dynamo growth of magnetic field is driven by the turbulent motions. They are essentially hydrodynamic over length scales larger than the equipartition scale, where the turbulent and magnetic energies are in equipartition, and are assumed to follow the Kolmogorov scaling.

In the partially ionized ISM, the dynamo evolution of magnetic field in the linear regime has its time dependence and growth rate strongly affected by the ion-neutral collisional damping (Kulsrud & Anderson 1992; XL16). For the propagation of a strong shock wave through the partially ionized ISM, it is necessary to take into account the partial ionization of the upstream medium for a realistic description of the magnetic field amplification in the preshock region.

2.1. Damping Effect on the Kinematic Dynamo

The damping rate of magnetic fluctuations due to ion-neutral collisions is given by Kulsrud & Pearce (1969) and Kulsrud & Anderson (1992) as

$$\begin{eqnarray}&&{\omega }_{d}={ \mathcal C }{k}^{2}{{ \mathcal E }}_{M},\,\,{ \mathcal C }=\displaystyle \frac{{\xi }_{n}}{3{\nu }_{{ni}}},\end{eqnarray} \tag{ 1 }$$

where ${{ \mathcal E }}_{M}$ is the magnetic-fluctuation energy per unit mass, ${\xi }_{n}={\rho }_{n}/\rho$ is the neutral fraction with the neutral mass density ρ_n and the total mass density ρ, and ν_ni = γ_dρ_i is the neutral-ion collision frequency with γ_d as the drag coefficient (see Shu 1992) and ρ_i as the ion mass density.

Magnetic fluctuations in strongly coupled neutrals and ions are also subject to the neutral viscous damping (Lazarian et al. 2004; Xu et al. 2015). The corresponding viscous scale is

$$\begin{eqnarray}&&{k}_{\nu }={L}^{-\tfrac{1}{4}}{V}_{L}^{\tfrac{3}{4}}{\nu }_{n}^{-\tfrac{3}{4}},\end{eqnarray} \tag{ 2 }$$

where the Kolmogorov scaling of turbulence is used, V_L is the turbulent velocity at the injection scale L, ${\nu }_{n}={v}_{\mathrm{th}}/({n}_{n}{\sigma }_{{nn}})$ is the kinematic viscosity in neutrals, with the neutral number density n_n, the neutral thermal speed v_th, and the cross-section of a neutral-neutral collision σ_nn. If the ion-neutral collisional damping rate at k_ν, ${\omega }_{d}({k}_{\nu })={ \mathcal C }{k}_{\nu }^{2}{{ \mathcal E }}_{M}$, is larger than the viscous damping rate,

$$\begin{eqnarray}&&{\omega }_{v}({k}_{\nu })={k}_{\nu }^{2}{\nu }_{n}={L}^{-\tfrac{1}{2}}{V}_{L}^{\tfrac{3}{2}}{\nu }_{n}^{-\tfrac{1}{2}},\end{eqnarray} \tag{ 3 }$$

that is

$$\begin{eqnarray}&&{{ \mathcal E }}_{M}\gt {{ \mathcal C }}^{-1}{\nu }_{n},\end{eqnarray} \tag{ 4 }$$

ion-neutral collisions dominate over the neutral viscosity, leading to a damping scale of magnetic fluctuations larger than the viscous scale.

The turbulent eddies at the damping scale k_d are mainly responsible for driving the dynamo growth of magnetic energy, at a rate comparable to the eddy-turnover rate,

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{d}={k}_{d}{v}_{d}={L}^{-\tfrac{1}{3}}{V}_{L}{k}_{d}^{\tfrac{2}{3}},\end{eqnarray} \tag{ 5 }$$

where the turbulent velocity at k_d is,

$$\begin{eqnarray}&&{v}_{d}={V}_{L}{({k}_{d}L)}^{-\tfrac{1}{3}}.\end{eqnarray} \tag{ 6 }$$

From the equalization between Γ_d and ω_d at k_d, we find the expression of k_d as (Equations (1) and (5))

$$\begin{eqnarray}&&{k}_{d}={{ \mathcal C }}^{-\tfrac{3}{4}}{L}^{-\tfrac{1}{4}}{V}_{L}^{\tfrac{3}{4}}{{ \mathcal E }}_{M}^{-\tfrac{3}{4}}.\end{eqnarray} \tag{ 7 }$$

We see that the damping scale increases with the growing magnetic energy ${{ \mathcal E }}_{M}$.

