Electron acceleration and heating influenced by whistler wave packets at quasi-parallel shock waves

Authors


Abstract

[1] The acceleration and heating of electrons at quasi-parallel shock waves are studied by means of a one-dimensional full particle computer simulation. Our simulation shows that the ion beam instability due to the anomalous cyclotron resonance excites whistler mode waves in the upstream region. When the Mach number becomes large beyond a critical value, the whistler wave packets do not appear. The electron acceleration parallel to the magnetic field results from the parallel electric fields caused by both the whistler mode waves and the electrostatic shock potential. The potential concerning the parallel electric field increases with Mach number below the critical Mach number but is relatively independent of the Mach number beyond the critical Mach number. This verifies that the contribution of the whistler waves to the parallel acceleration is as important as that of the electrostatic shock potential below the critical Mach number. Also, the spatial profile of the potential concerning the parallel electric field is clearly correlated with the magnetic field profile. Downstream, the electron temperature is anisotropic such that the parallel temperature is larger than the perpendicular temperature. The Mach number dependence of the electron parallel temperature can be evaluated from the viewpoint of a wave-particle interaction (current-driven instability).

1. Introduction

[2] Our understanding of physical phenomena at collisionless shock waves in plasmas, such as the Earth's bow shock, has been developed rapidly for the past few decades, by both spacecraft observations and computer simulation studies. The acceleration and heating mechanisms associated with energy dissipation of plasma particles, and the generation mechanism for plasma waves due to the wave-particle interactions are thought to be fundamental problems in collisionless shocks. In particular, many questions concerning quasi-parallel shock waves (for which θBn < 45°, where θBn is the angle between the normal to the shock front and the external magnetic field) still remain to be settled.

[3] The macroscopic structure of shock waves can be generally evaluated in terms of the scale length of ion motions. Numerous hybrid simulations, in which ions are treated as particles and electrons as a fluid, have been made to examine the shock structure and plasma waves related to the ion dynamics. In these studies, the cyclic behavior of quasi-parallel shock waves [Burgess, 1989; Thomas et al., 1990], the re-formation process of shock fronts [Scholer and Terasawa, 1990; Winske et al., 1990; Lyu and Kan, 1990], and the Mach number dependence of shock structures [Kan and Swift, 1983; Omidi et al., 1990; Krauss-Varban and Omidi, 1991] have been examined. Furthermore, several articles have examined the relation between ion motion and plasma waves in the upstream region [Mandt and Kan, 1985; Omidi and Winske, 1990; Onsager et al., 1991], and downstream ion heating [Quest, 1988; Mandt and Kan, 1990]. It is well known that the whistler mode waves are observed not only at the Earth's bow shock [Hoppe and Russell, 1980; Tsurutani and Rodriguez, 1981] but in a cometary environment as well [Tsurutani and Smith, 1986]. Also, computer simulations based on the hybrid model have succeeded in reproducing these whistler mode wave packets.

[4] Recent spacecraft observations have measured the electron temperature and the velocity distribution at the Earth's bow shock. These studies have examined the dependence of the electron temperature on other physical quantities, such as the shock potential [Schwartz et al., 1988; Hull et al., 2000] and the speed of the solar wind [Thomsen et al., 1987, 1993]. It has been also found that the electron velocity distribution parallel to the ambient magnetic field is flat-topped [Feldman et al., 1983; Thomsen et al., 1985]. On the other hand, the plasma wave receiver with high time resolution onboard Geotail spacecraft has revealed that the intense plasma waves observed around the bow shock region are closely related to electron dynamics [Matsumoto et al., 1997]. Those measurements suggest that there exist significant interactions between electrons and plasma waves in the bow shock region. It is, therefore, essential to use full particle simulation methods rather than hybrid simulation methods in order to investigate physical mechanisms for electron acceleration and heating, and wave-particle interactions associated with electron dynamics. Only a few full particle simulation studies have so far been made to examine properties of quasi-parallel shock waves. By means of a one-dimensional particle simulation, Pantellini et al. [1992] verified that electrons scarcely affect the re-formation process of shock waves. Quest et al. [1983] found that electrons are heated adiabatically across low Mach number quasi-parallel shock waves.

