Electron Acceleration and Heating during Magnetic Reconnection in the Earth's Quasi-parallel Bow Shock

Figure 1 shows the time evolution of the shock between Ω_i t = 18.75 to 21.88, where Ω_i is the ion cyclotron frequency based on B₀. The black curves in panels (a), (b), and (c) are magnetic field lines projected on the x–y plane, and the color indicates the out-of-plane current density J_z . As time progresses, many kinetic-scale (up to several d_i ) magnetic islands are generated in the shock transition region (the region right of the white vertical line). The white vertical line in each plot is the position of the maximum of 〈B_y 〉 (see the arrows in panel (d)), where 〈B_y 〉 is the magnetic field B_y averaged over the y-direction. Based on the propagation of the peak of 〈B_y 〉, we determine that the shock speed during this time interval is 1.5v_A0 in the simulation frame. Considering the upstream drift speed 9v_A0, the shock speed v_sh in the upstream rest frame is 10.5v_A0, i.e., the Alfvén Mach number M_A is 10.5, and the fast mode Mach number M_f = v_sh/v_f is 7.5. At Ω_i t = 18.75 (see panel (a)), there are several ion-scale magnetic islands whose size is several d_i , for example, around (x, y) = (49d_i , 0), (47d_i , 12d_i ), and (46d_i , 22d_i ), each of which is denoted by 1, 2, and 3, respectively, in the plot. They are generated because of the nonresonant ion–ion beam instability (Sentman et al. 1981; Gary et al. 1984; Gary 1991; Bessho et al. 2020; Chen et al. 2021, 2022). More details about the nonresonant wave as well as the resonant wave observed in the simulation are summarized in Appendix A. Many thin current sheets are generated, with both positive and negative J_z . These ion-scale magnetic islands grow further as time elapses, and at Ω_i t = 20.31 (panel (b)), the size of islands at (x, y) = (45d_i , 7d_i ) (island 1) and (x, y) = (46d_i , 20d_i ) (island 2) becomes greater than that at Ω_i t = 18.75. Note that the magnetic islands 1, 2, and 3 at Ω_i t = 20.31 are the same islands 1, 2, and 3 at Ω_i t = 18.75, respectively. At Ω_i t = 20.31, around those ion-scale islands, there are many sub-d_i scale (several d_e -scale, where d_e is the electron skin depth) magnetic islands. These are electron-only reconnection sties, in which electron jets form but ions just pass through the region without significant acceleration because of the small scale of magnetic gradients in these thin current sheets. The formation of these electron-only reconnection regions was discussed in the previous paper (Bessho et al. 2020). At Ω_i t = 21.88 (panel (c)), the ion-scale larger islands have already been dissipated (i.e., they merge surrounding fields, and the X lines associated with those islands disappear), and only smaller electron-only reconnection areas remain in a region 46d_i < x < 50d_i .

Zoom In Zoom Out Reset image size

Panel (f) shows 1D cuts of B_y along two lines, y = 20d_i and y = 18d_i at Ω_i t = 20.31 (see also panel (e)). The line y = 20d_i crosses the ion-scale island (island 2), while the line y = 18d_i passes through two sub-d_i scale islands. In panel (f), the thick blue curve is the 1D cut along y = 20d_i . B_y is positive in the left side of the island (x < 46d_i ), and B_y reverses its sign across the center of the island (x = 46d_i ) and becomes negative in the right side of the island (46d_i < x). This island covers the region 44.5d_i < x < 47.3d_i (see the blue arrow), and the size of this island (diameter) is around 3d_i . Across the island, B_y changes from 7B₀ to −5B₀. The black curve in panel (f) shows the 1D cut along y = 18d_i , which crosses the two sub-d_i scale islands. The two sub-d_i scale islands are denoted by black arrows in panel (f) (see also panel (e)). Across these two islands, B_y changes its sign, and the change of B_y is from 6B₀ to −4B₀, which is similar to the change of B_y in the ion-scale island. The size of these islands is smaller than d_i . Left of these sub-d_i scale islands, there is a reconnection X line (see also panel (e), where the X line is marked by "X"), and B_y also changes its sign across the X line region (i.e., a current sheet). The current sheet thickness is also around d_i . In panel (f), the thin blue curve represents the y-averaged value, 〈B_y 〉 (the same blue curve as in panel (d)). Compared with 〈B_y 〉, the local 1D cuts show larger-amplitude fluctuations.

