Electron Preacceleration in Weak Quasi-perpendicular Shocks in High-beta Intracluster Medium

The structures and time variations of collisionless shocks are primarily governed by the dynamics of reflected ions and the waves excited by the relative drift between reflected and incoming ions. In theories of collisionless shocks, therefore, a number of critical shock Mach numbers have been introduced to describe ion reflection and upstream wave generation; for a review, see Balogh & Truemann (2013). Although the main focus of this paper is the electron acceleration at Q_⊥-shocks, we here present a brief review of the "shock criticalities" due to reflected ions.

The reflection of ions has been often linked to the "first critical Mach number," ${M}_{{\rm{f}}}^{* }(\beta ,{\theta }_{\mathrm{Bn}});$ it was found for u_n2 = c_s2 by applying the Rankine-Hugoniot jump relation to fast MHD shocks, i.e., the condition that the downstream flow speed normal to the shock surface equals the downstream sound speed (e.g., Edmiston & Kennel 1984). In supercritical shocks with ${M}_{{\rm{f}}}\gt {M}_{{\rm{f}}}^{* }$, the shock kinetic energy cannot be dissipated sufficiently through resistivity and wave dispersion, and hence a substantial fraction of incoming ions should be reflected upstream in order to sustain the shock transition from the upstream to the downstream. In subcritcal shocks below ${M}_{{\rm{f}}}^{* }$, on the other hand, the resistivity alone can provide enough dissipation to support a stable shock structure. In collisionless shocks, however, the reflection of ions occurs at the shock ramp, due to the magnetic deflection and the cross shock potential drop—the physics of which are beyond the fluid description. Hence, it should be investigated with simulations resolving kinetic processes.

In Q_∥-shocks, the first critical Mach number also denotes the minimum Mach number, above which kinetic processes trigger overshoot/undershoot oscillations in the density and magnetic field, and the shock structures may become nonstationary under certain conditions. The reflection of ions is mostly due to the deceleration by the shock potential drop, and resonant and nonresonant waves are excited via streaming instabilities induced by reflected ions (e.g., Caprioli & Spitkovsky 2014a, 2014b). Such processes depend on shock parameters. For instance, in shocks with higher M_s and consequently higher shock kinetic energies, the structures tend to more easily fluctuate and become unsteady. In high-β plasmas, on the other hand, shocks could be stabilized against certain instabilities owing to fast thermal motions, which can subdue the relative drift between reflected and incoming particles; thus, theoretical analyses based on the cold plasma assumption could be modified in high-β plasmas. In Paper I, we found that ${M}_{{\rm{f}}}^{* }\approx 2.3$ for Q_∥-shocks in ICM plasmas with β ≈ 100, which is higher than the fluid prediction by Edmiston & Kennel (1984). In Q_∥-shocks, the kinetic processes involved in determining ${M}_{{\rm{f}}}^{* }$ are also part of the preacceleration of ions, and hence the injection to the Fermi-I process of DSA.

In Q_⊥-shocks, both ions and electrons are reflected through the magnetic deflection; the two populations are subject to deceleration by the magnetic mirror force due to converged magnetic field lines at the shock transition. In addition, the shock potential drop decelerates incident ions while it accelerates electrons toward the downstream direction. The reflected particles gain energy through the gradient drift along the motional electric field at the shock surface (SDA). Most of reflected ions, however, are trapped—mostly at the shock foot—before they advect downstream with the background magnetic field after about one gyromotion. As a result, streaming instabilities are not induced in the upstream, and hence the ensuing CR proton acceleration is ineffective, as previously reported with hybrid and PIC simulations (e.g., Caprioli & Spitkovsky 2014a, 2014b; Paper I). However, still the dynamics of reflected ions is primarily responsible for the main features of the transition zone of Q_⊥-shocks (e.g., Treumann & Jaroschek 2008; Treumann 2009). For instance, the current due to the drift motion of reflected ions generates the magnetic foot, ramp, and overshoot, and the charge separation due to reflected ions generates the ambipolar electric shock potential drop at the shock ramp.

In Q_⊥-shocks, the accumulation of reflected ions at the upstream edge of the foot may lead to the cyclic self-reformation of shock structures over ion gyroperiods and result in the excitation of low-frequency whistler waves in the shock foot region (e.g., Matsukiyo & Scholer 2006; Scholer & Burgess 2007). This leads to the so-called "second or whistler critical Mach number," ${M}_{{\rm{w}}}^{* }\approx (1/2)\sqrt{{m}_{i}/{m}_{e}}\cos {\theta }_{\mathrm{Bn}}$ in the β ≪ 1 limit (Kennel et al. 1985; Krasnoselskikh et al. 2002). In subcritical shocks with ${M}_{{\rm{f}}}\lt {M}_{{\rm{w}}}^{* }$, linear whistler waves can phase-stand in the shock foot upstream of the ramp. Dispersive whistler waves were found far upstream in interplanetary, subcritical shocks (e.g., Oka et al. 2006). Those waves interact with the upstream flow and contribute to the energy dissipation, effectively suppressing the shock reformation. Above ${M}_{{\rm{w}}}^{* }$, stationary linear wave trains cannot stand in the region ahead of the shock ramp.