[5] In this paper, using a one-dimensional full particle simulation code, we examine the acceleration and heating mechanisms of electrons at quasi-parallel shock waves and their dependence on the Alfvén Mach number from MA ≃ 2 to MA ≃ 11. We express the strength of shock waves in terms of the Alfvén Mach number MA defined by the ratio of the speed of the shock front to the Alfvén speed VA in the upstream region (VA = B0/equation image where B0, μ0, mi, and n0 are the background magnetic field, the magnetic permeability of vacuum, the ion mass, and the ion density, respectively). One of our significant results is that the whistler wave packets excited in the upstream region play an important role in the electron acceleration parallel to the magnetic field. While it has been pointed out that the wave packets can accelerate some electrons up to suprathermal velocities [Pantellini et al., 1992], we give a more detailed explanation of the physical mechanism for the electron acceleration due to the whistler wave packets in this paper.

[6] The outline of this paper is as follows. We describe our simulation model in Section 2. In Section 3, our results are presented in the following order: First, generation of electromagnetic wave packets in the upstream region is discussed. Next, electron acceleration parallel to the local magnetic field in the upstream region is demonstrated. We look closely at the origin of the electric field parallel to the local magnetic field. Then, electron heating through quasi-parallel shock waves is investigated. Our results are summarized in Section 4.

2. Simulation Model

[7] Our electromagnetic simulation code is based on the KEMPO (Kyoto university ElectroMagnetic Particle cOde), which has been developed by the space group at the Radio Science Center for Space and Atmosphere, Kyoto University [Matsumoto and Omura, 1985; Omura and Matsumoto, 1993]. The equations solved by our code are the equations of motion of plasma particles (electrons and ions), and Maxwell's equations. We use an explicit simulation scheme based on the particle-in-cell (PIC) method for time advancing of these equations. In this study, we carry out a one-dimensional calculation which has one space and three velocity components.

[8] We set up a simulation region bounded by x = 0, L2 (Figure 1). Initially, the plasma particles have a uniform density distribution in the region between x = L1 and L2, where L2 is much larger than L1. The region 0 ≤ xL1 is vacuum. The uniform external magnetic field lies on x – y plane [B0 = (B0x, B0y, 0)]. We drive shock waves in the plasma region by using a magnetic piston method [Forslund et al., 1984; Lembege and Dawson, 1986]. A large current added in the z direction at the position x = Lp drives large amplitude magnetosonic waves which propagate along the x axis. The plasma particles perform an E × B drift in the x direction, which is caused by the electromagnetic field of the waves. As the waves propagate, particles are pushed in turn at the leading edge of the magnetosonic wave front. A shock ramp, which propagates in the positive x direction, can be formed in front of the driven magnetosonic waves. We set artificial damping regions at both edges (x = 0, L2) of the calculational region, in which all waves are damped numerically. The plasma particles are perfectly reflected at the positions, x = L1 and L2.

Figure 1.

Schematic plot of the simulation system. The region, 0 ≤ xL1, is vacuum and plasma particles are distributed uniformly in the region L1xL2 at t = 0. A large current is added at x = Lp to excite magnetosonic waves with large amplitude.

[9] We assume the positions are L1 = 2.56(cpi), L2 = 81.92(cpi), and Lp = 1.26(cpi), where c is the speed of light and ωpi is the ion plasma frequency defined as $\sqrt{q_{i}^{2}n_{0}/\epsilon_{0}m_{i}}$ (qi and ϵ0 are the ion charge and the dielectric constant of vacuum, respectively). The simulation domain (0 ≤ xL2) is divided into 8192 grids and the uniform grid separation is Δx = 10−2(cpi). The loaded number of superparticles is 4 × 106 for ions and electrons. The initial velocity distribution of each species is assumed to be Maxwellian with spatially constant and isotropic temperature. The time step for integrating the fundamental equations is Δt = 0.004ωpi−1. The angle between the shock normal (x direction) and the external magnetic field can be expressed by θBn = tan−1(B0y/B0x). Especially, we investigate quasi-parallel shock waves with the angle θBn = 30° in this paper.

3 Results

[10] The typical simulation parameters are as follows: the mass ratio is mi/me = 100, the charge ratio is qi/e = −1, the ratio of the ion to electron temperature is Ti/Te = 1, the electron thermal velocity is vTe = 0.1c, and the electron cyclotron frequency defined by the external magnetic field B0 is Ωe = 2.5ωpi. Under these parameters, the Alfvén speed and electron beta are VA = 0.025c and βe = 0.32 [βe = n0kBTe/(B02/2μ0, where kB is the Boltzmann constant], respectively. The Mach number of shock waves we treat in this paper ranges from MA ≃ 2 to MA ≃ 11. By analyzing both the magnetosonic speed and the bulk speed of plasma in the downstream region in our simulation, we have confirmed that all of the shock waves, of which the Mach numbers are in this range, are supercritical shocks under our simulation parameters.