Figure 2 shows the y-component of the ion and electron fluid velocities V_iy and V_ey at Ω_i t = 20.31, near the region of an ion-scale magnetic island (island 2). We focus on two X lines: at (x, y) = (45.475d_i , 18.45d_i ) (see the blue X mark) and (x, y) = (46.825d_i , 18.925d_i ) (see the red X mark). The blue X line is in the reconnection region that generates the ion-scale magnetic island (island 2), where ion-coupled reconnection occurs. In panel (a), we see that an ion outflow jet (marked by the black oval) is generated because of reconnection with the blue X line, while panel (b) shows that an electron outflow jet is also produced along the separatrix. Both the ion and electron outflows are generated on the upper-left side of the separatrix with V_iy > 0 and V_ey > 0, and there are no downward (with V_iy < 0 and V_ey < 0) outflows in the lower right side of the separatrix. Formation of such a one-sided outflow jet (in this case, a flow with positive-y velocity) is a characteristic in turbulent reconnection in a shock (Bessho et al. 2022), and the ion outflow velocity (the velocity measured in the simulation frame, not the velocity relative to the plasma surrounding the jet) exceeds 6v_A0, while the electron outflow reaches 10v_A0, which is close to the electron Alfvén speed v_Ae0 under the mass ratio 200 (v_Ae0 = 14.4v_A0). The local values of magnetic fields (let us denote them as B₁ and B₂) and densities (n₁ and n₂) in the two sides across the current sheet around the blue X line are: B₁ = 5.6B₀, B₂ = 2.7B₀, n₁ = 3.0n₀, and n₂ = 2.7n₀, which give the local Alfvén speed, v_A-local = 3.2v_A0 (using B₁ and n₁) and 1.6v_A0 (using B₂ and n₂), respectively. The theoretical estimate of the ion outflow speed based on the asymmetric reconnection model (Cassak & Shay 2007), ${V}_{\mathrm{out}}={[{B}_{1}{B}_{2}({B}_{1}+{B}_{2})/4\pi {m}_{i}({n}_{1}{B}_{2}+{n}_{2}{B}_{1})]}^{1/2}$, is obtained as 2.3v_A0, which is close to the local Alfvén speeds, but is much smaller than the observed ion outflow speed ∼6v_A0. Note that the ion outflow generated by reconnection in the shock transition region can exceed the Alfvén speed, reaching of the order of 10v_A0, because of the super-Alfvénic background ion flows in the shock transition region (Bessho et al. 2022). In contrast, the red X line at (x, y) = (46.825d_i , 18.925d_i ) is a site of electron-only reconnection, and the ion plot (panel (a)) does not have an ion outflow jet from the red X line, while the electron plot (panel (b)) shows a rightward electron outflow jet that exceeds 10v_A0 (see the black oval around the red X line).

Zoom In Zoom Out Reset image size

Panels (c) and (d) display the in-plane speeds of the ions and the electrons. Panel (c) shows the ion in-plane speed, ${({V}_{{ix}}^{2}+{V}_{{iy}}^{2})}^{1/2}$, and only the regions with V_iy > 2v_A0 are shown in grayscale. Let us denote the two X lines as (x, y) = (x_X1, y_X1) (blue X line) and (x, y) = (x_X2, y_X2) (red X line). Around the blue X line, ions are flowing into the reconnection site from the region y < y_X1. The inflow speed at the blue X line is around 5v_A0, and in the outflow region (within the blue oval), the maximum speed exceeds 6.5v_A0. In contrast, the ion in-plane speed around the red X line does not constitute the outflow jet. Around the red X line, ions are passing through the red X line from the region x_X2 < x and y < y_X2 with velocities V_ix < 0 and V_iy > 0 (see the vector plot with magenta arrows). The ion in-plane flows are perpendicular to the magnetic field reversal (i.e., the current sheet), and they are decelerated (see the change of the color from gray to white after passing through the magnetic field reversal). After passing through this magnetic field reversal region, the ion flows are deflected by the magnetic field gradient in the ion-scale island, and the flow velocity changes to V_ix ∼ 0 and V_iy > 0. During this deflection, the ion in-plane flow speed is a bit enhanced (see the gray region around x = 47d_i and x = 20d_i ), and this increase may be due to the local electric fields (the detailed acceleration mechanism is beyond the scope of this paper). However, there is no collimated ion jet structure produced from the red X line, indicating that the increase of the ion speed is not due to the formation of a reconnection outflow jet. In contrast, the ion jet forms from the blue X line, and it shows a conspicuous collimated jet structure. The size of island 2, which is generated by the reconnection region with the blue X line, is larger than 2d_i , while the size of the magnetic islands generated by electron-only reconnection with the red X line is less than d_i . Ions may be slightly affected by the small-size islands in the electron-only reconnection region, but the change of ion speed is not substantial, and no ion jet forms. In contrast, around the red X line, a strong electron jet is seen whose speed exceeds 10v_A0 (panel (d)). More details for ion-coupled reconnection regions and electron-only reconnection regions in 2D PIC simulations are discussed in Bessho et al. (2019, 2020, 2022).