The "third or nonlinear whistler critical Mach number," ${M}_{\mathrm{nw}}^{* }\approx \sqrt{{m}_{i}/2{m}_{e}}\cos {\theta }_{\mathrm{Bn}}$ in the β ≪ 1 limit, was introduced to describe the nonstationarity of shock structures. Krasnoselskikh et al. (2002) predicted that, in supercritical shocks with ${M}_{{\rm{f}}}\gt {M}_{\mathrm{nw}}^{* }$, nonlinear whistler waves turn over because of the gradient catastrophe, leading to the nonstationarity of the shock front and quasi-periodic shock-reformation (see Scholer & Burgess 2007). However, Hellinger et al. (2007) used 2D hybrid and PIC simulations to show that phase-standing oblique whistlers can be emitted in the foot even in supercritical Q_⊥-shocks, so the shock-reformation is suppressed in 2D. In fact, the nonstationarity and self-reformation of shock structures are important, long-standing problems in the study of collisionless shocks, which have yet to be fully understood (e.g., Scholer et al. 2003; Lembeǵe et al. 2004; Matsukiyo & Scholer 2006).

In the β ≪ 1 limit (i.e., in cold plasmas), ${M}_{{\rm{w}}}^{* }=2.3$ and ${M}_{\mathrm{nw}}^{* }=3.2$ for the fiducial parameter values adopted for our PIC simulations (m_i/m_e = 100 and θ_Bn = 63°). Hence, in some of the models considered here, ${M}_{{\rm{w}}}^{* }\lt {M}_{{\rm{s}}}\lt {M}_{\mathrm{nw}}^{* }$, so the whistler waves induced by reflected ions could be be confined within the shock foot without overturning. However, these critical Mach numbers increase to ${M}_{{\rm{w}}}^{* }=9.7$ and ${M}_{\mathrm{nw}}^{* }=13.8$ for the true ratio of m_i/m_e = 1836. Therefore, in the ICM, weak Q_⊥-shocks are expected to be subcritical with respect to the two whistler critical Mach numbers, and so they would not be subject to self-reformation. The confirmation of these critical Mach numbers, or improved estimations for β ≳ 1, through numerical simulations is very challenging, as noted above. The excitation of oblique whistler waves and the suppression of shock-reformation via surface rippling require at least 2D simulations (e.g., Lembeǵe & Savoini 2002; Burgess 2006). The additional degree of freedom in higher-dimensional simulations tends to stabilize some instabilities revealed in lower-dimensional simulations (e.g., Matsukiyo & Matsumoto 2015). Moreover, simulation results are often dependent on m_i/m_e, as well as the magnetic field configuration, i.e., whether B₀ is in-plane or off-plane (Lembeǵe et al. 2009). Furthermore, adopting the realistic ratio of m_i/m_e in PIC simulations is very computationally expensive, as pointed in the previous section.

As mentioned in the introduction, GSN14a and GSN14b showed that in high-β, Q_⊥-shocks, electrons can be preaccelerated via multiple cycles of SDA, due to the scattering by the upstream waves excited via the EFI. Here, we introduce the additional "EFI critical Mach number," ${M}_{\mathrm{ef}}^{* }$, above which the electron preacceleration is effective. We seek it in the next section, along with the relevant kinetic processes involved.

The space physics and ISM communities have been mainly interested in shocks in low-β plasmas (β ≲ 1), and hence the analytic relations simplified for cold plasmas are often quoted (e.g., the dispersion relation for fast magnetosonic waves used by Krasnoselskikh et al. (2002)). In such works, M_A is commonly used to characterize shocks. However, in hot ICM plasmas, shocks have M_s ≈ M_f ≪ M_A, and magnetic fields play dynamically less important roles. Moreover, the ion reflection at the shock ramp is governed mainly by M_s rather than M_A (e.g., Paper I). Thus, in the rest of this paper, we will use the sonic Mach number M_s to characterize shocks.

As mentioned in the introduction, GSN14a discussed the relativistic SDA theory for electrons in Q_⊥-shocks, which involves the electron reflection at the shock and the energy gain due to the drift along the motional electric field. In a subsequent paper, GSN14b showed that these electrons can induce the EFI, which leads to the excitation of oblique waves. The electrons return back to the shock due to the scattering by those self-excited upstream waves, and are further accelerated through multiple cycles of SDA (Fermi-like process). Below, we follow these previous papers to discuss how the physical processes depend on parameters such as M_s, θ_Bn, and T₁ for the shocks considered here (Table 1). Inevitably, we cite below some of equations presented in GSN14a and GSN14b.

3.2.1. Shock Drift Acceleration

GSN14a derived the criteria for electron reflection by considering the dynamics of electrons in the so-called de Hoffmann–Teller (HT, hereafter) frame, in which the flow velocity is parallel to the background magnetic field—and hence the motional electric field disappears both upstream and downstream of the shock (de Hoffmann & Teller 1950). In the HT frame, the upstream flow has ${u}_{t}={u}_{\mathrm{sh}}\sec {\theta }_{\mathrm{Bn}}$ along the background magnetic field. Hereafter, v_∥ and v_⊥ represent the velocity components of incoming electrons, parallel and perpendicular to the background magnetic field, respectively, in the upstream rest frame, and ${\gamma }_{t}\,\equiv \,{(1-{u}_{t}^{2}/{c}^{2})}^{-1/2}$ is the Lorentz factor of the upstream flow in the HT frame.

