Electron Firehose Instabilities in High-$\beta$ Intracluster Shocks

In the current $\Lambda$CDM cosmology, galaxy clusters emerge through hierarchical clustering, and the ensuing supersonic flow motions generate shock waves in the intracluster medium (ICM) (e.g., Minati et al., 2000; Ryu et al., 2003; Pfrommer et al., 2006; Skillman et al., 2008; Vazza et al., 2009; Hong et al., 2014; Schaal & Volker, 2015). Among the ICM shocks, the merger shocks, induced during mergers of sub-clusters, are the most energetic; they form in almost virial equilibrium and hence are weak with low sonic Mach numbers of $M_{\rm s}\lesssim 4$ (e.g., Ha et al., 2018). As in most astrophysical shocks, cosmic ray (CR) protons and electrons are expected to be accelerated via diffusive shock acceleration (DSA) in these ICM shocks (e.g., Bell, 1978; Blandford & Ostriker, 1978; Drury, 1983; Brunetti & Jones, 2014). From observations of the so-called radio relics (e.g., van Weeren et al., 2010, 2012), in particular, the electron acceleration is inferred to operate in low Mach number, quasi-perpendicular ($Q_{\perp}$, hereafter) shocks with $\theta_{\rm Bn}\gtrsim 45^{\circ}$ in the hot ICM (see van Weeren et al., 2019, for a review). Here, $\theta_{\rm Bn}$ is the obliquity angle between the background magnetic field and the shock normal.

The preacceleration of electrons is one of the long-standing unsolved problems in the theory of DSA (e.g., Marcowith et al., 2016). Thermal electrons need to be energized to suprathermal energies ($p_{e}\gtrsim{\rm a~{}few}\times p_{\rm th,p}$) in order to be injected to the DSA process, because the full DSA requires the diffusion of CR electrons both upstream and downstream across the shock and the width of the shock transition region is comparable to the ion gyroradius. Here, $p_{\rm th,p}=(2m_{p}k_{B}T_{2})^{1/2}$ is the proton thermal momentum in the postshock gas of temperature $T_{2}$, $m_{p}$ is the proton mass, and $k_{B}$ is the Boltzmann constant. The electron injection, which is known to be effective mainly at $Q_{\perp}$-shocks (e.g., Gosling et al., 1980; Burgess, 2007), involves the following key elements: (1) the reflection of incoming ions and electrons at the shock ramp due to magnetic mirror forces, leading to backstreaming, (2) the energy gain from the motional electric field in the upstream region through shock drift acceleration (SDA) or shock surfing acceleration (SSA), and/or through interactions with waves, and (3) the trapping of electrons near the shock due to the scattering by the upstream waves, excited by backstreaming ions and electrons, which allows multiple cycles of SDA or SSA (Amano & Hoshino, 2009; Riquelme & Spitkovsky, 2011; Matsukiyo et al., 2011; Matsumoto et al., 2012; Guo et al., 2014a, b; Kang et al., 2019). The injection problem can be followed from first principles only through particle-in-cell (PIC) simulations, which fully treat kinetic microinstabilities and wave-particle interactions on both ion and electron scales around the shock transition.

Self-generated upstream waves play an important role in the electron preacceleration, because in the absence of scattering by the upstream waves, the energization of electrons would be terminated after one SDA cycle. Plethora of plasma instabilities can be destabilized in the shock foot, depending on the shock parameters, such as the plasma beta, $\beta=P_{\rm gas}/P_{\rm B}$ (the ratio of the gas to magnetic pressures), the sonic Mach number $M_{\rm s}$, the Alfvén Mach numbers, $M_{\rm A}$ ($M_{\rm A}=\sqrt{\beta\Gamma/2}M_{\rm s}$ where $\Gamma=5/3$ is the gas adiabatic index), the obliquity angle, $\theta_{\rm Bn}$, and the adopted ion-to-electron mass ration, $m_{p}/m_{e}$¹¹1We use the term ‘ion’ to represent the positively charged particle with a range of $m_{p}/m_{e}=100-1836$. But we use the subscript $p$ to denote the ion population, since the subscript $i$ is used for the coordinate component. (e.g. Matsukiyo & Scholer, 2006; Balogh & Truemann, 2013). Early PIC simulation studies centered on high Mach number shocks in $\beta\sim 1$ plasmas, characterizing the Earth bow shock in the solar wind and supernova blast waves in the interstellar medium. For instance, Amano & Hoshino (2009) and Matsumoto et al. (2012) found that at high $M_{\rm A}$ (specifically, $M_{\rm A}>(m_{p}/m_{e})^{2/3}$) $Q_{\perp}$-shocks in $\beta\approx 1$ plasmas, the drift between reflected ions and incoming electrons triggers the Buneman instability, which excites large-amplitude electrostatic waves. Then, backstreaming electrons are scattered by these waves and gain energies via multiple cycles of SSA at the leading edge of the shock foot. On the other hand, Riquelme & Spitkovsky (2011) argued that at low $M_{\rm A}$ (with $M_{\rm A}\lesssim(m_{p}/m_{e})^{1/2}$) $Q_{\perp}$-shocks in $\beta\approx 1$ plasmas, modified two-stream instabilities can excite oblique whistler waves and electrons gain energies via wave-particle interactions with those whistlers in the shock foot. Matsukiyo et al. (2011) also considered $M_{\rm A}\approx 5-8$, $Q_{\perp}$-shocks in $\beta\approx 3$ plasmas, and showed that reflected electrons are energized through SDA.

