RESEARCH ARTICLE Key Points: Electron acceleration (>100 kev) in magnetotail reconnection is due to perpendicular electric field Dependence of the parallel potential on physical parameters is derived Detail processes of pitch angle scattering in a reconnection exhaust are unveiled Correspondence to: N. Bessho, naoki.bessho@nasa.gov Citation: Bessho, N., L.-J. Chen, K. Germaschewski, and A. Bhattacharjee (2015), Electron acceleration by parallel and perpendicular electric fields during magnetic reconnection without guide field, J. Geophys. Res. Space Physics, 120, 9355 9367, doi:. Received 11 JUN 2015 Accepted 12 OCT 2015 Accepted article online 16 OCT 2015 Published online 4 NOV 2015 Electron acceleration by parallel and perpendicular electric fields during magnetic reconnection without guide field N. Bessho 1,2, L.-J. Chen 1,2, K. Germaschewski 3, and A. Bhattacharjee 4 1 Department of Astronomy, University of Maryland, College Park, Maryland, USA, 2 Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, Maryland, USA, 3 Space Science Center, University of New Hampshire, Durham, New Hampshire, USA, 4 Center for Heliophysics and Princeton Plasma Physics Laboratory, Princeton University, Princeton, New Jersey, USA Abstract Electron acceleration due to the electric field parallel to the background magnetic field during magnetic reconnection with no guide field is investigated by theory and two-dimensional electromagnetic particle-in-cell simulations and compared with acceleration due to the electric field perpendicular to the magnetic field. The magnitude of the parallel electric potential shows dependence on the ratio of the plasma frequency to the electron cyclotron frequency as (ω pe Ω e ) 2 and on the background plasma density as n 1 2. In the Earth s magnetotail, the parameter ω b pe Ω e 9 and the background (lobe) density can be of the order of 0.01 cm 3, and it is expected that the parallel electric potential is not large enough to accelerate electrons up to 100 kev. Therefore, we must consider the effect of the perpendicular electric field to account for electron energization in excess of 100 kev in the Earth s magnetotail. Trajectories for high-energy electrons are traced in a simulation to demonstrate that acceleration due to the perpendicular electric field in the diffusion region is the dominant acceleration mechanism, rather than acceleration due to the parallel electric fields in the exhaust regions. For energetic electrons accelerated near the X line due to the perpendicular electric field, pitch angle scattering converts the perpendicular momentum to the parallel momentum. On the other hand, for passing electrons that are mainly accelerated by the parallel electric field, pitch angle scattering converting the parallel momentum to the perpendicular momentum occurs. In this way, particle acceleration and pitch angle scattering will generate heated electrons in the exhaust regions. 2015. American Geophysical Union. All Rights Reserved. 1. Introduction Energetic electrons (> 100 kev) have been observed in the Earth s magnetotail during magnetic substorms. Wind spacecraft first detected 300 kev electrons in the diffusion region of magnetic reconnection [Øieroset et al., 2002]. The phase-space density of the energetic electrons had a peak near the center of the diffusion region, and the intensity decreased after the spacecraft passed through it. This suggested that an X point might be a source location of the observed electron acceleration (for simulation studies, see Fu et al. [2006], Pritchett [2006], and Huang et al. [2010]). Later, Imada et al. [2007] found by Cluster observations that electrons in the downstream exhaust region, where magnetic fields pile up, were more energetic than those in the vicinity of an X point. The observation was qualitatively consistent with the two-step acceleration model proposed by Hoshino et al. [2001], where electrons are first accelerated near an X point and are accelerated further in the magnetic field pileup region due to the grad B drift and the curvature drift parallel to the motional electric field. Huang et al. [2015] also studied electron acceleration in a magnetic pileup region (dipolarization front) by betatron acceleration. Other Cluster observations [Chen et al., 2008a; Wang et al., 2010a, 2010b] revealed electron energization in excess of 100 kev in magnetic islands that can be generated during magnetic reconnection. In principle, if islands contract in the direction of the reconnection outflows, magnetized electrons can be energized by Fermi acceleration due to multiple bounces in the islands [Drake et al., 2006]. Fu et al. [2006] also demonstrated that multiple bounces are important to energize electrons in a magnetic island with a similar mechanism as Fermi acceleration, even when electrons are unmagnetized and nonadiabatic in reconnection without a guide field. These mechanisms above are mostly due to work done by the out-of-plane electric field, which is basically perpendicular to the background magnetic field unless there is a strong guide field. BESSHO ET AL. ELECTRON ACCELERATION IN RECONNECTION 9355
Recently, Egedal et al. [2012] proposed that electrons can be accelerated in broad regions during reconnection by the parallel electric field E along the magnetic field in the exhaust region. They demonstrated by a particle-in-cell (PIC) simulation that the parallel electric potential φ, defined as the integral of E along a magnetic field line, becomes large in the exhaust region, and they argued that many electrons can be accelerated in the broad region that extends more than several tens of the ion skin depth from the X line. They used ω pe Ω e = 2 in their simulation, where ω pe and Ω e are the plasma frequency and the electron cyclotron frequency, respectively, defined with the density n 0 in the initial current sheet and the asymptotic magnetic field B 0. They showed that the magnitude of the parallel potential becomes eφ 0.7m e c 2 in their simulation, where e is the elementary charge, and m e c 2 is the electron rest mass energy. However, this value of eφ is unrealistically large for the Earth s magnetotail, since their value ω pe Ω e = 2 is