Stochastic Electron Acceleration by the Whistler Instability in a Growing Magnetic Field
Abstract
We use 2D particleincell (PIC) simulations to study the effect of the saturated whistler instability on the viscous heating and nonthermal acceleration of electrons in a shearing, collisionless plasma with a growing magnetic field, B. In this setup, an electron pressure anisotropy with naturally arises due to the adiabatic invariance of the electron magnetic moment ( and are the pressures parallel and perpendicular to B). If the anisotropy is large enough, the whistler instability arises, efficiently scattering the electrons and limiting (). In this context, taps into the plasma velocity shear, producing electron heating by the so called anisotropic viscosity. In our simulations, we permanently drive the growth of by externally imposing a plasma shear, allowing us to selfconsistently capture the longterm, saturated whistler instability evolution. We find that besides the viscous heating, the scattering by whistler modes can stochastically accelerate electrons to nonthermal energies. This acceleration is most prominent when initially , gradually decreasing its efficiency for larger values of (). If initially , the final electron energy distribution can be approximately described by a thermal component, plus a powerlaw tail with spectral index . In these cases, the nonthermal tail accounts for of the electrons, and for of their kinetic energy. We discuss the implications of our results for electron heating and acceleration in lowcollisionality astrophysical environments, such as lowluminosity accretion flows.
Subject headings:
plasmas – instabilities – particle acceleration – accretion disks1. Introduction
Nonthermal electrons are usually required to explain observations of various astrophysical systems where MHD turbulence is expected to be present. For example, nonthermal electrons are typically needed to explain the quiescent radio emission in some low luminosity AGNs (Liu & Wu, 2013) as well as in Sgr A*, the supermassive black hole at the center of the Milky Way (Özel et al., 2000; Yuan et al., 2003; Ball et al., 2016). Nonthermal electrons are also necessary to explain the NIR and Xray emission from Sgr A* flares (Yuan et al., 2004; Ponti et al., 2017). In the ICM, the presence of nonthermal electrons is commonly required to explain the extended radio synchrotron emission from some galaxy clusters (see Brunetti et al., 2014, for a review).
Several physical processes have been proposed to explain electron acceleration in these systems, including: diffusive shock acceleration (see Marcowith et al., 2016, for a review), magnetic reconnection (Sironi & Spitkovsky, 2014; Li et al., 2015), and various stochastic (second order Fermi) acceleration processes (Lynn et al., 2014; Zhdankin et al., 2017). In general, the disparity between MHD legthscales and the Larmor radii of particles makes it difficult for MHD turbulence to accelerate electrons via efficient pitchangle scattering, especially if their energy spectra are initially thermal. Previous works have proposed that this difficulty can be overcome by the resonant scattering provided by whistler waves, producing efficient stochastic electron acceleration in accreting systems (Dermer et al., 1996; Petrosian et al., 2004). In these works, however, the efficiency of the acceleration depends (amongst other parameters) on the spectrum of the whistler fluctuations, which is treated as an input of the models. In this work we use particleincell (PIC) simulations to show that pitchangle scattering by whistler modes can indeed produce efficient stochastic electron acceleration, in a context in which the whistler waves are consistently generated by magnetic field amplification due to a plasma velocity shear.
A key ingredient in our proposed mechanism is electron heating by the so called anisotropic viscosity. This viscous heating arises due to anisotropic pressure tapping into the energy contained in the plasma velocity shear. For an incompressible, homogeneous plasma with no heat flux, the electron internal energy density, , changes at a rate (Kulsrud, 1983; Snyder et al., 1997):
(1) 
where is the growth rate of the magnetic field (, with ) and is the difference between the electron pressure perpendicular and parallel to B. In a collisionless plasma, the difference between and is a consequence of the adiabatic invariance of the electron magnetic moment, , where is the electron velocity perpendicular to B. Thus, magnetic field amplification by plasma shear generically drives , giving rise to electron heating. This process, however, is expected to be limited mainly by the whistler instability (Gary et al., 1996), which is triggered when surpasses an instability threshold, although ionscale instabilities can also play a role (Riquelme et al., 2016).