3.1. Generation of Whistler Wave Packets

[11] In the present section, we focus on the plasma waves generated upstream of quasi-parallel shocks. Figure 2 shows the spatial profiles of the x component of the plasma bulk velocity, and each component of fields in the vicinity of the shock front at (a) MA = 5.2, t = 160ωpi−1, and (b) MA = 8.5, t = 200ωpi−1. The bulk speed of plasma is expected to change markedly at the shock ramp. In the case of low Mach number (Figure 2a), from the spatial profile of the bulk speed, the position of the shock ramp can be estimated at x ≃ 18(cpi), which moves in the positive x direction as time elapses. The left and right sides of the shock ramp correspond to the downstream and upstream regions, respectively. The field quantities in Figure 2 are drawn in the rest frame of the solar wind. Figure 2a shows that a monochromatic electromagnetic wave with large amplitude (|B|/|B0| ∼ 3) clearly appears in the upstream region. The amplitude of the wave tends to decay with distance from the shock ramp. We can recognize from the hodograph (not shown) of the electric field that this is a right-hand circularly polarized wave in the rest frame of the solar wind. The frequency and the wavelength of the wave are approximately ω ∼ 0.5ωpi and λ ∼ cpi, respectively. One can presume that these wave packets are whistler modes, which are often observed in the upstream region of the Earth's bow shock or near the comet Giacobini-Zinner. We do not see similar electromagnetic waves in the case of high Mach number (Figure 2b), where the position of the shock ramp is at x ≃ 34(cpi). There is another important point in Figure 2b. The compression ratio of the magnetic field in the downstream region becomes larger than 4 (|B|/|B0|>4) in our simulation for high Mach number shocks. This means that the high Mach number shocks in our calculations are not in the steady state described by Rankine-Hugoniot equations. In other words, our high Mach number shocks are in a turbulent-like state, and are dynamically evolving during the calculations. The physical quantities for high Mach number shocks tend to vary temporally and spatially. We determine some quantities plotted in figures of this paper as averaged values, and use error bars to represent the temporal and spatial variances.

Figure 2.

Spatial profiles of the x component of the plasma bulk velocity, the y and z component of the magnetic field By and Bz, and three components of the electric field Ex, Ey, and Ez at tωpi = 160, (a) MA = 5.2 and (b) 8.5.

[12] Figure 3 displays the stacked plots of the y component of the magnetic field at (a) MA = 5.2 and (b) MA = 8.5. The position of the shock front shown in each figure corresponds to the middle of the shock ramp. As we described earlier, the position is determined by the spatial variance of the bulk speed of plasma. The region with large amplitude on the left side in each figure (red-colored region), corresponds to the magnetic field of the magnetosonic wave for exciting shock waves. The downstream region of the shock wave is formed gradually in front of this region as time elapses. It is found in Figure 3a that the whistler wave packets propagating in the x direction grow in the upstream region as the shock front moves in the x direction. The downstream region is gradually formed as time elapses also in the case of high Mach number as shown in Figure 3b. We do not, however, see significant growth of waves with large amplitude in the upstream region of the shock given in Figure 3b.

Figure 3.

Stacked profiles of the magnetic field By at (a) MA = 5.2 and (b) 8.5. The spacing between profiles is tωpi = 4.

[13] A considerable number of studies have been made on the generation mechanism of these whistler wave packets [e.g., Goldstein and Wong, 1987; Brinca and Tsurutani, 1988; Gary and Madland, 1988; Kojima et al., 1989]. The wave packets in our simulation result from the nonlinear evolution of whistler waves excited by the ion-ion beam instability [Gary et al., 1984; Quest, 1988]. Figure 4 shows the temporal evolution of the phase space plots of the x component of the ion velocity in the same case as Figure 3a. The ions reflected back toward the upstream region interact with the background plasma (Figure 4a). When the speed of the reflected ions matches the anomalous cyclotron resonance condition, the ion beam instability grows and whistler modes are generated. The resonance condition is

equation image

where vr and Ωi denote the x component of the reflected ion velocity and the cyclotron frequency of background ions, respectively. We can recognize the ions are trapped by the whistler wave packets as the amplitude of the waves becomes large (Figures 4b4f).

