Electron Heating by the Ion Cyclotron Instability in Collisionless Accretion Flows.
II. Electron Heating Efficiency as a Function of Flow Conditions
In the innermost regions of low-luminosity accretion flows, including Sgr A at the center of our Galaxy, the frequency of Coulomb collisions is so low that the plasma is two-temperature, with the ions substantially hotter than the electrons. This paradigm assumes that Coulomb collisions are the only channel for transferring the ion energy to the electrons. In this work, the second of a series, we assess the efficiency of electron heating by ion velocity-space instabilities in collisionless accretion flows. The instabilities are seeded by the pressure anisotropy induced by magnetic field amplification, coupled to the adiabatic invariance of the particle magnetic moments. Using two-dimensional (2D) particle-in-cell (PIC) simulations, we showed in Paper I that if the electron-to-ion temperature ratio is , the ion cyclotron instability is the dominant mode for ion betas (here, is the ratio of ion thermal pressure to magnetic pressure), as appropriate for the midplane of low-luminosity accretion flows. In this work, we employ analytical theory and 1D PIC simulations (with the box aligned with the fastest growing wavevector of the ion cyclotron mode) to fully characterize how the electron heating efficiency during the growth of the ion cyclotron instability depends on the electron-to-proton temperature ratio, the plasma beta, the Alfvén speed, the amplification rate of the mean field (in units of the ion Larmor frequency) and the proton-to-electron mass ratio. Our findings can be incorporated as a physically-grounded sub-grid model into global fluid simulations of low-luminosity accretion flows, thus helping to assess the validity of the two-temperature assumption.
Subject headings:accretion, accretion disks – black hole physics – galaxies: clusters: general – instabilities – plasmas – radiation mechanisms: general – solar wind
In low-luminosity accretion flows, including the ultra-low-luminosity source Sagittarius A (Sgr A) at our Galactic Center (Narayan et al., 1995, 1998; Yuan et al., 2003; Xu et al., 2006; Mościbrodzka et al., 2012; Yuan & Narayan, 2014), the timescale for electron and ion Coulomb collisions is much longer than the inflow time in the disk, i.e., the plasma is collisionless. At distances less than a few hundred Schwarzschild radii from the black hole ( is the Schwarzschild radius, where is the black hole mass), ions and electrons are thermally decoupled and the plasma is two-temperature, with the ions substantially hotter than the electrons (Narayan & Yi, 1995; Yuan et al., 2003). This stems from the fact that (i) compressive heating favors non-relativistic ions over relativistic electrons, and that (ii) electrons copiously lose energy via radiative cooling.
Early work on two-temperature models of low-luminosity accretion flows (also known as ADAFs, or advection-dominated accretion flows) assumed that most of the turbulent viscous energy goes into the ions (Ichimaru, 1977; Rees et al., 1982; Narayan & Yi, 1995), and that only a small fraction goes into the electrons. There have been occasional attempts to estimate this fraction from microphysics, by considering magnetic reconnection (Bisnovatyi-Kogan & Lovelace, 1997; Quataert & Gruzinov, 1999), magnetohydrodynamic (MHD) turbulence (Quataert, 1998; Blackman, 1999; Medvedev, 2000), plasma waves (Begelman & Chiueh, 1988), or dissipation of pressure anisotropy in collisionless plasmas (Sharma et al., 2007). Currently, it appears that the electron-to-proton temperature ratio lies in the range , but this is by and large no more than a guess (Narayan & Yi, 1995; Yuan et al., 2003; Yuan & Narayan, 2014).111Note that the plasma in the fast solar wind and behind shocks in supernova remnants is also two-temperature (Marsch, 2012; Rakowski, 2005; Ghavamian et al., 2007; Morlino et al., 2012). The micro-physics of energy dissipation and electron heating in collisionless accretion flows cannot be captured in the MHD framework, but it requires a fully-kinetic description with first-principles particle-in-cell (PIC) simulations.