The dynamo is in the linear regime, i.e., kinematic dynamo, as long as the magnetic energy is below the turbulent kinetic energy at k_d,

$$\begin{eqnarray}&&{{ \mathcal E }}_{M}\lt \displaystyle \frac{1}{2}{v}_{d}^{2}.\end{eqnarray} \tag{ 8 }$$

Combining the above relation with Equations (6) and (7) yields

$$\begin{eqnarray}&&{{ \mathcal E }}_{M}\lt \displaystyle \frac{{ \mathcal C }}{4}{L}^{-1}{V}_{L}^{3}.\end{eqnarray} \tag{ 9 }$$

It can be further rewritten as

$$\begin{eqnarray}&&{R}_{\mathrm{ene}}\lt \displaystyle \frac{{\xi }_{n}}{6}{R}_{t},\end{eqnarray} \tag{ 10 }$$

with

$$\begin{eqnarray}&&{R}_{\mathrm{ene}}=\displaystyle \frac{{{ \mathcal E }}_{M}}{\tfrac{1}{2}{V}_{L}^{2}},\,\,{R}_{t}=\displaystyle \frac{{\nu }_{{ni}}^{-1}}{L/{V}_{L}}\end{eqnarray} \tag{ 11 }$$

as the ratio between ${{ \mathcal E }}_{M}$ and the turbulent energy at L and the ratio between the neutral-ion collision time and the largest eddy-turnover time, where the expression of ${ \mathcal C }$ in Equation (1) is used. When the magnetic energy satisfies both conditions in Equations (4) and (9), one should consider the damping regime for the kinematic dynamo growth of the magnetic field.

As illustrative examples, we use the typical ISM phases (see Draine 2011), e.g., the cold neutral medium (CNM) and molecular cloud (MC), as the preshock conditions for supernova remnants interacting with atomic and molecular clouds. Their typical parameters are listed in Table 1 (see Draine & Lazarian 1998), including the number densities of the atomic hydrogen n_H and electrons n_e, the temperature T. For the turbulence parameters, we adopt the characteristic scale ∼0.1 pc of the density structure in the CNM (Heiles & Troland 2003) and MC (Goodman et al. 1998; Motte et al. 2007) as the injection scale L of the upstream turbulence. The turbulent velocity at L depends on the density perturbation Δρ/ρ and the shock speed v_sh (Drury & Downes 2012). As an order-of-magnitude estimate, we adopt

$$\begin{eqnarray}&&{V}_{L}\sim \displaystyle \frac{{\rm{\Delta }}\rho }{\rho }{v}_{\mathrm{sh}}.\end{eqnarray} \tag{ 12 }$$

For illustrative purposes, we assume Δρ/ρ ∼ 1 and V_L on the order of 10³ km s⁻¹ in our following calculations. It is worthwhile to notice that the density perturbation can be vastly different in different ISM phases, e.g., local density enhancements with Δρ/ρ ≫ 1 but with a small volume-filling factor in clumpy MCs (see Stutzki et al. 1988; Hennebelle & Falgarone 2012), or the low-density contrast with Δρ/ρ ≪ 1 in a more diffuse medium. Thus the corresponding V_L can actually deviate significantly from v_sh.

Table 1.  Parameters in the Preshock Region

	nH (cm−3)	ne/nH	T (K)	B0 (μG)	kν−1 (pc)	${k}_{d0}^{-1}$ (pc)	tdyn (years)	Bdyn (μ G)
CNM	30	10−3	100	5	1.3 × 10−7	1.2 × 10−4	741.9	452.6
MC	300	10−4	20	5	1.7 × 10−8	1.6 × 10−6	749.7	7.7 × 103

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We find that, given the magnetic field strength B₀ in the ambient ISM, the initial magnetic energy is

$$\begin{eqnarray}&&{{ \mathcal E }}_{M0}=\displaystyle \frac{1}{2}{V}_{{\rm{A}}0}^{2}=\displaystyle \frac{{B}_{0}^{2}}{8\pi \rho },\end{eqnarray} \tag{ 13 }$$

where V_A0 is the Alfvén speed. It satisfies

$$\begin{eqnarray}&&{{ \mathcal C }}^{-1}{\nu }_{n}\lt {{ \mathcal E }}_{M0}\lt \displaystyle \frac{{ \mathcal C }}{4}{L}^{-1}{V}_{L}^{3},\end{eqnarray} \tag{ 14 }$$

or in terms of the ionization fraction ξ_i = ρ_i/ρ,

$$\begin{eqnarray}&&{\xi }_{i}\lt \displaystyle \frac{{\xi }_{n}{{ \mathcal E }}_{M0}}{3{\gamma }_{d}\rho {\nu }_{n}},\,{\xi }_{i}\lt \displaystyle \frac{{\xi }_{n}{V}_{L}^{3}}{12{\gamma }_{d}\rho L{{ \mathcal E }}_{M0}},\end{eqnarray} \tag{ 15 }$$

for both the CNM and MC, indicative of the dominant ion-neutral collisional damping and the kinematic dynamo growth of magnetic energy according to the above analysis (Equations (4) and (9)). In particular, we find R_ene ≈ 4 × 10⁻⁶, R_t ≈ 185 in the case of the CNM and R_ene ≈ 4 × 10⁻⁷, R_t ≈ 6 in the case of the MC (Equation (11)). Naturally, R_ene is very small as the turbulence in the precursor carries substantial kinetic energy originating from the supernova ejecta, while a large R_t suggests the weak coupling between neutrals and ions and thus strong damping. Here we also adopt m_i = m_n = m_H for the masses of ions and neutrals in the CNM, and m_i = 29m_H, m_n = 2.3m_H for those in the MC (Shu 1992), γ_d = 3.5 × 10¹³ cm³ g⁻¹ s⁻¹ (Draine et al. 1983), and σ_nn ≈ 10⁻¹⁴ cm² (Vranjes & Krstic 2013).