Figure 4.

Time sequential plots of the x–vx phase space of ions at (a) tωpi =40, (b) 64, (c) 80, (d) 104, (e) 120, and (f) 144 for the same case as Figure 3a.

[14] It must be noted that there exist three resonance frequencies in the whistler mode branch which are satisfied with the resonance condition (1). However, we neglect the highest resonance frequency which is almost equal to the electron cyclotron frequency since the wave with its frequency is strongly damped. Concerning the other two resonance points, we call the higher (H) and lower (L) frequency unstable modes as the RH and RL mode, respectively. The whistler wave packets in our simulation result from the nonlinear evolution of the RH mode. The excitation of the RL mode is not important because the growth rate of the RL mode is much smaller than that of the RH mode [Kojima et al., 1989].

[15] Figure 5 indicates the beam speed dependence of the maximum growth rate for the RH mode. The result is obtained by using KUPDAP (Kyoto University Plasma Dispersion Analysis Package) program, which numerically solves the Vlasov-Maxwell equations. We can see that the instability no longer evolves when the speed becomes large beyond a critical value. This means that there exists an upper bound of the beam speed for the resonance condition of the RH mode. In Figure 6, we show the Mach number dependence of the averaged speed of reflected ions obtained from our simulation results. It is clear that the speed of reflected ions tends to increase as the Mach number increases. From the results shown in Figures 5 and 6, we can estimate the critical Mach number (Mc) for the generation of the RH mode as Mc ≃ 6, which is equivalent to the critical resonance speed vr/c ≃ 0.17.

Figure 5.

Maximum growth rate for the ion beam instability as a function of ion beam speed.

Figure 6.

Dependence of the reflected ion speed vr/c on the Mach number MA.

[16] We can roughly estimate the range of the beam speed, in which the RH mode destabilizes, with a cold plasma approximation. The dispersion relation for whistler mode waves in cold plasmas is written as,

equation image

Using (1), we can remove k from (2) and obtain the equation as,

equation image

Under the assumption, vrc and memi cos θBn, we can obtain the condition that the Equation 3 has at least two solutions as follows,

equation image

This formula yields the following condition for the growth of the whistler wave packets in our simulation,

equation image

The upper bound of the beam speed in (4) is less than that obtained from the linear analysis (Figure 5) and the simulation result (Figure 6). However, taking into account that we have applied a cold plasma approximation to derive (3), one can say that the analytical result agrees well with the simulation result.

3.2 Electron Acceleration

[17] In this section, we investigate electron acceleration along the local magnetic field in the upstream region. Figure 7 displays the spatial profiles of the magnetic field |B| (top), the parallel electric field (second from the top), the potential of the parallel electric field (third from the top), and parallel component of electron velocity (bottom) for the same case as Figure 2, where the term “parallel component” means the component which is parallel to the local magnetic field. The parallel potential is defined as

equation image

which describes the spatial variation of the parallel electric field. We notice that there exists a clear correlation of the magnetic field and the parallel potential, i.e., |▽B| ∼ |▽ϕ|, in Figure 7. Satellite observations have verified the same property between the magnetic field intensity and de Hoffmann-Teller potential at the Earth's bow shock [Hull et al., 2000].

Figure 7.

Spatial profiles of the magnetic field (top), the electric field parallel to the magnetic field (second), and the parallel potential (third), and x − v phase space of electrons (bottom) for the same case as Figure 2.

[18] In the case of low Mach number (Figure 7a), we can see from the bottom panel that the ensemble averaged value of the electron drift velocity parallel to the magnetic field is zero,<v> = 0, in the far upstream region [x/(cpi) > 24], and it takes a positive value in the region between the leading edge of the wave packet and the shock front [18 ≤ x/(cpi) ≥ 24]. This means that there exist electrons with large positive parallel velocity, which correspond to the electrons accelerated along the local magnetic field in the region where the whistler wave packet exists. Since the increment in the parallel velocity of the electrons results from the parallel electric field, we can determine the region, in which the acceleration occurs, from the spatial profile of the potential ϕ. It varies gradually through the broad region along the x axis (identified as R1 in Figure 7a). Also in Figure 7b, we can recognize the electron acceleration along the magnetic field in the vicinity of x ≃ 35(cpi). The region, in which the electron acceleration occurs, is narrower (identified as R2 in Figure 7b) than that in Figure 7a.