In this work, the second of a series, we study with PIC simulations the efficiency of electron heating by ion velocity-space instabilities, in the context of low-luminosity accretion disks. Pressure anisotropies are continuously generated in collisionless accretion flows due to the fluctuating magnetic fields associated with the non-linear stages of the magnetorotational instability (MRI, Balbus & Hawley 1991, 1998), a MHD instability that governs the transport of angular momentum in accretion disks (see Riquelme et al., 2012; Hoshino, 2013, for a study of the collisionless MRI). If the magnetic field is amplified by the MRI, the adiabatic invariance of the magnetic moments of charged particles drives the perpendicular (to the magnetic field) pressure to be much greater than the parallel pressure , a configuration prone to velocity-space instabilities. The same instabilities are believed to play an important role in the solar wind (Kasper et al., 2002, 2006; Bale et al., 2009; Maruca et al., 2011, 2012; Matteini et al., 2007, 2013; Cranmer et al., 2009; Cranmer & van Ballegooijen, 2012) and in the intracluster medium (Schekochihin et al., 2005; Lyutikov, 2007; Santos-Lima et al., 2014).
In Paper I, we have developed a fully-kinetic method for studying velocity-space instabilities in a system where the field is continuously amplified. In this case, the anisotropy is constantly driven (as a result of the field amplification), rather than assumed as a prescribed initial condition, as in most earlier works (see Gary, 1993, for a review). In our setup, the increase in magnetic field is driven by compression (as in Hellinger & Trávníček, 2005)222Note that Hellinger & Trávníček (2005) employed a hybrid code — that treats the ions as kinetic particles, but the electrons as a massless charge-neutralizing fluid. Thus, the electron kinetic physics was not properly captured., mimicking the effect of large-scale compressive motions in ADAFs. However, our results hold regardless of what drives the field amplification, so they can be equally applied to the case where velocity-space instabilities are induced by incompressible shear motions (as in Riquelme et al., 2014; Kunz et al., 2014).
Most of the previous numerical studies of anisotropy-driven instabilities focused either on ion instabilities alone (with hybrid codes, see Gary 1993; Hellinger et al. 2006 for a review; or fully-kinetic simulations in electron-positron plasmas, Riquelme et al. 2014) or on electron modes alone (keeping the ions as a static neutralizing background, see Gary 1993 for a review). Neither approach can capture self-consistently the energy transfer from ions to electrons, which requires fully-kinetic simulations with mobile ions and a realistic mass ratio. This is the purpose of our work.
With fully-kinetic PIC simulations, we study the efficiency of electron heating that results from ion velocity-space instabilities driven by magnetic field amplification in collisionless accretion flows. We focus on the regime (here, and are the electron and ion temperatures, respectively) where, as demonstrated in Paper I, the dominant mode is the ion cyclotron instability (e.g., Gary et al., 1976, 1993; Hellinger et al., 2006), rather than the mirror instability (e.g., Hasegawa, 1969; Southwood & Kivelson, 1993; Kivelson & Southwood, 1996). Since the wavevector of the ion cyclotron instability is along the mean magnetic field, the relevant physics can be conveniently studied by means of 1D simulations, with the box aligned with the ordered field. Via 1D PIC simulations, we assess here the dependence of the electron heating efficiency on the initial ratio between electron and proton temperatures, the ion beta (namely, the ratio of ion thermal pressure to magnetic pressure), the Alfvén speed, the amplification rate of the mean field (in units of the ion Larmor frequency) and the proton-to-electron mass ratio. Eqs. (23)-(25) emphasize how the various contributions to electron heating depend on the flow conditions. Their sum gives the overall electron energy gain due to the growth of the ion cyclotron instability, and it represents the main result of this paper.
This work is organized as follows. In Section 2, we describe the physical conditions in the innermost regions of low-luminosity accretion flows, where the plasma is believed to be two-temperature. The setup of our simulations is discussed in Section 3. Section 4 summarizes the conclusions of Paper I and anticipates the main results of this work (consisting of Eqs. (23)-(25)), which are extensively substantiated by our findings in Section 5. We discuss the astrophysical implications of our work in Section 6.