In this work we show that, besides controlling the viscous heating of the electrons, the scattering by the whistler waves can accelerate a fraction of the electrons to energies significantly above thermal. For this, we use 2D particleincell (PIC) simulations of a plasma subject to a permanent shear motion, which continuously amplifies the background magnetic field. Given that this shear motion can be caused by the presence of incompressible MHD turbulence or by differential plasma rotation, our setup mimics a fairly generic physical situation in many turbulent astrophysical systems.
In order to optimize our computational resources, we concentrate exclusively on the electronscale physics. This is done by modeling the ions as infinitely massive particles that only provide a neutralizing electric charge (see §2). This strategy is supported by our previous study of an electronion plasma with (), where the electron anisotropy was mostly regulated by the electronscale whistler instability, with a moderate contribution from the ionscale mirror instability (Riquelme et al., 2016). For smaller values of , the effect of the mirror modes is expected to be even smaller (Riquelme et al., 2015). As we will see below, this low regime is the most interesting in terms of electron acceleration by whistler waves.
Our paper is organized as follows. §2 explains the setup of our simulations. §3 summarizes the physics of heating by anisotropic viscosity in our simulations. §4 describes the way this viscous heating, through the action of the whistler waves, gives rise to nonthermal electron acceleration. §4 also shows the effects of some plasma parameters on the acceleration efficiency. In §5 we summarize our results and discuss its astrophysical implications.
2. Simulation Setup
In this work we use the particleincell (PIC) code TRISTANMP (Buneman, 1993; Spitkovsky, 2005) in 2D. In order to make our simulations computationally efficient, we focus exclusively on the electronscale whistler instability. This is done by using “infinite mass ions” (the ions are immobile and only provide a neutralizing charge).
Our simulation boxes consist of a square domain in the  plane (as shown in Figure 1), which contains plasma with a homogeneous initial magnetic field . To amplify the field in an incompressible way, we impose a velocity shear so that the mean particle velocity is , where is the distance along and is the shear parameter, which has units of frequency.^{1}^{1}1The simulations are performed in the ‘shearing coordinate system’ described in Riquelme et al. (2012), where the shearing velocity of the plasma vanishes, and both Maxwell’s equations and the Lorentz force on the particles are modified accordingly. From flux conservation, the component of the mean field evolves as (throughout this paper, may represent an average over volume or over particles, depending on the context), implying a net growth of . This, due to conservation, drives during the whole simulation.
The varying physical parameters between our runs are their initial () and the electron magnetization, quantified by the ratio between the initial electron cyclotron frequency and the shear parameter of the plasma, . In our runs we use electron magnetizations that satisfy , but that are still much smaller than expected in real astrophysical settings.^{2}^{2}2 As a reference, at Schwarzschild radii from Sgr A*, the expected conditions in the accreting plasma imply (e. g., Ponti et al., 2017), where we have approximated as the Keplerian rotation period at that radius. Because of this, we made sure to reach the regime where is large enough to not to play any role in our final results.
Simulation  N  L/R  

S1  1200  1  15  80  110 
S2  1200  2  15  160  78 
S3  3000  2  15  160  78 
S4  1200  5  10  160  74 
S5  3000  5  10  160  74 
S6  1200  10  7  160  74 
Note. – Simulation parameters: the initial electron magnetization , , the electron skin depth ( is the grid point separation), the number of electrons per cell N, the box size in units of the initial electron Larmor radius . In all runs initially and , where is the simulation time step.
All of our simulations have initially (, , and are the Boltzmann’s constant, the electron temperature, and the electron mass), which implies ( is the electron plasma frequency). Their numerical parameters are: the number of macroelectrons per cell (N), the electron skin depth in terms of grid point spacing (), and the box size in terms of the initial electron Larmor radius (; , where ). Table 1 shows a summary of our key simulations. We ran a series of simulations ensuring that the numerical parameters do not affect our results. The runs used just for numerical convergence are not in Table 1.
3. The Physics of Electron Heating by Anisotropic Viscosity
In this section we describe the process of electron heating by anisotropic viscosity. As a fiducial case we use simulation S2 of Table 1.