Figure 3 displays the electric fields E_x , E_y , and E_z in the same region as Figure 1 at the three times. In the shock, there are three types of electric fields: one is the electrostatic field associated with the cross-shock potential (Woods 1969; Goodrich & Scudder 1984; Scudder 1995). This electric field is mainly in the x-direction, and E_x > 0. The second type of electric field is generated in ion-scale magnetic islands, and this electric field points toward the center of each island. The third type of electric field is the reconnection electric field, and in this 2D simulation, this electric field is in the z-direction.

Zoom In Zoom Out Reset image size

Overall, E_x (panels (a), (b), and (c)) in the shock transition region is positive, and this E_x > 0 causes the formation of the cross-shock potential. In addition to that, there are local enhancements of the magnitude of E_x due to the formation of turbulence and reconnection areas. In the ion-scale magnetic islands, E_x becomes positive in the left side of the islands, while E_x becomes negative in the right side of the islands. Panels (d), (e), and (f) for E_y show that the top part of each ion-scale island has negative E_y , while the bottom part has positive E_y . In summary, the in-plane electric fields in each ion-scale magnetic island are pointing toward the center. The magnitude of the in-plane electric field in those islands reaches B₀.

These strong in-plane electric fields inside the ion-scale islands are considered to be the Hall electric field, because they are generated by the convection effect, E _in-plane = − V_ez e _z × B /c, where e _z is the z-component unit vector. Since ∣V_iz ∣ ≪ ∣V_ez ∣ in these regions, J_z ∼ − enV_ez holds, and E _in-plane ∼ J_z e _z × B /(en_e c). As we will see later in Sections 3.2 and 3.3, these Hall electric fields in the ion-scale islands play an important role to energize electrons. Note that the Hall electric field E _Hall = J × B /c cannot be a source for a net energy increase in the total plasma, because J · E _Hall = 0. However, it can transfer the energy from one species to another, such as from ions to electrons. Let us estimate the magnitude of the in-plane electric field, considering the convection electric field E_x = V_ez B_y /c. Since electrons are accelerated to reach the electron Alfvén speed, V_ez in the ion-scale island also becomes of the order of v_Ae0. Therefore, V_ez /c ∼ v_Ae0/c = Ω_e /ω_pe = 1/4 in the simulation, and we obtain E_x ∼ V_ez B_y /c ∼ B₀ using B_y ∼ 4B₀, which is the magnetic field in the shock (see Figure 1(d)). In a spacecraft observation, if the magnetic field and the plasma density in the solar wind are B₀ = 5 nT and n₀ = 20 cm⁻³, respectively (consistent with typical observation values such as in Gingell et al. 2017, 2019; Wang et al. 2019), the electric field E_x in an island would be 4cB₀(Ω_e /ω_pe ) ∼ 20 mV m⁻¹.

In a 1D laminar shock model, the E_z component is a constant value across the shock, with its magnitude ${E}_{z}={v}_{d}{B}_{0}\sin \theta /c\,=\,0.067{B}_{0}$ under the initial parameters in this study. However, in the 2D simulation, the shock transition region becomes highly turbulent, and panels (g), (h), and (i) of Figure 3 show that there are locally enhanced positive and negative E_z in the shock transition region. At Ω_i t = 18.75 (panel (g)), there are positive and negative stripes of E_z mostly along the ion-scale magnetic islands. The convection electric field is given by E_z = − V_ex B_y /c + V_ey B_x /c, and the electron fluid velocities in this region are basically V_ex < 0 and V_ey > 0 (not shown, but see Figure 1 in Bessho et al. 2019). The top-left side of an ion-scale island (the second quadrant of the island with respect to the island center) has positive E_z , because B_x > 0 and B_y > 0. In contrast, the bottom-right side of the island (the fourth quadrant of the island) shows negative E_z , because B_x < 0 and B_y < 0. At Ω_i t = 20.31 (panel (h)), several sub-ion-scale islands are generated due to electron-only reconnection, and there are local enhancements of positive and negative E_z in those regions. The polarity of E_z varies from island to island, because the electron fluid velocities vary in each sub-ion-scale island. At Ω_i t = 21.88 (panel (i)), after those ion-scale islands are dissipated, the intensity of E_z becomes smaller than the previous times in panels (g) and (h).