2. Physical Conditions in Accretion Disks
For our investigation of velocity-space instabilities in collisionless accretion flows, we employ values of the ion temperature , the ion beta and the Alfvén velocity that are consistent with estimates taken from state-of-the-art general-relativistic magnetohydrodynamic (GRMHD) simulations of Sgr A, the low-luminosity accretion flow at our Galactic Center (e.g., Sa̧dowski et al., 2013). Here, and are the ion number density and the magnetic field strength, respectively.
GRMHD simulations can successfully describe the linear and non-linear evolution of the MRI. Yet, they cannot properly capture the dissipation of MRI-driven turbulence on small scales (the electron Larmor radius), which ultimately regulates the temperature balance between protons and electrons, in the innermost regions of ADAFs where the plasma is two-temperature. It follows that GRMHD simulations cannot predict the value of the electron-to-proton temperature ratio to be employed in our PIC experiments, so we will have to study the dependence of our results on this parameter.
We estimate the values of , and in the innermost regions of ADAFs (at distances less than a few hundred Schwarzschild radii from the black hole) from Fig. 1 of Sa̧dowski et al. (2013). The ion temperature decreases with distance from the black hole as , from at the innermost stable circular orbit (, for a non-rotating black hole) down to at . In units of the proton rest mass energy, it varies in the range , i.e., the ions are non-relativistic. On the other hand, if the electrons were to be in equipartition with the ions, they would have , i.e., the electrons can be relativistically hot.
The ion beta is nearly constant with radius, and along the disk midplane it ranges from to (Fig. 1 of Sa̧dowski et al. (2013)).333GRMHD simulations are only sensitive to the total plasma beta, including ions and electrons. For the estimates presented in this section, we implicitly assume that electrons do not contribute much to the total plasma pressure. Lower values of the plasma beta are expected at high latitudes above and below the disk, in the so-called corona, where the plasma might be magnetically dominated (i.e., ). In this work, we only focus on velocity-space instabilities triggered in the bulk of the disk, and we take as the lowest value we consider, arguing that our results can be applied all the way down to . We extend our investigation to higher values of , up to . Most likely, this is not directly relevant for low-luminosity accretion flows, but it might have important implications for the physics of the intracluster medium (Schekochihin et al., 2005; Lyutikov, 2007; Santos-Lima et al., 2014).
The value of the Alfvén velocity can be derived from the ion temperature and the ion beta, since . It follows that the Alfvén speed varies in the range , if and . Below, we show that our results are not sensitive to variations in the Alfvén velocity, in the range that we explore.
To fully characterize our system, we need to specify the rate of magnetic field amplification, in units of the ion gyration frequency . In accretion flows, we expect the ratio to be much larger than unity (, if is comparable to the local orbital frequency). For computational convenience, we employ smaller values of , exploring the range , and we show that our results converge in the limit .
Also, due to computational constraints, fully-kinetic PIC simulations are forced to employ a reduced value of the ion-to-electron mass ratio (e.g., Riquelme et al. 2014 primarily studied velocity-space instabilities in electron-positron plasmas, i.e., with ). Most of our results employ a reduced mass ratio ( or 64). However, we show that our conclusions can be readily rescaled up to realistic mass ratios (we extend our study up to ). This is extremely important for low-luminosity accretion disks, since only with a realistic mass ratio one can describe the case of non-relativistic ions and ultra-relativistic electrons that is of particular interest for ADAF models.
3. Simulation Setup
We investigate electron heating due to ion velocity-space instabilities in collisionless accretion flows by means of fully-kinetic PIC simulations. We have modified the three-dimensional (3D) electromagnetic PIC code TRISTAN-MP (Buneman, 1993; Spitkovsky, 2005; Sironi et al., 2013; Sironi & Spitkovsky, 2014; Sironi & Giannios, 2014) to account for the effect of an overall compression of the system.444Our method can also be applied to an expanding plasma, but in this work we only study compressing systems. Our model is complementary to the technique described in Riquelme et al. (2012) and employed in Riquelme et al. (2014), which is appropriate for incompressible shear flows.