Figure 1 shows an example of magnetic and electric field fluctuations at , which naturally arise as we continuously drive the growth of (we only show the components). The black arrows in Figure 1 represent the projection of on the  plane, showing that , consistent with magnetic flux freezing provided that initially . Figure 1 also shows that the field fluctuations are nearly parallel to and dominated by wavenumbers with ,^{3}^{3}3Throughout this paper, will be a time dependent quantity, which coincides with at , and takes into account the evolution of and of the rms electron velocity. both being features that roughly coincide with the characteristics of marginally stable whistler modes (Gary et al., 1996; Yoon et al., 2011).^{4}^{4}4The whistler wavenumber at marginal stability satisfies (Yoon et al., 2011), which means for run S2 at (see from Figure 2).
Figure 2 shows how the time evolution of the whistler instability in our shearing setup determines the pressure anisotropy as well as the viscous heating of the electrons. First, Figure 2 shows how the plasma shear makes (dottedred) grow while (dottedblack) is kept constant (in our runs is identically zero). The solid lines show the energy in the fluctuating field , separated into its component perpendicular (; green) and parallel (; blue) to . We see that there is an initial time (until ) where essentially does not change. During this time, increases linearly, as is shown by the green line in Figure 2. This evolution of is caused by the adiabatic invariance of , as can be seen from Figure 2.
This initial regime ends when reaches the threshold for the growth of the whistler instability. This can be seen by comparing the evolution with the level necessary for the whistler modes to grow at a rate, , equal to (blue line), obtained from Gary et al. (1996).^{5}^{5}5We chose the threshold criterion since this is the threshold that reproduced best the evolution in Riquelme et al. (2016), where . After this level is reached, the rapid anisotropy growth stops, and is maintained in a quasistationary state that evolves close to the whistler threshold. The end of the conserving regime can also be seen from the exponential growth of in Figure 2 at . This growth is dominated by , as expected for the nearly parallel whistler modes. After , the growth of saturates and its amplitude is maintained at a quasistationary level. At , decreases at a rate close to (see Figure 2), implying that the pitchangle scattering frequency provided by is of the order of .
Although the whistler anisotropy threshold and our obtained evolve similarly, there is still a factor discrepancy. One possible reason for this is the fact that the theoretical threshold is calculated using a biMaxwellian electron energy distribution while, as we will see below, in our simulations there is a significant departure from a biMaxwellian behavior. Indeed, the presence of a nonthermal, highenergy tail in the electron energy distribution has been shown to make the whistler instability less unstable in the large anisotropy regime relevant for this work (Mace et al., 2010). Also,
our mildly relativistic electrons () should increase somewhat the pressure anisotropy threshold considered in Figure 2 (Bashir et al., 2013), which is calculated for nonrelativistic electrons.
The electron pressure anisotropy is expected to give rise to viscous heating (see Equation 1), as has been found by previous PIC simulations (see, e.g., Riquelme et al., 2016). In order to check the importance of this heating mechanism, Figure 2 shows the volumeaveraged electron heating rate (black) for run S2: , and compares it with the expected electron heating rate due to anisotropic viscosity (green), which is obtained from volumeaveraging . We see that viscous heating accounts for essentially all of the electron heating.
4. Nonthermal Electron Acceleration
Figure 3 shows the electron energy spectra at different times for run S2, with the time of each spectrum shown by the color bar. After the whistler modes have reached saturation (), there is a rapid growth of a nonthermal tail. By the end of the simulation (), the tail can be approximated reasonably well by a powerlaw of spectral index .^{6}^{6}6 This spectral index is estimated using least squares fitting, considering only the range of marked by the black line in Figure 3 (), which gives . The magenta lines show that by the spectral index is still decreasing, although significantly slower than between and , when it experiences its fastest evolution. The peak of the spectrum (representing its ‘thermal’ component) also shifted to larger energies (by a factor ) during the simulation, consistent with the overall electron heating.
4.1. The Acceleration Mechanism
In order to understand the origin of the nonthermal tail, we calculated the average change in the electron Lorentz factors, , as a function of time for two different electron groups. One group represents the electrons from the powerlaw tail (hereafter, the ‘nonthermal’ electrons), which are picked so that at they have . The other group belongs to the bulk of the energy distribution (the ‘thermal’ electrons), and are chosen so that, at , . These nonthermal and thermal electrons are marked by the grey area and by the vertical red line in Figure 3, and their evolutions are shown by black lines in Figures 4 and 4, respectively. Besides , Figures 4 and 4 also show:

The work performed by the electric field associated with the whistler waves, hereafter the ‘electric work’ (EW), ^{7}^{7}7In a shearing plasma there is also an electric field associated to the bulk plasma motion. Since our simulations are performed in the ‘shearing frame’, this electric field vanishes, therefore the electric field in our runs corresponds entirely to the whistler waves. which is shown by the blue line.