2.2. Dynamo Evolution of the Magnetic Field

The kinematic dynamo is described by the Kazantsev theory, with ${{ \mathcal E }}_{M}$ determined by the integral of the Kazantsev spectrum (Kazantsev 1968; Kulsrud & Anderson 1992; Schekochihin et al. 2002; Brandenburg & Subramanian 2005) over the wavenumbers smaller than k_d,

$$\begin{eqnarray}{{ \mathcal E }}_{M} & = & \displaystyle \frac{1}{2}{\displaystyle \int }_{0}^{{k}_{d}}M(k,t){dk}\\ & = & \displaystyle \frac{1}{2}{\displaystyle \int }_{0}^{{k}_{d}}{M}_{0}\exp \left(\displaystyle \frac{3}{4}\displaystyle \int {{\rm{\Gamma }}}_{d}{dt}\right){\left(\displaystyle \frac{k}{{k}_{0}}\right)}^{\displaystyle \frac{3}{2}}{dk}\\ & = & \displaystyle \frac{1}{5}{M}_{0}{k}_{0}\exp \left(\displaystyle \frac{3}{4}\displaystyle \int {{\rm{\Gamma }}}_{d}{dt}\right){\left(\displaystyle \frac{{k}_{d}}{{k}_{0}}\right)}^{\displaystyle \frac{5}{2}},\end{eqnarray} \tag{ 16 }$$

where M(k, t) is the growing Kazantsev spectrum of magnetic energy, and M₀ is the initial magnetic energy spectrum centered about k₀. By combining Equations (5), (7), and (16), after some algebra we can reach the evolution law of the magnetic energy,

$$\begin{eqnarray}&&\sqrt{{{ \mathcal E }}_{M}}=\sqrt{{{ \mathcal E }}_{M0}}+\displaystyle \frac{3}{23}{{ \mathcal C }}^{-\tfrac{1}{2}}{L}^{-\tfrac{1}{2}}{V}_{L}^{\tfrac{3}{2}}t,\end{eqnarray} \tag{ 17 }$$

where ${{ \mathcal E }}_{M0}$ is the initial magnetic energy. This dynamo regime has been identified as the damping kinematic dynamo in XL16. Because $B=\sqrt{8\pi \rho {{ \mathcal E }}_{M}}$, it shows that the magnetic field strength grows linearly with time. The time evolution of k_d can also be obtained by inserting the above equation into Equation (7),

$$\begin{eqnarray}&&{k}_{d}={\left[{k}_{d0}^{-\tfrac{2}{3}}+\displaystyle \frac{3}{23}{L}^{-\tfrac{1}{3}}{V}_{L}t\right]}^{-\tfrac{3}{2}},\end{eqnarray} \tag{ 18 }$$

where k_d0 is the initial damping scale corresponding to ${{ \mathcal E }}_{M0}$.

The kinematic saturation happens when the magnetic energy grows to equipartition with the local turbulent energy, i.e.,

$$\begin{eqnarray}&&{{ \mathcal E }}_{M}=\displaystyle \frac{1}{2}{v}_{d}^{2}.\end{eqnarray} \tag{ 19 }$$

By using Equations (6) and (7), the above equation gives the critical damping scale k_d,cr corresponding to the kinematic saturation,

$$\begin{eqnarray}&&{k}_{d,\mathrm{cr}}={\left(\displaystyle \frac{{ \mathcal C }}{2}\right)}^{-\tfrac{3}{2}}{L}^{\displaystyle \frac{1}{2}}{V}_{L}^{-\tfrac{3}{2}}.\end{eqnarray} \tag{ 20 }$$

For the CNM and MC under consideration, there is

$$\begin{eqnarray}&&{k}_{d,\mathrm{cr}}L={\left(\displaystyle \frac{2L}{{ \mathcal C }{V}_{L}}\right)}^{\displaystyle \frac{3}{2}}\lt 1,\end{eqnarray} \tag{ 21 }$$

or equivalently,

$$\begin{eqnarray}&&{\xi }_{i}\lt \displaystyle \frac{{\xi }_{n}{V}_{L}}{6{\gamma }_{d}\rho L}.\end{eqnarray} \tag{ 22 }$$

It implies that the kinematic saturation cannot be reached during the entire dynamo process.