The ICM, which is composed of the hot gas of a few to several keV and the magnetic fields of the order of $\mu$G, on the other hand, contains plasmas of $\beta\sim 100$ (e.g., Ryu et al., 2008; Brunetti & Jones, 2014; Porter et al., 2015). Hence, although ICM shocks are weak with low sonic Mach numbers of $M_{\rm s}\lesssim 4$, they have relatively high Alfvén Mach numbers of up to $M_{\rm A}\lesssim 40$. To understand the electron acceleration at ICM shocks, Guo et al. (2014a, b) performed two-dimensional (2D) PIC simulations of $M_{\rm s}=3$, $Q_{\perp}$-shocks in $\beta=6-200$ plasmas with $k_{B}T=86$ keV. They argued that the temperature anisotropy ($T_{e\parallel}>T_{e\perp}$) due to reflected electrons, backstreaming along the background magnetic fields with small pitch angles, derives the electron firehose instability (EFI, hereafter), which excites mainly nonpropagating oblique waves in the shock foot. Here, $T_{e\parallel}$ and $T_{e\perp}$ are the electron temperatures, parallel and perpendicular to the background magnetic field, respectively. The SDA-reflected electrons²²2The SDA-reflected electrons mean those that are energized by SDA in the course of reflection. are scattered back and forth between the magnetic mirror at the shock ramp and the EFI-driven upstream waves, but they are still suprathermal and do not have sufficient energies to diffuse downstream across the shock transition. On the other hand, Trotta & Burgess (2019) and Kobzar et al. (2019) have recently shown through 2D and 3D plasma simulations of supercritcal $Q_{\perp}$-shocks that shock surface ripplings generate multi-scale perturbations that can facilitate the electron acceleration beyond the injection momentum.

Kang et al. (2019, KRH19, hereafter) revisited this problem with wider ranges of shock parameters that are more relevant to ICM shocks, $M_{\rm s}=2.0-3.0$, $\beta=50-100$, and $k_{B}T=8.6$ keV. They showed that the electron preacceleration through the combination of reflection, SDA, and EFI may operate only in supercritical, $Q_{\perp}$-shocks with $M_{\rm s}\gtrsim 2.3$. In addition, they argued that the EFI alone may not energize the electrons all the way to the injection momentum, $p_{\rm inj}\sim 130p_{\rm th,e}$ (where $p_{\rm th,e}=(2m_{e}k_{B}T_{2})^{1/2}$), unless there are pre-existing turbulent waves with wavelengths longer than those of the EFI-driven waves ($\lambda\gtrsim 20c/\omega_{pe}$, where $c$ is the speed of light and $\omega_{pe}$ is the electron plasma frequency). Analyzing self-excited waves in the shock foot, they found that (1) nonpropagating oblique waves with $\lambda\sim 15-20c/\omega_{pe}$ are dominantly excited, (2) the waves decay to those with longer wavelengths and smaller propagation angle, $\theta=\cos^{-1}(\mathbf{k}\cdot\mathbf{B}_{0}/kB_{0})$, the angle between the wavevector, $\mathbf{k}$, and the background magnetic field $\mathbf{B}_{0}$, and (3) the scattering of electrons by those waves reduces the temperature anisotropy. These findings are consistent with previous works on EFI using linear analyses and PIC simulations (e.g., Gary & Nishimura, 2003; Camporeale & Burgess, 2008; Hellinger et al., 2014).