The energy gain by anisotropic viscosity (AV), caused by the pressure anisotropy of the electrons,^{8}^{8}8For each electron, this energy gain is calculated by integrating in time the quantity , where is the component of the electron velocity. In a gyrotropic plasma, it can be shown that, when averaged over the electrons, this expression reproduces the heating rate given by Equation 1. which is shown by the green line.

The sum of the heating by anisotropic viscosity and the electric work (AV+EW), in red line.
Figure 4 shows that the work done by the whistlers’ electric field on the nonthermal electrons is more than two times the viscous heating rate. This implies that these electrons are mainly energized through scattering with the whistler waves. On the other hand, Figure 4 shows that the energy given by this electric field to the thermal electrons is negative. This implies that the scattering process on average makes the thermal electrons lose energy to the waves. The total heating of the thermal electrons is still positive, and is mainly caused by the dominant (and positive) viscous heating. The reasonably good correspondence between and the sum of the energization by anisotropic viscosity and by the electric work shows that these two processes account fairly well for the total heating of electrons. Also, the fact that the works done by the electric field on the nonthermal and thermal particles are positive and negative, respectively, is consistent with this electric field not affecting the average heating of the electrons, which is dominated by viscous heating (as shown in Figure 2).
In the case of the run S2 analyzed in this section, the nonthermal tail at contains of the electrons and carries of their energy. Notice that, for the purpose of measuring these numbers, we defined the nonthermal tail through the condition , where the spectrum is fairly well represented by a powerlaw tail.
Thus, in the presented acceleration scenario, the electric field of the whistler waves effectively substracts energy from the bulk of the electron distribution and pass it to a small population of electrons, which get accelerated to nonthermal energies. However, for this mechanism to be efficient, the resonant scattering by whistler modes must occur efficiently for the thermal particles, as well as for the nonthermal electrons in the tail. The resonant condition is
(2) 
where is the frequency of the modes, is the electron velocity parallel to B, and is the nonrelativistic electron cyclotron frequency. However, the marginally stable whistler modes satisfy (Yoon et al., 2011). Thus, for (see Figure 2), the resonant condition becomes . Thus, in order for this condition to be satisfied by the thermal electrons, one needs that . For the nonthermal electrons a similar requirement needs to be satisfied, but with being a factor smaller, since the typical value of for this populations is a factor larger than for the thermal electrons (which are the ones that define ).^{9}^{9}9Indeed, thermal particles in simulation S2 have on average , while for the nonthermal ones . This means that, for the nonthermal tail to grow, one requires a moderate widening of the whistler fluctuation power spectrum.
This is indeed what happens in simulation S2, as can be seen from Figure 5. This figure shows the power spectra of whistler fluctuations at different times ranges: right after the saturation of the instability (; Figure 5) and after the nonthermal tail is well developed (; Figure 5). After saturation, the spectra are rather narrowly peaked, with their maxima on average at . For the spectra widen significantly, having on average significant power between and 0.4. This result is thus consistent with resonant scattering happening for both thermal and nonthermal electrons. This appears to be a key ingredient to allow the transfer of energy from the thermal to the nonthermal particles. Also, since the whistler electric field is essentially set by the phase velocity of the waves, the proposed acceleration scenario can be interpreted as a stochastic (or second order Fermi) process, given that the scattering with the traveling whistler waves is what ultimately ends up accelerating the nonthermal electrons.
In the following sections we analyze how the efficiency of this acceleration depends on different plasma parameters.