When the damping scale increases up to L, we have the timescale of the damping kinematic dynamo in the partially ionized gas (Equation (18))

$$\begin{eqnarray}&&{t}_{\mathrm{dyn}}=\displaystyle \frac{23}{3}{L}^{\displaystyle \frac{1}{3}}{V}_{L}^{-1}({L}^{\displaystyle \frac{2}{3}}-{k}_{d0}^{-\tfrac{2}{3}}),\end{eqnarray} \tag{ 23 }$$

which is approximately 7.7 largest eddy-turnover times. At t_dyn, the magnetic energy is amplified to (Equations (7), (17), and (23))

$$\begin{eqnarray}&&{{ \mathcal E }}_{\mathrm{dyn}}={{ \mathcal C }}^{-1}{{LV}}_{L}.\end{eqnarray} \tag{ 24 }$$

The calculated t_dyn and the field strength corresponding to ${{ \mathcal E }}_{\mathrm{dyn}}$ in the CNM and MC are presented in Table 1. The results here only serve as an order-of-magnitude estimate. More rigorous calculations rely on a more realistic and detailed modeling of the shock, as well as the ambient environment. We caution that with the development of the strong magnetic field in front of the shock, the shock Alfvén Mach number M_sh,A = v_sh/V_A decreases. In the linear kinematic dynamo regime, the magnetic back-reaction on the shock propagation is usually unimportant. But in the case when the Alfvén speed corresponding to the amplified magnetic filed at a later time of the dynamo approaches the shock speed (e.g., in the case of MC), the strong shock jump condition will not be satisfied. Consequently, the turbulence driving and the dynamo growth also cease. In the self-regulated system in the realistic situation, the dynamo efficiency decreases with the arising of the dynamically important magnetic field.

Moreover, for the amplification of the upstream magnetic field, the turbulent dynamo can only operate within the precursor crossing time t_c = L_p/v_sh. The maximum size of the CR precursor L_p is determined by the diffusion of the highest-energy CRs,

$$\begin{eqnarray}&&{L}_{p}=\displaystyle \frac{{\kappa }_{\max }}{{v}_{\mathrm{sh}}},\end{eqnarray} \tag{ 25 }$$

where the CR diffusion coefficient κ should be computed self-consistently as they interact with the magnetic fluctuations amplified by the CR-driven turbulence.

2.3. Magnetic Field Structure and CR Diffusion

Figure 1(a) illustrates the magnetic energy spectrum M(k) during the damping kinematic dynamo. The Kazantsev spectrum extends over large scales down to the spectral peak at the damping scale, where the spectrum is cut off due to ion-neutral collisional damping. Since the magnetic energy is well below the turbulent kinetic energy over all scales, the nonlinear Lorentz back-reaction on turbulent motions is insignificant, and the entire dynamo process stays in the linear regime.

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The magnetic field resulting from the damping kinematic dynamo in the presence of ion-neutral collisions has a characteristic scale as the increasing damping scale. It means that the magnetic field has a tangled structure over larger scales, but is smooth over smaller scales with the magnetic fluctuations suppressed. The bending of field lines by the super-Alfvénic turbulent motions at k_d gives rise to an effective mean free path λ_mfp of CRs equal to k_d⁻¹ (see Brunetti & Lazarian 2007). Note that the Larmor radius r_L of CRs should be smaller than ${k}_{d}^{-1}$. The corresponding diffusion coefficient is

$$\begin{eqnarray}&&\kappa \approx \displaystyle \frac{c{\lambda }_{\mathrm{mfp}}}{3}=\displaystyle \frac{c}{3{k}_{d}},\end{eqnarray} \tag{ 26 }$$

where k_d is given by Equation (18). Suppose that the damping scale, as well as λ_mfp, can reach L, thus the precursor has the maximum thickness

$$\begin{eqnarray}&&{L}_{p}=\displaystyle \frac{{cL}}{3{v}_{\mathrm{sh}}},\end{eqnarray} \tag{ 27 }$$

and the crossing time is

$$\begin{eqnarray}&&{t}_{c}=\displaystyle \frac{c}{3{v}_{\mathrm{sh}}}\displaystyle \frac{L}{{v}_{\mathrm{sh}}}=\displaystyle \frac{c}{3{v}_{\mathrm{sh}}}\displaystyle \frac{{V}_{L}}{{v}_{\mathrm{sh}}}\displaystyle \frac{L}{{V}_{L}}.\end{eqnarray} \tag{ 28 }$$

If we adopt V_L ∼ v_sh ∼ 10³ km s⁻¹, then t_c is equal to 100 largest eddy-turnover times, which is much larger than t_dyn (Equation (23)) and certainly sufficient for the full development of the precursor turbulence and the damping kinematic dynamo. More conservatively, the turbulent velocity should satisfy V_L/v_sh > 7.7 × 10⁻² (see Equation (23)) for the dynamo to proceed to the outer scale of the turbulence.