significantly smaller than the observed value around 9 in the Earth s magnetotail [Chen et al., 2012]. They renormalized the magnitude of eφ ( 0.7m e c 2 ) by T eb, which is the electron temperature in the background lobe plasma, and their result showed that eφ 90T eb, which is consistent with the Cluster observation of the flat-top distribution that extends up to 14 kev, when one uses T eb about 200 ev. However, electron energization in excess of 100 kev cannot be accounted for by the mechanism, which relies on the parallel electric field. One of the questions we will address in this paper is how electrons can be accelerated to much higher energies than can be produced by the parallel electric potential. We will discuss the scaling of the parallel electric potential by the parameter ω pe Ω e and then compare the mechanisms of electron acceleration due to parallel and perpendicular electric fields. In addition to the ratio ω pe Ω e, the background lobe density n b is an important parameter for fast reconnection and particle acceleration. PIC simulations have demonstrated that reconnection in the low-density regime (n b 0.01n 0 ) proceeds much faster than that in the high-density regime (n b 0.1n 0 )[Bessho and Bhattacharjee, 2010a, 2010b; Wu et al., 2011]. As a result, a strong reconnection electric field E y 1B 0 v A0 c (comparing to a typical 0.1B 0 v A0 c), where v A0 is the Alfvén speed based on B 0 and n 0, and c is the speed of light, is formed during reconnection, and it can also enhance the parallel electric potential. In the Earth s magnetotail, the lobe density is around 0.01 to 0.1 cm 3 [Svenes et al., 2008], and the density in the current sheet before reconnection starts is around 1 cm 3 [Chen et al., 2012]; therefore, reconnection in the low-density regime is expected. Le et al. [2010] derived a formula for eφ as a function of the upstream electron beta β e ( n b T eb ) from their proposed equations of state. However, the physical origin of the dependence of φ on n b remains unclear. How large the parallel electric potential becomes in the low-density regime is an important question for electron acceleration in the magnetotail. This paper addresses the problem of electron acceleration in antiparallel reconnection (no guide field), due to components of the electric field both parallel and perpendicular to the background magnetic field by means of scaling arguments and two-dimensional (2-D) electromagnetic PIC simulations. Section 2 describes a theoretical estimation of the scaling of the parallel electric potential. Section 3 describes our simulation method and parameters. In section 4, we show simulation results. In section 5, we discuss the contribution of the parallel and perpendicular electric fields to electron acceleration. In section 6, we discuss pitch angle scattering during acceleration. In section 7, we summarize this study. 2. Scaling of the Parallel Electric Potential We consider 2-D antiparallel magnetic reconnection, where the reconnecting component of the magnetic field is B x, and z = 0 is a neutral plane. The parallel electric potential φ, defined as the integral of the parallel electric field E along a magnetic field line [Egedal et al., 2012], is given as x x φ = E dl = E dl, (1) where the infinitesimal length dl is along the field line and the integral is from infinity to a point x on the field line. In 2-D reconnection where y = 0, E y is the electromagnetic field due to (1 c) A y t, where A y is the y component of the vector potential; therefore, equation (1) contains both electrostatic and electromagnetic contributions. If we assume that the electromagnetic component of the electric field is dominated by E y (in other words, the electromagnetic component in E x and E z is negligible, which has been confirmed by our simulation results), we have y φ = φ E y dy l, (2) BESSHO ET AL. ELECTRON ACCELERATION IN RECONNECTION 9356
where the first termφ is the electrostatic potential, the integral in the second term is along the magnetic field line, y is the y component of the position x on the field line, and dy l is the infinitesimal length in the y direction along the field line. The parallel electric potential φ is composed of two terms by the electrostatic potential φ, which we assume comes from E x and E z, and the integral of the electromagnetic field E y. Let us discuss the scalings for both terms. The electrostatic potential φ acquires a negative value in the reconnection region. There is a bipolar electric field E z across z = 0 in the vicinity of the X line, and E z u ye B x c, where u ye is the y component of the electron fluid velocity, and the effect of the pressure tensor on E z is negligibly small [Chen et al., 2008b]. In the thin electron current layer, the current density J y neu ey, where n is density. Then Ampere s law B x z (4π c)j y and the relation E z u ye B x c give φ z (1 8πne) B 2 z. Therefore, we obtain the electrostatic potential x in the vicinity of the X point as ( ωpe0 φ m ) i 2e v2 A = m e c2 n 2 0, (3) 2e n Ω e0 where m i and m e are the ion mass and the electron mass, respectively, v A is the Alfvén speed based on the asymptotic magnetic field B 0 and the density in the current sheet n, and ω pe0 and Ω e0 are the plasma frequency based on n 0 (the initial density in the current sheet) and the electron cyclotron frequency based on B 0, respectively. Next, let us estimate the second term on the right-hand side of equation (2), y E y dy l. Although we cannot determine its exact scaling, we can establish the proportionality of this integral to (ω pe0 Ω e0 ) 2. Let us assume that E y is mostly localized in the diffusion region and in the magnetic pileup region (the characteristic dimension of the region with a large E y is of the order of several tens of the ion skin depth), and the overall magnitude of E y in the region considered is similar to that of the reconnection electric field (despite the presence of local intermittent structures), which is E y R r B 0 v A0 c, where R r is the reconnection rate and R r 0.1 to 1 depending on n b n 0 (where n b is the background lobe density) [Bessho and Bhattacharjee, 2010a; Wu et al., 2011], and v A0 is the Alfvén speed based on B 0 and n 0. If we integrate this field along a magnetic field line, E y is multiplied with the y projection of the field line length, which may be written as αd i0, where α represents a factor of order 10 and d i0 is the ion skin depth based on n 0. We obtain y E y dy l R r αd i0 B 0 v A0 c = R r α m e c2 e ( ωpe0 Ω e0 ) 2. (4) Therefore, both the first and the second terms in equation (2) are proportional to (ω pe0 Ω e0 ) 2, according to equations (3) and (4). In addition, as we will see in section 4, the electromagnetic component (equation (4)) dominates the parallel electric potential; therefore, φ is expected to depend on the reconnection rate R r, which also depends on the ratio (n b n 0 ). 