4.2. Dependence on
Figure 6 shows the final spectra (at ) of four simulations where the electrons have the same initial temperature and magnetization ( and ), but different . The four cases considered have 1 (black), 2 (red), 5 (blue), and 10 (green); the blackdotted line shows the case of a MaxwellBoltzmann distribution. We see that the hardness of the nonthermal tail gradually decreases as increases, with the and 2 cases being quite similar. In Figure 6 we compare the spectra for and 5 (solidred and solidblue, respectively) with powerlaws of spectral indices and 4.2 (black and green, respectively; ). We see that these powerlaw indices provide a reasonably good approximation to the nonthermal tails in these two cases (in the case of , ).
The softening of the nonthermal tail as increases is in line with the decrease in the overall electron heating, which can be seen from the location of the peak of the spectra in Figure 6. Indeed, given that the energy for the formation of the tail is ultimately provided by the viscous heating of the electrons (as we showed in Figure 2), it makes sense that the nonthermal tail gets less prominent as the viscous heating gets less efficient. This decrease in the viscous heating for larger is expected due to the reduction in provided by the whistler instability (Gary et al., 1996).
4.3. Dependence on
Although in our simulations the electron magnetization satisfies , the values used are still much smaller than the ones expected in realistic astrophysical scenarios. Because of this, we made sure that in our simulations is always large enough to not affect our results. We did this for the cases and 5, by comparing simulations with and 3000. Our results are shown in Figure 6. We see that both for and 5 (in red and blue, respectively) there is very little difference between cases with and 3000 (solid and dotted lines, respectively.). The fact that the spectra are essentially independent on is consistent with the fact that the time scale for the electron acceleration process is set by the scattering rate by whistler waves. Since in our shearing plasma setup this scattering rate is essentially set by the shear rate (see Riquelme et al., 2016), the expectation is that, at a given value of , the electron spectra should be about the same, regardless of the value of .
5. Summary and Discussion
In this work we used 2D particleincell (PIC) simulations to show that, in the context of a magnetic field amplified by shear plasma motion, the process of electron heating by anisotropic viscosity can give rise to a prominent electron nonthermal tail. Indeed, for and after the field is amplified by a factor , the electron energy spectrum (measured when , because of field amplification) contains a nonthermal powerlaw tail with spectral index (see Fig. 3). The nonthermal tail contains of the electrons, and of their energy. However, the nonthermal tail becomes progressively softer as grows (see Fig. 6). This behavior is in accordance with the decrease in the overall viscous heating of the electrons as increases, since this heating constitutes the ultimate source of energy for the tail formation.
The role of the whistler instability is key in the electron acceleration process. Firstly, the whistler waves regulate the overall electron energization by determining the efficiency of the electron heating by anisotropic viscosity. This is done by providing an electron pitchangle scattering rate comparable to the plasma shearing rate, . This keeps the electron pressure anisotropy at a quasisteady level, which is roughly determined by the whistler instability threshold for growth rate comparable to . Secondly, the whistler waves drive the formation of the nonthermal tail by transferring a significant part of their energy (ultimately gained through the heating by anisotropic viscosity) to a small fraction of electrons. Thus, effectively, the whistler waves subtract energy from the bulk of the distribution, and pass it to the electrons that end up forming the tail. This is inferred from the negative and positive work performed by the whistler electric field on the electrons of the thermal and nonthermal part of the energy spectrum, respectively (see Fig. 4).
The present work focused on mildly relativistic electrons, with initial temperature . Previous theoretical studies have shown that stochastic electron acceleration by whistler waves should be sensitive to the ratio (Dermer et al., 1996; Petrosian et al., 2004). This sensitivity can be seen as a dependence on , since, for a given value of , uniquely fixes .^{10}^{10}10Petrosian et al. (2004) showed that the stochastic acceleration of electrons by whistler waves is more efficient for larger values of , which, for a fixed , implies larger values of . We thus defer to a future work the study of the dependence of the presented acceleration process on . It will also be interesting to explore more extensively the regime , where the dominant whistler modes are expected to adopt a more oblique orientation, which may affect the way the proposed nonthermal acceleration works (Gary et al., 2011).
Our proposed acceleration process is likely relevant for lowluminosity accretion flows around black holes, like in Sgr A*. For the specific case of Sgr A*, mildly relativistic electron temperatures and nonthermal spectral components with powerlaw indices are favored by multiwavelength observations (Özel et al., 2000; Yuan et al., 2003; Ball et al., 2016). It is important to caution, however, that the proposed factor amplification of the local magnetic field may not be so common in the context of a typical turbulent accretion flow. Nevertheless, one could alternatively picture this acceleration to happen due to the cumulative effect of successive (smaller amplitude) growths and reductions of the ambient magnetic field in a turbulent medium. We will study this last possibility in a future work.