At the Bohm limit with r_L = L, the maximum energy of CRs with an effective diffusion that is governed by the amplified magnetic field is

$$\begin{eqnarray}&&{E}_{\mathrm{CR},\max }={{eB}}_{\mathrm{dyn}}L,\end{eqnarray} \tag{ 29 }$$

where e is the particle's charge. Given the field strength in Table 1, it is 4.2 × 10¹⁶ eV in the CNM and 7.1 × 10¹⁷ eV in the MC. It implies that the damping kinematic dynamo in the precursor can easily generate the magnetic field required for the acceleration of CRs up to the knee energy of ∼10¹⁵ eV and beyond.

The magnetic field structure and the related CR diffusion in the preshock region are critical for the DSA. The limits on the DSA arising from the partial ionization of the upstream medium were studied by e.g., O'C Drury et al. (1996) and Malkov et al. (2011), under the consideration of the ion-neutral collisional damping of Alfvén waves. In the situation with the precursor turbulence, the turbulent dynamo increases both the strength and the characteristic length scale of the magnetic field.³ Starting from the weak interstellar field strength and in the presence of the severe damping, the turbulent dynamo generates the damping-scale magnetic field. Without the pitch-angle scattering, CRs with a Larmor radii smaller than the damping scale gyrate about the field lines. The random change of the magnetic field orientation over the distance equal to the damping scale entails an effective diffusion of CRs. Unlike the resonant interaction, both low- and high-energy CRs undergo the same diffusion process in this dynamo-generated magnetic field.⁴

The arising of the precursor turbulence depends upon the density fluctuations in the upstream medium, which have been commonly observed in the ISM (Armstrong et al. 1995; Chepurnov & Lazarian 2010; Xu & Zhang 2017b), and especially in the cold and dense phases (Lazarian 2009; Hennebelle & Falgarone 2012) but with a small volume-filling factor (Tielens 2005; Haverkorn & Spangler 2013). In the case that the SNR shock propagates through a relatively uniform ambient medium, the above dynamo process would become less efficient in the mildly turbulent precursor.

SNR shocks efficiently heat the ISM gas, resulting in a high temperature in the postshock region. We assume that the partially ionized gas passing through the shock becomes fully ionized downstream. Compared with the partially ionized preshock medium, the magnetic fluctuations in the fully ionized postshock medium are only marginally damped by the resistivity, with the resistive scale much smaller than the viscous scale (see the Appendix). On the other hand, the equipartition scale with the local turbulent energy in balance with the initial magnetic energy is given by Lazarian (2006) as

$$\begin{eqnarray}{l}_{{\rm{A}}} & = & {{LM}}_{{\rm{A}}}^{-3}={{LV}}_{L}^{-3}{V}_{{\rm{A}}}^{3}\\ & = & 1.3\times {10}^{13}\left(\displaystyle \frac{L}{0.1\mathrm{pc}}\right){\left(\displaystyle \frac{{V}_{L}}{100\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{-3}\\ & & {\left(\displaystyle \frac{{n}_{i}}{10{\mathrm{cm}}^{-3}}\right)}^{-\tfrac{3}{2}}{\left(\displaystyle \frac{B}{5\mu {\rm{G}}}\right)}^{3}\,\mathrm{cm},\end{eqnarray} \tag{ 30 }$$

where M_A is the Alfvén Mach number, and the ion number density n_i = n_H. Because it is larger than the viscous scale of turbulence (Equation (37)), the turbulent dynamo falls in the nonlinear regime in the presence of the significant Lorentz back-reaction on the turbulent motions over smaller scales. The initial field strength adopted here is comparable to the interstellar value. When the precursor dynamo is also taken into account, a larger l_A is expected, and the postshock dynamo starts with a stronger magnetic field.

The magnetic energy spectrum M(k) during the nonlinear dynamo process is illustrated in Figure 1(b). The Kazantsev spectrum exists on scales above l_A where the turbulence is super-Alfvénic. The trans-Alfvénic MHD turbulence is developed over smaller scales, with the same Kolmogorov form for both the turbulent and magnetic energy spectra. Along with the magnetic energy growth, the equipartition scale l_A shifts to ever-larger scales, and eventually the MHD turbulence extends up to L. In the sub-viscous range, i.e., k > k_ν, M(k) is further prolonged to the resistive cutoff and follows the k⁻¹ profile as a result of the balance between the magnetic tension force and the viscous drag (Cho et al. 2002, 2003; Lazarian et al. 2004; XL16). The numerical evidence for the above scalings of M(k) can be found in dynamo simulations, e.g., Haugen et al. (2004) and Brandenburg & Subramanian (2005).