3. Simulation Method We have performed 2-D, electromagnetic, PIC simulations to study magnetic reconnection without guide field. The details of our simulation codes are given in Bessho and Bhattacharjee [2007] and Fox et al.[2012]. The simulation domain is in the x-z plane, L x 2 < x < L x 2, and L z 2 < z < L z 2. The initial magnetic field and the density have the following forms: B x = B 0 tanh(z w), (5) n = n 0 sech 2 (z w)+n b, (6) where w is the width of the current sheet. We use conditions B 2 0 8π = n 0 (T i +T e ), v di v de =(2c web 0 )(T i +T e ), and v di v de = T i T e, where T i and T e are the ion and electron temperatures, respectively, and v di and v de represent the drift speed of ions and electrons, respectively, in the y direction. We perform two types of simulations: the first type of simulation (Type A; see Table 1 for more details) is for a small domain, L x L z = 25.6d i0 25.6d i0. In this case, all four boundaries are open boundaries, where electromagnetic waves can pass through, particles are removed when they cross the boundaries, and new particles are injected at every time step [Daughton et al., 2006]. This open boundary condition is necessary for these runs to prevent accelerated particles from recirculating in the simulation region in the x direction. BESSHO ET AL. ELECTRON ACCELERATION IN RECONNECTION 9357
Table 1. Simulation Parameter Run Type L x L z m i m e n b n 0 ω pe0 Ω e0 T i T e Runs 1 3 A 25.6d i0 25.6d i0 25 0.3 2, 4, 8 1 Runs 4 6 A 25.6d i0 25.6d i0 100 0.3 2, 4, 8 1 Runs 7 9 A 25.6d i0 25.6d i0 100 0.15, 0.075, 0.0375 2 1 Runs 10 12 B 204.8d i0 25.6d i0 400 0.05 2, 4, 8 5 The second type of simulation (Type B in Table 1) is for a large system size, L x L z = 204.8d i0 25.6d i0, where the x boundaries are periodic and the z boundaries are conducting walls. The use of periodic boundaries is allowed for this large system size in the x direction, until the effects of the reconnection outflows reach the boundaries. We use the following parameters: for Type A simulations, the sheet width w = 0.5d i0, and the temperature ratio T i T e = 1. We use two mass ratios, m i m e = 25 and 100, and change the ratio of the plasma frequency (based on n 0 ) to the electron cyclotron frequency (based on B 0 ), ω pe0 Ω e0, from 2 to 8 (Runs 1 to 6). We also change the background density n b from 0.3n 0 to 0.0375n 0 (Runs 4 and 7 9). The grid numbers are 512 grids for 25.6d i0 (or 1024 grids for Run 6 with m i m e = 100 and ω pe0 Ω e0 = 8). We impose an initial perturbation on the equilibrium magnetic flux function ψ, of the form ψ 1 = 0.1B 0 d i0 cos(2πx L x ) cos(πz L z ). In Type B simulations (Runs 10 to 12), the parameters are the same as in Egedal et al. [2012]: the sheet width w = 0.5d i0, the temperature ratio for the current sheet plasma T i T e = 5, the temperature for the background plasma T ib = T i 3 and T eb = T e 3, the background density n b = 0.05n 0, and the mass ratio m i m e = 400. We change the ratio ω pe0 Ω e0 from 2 to 8. We use the grid numbers of 8192 in the x direction and 1024 in the z direction and impose an initial perturbation ψ 1 of the form ψ 1 = 0.2B 0 d i0 sech 2 (x 2w)sech 2 (z w). 4. Simulation Results 4.1. Dependence of the Parallel Electric Potential on ω pe0 Ω e0 First, we investigate the dependence of the parallel electric potential φ on the parameter ω pe0 Ω e0.aswe explained in section 2, we expect that φ is proportional to (ω pe0 Ω e0 ) 2. Figure 1 displays the parallel electric field E and the parallel electric potential φ in two runs (Runs 4 and 10). Figure 1a shows Run 4, a Type A simulation with the mass ratio 100, n b = 0.3n 0, and ω pe0 Ω e0 = 2 (equivalently, v A0 c = 1 20). Figure 1a (left) shows a contour plot of E at the time Ω i t = 20 (the reconnection rate attains the maximum value around Ω i t = 16). In the plot, the gray curves are magnetic field lines, and we choose the field line that corresponds to the separatrix in the quadrant (white curve). The integral to calculate φ (= x E dl) is from the right side of this white curve (at the right boundary) to a position x on this field line. In the contour of E for Run 4, multiple dipolar structures (alternating yellow (red) and blue stripes, consistent with previous studies [Egedal et al., 2012; Fujimoto, 2014]) are formed along separatrices, on top of the coherent larger structure. For example, we see a positive E in the exhaust region (in yellow color) as a large-scale structure in the first quadrant. This is because E =(E x B x + E y B y + E z B z ) B, and E y (> 0) contributes most to E in that region where the Hall magnetic field B y > 0. In contrast, in the thin electron current layer in the vicinity of the X point, E becomes negative in the first quadrant (in blue color), because E z (< 0) is very strong in the electron current layer. The black curve in Figure 1a (right) shows the profile of φ as a function of the distance l from the X point (l = 0). The maximum value of eφ is around 0.1m e c 2 0.4m i v 2 A0 at l 2.5d i0. The magnitude of φ is large in the diffusion region, and it decreases as the distance from the X line increases. In Figure 1a (right), the blue and red curves show contributions to φ from the electrostatic potential φ and the electromagnetic field E y, respectively. The contribution from φ is negative near the X line, and it increases to positive as we move from the X point (l = 0) to around l = 8d i0 ; however, the electromagnetic field E y is the most dominant contribution to φ near its peak around l = 2.5d i0, as well as near the X line. This is because the in-plane field (E x and E z ) is nearly perpendicular to the magnetic field line, and E is dominated by E y B y B in most of the region along the field line. Figure 1b shows Run 10, a Type B simulation with the mass ratio 400, and ω pe0 Ω e0 = 2 (equivalently, v A0 c = 1 40) atω i t = 48. In this simulation, clear localized structures of E are formed at this time, particularly along the separatrices. This localization is due to the localized initial perturbation (in the form of sech 2 (x 2w)sech 2 (z w)), which should be contrasted with the result by Egedal et al. [2012], where BESSHO ET AL. ELECTRON ACCELERATION IN RECONNECTION 9358