The EFI in homogeneous, magnetized, collisionless plasmas has been extensively studied in the space-physics community as a key mechanism that constrains the electron anisotropy in the solar wind (see Gary, 1993, for a review). It comes in the following two varieties: (1) the electron temperature-anisotropy firehose instability (ETAFI, hereafter), driven by a temperature anisotropy, $T_{e\parallel}>T_{e\perp}$ (e.g. Gary & Nishimura, 2003; Camporeale & Burgess, 2008; Hellinger et al., 2014), and (2) the electron beam firehose instability (EBFI, hereafter), also known as the electron heat flux instability, induced by a drifting beam of electrons (e.g. Gary, 1985; Saeed et al., 2017; Shaaban et al., 2018). In the EBFI, the bulk kinetic energy of electrons is the free energy that drives the instability. In the linear analyses of these instabilities, typically ions are represented by an isotropic Maxwellian velocity distribution function (VDF, hereafter) with $T_{\rm p}$, while electrons have different distributions, that is, either a single anisotropic bi-Maxwellian VDF with $T_{e\parallel}>T_{e\perp}$ for the ETAFI, or two isotropic Maxwellian VDFs (i.e., the core with $T_{\rm c}$ and the beam with $T_{\rm b}$) with a relative drift speed, $u_{\rm rel}$, for the EBFI.

The main findings of the previous studies of the ETAFI can be summarized as follows (see, e.g., Gary & Nishimura, 2003). (1) The threshold condition of the instability decreases with increasing $\beta_{\rm e}$, approximately as $(T_{e\parallel}-T_{e\perp})/T_{e\parallel}\gtrsim 1.3~{}\beta_{\rm e}^{-1}$. (2) The instability induces two branches, i.e., the parallel ($\theta\approx 0^{\circ}$), nonresonant, propagating ($\omega_{r}\neq 0$) mode and the oblique ($\theta\gg 0^{\circ}$), resonant, nonpropagating ($\omega_{r}=0$) mode. The propagating mode is left-hand (LH) polarized. The latter, nonpropagating mode has the growth rate higher than the former. (3) The perturbed magnetic field of the nonpropagating mode is dominantly along the direction perpendicular to both $\mathbf{k}$ and $\mathbf{B_{0}}$. (4) The oblique nonpropagating modes decay to the propagating modes of smaller wavenumbers and smaller angles. (5) The ETAFI-induced waves scatter electrons, resulting in the reduction of the electron temperature anisotropy and the damping of the waves.

For the case of the electron heat flux instability (i.e., the EBFI), the parallel-propagating ($\theta=0^{\circ}$) mode was analyzed before, and is known to have two branches, that is, the right-hand (RH) polarized whistler mode and the LH polarized firehose mode (Gary, 1993). The firehose mode becomes dominant at sufficiently large drift speeds (Gary, 1985). Although the oblique ($\theta\neq 0^{\circ}$) mode has not been sufficiently examined in the literature so far, it is natural to expect that the oblique mode would have the growth rate larger than the parallel mode, similarly as in the case of the ETAFI (Saeed et al., 2017). Recently, Shaaban et al. (2018) studied the electron heat flux instability driven by a drifting beam of anisotropic bi-Maxwellian electrons, but again only for the parallel propagation.

As mentioned above, Guo et al. (2014b) and KRH19 argued that the upstream waves in the shock foot in their PIC simulations have the characteristics consistent with the nonprogating oblique waves excited by the ETAFI. Considering that backstreaming electrons would behave like a drifting beam, however, it would have been more appropriate to interpret the operating instability as the EBFI. So we here consider and compare the two instabilities, in order to understand the nature of the upstream waves in $Q_{\perp}$-shocks in high-$\beta$ plasmas. Another reason why we study this problem is that the ETAFI and EBFI in high-$\beta$ plasmas have not been examined before. In particular, we study the instabilities at both parallel and oblique propagations through the kinetic Vlasov linear theory and 2D PIC simulations, focusing on the kinetic properties of the EFI in high-$\beta$ ($\beta_{\rm p}\approx 50$ and $\beta_{\rm e}\approx 50$) plasmas relevant for the ICM.

The paper is organized as follows. Section 2 describes the linear analysis of the ETAFI and EBFI. In Section 3, we present the nonlinear evolution of the EBFI in 2D PIC simulations in a periodic box. A brief summary is given in Section 4.