Our method, the first of its kind for fully-kinetic PIC simulations of compressing systems, has been extensively described in Paper I (there, see Section 2 and Appendix A). For completeness, we report here its main properties. We solve Maxwell’s equations and the Lorentz force in the fluid comoving frame, which is related to the laboratory frame by a Lorentz boost. In the comoving frame, we define two sets of spatial coordinates, with the same time coordinate. The unprimed coordinate system has a basis of unit vectors, so it is the appropriate coordinate set to measure all physical quantities. Yet, we find it convenient to re-define the unit length of the spatial axes in the comoving frame such that a particle subject only to compression stays at fixed coordinates. This will be our primed coordinate system.
The location of a particle in the laboratory frame (identified by the subscript “L”) is related to its position in the primed coordinate system of the fluid comoving frame by , where compression is accounted for by the diagonal matrix
which describes compression along the and axes. By defining the determinant , the two evolutionary Maxwell’s equations of the PIC method in a compressing box are, in the limit of non-relativistic compression speeds,
where the temporal and spatial derivatives pertain to the primed coordinate system (the reader is reminded that the primed and unprimed systems share the same time coordinate, so , whereas the spatial derivatives differ: ). We define and to be the physical electromagnetic fields measured in the unprimed coordinate system. The current density is computed by summing the contributions of individual particles, as we specify in Appendix A of Paper I.
The equations describing the motion of a particle with charge and mass can be written, still in the limit of non-relativistic compression speeds, as
where . The physical momentum and velocity of the particle are measured in the unprimed coordinate system (here, is the particle Lorentz factor). Yet, the particle velocity entering Eq. (5) refers to the primed coordinate system, where . Eqs. (4) and (5) hold for particles of arbitrary Lorentz factor.
A uniform ordered magnetic field is initialized along the direction. As a result of compression, Eq. (2) dictates that it should grow in time as , which is consistent with flux freezing (the particle density in the box increases at the same rate). From the Lorentz force in Eq. (4), the component of particle momentum aligned with the field does not change during compression, so , whereas the perpendicular momentum increases as . This is consistent with the conservation of the first () and second () adiabatic invariants.
Our computational method is implemented for 1D, 2D and 3D computational domains. We use periodic boundary conditions in all directions, assuming that the system is locally homogeneous, i.e., that gradients in the density or in the ordered field are on scales larger than the box size. In Paper I, we have demonstrated that if the initial electron temperature is less than of the ion temperature, the wavevector of the dominant instability is aligned with the ordered magnetic field. It follows that the evolution of the dominant mode can be conveniently captured by means of 1D simulations with the computational box oriented along , which we will be employing in this work. Yet, all three components of electromagnetic fields and particle velocities are tracked.
The focus of this work is to assess how the efficiency of electron heating depends on the properties of the flow. We vary the ion plasma beta
from up to 80. Here, is the particle number density at the initial time and the initial ion temperature (ions, as well as electrons, are initialized with a Maxwellian distribution). The electron thermal properties are specified via the electron beta
We vary the ratio from down to . The magnetization is quantified by the Alfvén speed
so that the initial ion temperature equals . We will show that our results are the same when varying the Alfvén velocity from 0.025 up to 0.1, with being our reference choice.
The ion cyclotron frequency, which is related to the ion plasma frequency by , will set the characteristic unit of time. In particular, we will scale the compression rate to be a fraction of the ion cyclotron frequency . In accretion flows, we expect the ratio to be much larger than unity. We explore the range , showing that our results converge in the limit . We evolve the system up to a few compression timescales.
Since we are interested in capturing the efficiency of electron heating due to ion instabilities, we need to properly resolve the kinetic physics of both ions and electrons. We typically employ 32,768 computational particles per species per cell, but we have tested that our results are the same when using up to 131,072 particles per species per cell. Such large values are of critical importance to suppress the spurious heating induced by the coarse-grained description of PIC plasmas (e.g., Melzani et al., 2013), and so to reliably estimate the efficiency of electron heating by ion velocity-space instabilities, which is the primary focus of this work.