Figure 1. Contours of E and the profiles of φ. E is formed in the diffusion region, and the parallel potential φ becomes large in the vicinity of the X line. (a) Run 4 at Ω i t = 20.(b)Run10atΩ i t = 48. In Figures 1a (left) and 1b (left), the gray curves are magnetic field lines, and the white curves are the field lines where φ is calculated. In Figures 1a (right) and 1b (right), black, blue, and red curves are φ, the contribution from φ, and the contribution from the integral of E y, respectively. many dipolar structures of E are seen along separatrices. The parallel electric potential, obtained by the integration along the field line (shown in white), becomes 0.51m e c 2 2m i v 2 (the black curve in A0 Figure 1b (right)). We also see that φ at this time is more than twice as large than that at Ω i t = 33 (the gray curve), around when the maximum reconnection rate is attained. This is because the region with large E y extends along with the extension of the diffusion region in the x direction with time. In this run, the magnitude of φ becomes larger than that in Run 4 (Figure 1a), and becomes mildly relativistic, even though there is no dipolar structure in E (see also Egedal et al. [2012] where eφ 0.7m e c 2, where there are many dipolar structures of E ); however, as we discuss below, this mildly relativistic energy occurs because we used a small value of ω pe0 Ω e0 = 2. We also see that φ is dominated by the contribution of E y (the red curve), and the contribution from the electrostatic potential (the blue curve) is small. Comparison between our result and the result by Egedal et al. [2012] suggests that the dipolar structures of E can contribute little to the magnitude of φ, because the integration of E cancels the contributions of a large positive value and a large negative value in the dipolar E. Figure 2. Dependence of the magnitude of φ on ω pe0 Ω e0, which shows φ (ω pe0 Ω e0 ) 2. (top) Profile of φ along a field line with different values of ω pe0 Ω e0 for Runs 1 3. (bottom) Dependence of φ on ω pe0 Ω e0. Figure 2 (top) shows the dependence of the profile of φ on ω pe0 Ω e0, for Type A simulations (Runs 1 3) with the mass ratio 25 and n b = 0.3n 0, at the same time Ω i t = 20. The maximum of φ decreases as ω pe0 Ω e0 increases. Figure 2 (bottom) shows the maximum of eφ m e c 2 as a function of ω pe0 Ω e0. The red and blue dots are for Type A BESSHO ET AL. ELECTRON ACCELERATION IN RECONNECTION 9359
Figure 3. Dependence of φ, the reconnection electric field E y, and the magnitude of bipolar E z on n b n 0, at the time of the maximum reconnection rate. Both φ and E y show the dependence close to (n b n 0 ) 1 2. runs, with the mass ratio 25 and 100, respectively, and the background density n b = 0.3n 0. The green dots are for Type B runs with the mass ratio 400 and the background density n b = 0.05n 0. The black line corresponds to a dependency (ω pe0 Ω e0 ) 2, and all the dots are consistent with the relation eφ m e c 2 (ω pe0 Ω e0 ) 2. In Type B simulations (green dots), the magnitudes of φ become much larger than those in Type A simulations. This is partially due to the smallness of n b, which will be discussed in the next section. 4.2. Dependence of the Parallel Potential on the Ratio n b n 0 Figure 3 shows the magnitude of φ (the red dots) at the time of the maximum reconnection rate, as a function of n b n 0,for runs with the mass ratio 100 and ω pe0 Ω e0 = 2 (Runs 4 and 7 9). As n b n 0 decreases, the magnitude of φ becomes large. This is because reconnection with lower n b n 0 proceeds faster than that with higher n b n 0 [Bessho and Bhattacharjee, 2010a, 2010b; Wu et al., 2011]. Since the dominant contribution to E is E y B y B, the increase of E y in reconnection with lower n b n 0 (larger reconnection rate R r in equation (4)) results in larger φ than with higher n b n 0. The blue dots in Figure 3 show the dependence of the reconnection electric field E y (measured at the X line) on n b n 0.Asn b n 0 becomes smaller, E y becomes larger, and both φ and E y are roughly proportional to (n b n 0 ) 1 2 (the dotted line). This dependence as well as the magnitude of eφ at this time is consistent with the formula in Le et al. [2010], in which eφ T e (1 2)[(4 β e ) 1 4 1 2] 2, where β e is the electron beta value in the upstream region. The formula was derived from their proposed equations of state for reconnection, but the physical interpretation about the dependence of φ on n b was not given in Le et al. [2010]. Our analysis in equation (4) explains why eφ (n b n 0 ) 1 2. The dominant term for φ (electromagnetic contribution given in equation (4)) is proportional to E y. Wu et al. [2011] demonstrated by PIC simulations that E y v A,up B up = B 2 up (4πm i n up )1 2, where the subscript up represents the upstream value and n up n b. Therefore, considering E y = R r B 0 v A0 c = R r B 2 0 [c(4πm i n 0 )1 2 ], we obtain that R r (n b n 0 ) 1 2. As a result, from equation (4), we conclude that φ (n b n 0 ) 1 2. In contrast, the in-plane component E z (the green dots in Figure 3, which are the maximum values of the bipolar structure of E z across the electron current layer) does not depend as strongly on n b n 0 as E y does. Therefore, the increase of φ when n b n 0 is small is due to the increase in the magnitude of the out-of-plane E y. Note that Figure 3 is for the time of the maximum reconnection rate, and the value of φ depends on time as well; the magnitude of φ tends to become larger as the diffusion region extends in the x direction, and we have observed that φ becomes a few times larger in the late stage than at the time of the maximum reconnection rate (see Figure 1b (right)). This increase of φ corresponds to the increase of the integral length in equation (4), αd i0. 