Unlike the kinematic dynamo, which is only subject to the microscopic magnetic diffusion, e.g., ambipolar diffusion in a partially ionized medium, the nonlinear dynamo mostly suffers the turbulent magnetic diffusion arising in the MHD turbulence. Because the turbulent diffusion rate is comparable to the dynamo growth rate, i.e., the eddy-turnover rate (Lazarian & Vishniac 1999), the nonlinear dynamo is inefficient. The evolution law of the magnetic energy in the nonlinear regime with the turbulent diffusion taken into account was analytically derived by XL16,

$$\begin{eqnarray}&&{{ \mathcal E }}_{M}={{ \mathcal E }}_{M0}+\displaystyle \frac{3}{38}\epsilon t,\end{eqnarray} \tag{ 31 }$$

where

$$\begin{eqnarray}&&\epsilon ={L}^{-1}{V}_{L}^{3}\end{eqnarray} \tag{ 32 }$$

is the constant turbulent energy transfer rate. It reveals a linear-in-time growth of magnetic energy with the growth rate as a small fraction of , which is consistent with direct numerical measurements, e.g., Cho et al. (2009), Beresnyak (2012).

When the dynamo saturation is achieved at L, the magnetic energy is equal to the kinetic energy of the largest turbulent eddy,

$$\begin{eqnarray}&&{{ \mathcal E }}_{M}=\displaystyle \frac{1}{2}{V}_{L}^{2}.\end{eqnarray} \tag{ 33 }$$

The timescale of the nonlinear dynamo is (Equation (31))

$$\begin{eqnarray}&&{\tau }_{\mathrm{nl}}=\displaystyle \frac{38}{3\epsilon }({{ \mathcal E }}_{M}-{{ \mathcal E }}_{M0}).\end{eqnarray} \tag{ 34 }$$

To examine the applicability of the above analysis on the magnetic field amplification in the postshock region, we next carry out a comparison between our theoretical predictions and the numerical results in Inoue et al. (2009, hereafter I09). They performed MHD simulations of SNR shocks propagating through the inhomogeneous ISM. We only consider parallel shocks (Models 2, 3, and 4 in I09) and disregard additional magnetic field amplification due to the shock compression. The parameters that we use are listed in Table 2.

Starting from the initial field strength B₀, the temporal evolution of the magnetic field is calculated according to Equation (31). As shown in Figure 2(a), our analytical results can well match the numerical findings. The saturated field strength is determined by the turbulent velocity V_L (Equation (33)). It approaches 1 mG given the settings of Model 2.

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We can also calculate the distance between the location where the magnetic field reaches saturation and the shock front, which is given by

$$\begin{eqnarray}&&{l}_{\mathrm{sat}}\approx {v}_{\mathrm{down}}{\tau }_{{nl}},\end{eqnarray} \tag{ 35 }$$

where the downstream bulk velocity is v_down = 1/4v_sh for a strong shock, and v_sh = 1289 km s⁻¹ is the shock velocity (Model 2 in I09). The timescale of the nonlinear dynamo τ_nl is provided by Equation (34). Figure 2(b) shows that our estimate (the vertical dashed line) coincides with the peak position of the magnetic field profile numerically produced in I09.

Because of the inefficiency of the nonlinear turbulent dynamo and its prolonged timescale, the peak field strength is developed at a significant distance, rather than the immediate vicinity, behind the shock front. This feature naturally explains the location of the X-ray hot spots detected at more than 0.1 pc downstream of SNR shocks (I09; Uchiyama et al. 2007; Uchiyama & Aharonian 2008).

The above comparison confirms the predictive power of the XL16 model for the nonlinear turbulent dynamo. It provides an analytically tractable means of studying the evolution and distribution of the magnetic field in the postshock region. Notice that, in the case of efficient precursor dynamo starting from a relatively high magnetization, the timescale of the postshock nonlinear dynamo can be much shortened, and the growth of the magnetic energy is insignificant.

The amplified magnetic field affects the diffusion of CRs behind the shock. For lower-energy CRs with E_CR < eB(l_A)l_A, where B(l_A) is the field strength reached at the energy equipartition, their mean free path is independent of energy and given by l_A (see BJL09). The corresponding diffusion coefficient is κ = cl_A/3. For higher-energy CRs with the Larmor radius exceeding l_A, the larger-scale magnetic fluctuations following the Kazantsev spectrum, i.e., B(k) ∝ k^5/4, are too weak to confine the CRs as eB(k)k⁻¹ ∝ k^1/4. But as l_A increases with time, more energetic CR particles are influenced by the amplified magnetic field during the nonlinear turbulent dynamo. At the distance l_sat (Equation (35)) behind the shock front, l_A reaches the outer scale of turbulence L at the nonlinear saturation, where CRs with E_CR < eB(L) L have κ = cL/3.