[19] Figure 8 displays three contour plots of electron velocity distributions in the v–|v| plane for the case of (a) MA = 3.0, (b) 5.2, and (c) 11.0, respectively, where v denotes the velocity component perpendicular to the local magnetic field. Each electron distribution is obtained at the position of the shock front at t = 160ωpi−1, when the shock structure is well developed, where we define the shock front as the middle of the shock ramp. We construct the distribution functions by using ten cells, which include approximately 500 particles. In each panel, we can see the deviation of the position of the peak in the positive v direction. This deviation results from the electron acceleration parallel to the magnetic field. Comparing the parallel drift speed in Figure 8a with that in Figure 8b, it is found that it tends to be large with increasing Mach number. On the other hand, comparison of Figure 8b with Figure 8c shows that the electron acceleration parallel to the magnetic field is suppressed in spite of the Mach number increment.

Figure 8.

Contour plots of the electron velocity distribution as a function of v and |v| at tωpi = 160 and three different Mach numbers, (a) MA = 3.0, x/(cpi) = 16, (b) MA = 5.2, x/(cpi) = 20, and (c) MA = 11.0, x/(cpi) = 48.

[20] In Figure 9, we plot the Mach number dependence of the difference between the electric potential in the upstream region and that in the downstream region. The potential difference shown in Figure 9 is equivalent to the net gain of the parallel kinetic energy of electrons which travel from the upstream to downstream region. The parallel drift speed of electrons in the downstream region can be estimated by

equation image

Using the result in Figures 9 and (7), we can evaluate the Mach number dependence of the electron drift speed. As we may expect, the parallel velocities at the peaks of electron distributions shown in Figure 8 almost coincide with the values obtained from Figure 9. The important point is that the Mach number dependence definitely tends to change its tendency in the vicinity of MA = Mc ≃ 6. As we mentioned in the previous section, this value of Mc corresponds to the critical value for the generation of whistler mode waves in the upstream region. In the case of low Mach number (MA < Mc), which corresponds to the parameter range for the generation of whistler mode waves, the potential difference tends to become large as the Mach number increases in Figure 9. This means that the electron acceleration is enhanced by the increase of the Mach number. On the other hand, in the case of high Mach number (MA > Mc), the potential difference becomes relatively independent of the Mach number.

Figure 9.

Dependence of the potential difference eΔϕ/mec2 on the Mach number MA.

[21] Let us discuss the above Mach number dependence of the potential difference by concentrating on the parallel electric field. We can divide the parallel electric field into two components, one containing only the x component of the electric field and the other containing both the y and z components of the electric field. This division is written as

equation image

Figure 10 shows the spatial profiles of these two terms for the same case as Figure 2, where each line corresponds to the total E (Line 1), the first term of (8) including Ex (Line 2), and the second term of (8) including both Ey and Ez (Line 3), respectively. Also plotted in this Figure are the potential profiles of the parallel electric field (top panels). Because of the existence of the large amplitude whistler wave packets, which propagate in the oblique direction, in the upstream region, the y and z components of the electric field in the upstream region are caused by the electric field of the whistler waves, and the x component consists of the parallel component of the whistler mode wave packets and the static electric field component which corresponds to the electrostatic potential structure in the vicinity of the shock ramp region. The origin of the electrostatic potential at quasi-parallel shock waves is still in controversy. In our simulation, it obviously results from the density difference between electrons and ions.

Figure 10.

(a, b) Spatial profiles of the parallel potential (top) and each term in (8) (bottom), where each line represents the total E (solid line 1), the term including Ex (dotted line 2), and the term including Ey and Ez (broken line 3) for the same case as Figure 2.

[22] In Figure 10a, we can see the phases of spatial oscillations of two terms on the right-hand side in (8) are opposite to each other. The amplitude of the oscillations tends to decay with distance from the shock ramp. It seems reasonable to suppose that the oscillating part of the electric field mainly consists of whistler wave packets. Due to this oscillational damping behavior, the spatial profile of the parallel potential in Figure 10a gradually increases with oscillating through the region where the whistler wave packets exist (identified as P1 in Figure 10a). The parallel velocity of electrons, therefore, begins to increase at the leading edge of the wave packets in the upstream region. It can be expected that the electrostatic field contributes to the parallel electric field only in the vicinity of the shock front, since the thickness of the shock front can be estimated to be of the order of the ion Larmor radius. In fact, in the case of high Mach number, we can recognize the increase of the potential only in the narrow upstream region (identified as P2) in Figure 10b. This increment in the parallel potential is mainly due to the shock potential.