5. Electron Acceleration by the Parallel and the Perpendicular Electric Fields As we have seen in section 4, eφ m e c 2 (ω pe0 Ω e0 ) 2. This relation is crucial, because it limits electron acceleration by E. With PIC simulations, Egedal et al. [2012] demonstrated that electrons can be accelerated due to eφ in the broad extent of the reconnection exhaust region. Their simulations used ω pe0 Ω e0 = 2.0, and the maximum of eφ 0.7m e c 2. This does not necessarily mean that electrons can be accelerated to 350 kev (note that m e c 2 511 kev); in fact, Egedal et al. [2012] discussed the electron acceleration using a different normalization eφ T eb 90, which corresponds to 18 kev when we use T eb = 200 ev and which can explain the flat-top part of the electron distribution function up to 14 kev observed by Cluster. The discrepancy between the simulation value eφ 0.7m e c 2 and the observed value 14 kev by Cluster in the Earth s magnetotail can be explained by the scaling eφ m e c 2 (ω pe0 Ω e0 ) 2. In the Earth s magnetotail, ω pe0 Ω e0 9 [Chen et al., 2012], and this suggests that eφ in the Earth s magnetotail becomes smaller by nearly a factor of 20 than 0.7m e c 2 350 kev, that is, about 20 kev. While this is consistent with the discussion by Egedal et al. [2012], it is much smaller than observed energetic electrons that have energies in excess of 100 kev [Øieroset et al., 2002; Imada et al., 2007; Chen et al., 2008a; Retinò et al., 2008]. Even if we consider a possible case with n b n 0 = 0.01, which we expect enhances the magnitude of the potential (0.05 0.01) 1 2 2.2 times larger than BESSHO ET AL. ELECTRON ACCELERATION IN RECONNECTION 9360
the value obtained by Egedal et al. [2012], where n b = 0.05n 0, the parallel electric potential only accounts for electrons up to around 40 kev. This indicates that we need to identify different electron acceleration mechanisms than by E. Let us begin by considering two electron populations: the first one consists of passing electrons that pass through the z = 0 plane far from an X line, and they do not interact with the X line. The second one consists of accelerated electrons that enter the vicinity of the X line and are accelerated there. The passing electrons, which enter the exhaust region from the upstream region along magnetic field lines, obtain energy due to positive values of eφ at z = 0 (see equation (9) in the later discussion); however, as they pass through the z = 0 plane and move away along the field lines, they lose energy, because eφ is symmetric with respect to the z = 0 plane and becomes small in the downstream region. On the other hand, for the second population of electrons (accelerated electrons), φ enhances their energy as they approach the X line, and they are further accelerated in the vicinity of the X line. However, in this phase of acceleration, φ does not play a significant role. To see this point, let us consider the energy equation for an electron: y p2 e2 e1 = e(φ 2 φ 1 ) e E y dy p, (7) y p1 where e is the energy of the electron, the subscripts 1 and 2 represent the values at the time t = t 1 and t = t 2 (t 2 > t 1 ), respectively, and the integral in the y direction is along the trajectory of the particle, represented by y p. The second term on the right-hand side of equation (7) is decomposed as t 2 y p2 B y e E y dy p = e E y p1 y v t 1 B dt e E y v dy dt e E t 1 y v ndy dt, (8) t 1 where v, v dy, and v ndy represent the parallel velocity, the y component of the total drift velocity, and the nondrift velocity perpendicular to the magnetic field, respectively. The first term is due to the parallel motion, the second term is due to the drift speed when the particle is magnetized, and the third term is due to nondrift perpendicular motion, which includes both gyromotion (when the particle is magnetized) and unmagnetized motion. In the following, when a particle is unmagnetized, we set v dy, which is due to the guiding center drift motion, to be zero, and we only consider v ndy for the perpendicular velocity. Let us consider the following two phases: the incoming phase (from the upstream region at t = t o until the electron reaches the vicinity of the X line at t = t X ) and the outgoing phase (after t = t X until t = t n when the particle reaches infinity (in other words, a far downstream region)). In the incoming phase, the electron is magnetized and moving along a magnetic field line from the upstream region (at t = t o ) to the vicinity of the X line (at t = t X ). In this phase, let us assume that the drift velocity (owing to E B, B, and curvature drifts) is small and the condition v v d is satisfied, where v d is the magnitude of the drift velocity. In this case, the first term on the right-hand side of equation (8) becomes e t 2 E t 1 y v (B y B)dt e y 2 E y 1 y dy l.ifwe use equations (2) and (8), equation (7) becomes t X ex eo = eφ X e E y v dy dt, (9) t o where we assume that the potential φ o at the upstream region is zero, and we averaged the equation during gyromotion (the effect of the term with v ndy is canceled because of this averaging). The electron in the incoming phase gains energy from the parallel electric potential, as well as due to the effect of the guiding center drift, represented by the second term on the right-hand side, which includes acceleration due to B and curvature drifts [Hoshino et al., 2001] and Fermi acceleration due to the curvature drift [Drake et al., 2006]. For a passing electron that passes through the z = 0 plane, the electron eventually loses the same parallel potential energy in the outgoing phase, because the passing electron continues to be magnetized and moves along the same field line to the downstream region. However, the second term remains positive as the drift velocity is in the negative y direction; therefore, only the effects of acceleration due to B drift and Fermi acceleration (due to the curvature drift) remain in the outgoing phase in the far downstream region. In contrast, for an electron that interacts with fields near the X line (accelerated electron), in the outgoing phase, the electron is unmagnetized and accelerated in the vicinity of the X line and continues to move in the negative y direction to gain further energy. Using equations (2) and (8), we obtain the energy increase from the vicinity of the X point (at t = t X ) to infinity (at t = t n )as y ln en ex = eφ X + e E y dy l e y px t X t n t 2 t 2 t n B y E y v B dt e E y v ndy dt, (10) t X BESSHO ET AL. ELECTRON ACCELERATION IN RECONNECTION 9361