We consider the ETAFI and EBFI in a homogeneous, collisionless, magnetized plasma, which is specified by the density and temperature of ions and electrons, $n_{p}$, $n_{e}$, $T_{p}$, $T_{e}$, and the background magnetic field of $\mathbf{B}_{0}$. For the case of the ETAFI, the anisotropic bi-Maxwellian distribution of electrons is described by $T_{e\parallel}$ and $T_{e\perp}$, and the temperature anisotropy parameter is given as $\mathcal{A}=T_{e\parallel}/T_{e\perp}$. The ion population is described with a single temperature. For the case of the EBFI, the core and beam populations of electrons with the drift speeds of $u_{c}$ and $u_{b}$ along the direction of the background magnetic field are assumed³³3The subscripts $c$ and $b$ stand for the core and beam populations.. The ion population is on average at rest with zero drift speed. The adopted ion-to-electron mass ratio includes the realistic ratio, $m_{p}/m_{e}=1836$, of the proton to electron mass and a reduced one, $m_{p}/m_{e}=100$. The reduced mass ratio is considered for comparison with the PIC simulations in the next section where $m_{p}/m_{e}=100$ and 400 are adopted.

The VDF of a drifting bi-Maxwellian population can be written in the general form,

The subscript $a$ can denote the population of core electrons ($c$), beam electrons ($b$), or ions ($p$). Here, $n_{a}$ is the number density of the particle species $a$, and $n_{0}$ is the number density of electrons and ions, which satisfy $n_{0}=n_{c}+n_{b}=n_{p}$, the charge neutarlity condition; $u_{a}$ is the drift speed directed along the background magnetic field, and satisfies $n_{c}u_{c}+n_{b}u_{b}-n_{p}u_{p}=0$, the zero net current condition. The thermal velocities are $\alpha_{a\parallel}=\sqrt{2k_{\rm B}T_{a\parallel}/m_{a}}$ and $\alpha_{a\perp}=\sqrt{2k_{B}T_{a\perp}/m_{a}}$, respectively. Throughout the paper, the plasma beta, $\beta_{a}=8\pi n_{0}k_{B}T_{a}/B_{0}^{2}$, the plasma frequency, $\omega_{pa}^{2}=4\pi n_{0}e^{2}/m_{a}$, and the gyro-frequency, $\Omega_{a}=eB_{0}/m_{a}c$, for electrons and ions are used. The Alfv$\rm{\acute{e}}$n speed, given as $v_{A}=(B_{0}^{2}/4\pi n_{0}m_{p})^{1/2}$, is also used. Note that for the ion (proton) population, $T_{p\parallel}=T_{p\perp}=T_{p}$ in the ETAFI analysis, while $u_{p}=0$ in the EBFI analysis in the following subsections.

The linear dispersion relation of general electromagnetic (EM) modes for the ETAFI and EBFI can be derived from the normal mode analysis with the linearized Vlasov-Maxwell system of equations for plasmas of multi-species. The derivation can be found in standard textbooks on plasma physics (e.g., Stix, 1992; Brambilla, 1998). The dispersion relation is given as

with the dielectric tensor, $\epsilon_{ij}$, where $k_{i}$ and $k_{j}$ are the components of the wavevector $\mathbf{k}$. Then, the complex frequency, $\omega=\omega_{r}+i\gamma$,⁴⁴4The quantity $i$ is the imaginary unit, not the coordinate component. can be calculated as a function of the wave number, $k$, and the propagation angle, $\theta$. The dielectric tensor for the general VDF is given in the appendix. Setting that $\parallel$ is the $z$-direction and $\perp$ is the $x$-direction without loss of generality, that is, $\mathbf{B}_{0}=B_{0}{\hat{z}}$ and $\mathbf{k}$ are in the $z-x$ plane, the components of $\epsilon_{ij}$ for the VDF in Equation (1) is written as

where

Here, $I_{n}(\lambda)$ denotes the modified Bessel function of the first kind and $Z(\zeta)$ is the plasma dispersion function (see the appendix). The prime in $\Lambda_{n}^{\prime}(\lambda)$ and $Z^{\prime}(\zeta)$ indicates the derivative with respect to the argument.