[23] Accordingly, it is clear that the parallel component of the electric field of the whistler wave packets contributes to the increment in the parallel potential, and accelerates electrons along the magnetic field in the upstream region. Therefore, we conclude that electrons in the vicinity of the shock front are accelerated by the parallel electric fields of the whistler mode wave packets and the electrostatic shock potential. For lack of the acceleration due to the whistler mode waves beyond MA > Mc, the Mach number dependence of the potential difference at MA > Mc exhibits a moderate change shown in Figure 9.

3.3. Electron Heating

[24] In this section, we examine the properties of electron heating at quasi-parallel shock waves. In Figure 11, the magnitude of the magnetic field (top panel), the electron temperature parallel and perpendicular to the local magnetic field (middle panel), and the ratio of the parallel to perpendicular electron temperature (bottom panel) are plotted as a function of position at tωpi = 200 for (a) MA = 4.4 and (b) 8.5.

Figure 11.

(a, b) Spatial profiles of the magnetic field (top), parallel T and perpendicular T component of electron temperature (middle), and the temperature ratio T/T (bottom) at tωpi = 200 for two different values of MA: (a) MA = 4.4, there exists wave packets, and (b) MA = 8.5, there exists no wave packets in the upstream region.

[25] Here we make a few remarks concerning the numerical heating in the simulation code, KEMPO. In this code, a lack of energy conservation or numerical heating results from the discretization of a physical model in space and time. The heating rate is inversely proportional to the particle number per grid [Ueda et al., 1994]. Taking this into account, the variation of electron kinetic energy, which is defined by the ratio of the kinetic energy increment in unit time to the initial energy, due to the numerical heating can be estimated to be the order of 10−6 under our simulation parameters. Figure 11 shows that the spatial variation of electron temperature from upstream to downstream shows a larger increment than that expected by the numerical heating. It is, therefore, clear that we can discuss the physical heating of electrons quantitatively by using the simulation code, KEMPO.

[26] To begin with, we focus on the low Mach number case depicted in Figure 11a, where the wave packets appear in the upstream region. The electron temperature in the far upstream region [x/(cpi) > 30] is isotropic, and the perpendicular component tends to be larger than the parallel component in the region where the whistler mode waves with large amplitudes exist [20 ≤ x/(cpi) ≤ 29]. The clue to explain this increase of the perpendicular temperature is in the fact that the spatial profile of the perpendicular temperature exhibits a similar trend with the magnetic field. It is likely that the increase of perpendicular temperature is caused by the adiabatic heating. In other words, the perpendicular component of electron kinetic energy becomes larger than the parallel one because the electrons encounter strong magnetic fields in this region. In the vicinity of the shock ramp, the parallel temperature begins to increase. The parallel component tends to be larger than the perpendicular component towards the downstream region [x/(cpi) ≤ 24]. Although the downstream temperature ratio does not remain steady in Figure 11a, it is clear that T/T > 1 in the downstream region.

[27] Figure 12 displays three-dimensional plots of the velocity distribution f(v, |v|) of electrons at three different positions for the same case as Figure 11a. The positions, (a) x/(cpi) = 18, (b) 22, and (c) 26, correspond to the downstream region, the region where the wave packets exist, and the upstream region, respectively. The electron velocity distribution for the parallel component has a flat-topped form in the downstream region (Figure 12a), whereas f(|v|) is nearly Maxwellian-like everywhere downstream. This anisotropic characteristic of the distribution function is consistent with the anisotropy of the electron temperature shown in Figure 11a. The beam component shown in Figure 12b corresponds to accelerated electrons, which we have discussed in the previous section. Upstream, f(v) is an isotropic Maxwellian (Figure 12c).

Figure 12.

Three-dimensional plot of the electron velocity distribution as a function of v and |v| at tωpi = 160 and MA = 5.2. The positions are (a) x/(cpi) = 18 (downstream), (b) 22 (upstream in the vicinity of the shock front), and (c) 26 (upstream).