We resolve the electron skin depth with 5 cells, which is sufficient to capture the physics of the electron whistler instability (Kennel & Petschek, 1966; Ossakow et al., 1972b, a; Yoon & Davidson, 1987; Yoon et al., 2011). On the other hand, our computational domain needs to be large enough to include at least a few wavelengths of ion-driven instabilities, i.e., a few ion Larmor radii
Most of our results employ a reduced mass ratio ( or 64), for computational convenience. However, we show that our conclusions can be readily rescaled up to realistic mass ratios (we extend our study up to ). For and , which will be our reference case, we employ a 1D computational box with . When changing or , we ensure that our computational domain is scaled such that it contains at least , so that the dominant wavelength of ion-driven instabilities is properly resolved.
4. Electron Heating by the Ion Cyclotron Instability
In this Section, we first summarize the main conclusions of Paper I. Then, we describe how the efficiency of electron heating depends on the flow conditions.
By means of 2D simulations, we have demonstrated in Paper I that, if the initial electron-to-ion temperature ratio is , the ion cyclotron instability dominates the relaxation of the ion anisotropy, over the competing mirror mode. Since the wavevector of the ion cyclotron instability is aligned with the mean magnetic field, the relevant physics can be conveniently studied by means of 1D simulations, with the box aligned with the ordered field. Thanks to the greater number of computational particles per cell allowed by 1D simulations, as opposed to 2D, the efficiency of electron heating by the ion cyclotron instability can be reliably estimated.
Since we are interested in the net heating of electrons by the ion cyclotron instability, and not in the straightforward effect of compression, we have chosen to quantify the efficiency of electron heating by defining the ratio
where and are the electron and ion momenta. The ratio remains constant before the growth of anisotropy-driven instabilities, since in a compressing box the particle parallel momentum does not change, whereas the perpendicular momentum increases as , so the total momentum increases as , for both electrons and ions. It follows that the parameter is a good indicator of the electron energy change occurring as a result of the ion cyclotron mode. The choice of the pre-factor is such that for non-relativistic particles, having and , the parameter reduces to , i.e., to the ratio of the average kinetic energies of electrons and ions.
As described in Paper I, the development of the ion cyclotron instability causes strong electron heating. The resulting change in the electron Lorentz factor , averaged over all the electrons in the system, gives a corresponding increase in the parameter of
where we have neglected higher order terms in , as we have motivated in Paper I. In turn, the mean energy gain at the end of the exponential growth of the ion cyclotron mode can be written as a sum of different terms (see Paper I)
where we have subtracted on the left hand side, since it describes the electron energy increase due to compression alone, which would be present even without any instability. The four terms on the right hand side of Eq. (12) can be written as a function of the plasma properties at the end of the exponential phase of ion cyclotron growth (as indicated by the subscript “exp” below)
where Eq. (13) accounts for the sum of the energy gain due to the magnetic field growth and the energy loss associated with the curvature drift; Eq. (14) describes the effect of the grad-B drift; and Eq. (15) accounts for the energy gain associated with the E-cross-B velocity. We refer to Section 4.1 of Paper I for a complete characterization of the different terms.
In Eqs. (13)-(15), and are the electron temperatures perpendicular and parallel to the mean field , and we have defined the space-averaged fields and . From Maxwell’s equation in Eq. (2), the electric field energy is related to the magnetic field energy by , which we have used in Eq. (15). Here, is the phase speed of ion cyclotron waves. At the condition of marginal stability, the ion anisotropy approaches (e.g., Gary et al., 1976; Gary & Lee, 1994; Gary et al., 1994, 1997; Hellinger et al., 2006; Yoon & Seough, 2012; Seough et al., 2013, see also Paper I)
The oscillation frequency and the dominant wavevector of the ion cyclotron mode at marginal stability are (Kennel & Petschek, 1966; Davidson & Ogden, 1975; Yoon, 1992; Yoon et al., 2010; Schlickeiser & Skoda, 2010)
so that the phase speed to be used in Eq. (15) reduces to . Finally, since for , as employed in this work, the expressions above can be simplified to give and , so that . Given the scaling , it follows that the ratio of the dominant wavelength to the ion Larmor radius should be nearly a constant, as we indeed confirm in Section 5.2.
In order to compute the various heating terms in Eqs. (13)-(15), we need an estimate of the ratios and at the end of the exponential phase of the ion cyclotron instability. Moreover, for Eq. (13) we will need the value of at the saturation of the instability. All the other ingredients (e.g., in Eq. (14) and in Eq. (15)) can be estimated from their initial values, using the scalings expected as a result of compression alone.