Figure 4. Profile of φ in the x direction and energy distribution functions of electrons. We observed many energetic electrons that have energy much larger than the magnitude of eφ. (first panel) Magnetic field lines for Run 9 at Ω i t = 14.5. (second panel) Profile of φ at z = 0 as a function of x. The regions between the dashed lines are where magnetic islands form. (third panel) Electron energy distribution functions for particles in z < 1.5d i0 and three regions in the three different colors (light blue, magenta, and yellow) shown in the x axis in Figure 4 (second panel). The black curves are the distribution functions at Ω i t = 14.5, and the gray curves are those at t = 0. (fourth panel) Counts of particles as a function of the work done by the parallel electric field (red) and the work due to the perpendicular motion (black, blue, and green), during 8.7 < Ω i t < 14.5, for electrons in the area ( 1.25d i0 < x and z < 1.5d i0 )atω i t = 14.5. where the integral in the second term is along the field line that passes through the point of the particle at t = t X and y ln is the y position of the magnetic field line at infinity, and we assume that φ at infinity (φ n )iszero. In this phase, v ndy owing to the nongyromotion (unmagnetized motion) becomes dominant, and the effect of the v ndy is not negligible compared with v. Therefore, there is no complete cancelation between the second term and the third term in the right-hand side. In section 4.1 and Figure 1, we discussed that φ is dominated by the integral of E y along the field line. Therefore, as a rough approximation, the first and the second terms in the right-hand side cancel each other, and the rest of the terms that are due to acceleration by E y are dominant. This means that for the outgoing electron, φ does not play an important role in accelerating electrons. We will demonstrate by simulation in Figure 5 that the last term on the right-hand side of equation (10) is the most dominant term for an accelerated electron. Figure 4 (second panel) plots the profile of eφ along the z = 0 line for Run 9 (n b = 0.0375n 0 and ω pe0 Ω e0 = 2) at Ω i t = 14.5. This is around the time of the maximum reconnection rate. In the plot, the two gaps (between the dashed lines) are the regions where magnetic islands form, and no φ is defined in such closed field lines. The maximum of eφ is around 0.31m e c 2 near the X line at x = 1.25d i0, which suggests that the incoming BESSHO ET AL. ELECTRON ACCELERATION IN RECONNECTION 9362
Figure 5. Electron trajectories for a passing electron and an accelerated electron. The passing electron gains energy by acceleration due to E near z = 0, but it loses energy after it passes through z = 0. On the other hand, the accelerated electron gains energy in the vicinity of the X line by acceleration due to E y and is ejected from the acceleration region without losing energy. (top row) Trajectories of electrons in the x-z plane (first panel) and the x-y plane (second panel). The gray curves in Figure 5 (top row) are the separatrices at Ω i t = 11.6. (top left) Passing electron. (top right) Accelerated electron. (bottom row) The time history of the work done by the parallel electric field (red) and the work due to the perpendicular motion (black, blue, and green). The light blue curves represent the total energy gain. electrons will increase their Lorentz factor γ by the amount of 0.31. However, in the exhaust region, there are electrons with energies significantly exceeding the magnitude of eφ. Figure 4 (third panel) shows the energy distribution functions of electrons in three different areas. The electrons in z < 1.5d i0 and within the range of x in different colors in Figure 4 (second panel) are used to make these plots. The maximum of γ 1 is around 5, and there are many electrons with increases of γ much larger than 0.31 (see the gray curves: the energy distribution functions at t = 0), which cannot be explained by acceleration due to φ. Figure 4 (fourth panel) shows the distribution (counts of particles) of work done by the parallel electric field, e E v dt, and the work due to the perpendicular motion, e E v dt, during 8.7 < Ω i t < 14.5, for particles in the area ( 1.25d i0 < x and z < 1.5d i0 )atω i t = 14.5. The contribution to electron energization by the parallel electric field (the red curve) is limited to low-energy particles (Δγ<1), and the most dominant work to produce high-energy particles (Δγ > 1) is e E y v y dt. Note that the sum of all the integrals for the perpendicular velocity (the black, blue, and green curves), e E v dt, is the work done by the perpendicular electric field, because E v = E v. We chose some electrons and traced their trajectories from Ω i t = 8.7 to 14.5 to investigate the contribution of work done by E and the electric field perpendicular to the magnetic field on particle acceleration. Figure 5 displays two examples of trajectories: Figure 5 (left column) shows a passing electron that passes through the z = 0 line mostly along the magnetic field line, and Figure 5 (right column) shows an electron accelerated in the vicinity of the X line (near the origin). The passing electron (Figure 5, left column) gains energy when it passes through the z = 0 plane around Ω i t = 13.8, and the most dominant contribution to the energization is from e E v dt (the red curve in Figure 5, bottom row). Note that as long as the particle is moving along the magnetic field line, e E v dt = eφ. After this particle passes the current sheet, the contribution from e E v dt decreases, and the total energy gain at Ω i t = 14.5 is mainly due to e E y v y dt (the blue curve in Figure 5, bottom row). On the other hand, the accelerated electron (Figure 5, right column) gains energy from eφ (= e E v dt) around Ω i t = 12.6 (during the incoming phase, discussed in equation (9)) and loses energy after it passes through the point where eφ takes the maximum. After then (the outgoing phase), BESSHO ET AL. ELECTRON ACCELERATION IN RECONNECTION 9363