To see the overall structures in the electric fields and magnetic fields in the shock transition region, let us discuss quantities averaged over y and examine their 1D structures. In the following, a quantity 〈Q〉 represents the field quantity Q averaged over y. Panels (a)–(c) in Figure 4 display the three components of the electric field, 〈E_x 〉, 〈E_y 〉, and 〈E_z 〉, as a function of x, while panel (d) shows the magnetic field 〈B_z 〉. For 〈B_y 〉, see Figure 1(d). Note that 〈B_x 〉 is a constant, because 〈∇ · B 〉 = 〈dB_x /dx〉 = 0, and 〈B_x 〉 is not plotted in Figure 4. Panel (a) shows 〈E_x 〉, and at Ω_i t = 18.75, the peak is around 0.25B₀ at x = 44d_i . At Ω_i t = 20.31, the peak increased a lot, and the value is around 0.45B₀ at x = 48d_i . There is a secondary peak 0.38B₀ at x = 45d_i , and the dip between the highest and the secondary peaks is due to the growth of ion-scale magnetic islands. At Ω_i t = 21.88, the peak moves farther to the right, and the peak value is around 0.3B₀ at x = 49.5d_i . The smaller peak value is because the ion-scale islands have already been dissipated at Ω_i t = 21.88. Overall, 〈E_x 〉 > 0 in the shock transition region, which makes the cross-shock potential positive. Panel (b) is for 〈E_y 〉, and it is positive in the region where 〈E_x 〉 is very small (such as x > 50d_i ), while 〈E_y 〉 is negative in the region where 〈E_x 〉 becomes large. 〈E_y 〉 is mostly determined by the convection electric field, E_y = − V_ez B_x /c + V_ex B_z /c. In the region with strong 〈E_x 〉, electrons move with the E_x × B_y drift in the positive z-direction; therefore, the first term, −V_ez B_x /c, dominates, and E_y becomes negative. In contrast, in the region where 〈E_x 〉 is small (x > 50d_i ), there is a finite negative 〈B_z 〉 (see panel (d)), and the second term, V_ex B_z /c, becomes large because V_ex < 0 and B_z < 0, which makes E_y positive.

Zoom In Zoom Out Reset image size

Panel (c) shows that overall, 〈E_z 〉 is positive because of the motional electric field $-{v}_{d}{B}_{0}\sin \theta /c\,=\,0.067{B}_{0};$ however, in the shock transition region, 〈E_z 〉 fluctuates because of waves and generation of magnetic islands. In the far-upstream region in x > 70d_i (not shown), 〈E_z 〉 asymptotes to the value 0.067B₀. Panel (d) demonstrates that 〈B_z 〉 becomes negative, and this is because the electron drift V_ez > 0 drags the magnetic field line in the shock transition region. Therefore, a large-magnitude 〈B_z 〉 ∼ − 4B₀ is generated, and reconnection in this region becomes guide field reconnection in 2D simulations (Bessho et al. 2019, 2020, 2022). Note that in 3D cases (see Ng et al. 2022), the reconnection plane does not have to be in the x–y plane; therefore, reconnection can occur with a small guide field if the reconnection plane rotates and it becomes close to the x–z plane.

These electric fields, both the in-plane fields (E_x and E_y ) and the out-of-plane field E_z (partially due to the reconnection electric field), are important to energize and heat electrons in the shock. In the next subsection, we will discuss electron acceleration and heating.