As we demonstrate in Section 5 and justify analytically in Appendix A, the following scalings can be employed to fully characterize the efficiency of electron heating by the ion cyclotron instability:
where . The anisotropy should be evaluated at the threshold of marginal stability for the ion cyclotron mode, following Eq. (16). In the same way, the electron anisotropy in Eq. (21) is to be computed at the threshold of marginal stability for the electron whistler instability, given by (Gary & Wang, 1996; Gary & Karimabadi, 2006, see also Paper I)
The coefficients in Eq. (19) and Eq. (21) have been fitted using simulations with . As we argue in Section 5.4, in the astrophysically-relevant limit , the coefficients asymptote to a value that is just a factor of a few smaller.
The scalings in Eqs. (19)-(21) — which will be extensively tested in the following sections — provide, together with Eqs. (13)-(15), a complete characterization of the efficiency of electron heating by the ion cyclotron instability in accretion flows. In particular, Eq. (21) shall be used to estimate the electron anisotropy that enters Eq. (13). Using Eqs. (19)-(21), we shall now estimate how the different contributions to electron heating in Eqs. (13)-(15) depend on the flow conditions.
We assume that the exponential growth of the ion cyclotron instability terminates at , which is supported by our findings in Section 5. The plasma conditions at this time can be estimated from their initial values, using the scalings resulting from compression (e.g., ; also, and for non-relativistic ions). This allows us to emphasize how the various contributions to electron heating depend on the initial flow properties:
From these equations, the resulting increase in the parameter of Eq. (11) can be easily obtained. For the sake of simplicity, let us now neglect any explicit time dependence (yet, we include all the appropriate powers of in the numerical estimates of that we present in the following sections). With this approximation, we find that , since the ions are non-relativistic. Also, in the numerator of Eq. (11), we can approximate , where is the mean electron Lorentz factor at the initial time, which reduces to for (non-relativistic electrons) and to for (ultra-relativistic electrons). The increase in the parameter
can then be computed for a range of flow conditions.
5. Dependence on the Flow Conditions
In this section, we discuss how the efficiency of electron heating depends on the physical conditions of the flow. We explore the role of the electron-to-ion temperature ratio in Section 5.1, of the ion plasma beta in Section 5.2, of the Alfvén velocity in Section 5.3, of the compression timescale (in units of ) in Section 5.4 and of the ion-to-electron mass ratio in Section 5.5. In doing so, we will confirm the scalings anticipated in Eqs. (19)-(21), which are at the basis of our heating model.
In Figs. 1–104, we present the temporal evolution of our system, for different initial conditions. In each case, we show in the top row the time evolution of the particle anisotropy (panel (a); solid lines for ions, dotted for electrons), of the electron momentum (panel (b); solid for the parallel component, dotted for the perpendicular component) and of the electron heating efficiency (panel (c)), quantified by , where the parameter is defined in Eq. (10) and is its value at the initial time. In the bottom row, we show the time evolution of the trasverse magnetic energy (panel (d)) and of the transverse electric energy (panel (e)). In addition, we show in panel (f) the spatial pattern (with as our unit of length) of the component of the unstable waves.
Below, we focus on a few representative cases, but we remark that the conclusions presented below have been checked across the whole parameter space explored in this work, i.e., for , for , for , for and up to a mass ratio .
5.1. Dependence on the Temperature Ratio
In this section, we illustrate the dependence on the electron-to-ion temperature ratio , for a representative case with , , and . We vary the electron-to-ion temperature ratio from up to .
In the range explored in
he development of the ion cyclotron instability is not sensitive to the electron thermal content. In fact,
ows that the time of growth of the ion cyclotron instability () and the saturation values of the magnetic and electric energy ( in
) and in
), respectively) are all insensitive to the electron-to-ion temperature ratio. Moreover, the secular evolution that follows the exponential phase of the ion cyclotron instability (at ) is the same, regardless of . Once the energy of the ion cyclotron waves reaches , efficient pitch-angle scattering brings the ion anisotropy (
)) back to the threshold of marginal stability in Eq. (16) (which is indicated in
) with a dashed black line). At , the ion anisotropy follows the same track of marginal stability, regardless of the initial electron temperature. In short, the ion physics does not depend on the electron thermal content.