The reflection criteria can be written as

$$\begin{eqnarray}&&{v}_{\parallel }\lt {u}_{t}\end{eqnarray} \tag{ 4 }$$

(Equation (19) of GSN14a), and

$$\begin{eqnarray}\begin{array}{rcl}{v}_{\perp } & \gtrsim & {\gamma }_{t}\tan {\alpha }_{0}\cdot \left[{\left({v}_{\parallel }-{u}_{t}\right)}^{2}+2{c}^{2}{\cos }^{2}{\alpha }_{0}{\rm{\Delta }}\phi \cdot G\cdot F\right.\\ & & +\ {\left.\{{c}^{2}{\cos }^{2}{\alpha }_{0}\cdot G-{\left({v}_{\parallel }-{u}_{t}\right)}^{2}\}{\rm{\Delta }}{\phi }^{2}\right]}^{1/2},\end{array}\end{eqnarray} \tag{ 5 }$$

assuming that the normalized cross-shock potential drop is ${\rm{\Delta }}\phi (x)\,\equiv \,e[{\phi }^{\mathrm{HT}}(x)-{\phi }_{0}^{\mathrm{HT}}]/{m}_{e}{c}^{2}\ll 1$. Here, $G\,\equiv \,{(1-{v}_{\parallel }{u}_{t}/{c}^{2})}^{2}$, $F\,\equiv \,{[1-{({v}_{\parallel }-{u}_{t})}^{2}/({{Gc}}^{2}{\cos }^{2}{\alpha }_{0})]}^{1/2}$, and ${\alpha }_{0}\,\equiv \,{\sin }^{-1}(1/\sqrt{b})$ with the magnetic compression ratio $b\,\equiv \,B{(x)}^{\mathrm{HT}}/{B}_{0}^{\mathrm{HT}}$. The superscript HT denotes the quantities in the HT frame. Note that, for Δϕ(x) = 0, Equation (5) becomes the same as Equation (20) of GSN14a.

In Figure 1(a), the red and blue solid lines mark the boundaries of the reflection criteria in Equations (4) and (5) for the M2.0 and M3.0 models, respectively. The red and blue dashed curves to the right of the vertical lines are the post-reflection velocities calculated via Equations (25) and (26) of GSN14a by inserting the boundary values of Equations (4) and (5). For b and Δϕ, the values estimated at the shock surface from simulation data were used. The solid black half-circle shows $v\,\equiv \,{({v}_{\parallel }^{2}+{v}_{\perp }^{2})}^{1/2}=c$, while the dashed black half-circle shows v = v_th,e, where ${v}_{\mathrm{th},e}=\sqrt{2{k}_{{\rm{B}}}{T}_{e1}/{m}_{e}}$ is the electron thermal speed of the incoming flow.

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As in GSN14b, we semi-analytically estimated the amount of the incoming electrons that satisfy the reflection condition, i.e., those bounded by the colored solid curves and the colored vertical lines together with the black circle in Figure 1(a). In Figure 1(b), the fraction of the reflected electrons, R, estimated for Q_⊥-shocks with θ_Bn = 63° is shown by the black filled circles connected by a black solid line, while R for Q_∥-shocks with θ_Bn = 13° is shown by the black filled circles connected by a black dashed line. In Q_⊥-shocks, the reflection fraction, R, is quite high and increases with M_s, ranging ∼20–25% for 2 ≤ M_s ≤ 3. In Q_∥-shocks, R is also high, ranging ∼17–20%, for 2.15 ≤ M_s ≤ 3, but it drops sharply at M_s = 2.

We should point out that the electron reflection becomes ineffective for superluminal shocks with large obliquity angles (i.e., ${u}_{\mathrm{sh}}/\cos {\theta }_{\mathrm{Bn}}\geqslant c$), because the electrons streaming upstream along the background field cannot outrun the shocks (see GSN14b). The obliquity angle for the superluminal behavior is ${\theta }_{\mathrm{sl}}\,\equiv \,\arccos ({u}_{\mathrm{sh}}/c)=86^\circ$ for the shock in M3.0 with m_i/m_e = 100 and T₁ = 10⁸ K, and it is larger for smaller M_s. This angle is larger than θ_Bn of our models in Table 1, and hence all the shocks considered here are subluminal.

For given T₁ and M_s, the reflection of electrons is basically determined by b(x) and Δϕ(x), which quantify the magnetic deflection and the acceleration at the shock potential drop. Both b(x) and Δϕ(x) increase with increasing θ_Bn. Larger b enhances the electron reflection (positive effect), while larger Δϕ(x) suppresses it (negative effect). In GSN14b, shocks are semi-relativistic with Δϕ ∼ 0.1–0.5, and hence the negative effect of the potential drop is substantial. However, in our models, shocks are less relativistic because of the lower temperature adopted, and ${\rm{\Delta }}\phi \sim {m}_{i}{u}_{\mathrm{sh}}^{2}/2{m}_{e}{c}^{2}\ll 1$. As a result, the magnetic deflection dominates over the acceleration by the cross-shock potential, leading to higher R at higher θ_Bn. GSN14b showed that SDA becomes inefficient for u_t ≳ v_th,e ($\cos {\theta }_{\mathrm{Bn}}\lesssim \cos {\theta }_{\mathrm{limit}}={M}_{{\rm{s}}}\sqrt{{m}_{e}/{m}_{i}}$), which is more stringent than the superluminal condition (u_t > c). Thus, the electron reflection fraction begins to decrease for θ_Bn ≳ 60° in their models. Although not shown here, in our models, R monotonically increases with the obliquity angle for a given M_s, because the adopted θ_Bn (≤73°) is smaller than the limiting obliquity angle, θ_limit, for M_s = 2–3 and m_i/m_e = 100.