[28] Turning now to the high Mach number case (Figure 11b), the profiles of temperature components indicate that the temperature anisotropy appears not only in the downstream region but also in the upstream region at high Mach number shocks. As we have already shown in Figure 7b, the electron distribution in phase space becomes broader in the upstream region than the initial distribution as the shock develops. This corresponds to the fact that there exist electrons transmitted steadily from the shock front to the upstream region in the case of high Mach number. Although the origin of these electrons is unclear, we have confirmed that the speed of these electrons tends to increase with increasing Mach number. Thus, the parallel temperature of electrons becomes larger than the perpendicular temperature in the upstream region. In addition to the anisotropy of the electron temperature in the downstream region, we can see in Figure 11b that the spatial profile of the perpendicular component of the electron temperature exhibits a similar trend with the profile of the magnetic field, as we have already seen in Figure 11a.

[29] Figure 13 shows the Mach number dependence of electron temperature component jumps between upstream and downstream, where the temperatures are normalized by the initial electron temperature Te0. The open circles and the closed circles designate the temperature component parallel and perpendicular to the local magnetic field, respectively. We focus on the parallel component of electron temperature. The temperature increases sharply with increase of the Mach number until MA ∼ 6, and remains almost steady in the range of MA > 6. A considerable number of studies have been made on the temperature change of electrons at perpendicular shock waves [Scudder et al., 1986a; Scudder, 1995; Hull et al., 1998]. These researches have revealed that the temperature change can be explained completely by the adiabatic motion of electrons in macroscopic fields. Krauss-Varban et al. [1995] confirmed the adiabatic behavior of electrons at perpendicular shock waves by means of an implicit particle simulation. Furthermore, it was demonstrated that the macroscopic electric field primarily affects the parallel temperature at perpendicular shocks [Scudder et al., 1986b]. From our simulation results of quasi-parallel shocks, it must be noted that there is a difference concerning the Mach number dependence between the temperature jump and the macroscopic potential jump, comparing Figure 13 with Figure 9. Although the potential jump has a sharp peak at MA ∼ 9 in Figure 9, the temperature jumps of both parallel and perpendicular components have no such peaks in Figure 13.

Figure 13.

Dependence of the electron temperature difference ΔTe/Te0 on the Mach number.

[30] Feldman et al. [1982] suggested that the parallel temperature of electrons is determined by microinstabilities due to the accelerated electrons parallel to the magnetic field. Our simulations support this picture because, as shown in the previous section, electrons are accelerated along the magnetic field near the shock front. Thus, there appears a difference of the parallel drift speed between electrons and ions in this region. This velocity difference can trigger the current-driven instability, and ion-acoustic waves are excited in our simulations [Nakao, 1999].

[31] In order to discuss the Mach number dependence of electron temperature shown in Figure 13, we use the KUPDAP program and carry out a linear analysis concerning the current-driven instability. We focus on the region where the difference of drift speed between electrons and ions (e.g., 21 < x/(cpi)< 23 in Figure 7a), and assume the distribution functions are a shifted Maxwellian with the drift speed vd for electrons, and a Maxwellian distribution at rest for ions. Also, it is assumed that the electron and ion temperatures take the upstream value. Figure 14 shows the electron drift speed dependence of the linear growth rate for the current-driven instability. From the potential difference shown in Figure 9 with (7), we can estimate the drift speed electrons obtain across shock waves. It is found that the electron drift speed varies from vd/c ∼ 0.2 to 0.8 with varying the Mach number from MA ∼ 2 to MA ∼ 11. This range of the drift speed corresponds to a range of growth rates from γ/ωpi ∼ 0.25 to 1.25 in Figure 14. Considering the wavelength of ion-acoustic waves excited by the current-driven instability to be λ ∼ cpi in the linear analysis, we can say that electrons spend enough time λ/vd ∼ 5 > 1/γ in the transition region to destabilize ion-acoustic waves.

Figure 14.

Maximum growth rate for current-driven instability as a function of electron drift speed.

[32] The important point in Figure 14 is that the growth rate does not significantly change at vd/c > 0.5. The potential difference which is equal to the electron kinetic energy of the velocity vd/c ∼ 0.5 can be estimated at eΔϕ/mec2 ∼ 0.12 from Equation 7. As shown in Figure 9, the value of the potential difference becomes larger than 0.12 at MA > 5. Accordingly, we can expect that the growth rate of the current-driven instability hardly changes at MA > 5. Thus, we may say that the electron heating due to the current-driven instability tends to be independent of the Mach number at MA > 5 as shown in Figure 13.