The choice of initial electron temperature can affect the electron physics before the onset of the ion cyclotron instability. For cold electrons ( in green and in orange), the ion cyclotron instability starts to grow when the electron anisotropy has not reached yet the threshold for the electron whistler instability in Eq. (22) (corresponding to for and to for ). In this case, the electron anisotropy is not strong enough to trigger anisotropy-driven instabilities on electron scales, and no signs of electron physics are present. In particular, the temporal evolution of the magnetic and electric energy (
) and (e), respectively) is the same at all times, regardless of the electron temperature. The only dependence on in this regime of cold electrons () appears in the value of electron anisotropy after the growth of the ion cyclotron instability (compare the green and orange dotted lines in
) at ). The degree of electron anisotropy is regulated by the mechanism of electron heating during the ion phase, as we detail below.555As we have described in Paper I (Section 4.2.1), the electron distribution for is not uniform in space, due to the local nature of the heating process (which is dominated by the term in Eq. (15)). The electron distribution resembles a Maxwellian drifting with the local E-cross-B velocity, so the degree of electron anisotropy shown by the green dotted line in a measure of the E-cross-B speed, rather than of the genuine electron anisotropy in the fluid frame.
At higher electron temperatures ( in blue and in red), the threshold for the whistler instability — which corresponds to for and to for — is reached before the onset of the ion cyclotron mode. So, the whistler instability can grow, generating transverse electromagnetic waves (see Appendix B in Paper I for further details), and producing a bump at early times () in the magnetic and electric energy of
) and (e), respectively.666As discussed in Paper I, our 2D simulations show that the wavevector of the fastest growing whistler mode is aligned with the mean magnetic field, so 1D simulations can properly capture the development of the whistler instability. In Appendix A, we estimate analytically the saturation values of the magnetic and electric energy produced by the whistler instability, as a function of the flow conditions. Our estimates are in good agreement with the results of our simulations, regarding the dependence on both (investigated in
During the whistler phase at , the electron anisotropy remains at the threshold of marginal stability, see the dotted blue and red lines in
). In contrast, the ion anisotropy still follows the track expected from compression alone (indicated by the triple-dot-dashed line in
)), suggesting that ions do not participate in the whistler instability. We expect the electron anisotropy during the whistler phase () to scale as , as in Eq. (22). Once we multiply the electron anisotropy by a factor of , we find that the temporal tracks for and almost overlap (blue and red dotted lines in the subpanel of
)), indicating that the condition of marginal stability in Eq. (22) indeed regulates the level of electron anisotropy after the onset of the whistler instability. While it is not suprising that the electron anisotropy stays at the threshold during the electron phase at , we remark that a similar degree of anisotropy is preserved after the growth of the ion cyclotron instability at , which justifies our ansatz in Eq. (21) (see the first term in the square brackets).
The electron whistler instability generates transverse electromagnetic waves on electron scales. At marginal stability, the characteristic wavelength and oscillation frequency of the electron whistler mode are respectively (Kennel & Petschek, 1966; Yoon & Davidson, 1987; Yoon et al., 2011; Bashir et al., 2013)
Since we account for the possibility that electrons are ultra-relativistic, the proper definitions of the electron cyclotron frequency and plasma frequency are
), we show the spatial pattern of the component of whistler waves, measured at the time indicated with the blue and red dashed vertical lines in all the other panels (for and , respectively). In units of the ion Larmor radius , the characteristic wavelength of the whistler instability is
e employ and . For , taking from the dotted bue line in
), we expect from Eq. (29) that , which is in good agreement with the spatial periodicity of the blue line in
). Similarly, for , taking from the dotted red line in
), we expect , which agrees with the spatial pattern of the red line in
). We have extensively checked the scalings in Eq. (29), most importantly regarding the dependence on and . In the limit of non-relativistic electrons (so, ), we find from Eq. (29) that , since . This scaling is consistent with the numerical findings of Gary & Karimabadi (2006).