The reflected electrons gain energy via SDA. We estimated the energy gain from a single SDA cycle as

$$\begin{eqnarray}&&{\rm{\Delta }}\gamma \,\equiv \,{\gamma }_{r}-{\gamma }_{i}=\displaystyle \frac{2{u}_{t}({u}_{t}-{v}_{\parallel })}{{c}^{2}-{u}_{t}^{2}}{\gamma }_{i},\end{eqnarray} \tag{ 6 }$$

where γ_i and γ_r are the Lorentz factors for the pre-reflection and post-reflection electron velocities, respectively (Equation (24) of GSN14a). For given T₁ (or given c_s1), u_t and Δγ depend on M_s and θ_Bn. For the shocks considered here, γ_i ≈ 1 and u_t ≪ c, so ${\rm{\Delta }}\gamma \approx 2[{({u}_{t}/c)}^{2}-{u}_{t}{v}_{\parallel }/{c}^{2}]$. In Figure 1(b), the red filled circles connected with the red solid line show the average energy gain, $\langle {\rm{\Delta }}\gamma \rangle$, in units of ${m}_{e}{c}^{2}/{k}_{{\rm{B}}}{T}_{e}$, estimated for Q_⊥-shocks with θ_Bn = 63°. The red filled circles with the red dashed line show the quantity for Q_∥-shocks with θ_Bn = 13°. Here, the average was taken over the incoming electrons of Maxwellian distributions, so the $\langle {\rm{\Delta }}\gamma \rangle$ shown are representative values during the initial development stage of suprathermal particles.

In addition, the product of R and $\langle {\rm{\Delta }}\gamma \rangle$ is plotted as a function of θ_Bn for different M_s in Figure 1(c). For the models in Table 1, R was calculated using b and Δϕ estimated at the shock surface from simulation data, as mentioned above. For the rest, the values of b and Δϕ for the models with θ_Bn = 13° presented in Paper I were adopted for Q_∥-shocks, while the values for the models with θ_Bn = 63° presented in this work were adopted for Q_⊥-shocks. Panels (b) and (c) of Figure 1 show that more electrons are reflected and higher energies are achieved at higher M_s and larger θ_Bn.

3.2.2. Electron Firehose Instability

GSN14b performed periodic-box simulations with beams of streaming electrons, in order to isolate and study the EFI due to the reflected and SDA-energized electrons. They found the following: nonpropagating (ω_r ≈ 0), oblique waves with wavelengths ∼(10–20)c/w_pe are excited dominantly, δB_z is stronger than δB_x and δB_y (the initial magnetic field is in the x–y plane), and both the growth rate and the dominant wavelength of the instability are insensitive to the mass ratio m_i/m_e. These results are consistent with the expectations from the previous investigations of oblique EFI (e.g., Gary & Nishimura 2003).

The EFI criterion in weakly magnetized plasmas can be defined as

$$\begin{eqnarray}&&I\equiv 1-\displaystyle \frac{{T}_{e\perp }}{{T}_{e\parallel }}-\displaystyle \frac{1.27}{{\beta }_{e\parallel }^{0.95}}\gt 0,\end{eqnarray} \tag{ 7 }$$

where ${\beta }_{e\parallel }\,\equiv \,8\pi {n}_{e}{k}_{{\rm{B}}}{T}_{e\parallel }/{B}_{0}^{2}$ is the electron beta parallel to the initial magnetic field (Equation (10) of GSN14b). Equation (7) indicates that the instability parameter, I, is larger for higher β_e∥ for a given value of T_e⊥/T_e∥. For higher M_s, R is larger and T_e⊥/T_e∥ is smaller, leading to larger I.

Figure 1(d) shows the instability parameter of shocks with θ_Bn = 63°, as a function of M_s, estimated using the velocity distributions of the electrons that are located within (0–1) r_L,i (r_L,i is the ion Larmor radius with the upstream field B₀) upstream from the shock position in simulation data. For the M_s = 2.0 model, I ≲ 0 with almost no temperature anisotropy, so the upstream plasma should be stable against the EFI. This finding, which will be further updated with simulation results in the next section, suggests that the preacceleration of electrons due to the EFI may not operate effectively in very weak shocks. For M_s > 2, on the other hand, the EFI criterion is satisfied and I increases with increasing M_s, implying that larger temperature anisotropies (T_e∥ > T_e⊥) at higher M_s shocks induce stronger EFIs. The figure also indicates that I increases steeply around M_s ≈ 2.2–2.3. Additional data points, marked using the blue triangles connected by the blue dashed line (θ_Bn = 53°) and the red squares connected with the red dashed line (θ_Bn = 73°), show that Q_⊥-shocks with higher obliquity angles are more unstable to the EFI.