this particle is not magnetized, and the parallel electric field does not contribute significantly to the particle energization. In this phase, e E v dt eφ. This particle is mainly accelerated by e E y v y dt near the X line at Ω i t = 13.4, as discussed in equation (10). After a few meandering orbits, the particle is ejected to the exhaust region. The contribution of e E v dt (the red curve in Figure 5, bottom row) is much smaller than that of the work done by the perpendicular electric field e E vdt (the sum of the black, blue, and green curves in Figure 5, bottom row) during the outgoing phase. After the acceleration in the vicinity of the X line, this particle drifts further in the negative y direction to gain further energy by E y ; however, E x points outward from the X line and decelerates the particle; therefore, the acceleration (the energy gain; see the light blue curve in Figure 5, bottom row) during the time of the drift motion in the magnetic pileup region (after Ω i t = 13.7) is not very significant. If particles are magnetized, this cancelation is due to E B drift, which does not energize particles. We have also studied the contribution from E and E in a run that has the same parameter as Egedal et al. [2012], except that the mass ratio is 100 and the system size is L x L z = 204.8d i 51.2d i (similar to Run 10, but the mass ratio is different). We have confirmed that the contribution of E is not significant for high-energy electrons but that E is the dominant energy source for those electrons. Therefore, we conclude that E contribution is not very large for the most energetic electrons accelerated during reconnection. Note that the results shown in Figures 4 and 5 are based on the parameter ω pe0 Ω e0 = 2; therefore, if we consider the scaling of energy (ω pe0 Ω e0 ) 2 down to the observed value ω pe0 Ω e0 = 9 in the Earth s magnetotail, energy of accelerated electrons in the Earth s magnetotail would be (9 2) 2 = 20.25 times smaller than the energy shown in Figures 4 and 5. Therefore, the maximum energy seen in Figure 4, γ 1 5 2.5 MeV, should be scaled down to around 125 kev, which is consistent with the energy range observed in the Earth s magnetotail. Note that the parallel potential takes the maximum value in the diffusion region, and its magnitude decreases in the downstream region as the distance from the X line increases. Therefore, the parallel acceleration (energization) up to 40 kev discussed in this study is the upper limit, and the energization due to parallel acceleration in the downstream is less than 40 kev. 6. Electron Acceleration and Pitch Angle Scattering Anisotropic electron distributions (T > T ) in the inflow region are often observed in simulations [Cattell et al., 2005; Chen et al., 2008b; Egedal et al., 2012] and satellite observations [Chen et al., 2008b; Egedal et al., 2010, 2012]. In contrast, in the exhaust region, almost isotropic heated electrons are observed [Chen et al., 2008b]. In this section, we demonstrate that isotropic heated electrons in the exhaust region are due to particle acceleration and pitch angle scattering. Egedal et al. [2012] argued that after electrons are accelerated by eφ and gain large parallel velocities, pitch angle scattering can scatter electrons in the passing region in p -p plane (with large p ) to the trapped region (with small p and large p ), where p and p are the parallel and the perpendicular momentum, respectively, and eventually an almost isotropic, flat-top distribution forms after some instabilities scatter electrons to lower energies. In this scenario, most of the electrons are first accelerated due to the parallel electric potential, and then pitch angle scattering will convert p to p. However, this scenario limits the maximum increase of electron energy to the maximum of the parallel electric potential eφ, because pitch angle scattering itself will not enhance the particle s energy. Here we show that the pitch angle scattering from p to p works as Egedal et al. [2012] suggested, but in addition to that, there is another type of pitch angle scattering from p to p, which occurs for electrons accelerated perpendicular to the magnetic field. Figure 6a shows the trajectories of two electrons in the p -p plane and in the x-z plane. The black curves are for the same passing particle in Figure 5 (left column). This electron enhances the parallel momentum p. After the acceleration by eφ,uptop 0.7m e c at Ω i t = 13.8, it slightly enhances its p while it moves in the y direction and gains energy by E y. After it reaches the maximum p, it starts to lose its parallel momentum again. This is because the electron passes the neutral plane z = 0 after Ω i t =13.8 (see Figure 6a (right) for the x-z plane), and the electron is decelerated by E. There is no significant pitch angle scattering for this particle. On the other hand, the blue curves represent the trajectories of an electron that shows strong pitch angle scattering from p to p. This electron moves toward the vicinity of the X line until Ω i t = 14.0, and it increases its parallel momentum up to p = 0.7m e c. After a small interaction with the fields in the vicinity of the X point, the pitch angle scattering converts p to p around Ω i t = 14.1. After then, this electron is no BESSHO ET AL. ELECTRON ACCELERATION IN RECONNECTION 9364