Figure 5 illustrates the time evolution of the electron temperature T_e . In the shock upstream region (x > 60d_i , not shown in the figure), the electron temperature is ${T}_{e-\mathrm{up}}=0.5{m}_{e}{v}_{{\rm{A}}e0}^{2}$. In Figure 5, the shock transition region is shown, and T_e already increases from the far-upstream value. As time progresses, the number of reconnection sites (also the number of magnetic islands) is getting larger and larger. As a result, the electron temperature is also increasing with time in the region 44 < x/d_i < 50. Both X lines and magnetic islands show the temperature enhancement, and the temperature in the entire region 40 < x/d_i < 50 also increases (the background color changes from green, ${T}_{e}/{m}_{e}{v}_{{\rm{A}}e0}^{2}\sim 2.0$, to yellow, ${T}_{e}/{m}_{e}{v}_{{\rm{A}}e0}^{2}\sim 2.5$). Note that T_e is defined as $\left[1/(3{n}_{e})\right]{m}_{e}{\sum }_{j}\int {f}_{e}{({v}_{j}-{V}_{ej})}^{2}{d}^{3}v$, where n_e is the electron density, f_e is the distribution function, v_j is the j-component of an electron velocity, and the sum is over the three components j = x, y, and z. The temperature T_e represents the thermal energy based on the entire distribution function f_e ; however, as discussed in Goldman et al. (2020), when there are multiple beam components in f_e , this definition of T_e represents a mixture of real thermal energy (based on the random speed with respect to each beam velocity) and a pseudo-thermal energy (the energy due to a relative velocity with respective to the bulk fluid velocity, and this can be nonzero even in the limit of cold beams). The temperature T_e can increase either due to the production of multiple beams, or the actual heating associated with the increase of random thermal spread, or both.

Zoom In Zoom Out Reset image size

At Ω_i t = 18.75 (panel (a)), the electron temperature is mostly uniform (green color) throughout the region; however, an ion-scale island (denoted by 2 in the plot) and a small-size island (indicated by a blue arrow) have local enhancements of T_e (yellow and red color). Also, around x = 44d_i , there are several stripes of red and yellow colors. Both island 1 and island 3 show no temperature enhancement yet. At Ω_i t = 20.31 (panel (b)), there are three ion-scale magnetic islands (islands 1, 2, and 3) with significantly enhanced T_e . Small-scale islands, which are produced due to electron-only reconnection, also show high T_e (red color), but some small-scale islands still show lower temperatures (yellow color). At this time, the ion-scale large islands are the dominant electron energization sites. In contrast, at Ω_i t = 21.88 (panel (c)), the large ion-scale islands have already been dissipated, and the red regions with high temperatures spread throughout the y-direction in the region 46d_i < x < 49d_i . Later (in Figures 7 and 8), we will see that the increase of T_e in magnetic islands is associated with multiple beams.

Figures 5(d) and (e) display the electron energy spectra at the three different times. Panel (d) is a log–log plot, while panel (e) is a linear-log plot, using the same distribution functions. Green, blue, and red curves are the energy spectra obtained at three different times, plot (a), (b), and (c), respectively, in the region marked by the red bar in the horizontal axis in each panel. The marked spatial interval at each time covers regions with significant temperature enhancements. At Ω_i t = 18.75, nonthermal electrons are produced (see the green curve in panel (e), and compare with the black curve, which is the upstream electron distribution at x = 500d_i ). In panel (d), the energy spectrum at Ω_i t = 18.75 (green) already shows a power-law-like distribution, and if we approximate the green curve as a power-law function, the power-law index is close to 6.0. Let us define ε_K as ε_K = γ − 1, where γ is the Lorentz factor, and ε_K represents the kinetic energy normalized by m_e c². The average of ε_K at Ω_i t = 18.75 is 0.32 (this value, ε_K = γ − 1 = 0.32, corresponds to $v/c={(1-1/{\gamma }^{2})}^{1/2}=0.65$, and v/v_Te-up = 2.6, where v is a speed, and v_Te-up is the upstream electron thermal speed ${(2{T}_{e-\mathrm{up}}/{m}_{e})}^{1/2}$). At Ω_i t = 20.31, when the ion-scale islands show the most significant electron temperature enhancements, the average of ε_K becomes 0.47, and panels (d) and (e) show that the electrons at Ω_i t = 20.31 (blue) are significantly energized compared with the electrons at Ω_i t = 18.75 (green). At Ω_i t = 21.88, when the ion-scale islands were already dissipated, the maximum energy in the spectrum (red) becomes smaller than that at the previous time Ω_i t = 20.31, suggesting that the most significant energization occurs at Ω_i t = 20.31 because of the ion-scale islands. In contrast, the average of ε_K at Ω_i t = 21.88 is 0.52, and it continuously increases from Ω_i t = 20.31 to Ω_i t = 21.88, suggesting that electrons are continuously accelerated during the dissipation of the ion-scale islands and due to electron-only reconnection that generates small-scale islands. All three curves (green, blue, and red) in panel (d) show that the power-law indices do not significantly change between Ω_i t = 18.75 and 20.31, and they are close to 6.