As discussed in Section 3.1, the criticality defined by the first critical Mach, ${M}_{{\rm{f}}}^{* }$, primarily governs the structures and time variations of collisionless shocks. In subcritical shocks, most of the shock kinetic energy is dissipated at the shock transition, resulting in relatively smooth and steady structures. In supercritical shocks, on the other hand, reflected ions induce overshoot/undershoot oscillations in the shock transition and ripples along the shock surface (e.g., Lowe & Burgess 2003, Krasnoselskikh et al. 2013, and Trotta & Burgess 2019). Q_∥-shocks with ${M}_{{\rm{f}}}\gt {M}_{{\rm{f}}}^{* }$ may undergo quasi-periodic reformation owing to the accumulation of upstream low-frequency waves. Q_⊥-shocks are less prone to reformation, because reflected ions mostly advect downstream after about one gyromotion.

Figure 2 compares the magnetic field structure for Q_⊥ and Q_∥-shocks with different M_s. In the Q_⊥-shocks, the overshoot/undershoot oscillation becomes increasingly more evident for higher M_s, but the shock structure seems to be quasi-stationary without any signs of reformation. This is consistent with the fact that the nonlinear whistler critical Mach number for our fiducial models is ${M}_{\mathrm{nw}}^{* }=3.2$. On the other hand, the Q_∥-shock in the M3.2-2D model exhibits quasi-periodic reformations.

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According to the fluid description of Edmiston & Kennel (1984), ${M}_{{\rm{f}}}^{* }\approx 1$ for Q_⊥-shocks in high-β plasmas, so the fraction of reflected ions is expected be relatively high in all the shock models under consideration. As can be seen in the phase-space distribution of protons in Figure 3(a)–(c), the back-streaming ions turn around mostly within about one ion gyroradius in the shock ramp (x − x_sh ≲ 60c/w_pe). Note that, with m_i/m_e = 100 in the M3.0 model, the shock ramp corresponds to the region of x − x_s < 60c/ω_pe, while the foot extends to x − x_s ≈ r_L,i ≈ 200c/ω_pe (e.g., Balogh & Truemann 2013). In the M2.0 model, r_L,i is smaller and so the characteristic widths of the ramp and foot are accordingly smaller as well. Figure 3(d) shows that the ion reflection fraction, ${\alpha }_{\mathrm{ref},i}={n}_{\mathrm{ref},i}/{n}_{i}$, increases with increasing M_s, and such a trend is almost independent of the mass ratio m_i/m_e. Here, ${n}_{\mathrm{ref},i}$ was calculated as the number density of ions with v_x > 0 in the shock rest frame in the ramp region. Because ${\alpha }_{\mathrm{ref},i}$ increases abruptly at M_s ≈ 2.2–2.3, we may regard ${M}_{{\rm{f}}}^{* }\approx 2.3$ as an effective value for the first critical Mach number, above which high-β Q_⊥-shocks reflect a sufficient amount of incoming ions and become supercritical. From Figure 2, we can see that the ensuing oscillations in shock structures appear noticeable in earnest only for M_f ≳ 2.3. Our estimation of ${M}_{{\rm{f}}}^{* }$ is higher than the prediction of Edmiston & Kennel (1984). This might be partly because, in high-β plasmas, kinetic processes due to fast thermal motions could suppress some of microinstabilities driven by the relative drift between backstreaming and incoming ions, as mentioned before.

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Reflected electrons are energized via SDA at the shock ramp. The consequences of this can be observed in the phase-space distribution of electrons shown in Figure 4: see panels (a)–(c) for the M2.0 model and panels (e)–(g) for the M3.0 model. Because B₀ is in the x−y plane, electrons initially gain the z-momentum, p_ez, through the drift along the motional electric field, E₀ = $-{{\boldsymbol{v}}}_{0}/c$ × B₀, and then the gain is distributed to p_ex and p_ey during gyration motions. In addition, reflected electrons streaming along the background magnetic field with small pitch angles in the upstream region have larger positive p_y than p_x. Panels (d) and (h) of Figure 4 also show the distributions of electron density n_e (black curve) and B_y (red curve) around the shock transition.

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If electrons are accelerated via the full Fermi-I process (i.e., DSA) and in the test-particle regime, the momentum distribution follows the so-called DSA power-law:

$$\begin{eqnarray}&&f(p)\approx {f}_{N}{\left(\displaystyle \frac{p}{{p}_{\mathrm{inj}}}\right)}^{-q}\exp \left[-{\left(\displaystyle \frac{p}{{p}_{\max }}\right)}^{2}\right],\end{eqnarray} \tag{ 8 }$$

where f_N is the normalization factor and $q({M}_{{\rm{s}}})=3r/(r-1)$ is the slope (Drury 1983; Kang & Ryu 2010). Here, p_max is the maximum momentum of accelerated electrons that increases with the shock age before any energy losses set in. The injection momentum, p_inj, is the minimum momentum with which electrons can diffuse across the shock and be injected into the full Fermi-I process as described in the introduction. It marks the approximate boundary between the thermal and nonthermal momentum distributions. The momentum spectrum in Equation (8) can be transformed to the energy spectrum in terms of the Lorentz factor as

$$\begin{eqnarray}&&4\pi {p}^{2}f(p)\displaystyle \frac{{dp}}{{dE}}\propto \displaystyle \frac{{dN}}{d\gamma }\propto {\left(\gamma -1\right)}^{-s},\end{eqnarray} \tag{ 9 }$$

where the slope is s(M_s) = q(M_s) − 2. For instance, s = 2.5 for M_s = 3.0, while s = 2.93 for M_s = 2.0.