We conclude by discussing the efficiency of electron heating, as a function of the electron-to-ion temperature ratio. In
), we plot the temporal evolution of the parameter defined in Eq. (10) relative to its initial value (solid lines if , dotted if ). The filled circles in
ow, for different choices of , our analytical prediction for the change in the parameter — as derived from the theory outlined in Section 4 — at the end of the exponential phase of the ion cyclotron instability. Regardless of the initial electron temperature, the growth of the ion cyclotron waves at leads to substantial electron heating, whose magnitude is properly captured by our analytical model (compare the lines with the filled circles of the same color). The increase in the parameter is more pronounced for higher values of . This can be easily understood from the scalings in Eqs. (23)-(25), as we now explain.
At small values of (green line in
), for ), electron heating by the ion cyclotron instability is dominated by the E-cross-B term in Eq. (25), which is insensitive to the electron temperature. In Paper I, we have demonstrated that the E-cross-B contribution dominates if . In this case, Eq. (25) together with Eq. (26) show that the increase in the parameter is independent of (as long as ), which we have confirmed by running dedicated simulations with even smaller values of the electron-to-ion temperature ratio, and (not shown in
For , which corresponds to for the parameters adopted in
he physics of electron heating is controlled by the terms in Eq. (23) and Eq. (24), with the former that typically dominates, due to its weaker dependence on . While the E-cross-B term accounts for as much as of the mean electron energy gain for , its contribution falls down to for , and it is even smaller for higher values of (see the rightmost column in Table 1). In parallel, the combination of and results in a fractional contribution to electron heating that increases from at up to at and is even larger for higher (see the leftmost column in Table 1). If the term in Eq. (23) dominates over the contribution in Eq. (24), as it happens for all the values of investigated in
ompare the first and second column in Table 1), we expect the electron fractional energy gain to increase initially as , as long as the second term in the square brackets of Eq. (23) is smaller than the first one, and then as . While the former scaling is the same as in the grad-B term of Eq. (24), the latter is shallower, which explains why for high values of the fractional contribution to electron heating by the grad-B term becomes increasingly larger, at the expense of the term in Eq. (23) (compare first and second columns in Table 1 between and ).
Finally, we point out that our model tends to overestimate the actual increase in the parameter observed for high values of (compare the red line and the filled red circle in
)). As we discuss in Appendix B, this is due to the the early decrease in the parameter that accompanies the growth of electron whistler waves. The value of becomes as low as at (dotted red line in
)), so that the subsequent increase driven by the ion cyclotron instability falls short of our analytical prediction, which does not take into account the degree of electron cooling during the whistler phase. Extrapolating this argument to even higher values of the electron-to-ion temperature ratio, one might be tempted to infer an upper limit on , where cooling by the whistler instability balances heating by the ion cyclotron mode. However, the following caveats should be considered.
As we demonstrate in Section 5.4, as increases, the ion cyclotron instability appears earlier, and in the astrophysically-relevant limit it will grow soon after the ion anisotropy exceeds the threshold in Eq. (16). Then, the condition required for the whistler instability to have a smaller threshold than the ion cyclotron mode, such that to precede the ion cyclotron growth, can be recast as a lower limit on the electron-to-ion temperature ratio
for the range of expected in accretion flows. This is already in the regime where oblique mirror modes cannot be neglected in the evolution of the system (see Paper I), so one needs to perform 2D simulations. A detailed study of electron heating in the regime — which must be performed with 2D simulations having — will be presented elsewhere.
|Note: We fix , , and .|
5.2. Dependence on the Ion Plasma Beta
In this section, we examine the dependence of our results on the ion plasma beta , for a representative case with fixed , , and . We explore the range , but we argue that our results apply down to .
e show that the level of magnetic and electric energy (panel (d) and (e), respectively) resulting from the ion cyclotron instability increases monotonically with (from black to red as varies from 5 up to 80). Once normalized to the mean field energy — whose temporal evolution is shown with a dot-dashed black line in
) — we find that for (green line in
)), the magnetic energy in ion cyclotron waves reaches