4. Summary

[33] We have investigated the mechanisms for the electron acceleration and heating at quasi-parallel shock waves by means of a one-dimensional full particle simulation. The Alfvén Mach numbers of shock waves observed in our simulations are in the range from MA ≃ 2 to MA ≃ 11. The main results are summarized as follows:

  1. In the case of low Mach numbers (MA < Mc), large amplitude whistler wave packets are generated upstream by the reflected ions which satisfy the anomalous cyclotron resonance condition ω = kvr − Ωi.
  2. In the case of high Mach numbers (MA > Mc), whistler wave packets do not appear upstream since the speed of reflected ions becomes too high to satisfy the anomalous cyclotron resonance condition.
  3. The electric field of both the whistler mode and the electrostatic potential gives rise to electron acceleration parallel to the local magnetic field in the vicinity of the shock front. The Mach number dependence of the electron acceleration no longer increases on reaching the critical Mach number (MA = Mc).
  4. In the downstream region, the electron temperature tends to be anisotropic with the parallel temperature larger than the perpendicular temperature.

[34] Finally, we shall compare our simulation results with observations. As we have already mentioned in Section 3.2, our results show that the spatial profile of the parallel potential is closely correlated with that of the magnetic field. A similar trend can be seen also in results observed at perpendicular shock waves [Hull et al., 2000], where the trend between the de Hoffmann-Teller potential and the magnetic field has been pointed out. The de Hoffmann-Teller potential is essentially the same quantity as the parallel potential in our simulation results. The important point is that there is a correspondence between our simulation (parallel shocks) and the observation (perpendicular shocks), regardless of the difference of the shock property.

[35] Thomsen et al. [1987] have found a clear correlation between the electron temperature jump ΔTe and ΔV2, where ΔV2 is the total change in the bulk flow energy across the shocks, from observations mainly on perpendicular shock waves. The flow jump ΔV2 tends to be large as the Mach number increases in our simulation. However, as we have discussed in Section 3.3, the same dependence on the Mach number is not true for ΔTe. When the Mach number increases beyond the critical value, the electron temperature hardly changes in Figure 13.

[36] Also, the relation between the potential jump in the de Hoffmann-Teller frame and ΔTe has been discussed in the observational results on perpendicular shock waves [Schwartz et al., 1988; Hull et al., 2000]. These researches reveal that the potential jump is clearly proportional to ΔTe. Figure 15 indicates the correlation between Δϕ and ΔTe in our simulation. The dotted line in Figure 15 has a slope of 2, which corresponds to the proportionality factor clarified in the observational results [Hull et al., 2000]. It is found also in our simulations that there appears a positive correlation between Δϕ and ΔTe. However, we cannot exactly say that it has a proportional tendency. As we have discussed in Section 3.3, the electron heating due to the current-driven instability can explain the Mach number dependence of the parallel electron temperature change shown in Figure 13. On the other hand, the electrostatic shock potential mainly affects the electron heating in the case of perpendicular shock waves [Scudder et al., 1986b]. Thus, it might be natural that there appears the difference between our simulation and the observational result. We can see in Figure 15 that the proportionality factor of our results becomes close to the value of the observation as the potential jump increases. This possibly means that the contribution of the macroscopic electric potential to the electron heating might become dominant rather than the wave-particle interactions if the potential jump becomes larger than the values we treated in this paper. However, we need more accurate calculations to answer the question as to which primarily affects the electron heating at shock waves, wave-particle interactions or macroscopic fields. It should be discussed in detail in the future research.

Figure 15.

Scatterplots of the potential difference eΔϕ/mec2 versus the temperature difference kBΔTe/mec2.

Acknowledgments

[37] The authors gratefully acknowledge helpful discussions with Y. Omura, K. Hashimoto, H. Usui, Y. Kasaba, and M. S. Nakamura, as well as useful comments from S. P. Gary. This work has been carried out under the COE program granted for one of the authors (K.N.). Computation in the present study was performed with the KDK system of Radio Science Center for Space and Atmosphere (RASC) at Kyoto University as a collaborative research project.

[38] Janet G. Luhmann thanks David Schriver and another referee for their assistance in evaluating this paper.

Ancillary