The injection momentum, which can be estimated as ${p}_{\mathrm{inj}}\sim 3{p}_{\mathrm{th},i}$ (e.g., Kang et al. 2002; Caprioli et al. 2015, Paper I), marks the approximate boundary between the suprathermal and nonthermal momentum distributions and it is well beyond the highest momentum that electrons can achieve in our PIC simulations. In the M3.0 model, for example, p_inj corresponds to γ_inj ≈ 10, while electrons of the highest momenta reach only γ ≲ 2 (see Figure 5). In other words, our simulations could follow only the preacceleration of suprathermal electrons, which are not energetic enough to diffuse across the shock. Thus, the DSA slope, s(M_s), is not necessarily reproduced in the energy spectra of electrons. However, the development of power-law tails with s(M_s) may indicate that the preaccelerated electrons have undergone a Fermi-like process, as proposed by GSN14a and GSN14b. Although not shown here, the trajectories of suprathermal electrons in our simulations demonstrated that, in supercritical shocks, the electrons gain energies via multiple cycles of SDA through scattering between the shock ramp and upstream waves, just like Figure 8 of GSN14a.

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The upper panels of Figure 5 compare the electron energy spectra, (γ − 1)dN/dγ, taken from the upstream region of (0–1)r_L,i, ahead of the shock, at tΩ_ci = 10 (blue lines) and 30 (red lines), in the models with different M_s. In the case of the M3.0 model, the simulation is performed longer and the spectrum at tΩ_ci = 60 is also shown with the green line (which almost overlaps with the red line). As described in Section 3.1, reflected electrons gain energy initially via SDA, and may continue to be accelerated via a Fermi-like process and multiple cycles of SDA, if oblique waves are excited by the EFI. Two points are evident here. (1) In the M2.0 model, the blue and red lines almost coincide, indicating almost no change of the spectrum from tΩ_ci = 10 to 30. The spectrum is similar to that of the electrons energized by a single cycle of SDA, which was illustrated in Figure 7 of GSN14a. Thus, the Fermi-like process followed by the EFI may not operate efficiently in this model. (2) The M3.0 model, on the other hand, exhibits a further energization from tΩ_ci = 10 to 30, demonstrating the presence of a Fermi-like process. However, there is no difference in the spectra of tΩ_ci = 30 and 60. As a matter of fact, the energy spectrum of suprathermal electrons seems to saturate beyond tΩ_ci ≈ 20 (not shown in the figure). This should be due to the saturation of the EFI and the lack of further developments of longer wavelength waves (see the next subsection for further discussions).

The middle and lower panels of Figure 5 show the electron energy spectra in models with different parameters. The models with m_i/m_e = 400 were followed only up to t_end Ω_ci = 10 (blue lines), because longer computing time is required for larger m_i/m_e. Comparison of the two sets of models with different values of m_i/m_e confirms that the EFI is  almost independent of m_i/m_e for sufficiently large mass ratios, as previously shown by Gary & Nishimura (2003) and GSN14b, and so is the electron acceleration. Figure 5(d) for the M2.3-θ73 model indicates that SDA—and hence the EFI—is more efficient at higher obliquity angles, which is consistent with Figure 1(c). Panels (e) and (f) of Figure 5, displaying the models with β = 50, demonstrate that the EFI is more efficient at higher β. All the models with M_s ≳ 2.3 show marginal power law–like tails beyond the spectra energized by a single cycle of SDA.

With the M2.3-r2 and M2.3-r0.5 models, we examined how the electron energy spectrum depends on the grid resolution, although the comparison plots are not shown. Our simulations with different Δx produced essentially the same spectra, especially for the suprathermal part.

In Paper I, we calculated the injection fraction, ξ(M_s, θ_Bn, β), of nonthermal protons with p ≥ p_inj for Q_∥ shocks, as a measure of the DSA injection efficiency. Because the simulations in this paper can follow only the preacceleration stage of electrons via an upstream Fermi-like process, we define and estimate the "fraction of suprathermal electrons" as follows:

$$\begin{eqnarray}&&\zeta \,\equiv \,\displaystyle \frac{1}{{n}_{2}}{\int }_{{p}_{\mathrm{spt}}}^{{p}_{\max }}4\pi \langle f(p)\rangle {p}^{2}{dp},\end{eqnarray} \tag{ 10 }$$

where $\langle f(p)\rangle$ is the electron distribution function, averaged over the upstream region of (0–1) r_L,i, ahead of the shock. For the "suprathermal momentum," above which the electron spectrum changes from a Maxwellian to a power law–like distribution, we use p_spt ≈ 3.3p_th,e. Note that ${p}_{\mathrm{spt}}\approx {p}_{\mathrm{inj}}{({m}_{i}/{m}_{e})}^{-1/2}$. For the M3.0 model, for instance, p_spt corresponds to γ ≈ 1.25. Different choices of p_spt result in different values of ζ, of course, but the dependence on parameters such as M_s and θ_Bn does not change much.