References
 Ball et al. (2016) Ball, D., Özel, F., Psaltis, D. & Chan, C. 2016, ApJ, 826, 77
 Bashir et al. (2013) Bashir, M., Zaheer, S., Iqbal, Z. & Murtaza, G. 2013, Phys. Lett. A, 377, 2378
 Brunetti et al. (2014) Brunetti, G. & Jones, T. 2014, Int. J. Mod. Phys. D 23, 1430007
 Buneman (1993) Buneman, O. 1993, “Computer Space Plasma Physics”, Terra Scientific, Tokyo, 67
 Dermer et al. (1996) Dermer, C., Miller, J., & Li, H. 1996, ApJ, 456, 106
 Gary et al. (1996) Gary, S. P. & Wang, J. 1996, J. Geophys. Res., 101, 10749
 Gary et al. (2011) Gary, P., Liu, K., & Winske, D. 2011, Phys. Plasmas, 18, 082902
 Kulsrud (1983) Kulsrud, R. M. 1983, in Handbook of Plasma Physics, ed. M. N. Rosenbluth & R. Z. Sagdeev (Amsterdam: North Holland), 115
 Li et al. (2015) Li, X., Guo, F., Li, H., & Li, G. 2015, ApJL, 811, L24
 Liu & Wu (2013) Liu, H., & Wu, Q. 2013, ApJL, 764, 17
 Lynn et al. (2014) Lynn, J., Quataert, E., & Chandran, B., & Parrish, I. 2014, 791, 71
 Mace et al. (2010) Mace, R. & Sydora, R. 2010, J. Geophys. Res., 115, A07206
 Marcowith et al. (2016) Marcowith, A., Bret, A., Bykov4, A., Dieckman, M., Drury, L., Lembege,B., Lemoine, M., Morlino, G., Murphy, G., Pelletier, G., Plotnikov, I., Reville, B., Riquelme, M., Sironi, L., & Stockem Novo, A. 2016, Rep. Prog. Phys., 79, 046901
 Özel et al. (2000) Özel, F., Psaltis, D., & Narayan, R. 2000, ApJ, 541, 234
 Petrosian et al. (2004) Petrosian, V. & Liu, S. 2004, ApJ, 610, 550
 Ponti et al. (2017) Ponti, G., George, E., Scaringi, S., Zhang, S., Jin, C., et al. 2017, MNRAS, 468, 2447
 Riquelme et al. (2012) Riquelme, M. A., Quataert, E., Sharma, P., & Spitkovsky, A. 2012, ApJ, 755, 50
 Riquelme et al. (2015) Riquelme, M. A., Quataert, & Verscharen, D. 2015, ApJ, 800, 27
 Riquelme et al. (2016) Riquelme, M. A., Quataert, & Verscharen, D. 2016, ApJ, 824, 123
 Sharma et al. (2007) Sharma, P., Quataert, E., Hammett, G. W., & Stone, J. 2007, ApJ, 667, 714
 Sironi & Spitkovsky (2014) Sironi, L., & Spitkovsky, A. 2014, ApJL, 783, L21
 Snyder et al. (1997) Snyder, P. B., Hammett, G. W., & Dorland, W. 1997, Phys. Plasmas, 4, 3974
 Spitkovsky (2005) Spitkovsky, A. 2005, AIP Conf. Proc, 801, 345, astroph/0603211
 Yoon et al. (2011) Yoon, P., Seough, J., Lee, J., An, J., & Lee, J. O. 2011, Phys. Plasmas, 18, 102103
 Yuan et al. (2003) Yuan, F., Quataert, E., & Narayan, R. 2003, ApJ, 598, 301
 Yuan et al. (2004) Yuan, F., Quataert, E., & Narayan, R. 2004, ApJ, 606, 894
 Zhdankin et al. (2017) Zhdankin, V., Werner, G., Uzdensky, D., & Begelman, M. 2017, Phys. Rev. Lett. 118, 055103