Figure 6. Trajectories of electrons in the p -p plane and in the x-z plane. We observe two types of pitch angle scattering: from p to p and from p to p. (a) The same electron as shown in Figure 5 as passing electron (black ) and another example of electron that shows pitch angle scattering from p to p (blue). (b) The same electron as shown in Figure 5 as accelerated electron, which shows pitch angle scattering from p to p. (c) Another particle that shows pitch angle scattering from p to p. The gray curves in Figures 6a 6c (right) are the separatrices at Ω i t = 11.6. more a passing particle, and it is continuously accelerated near the X line, and its perpendicular momentum gradually increases by the end of the simulation. Figures 6b and 6c show examples of the other type of pitch angle scattering, from p to p. Figure 6b plots the trajectories of the same accelerated particle in Figure 5 (right column). This particle obtains the parallel momentum up to 0.6m e c before Ω i t = 13.2, and the first pitch angle scattering (p to p ) occurs around Ω i t = 13.2. In the main acceleration phase (after Ω i t = 13.2), this particle enhances its perpendicular momentum up to p 1.9m e c (at Ω i t = 13.7). After Ω i t = 13.8, there is a second pitch angle scattering that converts p to p, and the particle attains p up to 1.6m e c at Ω i t = 14.0. Eventually, the momentum returns from high p to high p at Ω i t = 14.5. Figure 6c shows another trajectories of an accelerated electron. This particle shows the increase of p until Ω i t = 14.0 because of the acceleration by E y.afterω i t = 14.0, pitch angle scattering occurs, and p is converted to p 1.6m e c at Ω i t = 14.3 during the ejection phase of the electron. The pitch angle scattering from p to p can occur when electrons flow toward the X line, along the magnetic field line with acceleration by E. When p becomes large due to the acceleration by E, those electrons become unmagnetized near the X line, where the spatial size of B and the radii of curvature in magnetic field lines are small. When p is large enough, the electron inertia due to the high p prevents the electron from following the same field line with a small magnetic curvature; therefore, significant pitch angle scattering is possible from p to p. On the other hand, the pitch angle scattering from p to p can occur when unmagnetized electrons accelerated in the vicinity of the X line are ejected from the acceleration region and become magnetized again. Since there is a large B z near the end of the electron-diffusion region, the Lorentz force due to v y B z will convert the momentum of the accelerated electron from the y direction to the x direction. Therefore, eventually, the momentum of the ejected electron becomes mostly parallel to the magnetic field in which B x is the dominant component; that is, a conversion to a large p occurs. BESSHO ET AL. ELECTRON ACCELERATION IN RECONNECTION 9365
Figure 7. Transition from an anisotropic electron distribution function (near the X line) to an isotropic electron distribution function (in the exhaust region). Pitch angle scattering isotropizes an anisotropic distribution function due to parallel and perpendicular acceleration near the X line, and we observe an isotropic distribution function in the exhaust region. (top) Magnetic field lines (black curves) and the regions where electron distribution functions are taken, for Run 9 at Ω i t = 14.5. (middle) Electron distribution functions in the p -p plane (regions a c). (bottom) Electron distribution functions for particles that are initially in the background plasma and in the domain z > d i0. Wang et al. [2010c] have shown by simulations that pitch angles for accelerated particles vary rapidly due to stochastic motion when the gyroradii of those particles are comparable to the spatial scale of the magnetic curvature near the X line, and they argued that the process isotropizes the electron distribution function for high-energy particles. We have observed similar stochastic motion and pitch angle variations. Our PIC simulations and Figure 6 have revealed more detail in the process of isotropization in the exhaust region. Figure 7 displays electron distribution functions in the p -p plane at Ω i t = 14.5 for the same run, in the regions denoted by the red rectangular boxes in Figure 7 (top). In the panels of distribution functions, the top panels are the distribution functions of all the electrons in each box, while the bottom panels are the distribution functions of the particles that are initially in the background plasma and in the domain z > d i0, which are not in the current sheet initially. In region a, there are accelerated particles with high p, and there are two arcs extending from high p (with small p ) to high p, which suggests pitch angle scattering from p to p. Box b shows a distribution function with strong beam components (large p ). The beam components extend from small p ( 0) to a little larger p, which suggests pitch angle scattering from p to p.the top panel for the box c appears to exhibit almost isotropic electrons; however, the bottom panel shows a clear arc structure, which suggests pitch angle scattering either from p to p or from p to p. 7. Conclusion We have demonstrated by theory and PIC simulations that the magnitude of the parallel electric potential during magnetic reconnection is proportional to (ω pe0 Ω e0 ) 2 and (n b n 0 ) 1 2. The magnitude of the parallel potential takes the maximum value in the diffusion region, which determines the upper limit of the BESSHO ET AL. ELECTRON ACCELERATION IN RECONNECTION 9366
acceleration due to the parallel electric field, and the energization due to the parallel acceleration becomes smaller in the downstream exhaust than that in the diffusion region. In the Earth s magnetotail where ω pe0 Ω e0 9, acceleration by φ can explain electron energization up to a few tens of kev, but to understand electron energization in excess of 100 kev, we must consider other acceleration mechanisms. We have traced electron trajectories and demonstrated that the contribution of the work done by the parallel electric field e E v dt for high-energy electrons is much smaller than the work done by the perpendicular electric field e E vdt (in particular, e E y v y dt). The effect of pitch angle scattering has also been studied. For the electrons that are accelerated by E, pitch angle scattering converts p to p, while the electrons that are accelerated by the perpendicular electric field, pitch angle scattering converts p to p. 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