In Figure 6, the circles connected with solid lines show the suprathermal fraction, ζ(M_s), for the fiducial models with θ_Bn = 63° at tΩ_ci = 10–30. This fraction is expected to increase with increasing M_s, because the EFI parameter, I, is larger for higher M_s (see Figure 1(d)). Moreover, it increases with time until tΩ_ci ≈ 20, due to a Fermi-like process, as shown in Figure 5—except for the M2.0 model where the increase in time is insignificant. However, ζ seems to stop growing for tΩ_ci ≳ 20, indicating the saturation of electron preacceleration. This is related to the reduction of temperature anisotropy via electron scattering and the ensuing decay of EFI-induced waves, which will be discussed more in the next section.

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The red solid line in Figure 6 is represented roughly by $\zeta \propto {M}_{{\rm{s}}}^{4}$ in the range of 2.3 ≲ M_s ≤ 3, but it drops rather abruptly below 2.3, deviating from the power-law behavior. We note that the Mach number dependence of ζ is steeper than that of the ion injection fraction for Q_∥-shocks, which is roughly $\xi \propto {M}_{{\rm{s}}}^{1.5}$, as shown in Paper I. This implies that the kinetic processes involved in electron preacceleration might be more sensitive to M_s (see Section 3.2).

Figure 6 also shows ζ for models with θ_Bn = 53° (triangles) and θ_Bn = 73° (squares). For shocks with larger θ_Bn, the reflection of electrons and the average SDA energy gain are larger, resulting in larger I, as shown in Figure 1(c) and (d). Hence, ζ should be larger at a higher obliquity angle. However, for θ_Bn > θ_limit ≈ 73–78°, ζ should begin to decrease, as mentioned in Section 3.2.1.

Based on the above results, we propose that the preacceleration of electrons is effective only in Q_⊥-shocks with M_s ≳ 2.3 in the hot ICM, i.e., ${M}_{\mathrm{ef}}^{* }\approx 2.3$. We should point out that this is close to the first critical Mach number for ion reflection, ${M}_{{\rm{f}}}^{* }\approx 2.3$, estimated from the Mach number dependence of the fraction of reflected ions, ${\alpha }_{\mathrm{ref},i}$, shown in Figure 3(d). As shown in Figure 2, overshoot/undershoot oscillations develop in the shock transition, owing to a sufficient amount of reflected ions, in shocks with M_s ≳ 2.3; with larger magnetic field compression due to the oscillations, more electrons are reflected and energized via SDA (see Section 3.2.1). Hence, we expect that the electron reflection is directly linked with the ion reflection, so ${M}_{\mathrm{ef}}^{* }$ would be related with ${M}_{{\rm{f}}}^{* }$. Note that the critical Mach number, ${M}_{{\rm{f}}}^{* }\approx 2.3$, is also similar to the first critical Mach number for ion reflection and injection into DSA in Q_∥-shocks in high-β plasmas, ${M}_{{\rm{s}}}^{* }\approx 2.25$ (Paper I).

The nature and origin of upstream waves in collisionless shocks have long been investigated through both analytical and simulation studies, with the help of in situ observations of Earth's bow shock. In Q_∥-shocks with low Mach numbers, magnetosonic waves such as phase-standing whistlers and long-wavelength whistlers are known to be excited by backstreaming ions via an ion/ion-beam instability (e.g., Krauss Varban & Omidi 1991). This is especially so in supercritical Q_∥-shocks, where the foreshock region is highly turbulent with large-amplitude waves and the shock transition can undergo quasi-periodic reformation due to the nonlinear interaction of accumulated waves and the shock front (see Paper I and Figure 2(d)). In Q_⊥-shocks, a sufficient amount of incoming protons can be reflected at the shock, which in turn may excite fast magnetosonic waves. As discussed in Section 3.1, two whistler critical Mach numbers, ${M}_{{\rm{w}}}^{* }$ and ${M}_{\mathrm{nw}}^{* }$, are related with the upstream emission of whistler waves and the nonlinear breaking of whistler waves in the shock foot. In some of our models, ${M}_{{\rm{w}}}^{* }\lt {M}_{{\rm{s}}}\lt {M}_{\mathrm{nw}}^{* }$, and hence whistler waves are confined within the shock foot and shock reformation does not occur.

In supercritical shocks with M_s ≳ 2.3, we expect to see the following three kinds of waves: (1) nearly phase-standing whistler waves with kc/ω_pi ∼ 1 (kc/ω_pe ∼ 0.1) excited by reflected ions (e.g., Hellinger et al. 2007; Scholer & Burgess 2007), where ${w}_{\mathrm{pi}}=\sqrt{4\pi {e}^{2}n/{m}_{i}}$ is the ion plasma frequency (w_pi = 0.1w_pe for m_e/m_i = 100); (2) phase-standing oblique waves with kc/ω_pe ∼ 0.4 and larger θ_Bk (the angle between the wave vector k and B₀) excited by the EFI; and (3) propagating waves with kc/ω_pe ∼ 0.3 and smaller θ_Bk, also excited by the EFI (e.g., Hellinger et al. 2014).