As regards the CR diffusion in the direction perpendicular to the mean magnetic field, lower-energy CRs with E_CR < eB(l_A) l_A are characterized by the fast superdiffusion, i.e., $\langle {\sigma }_{\perp }^{2}\rangle \propto {t}^{\alpha }$, α > 1, where $\langle {\sigma }_{\perp }^{2}\rangle$ is the mean squared displacement in the perpendicular direction (Lazarian & Yan 2014). The superdiffusion dominates the perpendicular transport of CRs as l_A increases with the distance behind the shock front. The higher-energy CRs undergo the normal diffusion with $\langle {\sigma }_{\perp }^{2}\rangle$ as a linear function of time.

Compared with earlier studies on the turbulent dynamo in the shock precursor, e.g., BJL09, del Valle et al. (2016), based on the analytical results of turbulent dynamo in a partially ionized medium (XL16), we have demonstrated that the magnetic field amplification in a partially ionized medium can be drastically different from that in a fully ionized medium. In the CNM and MC, because of the severe damping, the dynamo remains in the linear regime with a linear-in-time growth of field strength, rather than an exponential growth expected for the kinematic dynamo in a highly ionized medium (Balsara & Kim 2005), or a linear-in-time growth of magnetic energy in the nonlinear regime (BJL09). Through comparison with the current-driven instability (Bell 2004), BJL09 suggested that the nonlinear turbulent dynamo is more favorable in amplifying the upstream magnetic fields. We present a more efficient dynamo regime arising in the partially ionized preshock medium compared to the nonlinear dynamo. Furthermore, different from the limited timescale considered in BJL09, the extended CR precursor in our scenario leads to ample time for the development of turbulence and magnetic field amplification.⁵ Our results support the notion that the turbulent dynamo in the shock precursor is the dominant process of magnetic field amplification.

It is important to stress that only when the ionization fraction is sufficiently high can the kinematic dynamo be considered unimportant as compared with the nonlinear dynamo stage. In the new damping regime identified in XL16, depending on the ionization fraction, the kinematic dynamo can be important through the turbulence inertial range up to the equipartition scale, or the outer scale as in the case discussed in this paper. Hence, in a weakly ionized medium, the kinematic dynamo can have astrophysically important applications. In the preshock turbulence, the characteristic scale of the magnetic field grows with the increasing damping scale of turbulence, independent of the CR Larmor radius. The resulting damping-scale magnetic field regulates the diffusion behavior and thus the acceleration of CRs from low energies up to very high energies. Different from the short-wavelength Bell mechanism, there is no need to invoke an additional inverse cascade process for transporting the magnetic energy to larger scales.

In the turbulent postshock medium, the observed year-scale X-ray variability of the compact hot spots suggests a largely amplified magnetic field of 1 mG and very efficient particle acceleration in the emitting regions (Uchiyama et al. 2007; Uchiyama & Aharonian 2008). Our results on the amplification of the postshock magnetic field show that the analytical expectations in XL16 agree well with the existing numerical simulations. Depending on the postshock turbulent velocity, the saturated field strength of the nonlinear dynamo can be on the order of 1 mG (Section 3). As pointed out by I09, the electrons responsible for the hot spots are unlikely to be accelerated at the shock front in view of the fast synchrotron cooling. The efficient in situ acceleration is attributable to the reverse shock (Uchiyama & Aharonian 2008) or secondary shocks arising from collisions of turbulent flows (I09). An alternative explanation could be the adiabatic non-resonant acceleration in MHD turbulence proposed by Brunetti & Lazarian (2016; see also Xu & Zhang 2017a). In MHD turbulence, magnetic field lines can be stretched due to the turbulent dynamo and shrink via the turbulent reconnection, which happens through every eddy turnover (Lazarian & Vishniac 1999). The CRs with Larmor radii much smaller than the size of the turbulent eddy are attached to field lines and confined within the eddy during its eddy-turnover time, bouncing back and forth between converging magnetic fields in the turbulent reconnection region. This process is similar to the first-order Fermi acceleration (de Gouveia dal Pino & Lazarian 2005). The resulting efficient acceleration within the eddy-turnover time may accommodate the year-scale variability in the synchrotron emission. A detailed study on this mechanism and its applicability in CR acceleration in the postshock turbulence is the subject of an upcoming paper.

Our results on the magnetic field amplification and its implications on the diffusion of energetic particles in both the preshock and postshock regions are important for further modeling the emission characteristics in comparison with multi-wavelength observations of SNRs (see e.g., Zeng et al. 2017).

By applying the turbulent dynamo theory formulated by XL16, we have investigated the magnetic field amplification in SNRs.

The dynamo evolution of magnetic fields in the preshock and postshock media are very different. The dynamo in the weakly ionized upstream medium, e.g., the CNM and MC, is characterized by a linear-in-time growth of field strength; that is, the magnetic energy grows quadratically with time. It is slower than the exponential growth in the linear regime without damping, but faster than the linear-in-time growth of magnetic energy in the nonlinear regime. In the extended CR precursor, the large damping-scale magnetic field formed at later times of the damping kinematic dynamo is beneficial for the confinement of high-energy CRs. This finding is important for a realistic treatment of the shock acceleration of CRs in the partially ionized ISM.