Recent studies for the electron preacceleration in weak $Q_{\perp}$-shocks in the high-$\beta$ ICM plasmas suggested that the temperature anisotropy of $T_{\parallel}>T_{\perp}$ due to SDA-reflected electrons generates oblique waves in the shock foot via the EFI, i.e., the ETAFI (Guo et al., 2014a, b, KRH19). The electrons can be effectively trapped between the shock ramp and these upstream waves, and hence continue to gain energy through multiple cycles of SDA. Those studies compared the properties of the excited upstream waves in shock simulations with the results of the linear analysis and PIC simulations of the ETAFI driven by anisotropic bi-Maxwellian electrons (e.g. Gary & Nishimura, 2003; Camporeale & Burgess, 2008). In the $Q_{\perp}$ ICM shocks, however, the instability due to SDA-reflected electrons is expected to be more like the electron heat flux instability driven by a drifting beam, i.e., the EBFI, since the electrons stream along the background magnetic field with small pitch angles, and hence they would behave similar to the electrons of a drifting beam rather than bi-Maxwellian electrons.

To describe the nature of the upstream waves excited in the shock foot, we here studied the EFI in two different forms: (1) the ETAFI induced by the electrons of a bi-Maxwellian VDF with the temperature anisotropy, $\mathcal{A}=T_{e\parallel}/T_{e\perp}>1$, and (2) the EBFI induced by the electrons of a drifting beam with an isotropic Maxwellian VDF and the relative drift speed, $u_{\rm rel}$. We carried out the kinetic linear analysis of both types of the EFI in Section 2 and the 2D PIC simulations of the EBFI in Section 3.

The main findings can be summarized as follows:

1. In the EBFI, an effective temperature anisotropy, $\mathcal{A}_{\rm eff}$, can be defined (see Section 2.2); $\mathcal{A}_{\rm eff}$ is larger for larger $u_{\rm rel}$. In the linear analysis, the characteristics of the EBFI are similar to those of the ETAFI, if $\mathcal{A}_{\rm eff}$ is similar to $\mathcal{A}$.

2. For both the EFI instabilities, the oblique modes with a large propagation angle $\theta$ grow faster, having a higher growth rate $\gamma$, than the parallel modes with a small $\theta$. For the Lu0.3 model, the fiducial model of the EBFI, for example, $ck_{m}/\omega_{pe}=0.38$ and $\theta_{m}=81^{\circ}$.

3. The growth rates of both the instabilities increase with the increasing plasma beta $\beta$ (see, e.g., Gary & Nishimura, 2003). For higher $\beta$, the fastest-growing mode occurs at smaller $k_{m}$ and larger $\theta_{m}$.

4. Naturally, the growth rate increases with increasing $\mathcal{A}$ or $u_{\rm rel}$. For larger $\mathcal{A}$ or $u_{\rm rel}$, the fastest-growing mode occurs at larger $k_{m}$ and larger $\theta_{m}$.

5. In both the instabilities, the growth rate of fast-growing oblique modes is insensitive to $m_{p}/m_{e}$, for a sufficiently large mass ratio (i.e., $m_{p}/m_{e}\gtrsim 100$), as shown in previous studies (e.g., Gary & Nishimura, 2003).

6. The fast-growing oblique modes excited by the ETAFI are nonpropagating with $\omega_{r}=0$ (zero real frequency), while those excited by the EBFI have $\omega_{r}\neq 0$, but $\omega_{r}<\gamma$. Hence, the fast-growing modes of the EBFI is nearly phase-standing, even though they are propagating.

7. The PIC simulations of the EBFI presented in Section 3 show that the time evolution of the magnetic field fluctuations induced by this instability is consistent with the prediction of the linear analysis given in Section 2.2 and also with the results for the ETAFI reported by Camporeale & Burgess (2008) and Hellinger et al. (2014). The oblique, almost nonpropagating modes inverse-cascade in time to the modes with smaller wavenumbers, $k$, and smaller propagation angles, $\theta$. The scattering of electrons by these waves reduces the beam strength, which in turn leads to the damping of the waves. As a result, the modes of long wavelengths with $\lambda\gg\lambda_{m}\sim 15-20c/\omega_{pe}$ are not produced by the EBFI.

As argued by KRH19, without longer waves that can scatter higher energy electrons, the Fermi I-like preacceleration in the shock foot may not proceed to all the way to $p_{\rm inj}$ at weak $Q_{\perp}$-shocks in the ICM. However, Trotta & Burgess (2019) and Kobzar et al. (2019) have recently shown that, if the simulation volume is large enough to include ion-scale perturbations, the shock surface rippling caused by the Alfvén ion cyclotron instability can generate multiscale waves, leading the electron injection to DSA. Additional elements, such as pre-existing fossil CR electrons and/or pre-exiting turbulence on kinetic plasma scales in the ICM, may also facilitate the electron injection to DSA.