Here, we focus on the waves excited by the EFI described in Section 3.2.2. Previous studies on the EFI and the EFI-induced waves showed the following characteristics (Gary & Nishimura 2003; Camporeale & Burgess 2008; Hellinger et al. 2014; Lazar et al. 2014, GSN14b). (1) The magnetic field fluctuations in the EFI-induced waves are predominantly along the direction perpendicular to both k and B₀, i.e., $| \delta {B}_{z}|$ is larger than $| \delta {B}_{x}|$ and $| \delta {B}_{y}|$ in our geometry. (2) Phase-standing oblique waves with almost zero oscillation frequencies (ω_r ≈ 0) have higher growth rates than propagating waves (${\omega }_{r}\,\ne \,0$). (3) Nonpropagating modes decay to propagating modes with longer wavelengths and smaller θ_Bk. (4) The EFI-induced waves scatter electrons, resulting in a reduction of the electrons' temperature anisotropy, which in turn leads to the damping of the waves.

Figure 7 shows the distribution of magnetic field fluctuations, $\delta {\boldsymbol{B}}$, in the upstream region for the M2.0 and M3.0 models. The epoch shown, tΩ_ci ≈ 7 (w_pe t ≈ 2.63 × 10⁴) is early—yet in the M3.0 model, waves are well-developed (see also Figure 9), while the energization of electrons is still undergoing (see Figure 5). For the supercritical shock of the M3.0 model, we interpret that there are ion-induced whistlers in the shock ramp region of 0 ≲ x − x_s ≲ 60c/ω_pe, while EFI-induced oblique waves are present over the whole region shown. As shown in GSN14b, the EFI-excited waves are oblique, with θ_Bk ∼ 60° and $| \delta {B}_{z}| \gt | \delta {B}_{x}|$ and $| \delta {B}_{y}|$. The increase in δB_y toward x − x_sh = 0 is due to the compression in the shock ramp. In the subcritical shock of the M2.0 model, on the other hand, the fractions of reflected ions and electrons are not sufficient for either the emission of whistler waves or the excitation of EFI-induced waves, so no substantial waves are present in the shock foot. This is consistent with the instability condition shown in Figure 1(d).

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Figure 8 compares δB_z in six different models at tΩ_ci ≈ 10. The wave amplitude increases with increasing M_s, and the EFI seems only marginal in the M2.3 models. This result confirms our proposal for the "EFI critical Mach number" ${M}_{\mathrm{ef}}^{* }\approx 2.3$, presented in Section 4.2.  Moreover, this figure corroborates our findings that the EFI is more efficient at larger θ_Bn and higher β. From δB_z of the M2.3 and M3.0 models in Figures 7 and 8, the dominant waves in the shock foot seem to have λ ∼ 15–20c/ω_pe, so they are consistent with the EFI-induced waves (GSN14b).

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Figure 9 shows the time evolution of the average of the magnetic field fluctuations, $\left\langle \delta {B}_{z}^{2}/{B}_{0}^{2}\right\rangle$, and the magnetic energy power, ${P}_{{Bz}}(k)\propto | \delta {B}_{z}(k){| }^{2}k$, of upstream waves for the M3.0 model. According to the linear analysis by Camporeale & Burgess (2008), the growth rate of the EFI peaks at k_maxc/ω_pe ∼ 0.4 for β_e∥ = 10 and T_e⊥/T_e∥ = 0.7. Thus, we interpret that the powers in the range of kc/ω_pe ∼ 0.2–0.3 are due to the oblique waves induced by the EFI, while those of kc/ω_pe ≲ 0.15 are contributed by the phase-standing whistler waves induced by reflected ions.

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Moreover, through periodic box simulations of the EFI, Camporeale & Burgess (2008) and Hellinger et al. (2014) demonstrated that initially nonpropagating oblique modes grow and then saturate, followed by the transfer of wave energy into propagating modes with longer wavelengths and smaller θ_Bk. Figure 8 of Camporeale & Burgess (2008) and Figure 4 of Hellinger et al. (2014) show that a cycle of the EFI-induced wave growth and decay occurs on timescales of roughly several hundreds of tΩ_ce. We suggest that the oscillatory behaviors of the excited waves with the timescales of ${{tw}}_{\mathrm{pe}}\sim 2\times {10}^{4}\mbox{--}4\times {10}^{4}$ (tΩ_ce ∼ 500–1000) shown in Figure 9 would be related to those characteristics of the EFI. Figure 9(c) illustrates such a cycle during the period of t w_pe ≈ (1.8–2.1) × 10⁵: excitation with k_maxc/w_pe ≈ 0.3 → inverse cascade with k_maxc/w_pe ≈ 0.2 → damping of waves.

Our results indicate that the EFI-induced waves do not further develop into longer wavelength modes with λ ≫ λ_max, where λ_max ≈ 15–20c/ω_pe is the wavelength of the maximum linear growth. Note that λ_max is close to the gyroradius of electrons with γ ≲ 2. Thus, the acceleration of electrons via resonant scattering by the EFI-induced waves is saturated. As a consequence, the energization of electrons stops at the suprathermal stage (γ < 2) and does not proceed all the way to the DSA injection momentum (γ_inj ≈ 10). We interpret that this result should be due to the intrinsic properties of the EFI, rather than the limitations or artifacts of our simulations,  as shown by the studies of Camporeale & Burgess (2008) and Hellinger et al. (2014). Hence, we conclude that the preacceleration via the EFI alone may not explain the injection of electrons to DSA in weak ICM shocks. However, the conclusion needs to be further verified through a more detailed study of the EFI and EFI-induced waves for high-β ICM plasmas, including kinetic linear analyses and numerical simulations, which we leave for a future work.