Importantly, we provide the criterion for the dominance of the ion-neutral collisional damping over the neutral viscous damping, and thus the emergence of the damping regime of the kinematic dynamo (Equation (4)). Then the conditions for the entire dynamo process to be in the damping regime are given by Equation (9) and (21).

The turbulent dynamo in the postshock region is in the nonlinear regime and drives a linear-in-time and inefficient growth of magnetic energy. Provided a weak initial field strength in the postshock region, the peak field strength at the dynamo saturation can only be reached in the farther downstream region. This explains the X-ray hot spots located far from the shock front. The consistency with the results of numerical simulations (e.g., I09) shows that the XL16 analytical model for the nonlinear turbulent dynamo can be used to quantify the evolution and distribution of downstream magnetic fields.

The postshock magnetic turbulence can serve as an alternative source to the shock front for efficient acceleration of CRs. The corresponding acceleration mechanism deserves more attention and detailed analysis in the future with updated observations.

We thank the anonymous referee for insightful comments. S.X. acknowledges the support for Program number HST-HF2-51400.001-A provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. A.L. acknowledges the support from grant NSF DMS 1622353.

In the presence of magnetic field, the ion viscosity is anisotropic. Even for the initial magnetization as low as the level in the ISM, the ion viscosity perpendicular to the field is very small (Simon 1955),

$$\begin{eqnarray}{\nu }_{i,\perp } & = & \displaystyle \frac{3{k}_{{\rm{B}}}T{\nu }_{{ii}}}{10{{\rm{\Omega }}}_{i}^{2}{m}_{i}}=2.0\times {10}^{7}\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{10}\right){\left(\displaystyle \frac{T}{{10}^{7}{\rm{K}}}\right)}^{-\tfrac{1}{2}}\\ & & \times \left(\displaystyle \frac{{n}_{i}}{10\,{\mathrm{cm}}^{-3}}\right){\left(\displaystyle \frac{B}{5\mu {\rm{G}}}\right)}^{-2}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 36 }$$

where c, m_e, e, $\mathrm{ln}{\rm{\Lambda }}$, k_B, ν_ii, Ω_i, m_i (=m_H), n_i are the speed of light, the electron mass and charge, the Coulomb logarithm, the Boltzmann constant, the ion collision and cyclotron frequencies, the ion mass (equal to the hydrogen atomic mass) and number density. It corresponds to a small viscous scale

$$\begin{eqnarray}{k}_{\nu } & = & {L}^{-\tfrac{1}{4}}{V}_{L}^{\tfrac{3}{4}}{\nu }_{i,\perp }^{-\tfrac{3}{4}}=2.5\times {10}^{-5}{\left(\displaystyle \frac{L}{0.1\mathrm{pc}}\right)}^{-\tfrac{1}{4}}{\left(\displaystyle \frac{{V}_{L}}{100\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{\tfrac{3}{4}}\\ & & \times {\left(\displaystyle \frac{{\nu }_{i,\perp }}{2.0\times {10}^{7}{\mathrm{cm}}^{2}{{\rm{s}}}^{-1}}\right)}^{-\tfrac{3}{4}}\,{\mathrm{cm}}^{-1}.\end{eqnarray} \tag{ 37 }$$

On the other hand, the resistivity is (Spitzer 1956),

$$\begin{eqnarray}&&\eta =\displaystyle \frac{{c}^{2}\sqrt{{m}_{e}}{e}^{2}\mathrm{ln}{\rm{\Lambda }}}{4{({k}_{{\rm{B}}}T)}^{\tfrac{3}{2}}}=3.0\times {10}^{2}\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{10}\right){\left(\displaystyle \frac{T}{{10}^{7}{\rm{K}}}\right)}^{-\tfrac{3}{2}}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 38 }$$

Then we have the magnetic Prandtl number,

$$\begin{eqnarray}&&{P}_{m}=\displaystyle \frac{{\nu }_{i,\perp }}{\eta }=6.7\times {10}^{4}\left(\displaystyle \frac{T}{{10}^{7}\,{\rm{K}}}\right)\left(\displaystyle \frac{{n}_{i}}{10\,{\mathrm{cm}}^{-3}}\right){\left(\displaystyle \frac{B}{5\mu {\rm{G}}}\right)}^{-2},\end{eqnarray} \tag{ 39 }$$

exceeding unity by orders of magnitude. The resistive scale is given by

$$\begin{eqnarray}&&{k}_{R}={P}_{m}^{\displaystyle \frac{1}{2}}{k}_{\nu },\end{eqnarray} \tag{ 40 }$$

which is much shorter than the viscous scale.