Choosing an X line and a magnetic island, we investigated the time evolution of the local temperatures and electron distribution functions. In Figure 6, the magenta closed circles are on the X line (position A), while the cyan circles are inside the island (position B). Both positions A and B demonstrate enhancements of the local temperatures as time elapses. At position A (X line) at Ω_i t = 18.75, the temperature is $2.0{m}_{e}{v}_{{\rm{A}}e0}^{2}$ (green color in panel (a)), and it increases to $3.0{m}_{e}{v}_{{\rm{A}}e0}^{2}$ (red color in panel (c)) at Ω_i t = 19.38 because of reconnection. After this time, the magnetic island above this X line (its island center is around x = 45.7d_i and y = 10d_i ) becomes smaller and smaller, and at Ω_i t = 19.69, the X line and the island have already disappeared. We chose two positions at Ω_i t = 19.69: position A and position A', which are not on an X line, but at positions in a red color region in panel (d), corresponding to the region where previously there an X line at Ω_i t = 19.38. The electron temperature T_e at position A (at (x, z) = (45.0d_i , 9.5d_i )) is $3.0{m}_{e}{v}_{{\rm{A}}e0}^{2}$, while T_e at position A' (at (x, z) = (45.0d_i , 10d_i )) is $2.7{m}_{e}{v}_{{\rm{A}}e0}^{2}$.

Zoom In Zoom Out Reset image size

Position B (magnetic island) also shows T_e increase as time elapses. At Ω_i t = 18.75, the temperature is $1.7{m}_{e}{v}_{{\rm{A}}e0}^{2}$. The temperature gradually enhances, and at Ω_i t = 19.38, it becomes $2.5{m}_{e}{v}_{{\rm{A}}e0}^{2}$. It continues to rise, and at Ω_i t = 19.69, T_e reaches $2.9{m}_{e}{v}_{{\rm{A}}e0}^{2}$. In the island, the temperature becomes large around the outer boundary of the island, and the center of the island has a smaller temperature than the outer region. Panel (d) shows a few ion-scale islands, and the electron temperature has a similar ring-like structure inside the islands, where the high-temperature region surrounds the central low temperature region.

Panels (e) and (f) in Figure 6 are the electron energy distribution functions at those two positions A and B. To compose these distribution functions, we collected electrons in a square region with a size 0.1d_i × 0.1d_i around each position. At position A (X line), the energy distribution function shows an increase in its width (the FWHM for the peak) until Ω_i t = 19.38. The highest ε_K ( = γ − 1) is around 1 at Ω_i t = 18.75 (black curve), while the highest energy ε_K exceeds 4 at Ω_i t = 19.69 (the highest ε_K is 5.1, not shown). Note that the spectrum for position A' at Ω_i t = 19.69 is similar to that for position A, and we only plotted the spectrum at position A in panel (e). At position B (magnetic island, panel (f)), significant electron energization occurs between Ω_i t = 19.38 (blue curve) and 19.69 (red curve). The energy distribution at Ω_i t = 19.69 (red curve) shows a slightly harder spectrum than those at position A. The highest ε_K at Ω_i t = 19.69 is 5.7 (not shown).

Figure 7 demonstrates the time evolution of the electron distribution function at position A. Panels (a)–(c) are reduced distribution functions in 2D velocity planes, integrated over the third direction (for example, panel (a) is the 2D reduced distribution function in the v_x –v_y plane, integrated over v_z ). At position A, at Ω_i t = 18.75 (T_e = 2.0m_e v_Ae0), the electron distribution function is nongyrotropic (panels (a)–(c)), where the white lines in panels (b) and (c) represent the direction of the magnetic field. Note that there is only B_z at the X line; therefore, panel (a) for the velocity plane v_x –v_y does not show the direction of the magnetic field, and this v_x –v_y plane is the velocity plane perpendicular to the magnetic field. The phase space densities in the regions of v_x > 0, v_y > 0, and v_z < 0 become high. As time evolves, reconnection generates heated electrons, and at Ω_i t = 19.06 (panels (d)–(f)), the gyrotropy of the distribution function increases and regions with high phase space density (red regions) spread. At this time, the temperature becomes $2.1{m}_{e}{v}_{{\rm{A}}e0}^{2}$. At Ω_i t = 19.38 (panels (g)–(i)), the radius of the distribution function becomes larger than that at the previous time, and the temperature increases to T_e ∼ 3.0m_e v_Ae0. At this time, multiple electron beams are seen in each velocity plane. In particular, panel (i) in the v_z –v_x plane shows two major components: the component with v_z < 0 and v_x > 0 and the component with v_z > 0 and v_x ∼ 0. After the X line is dissipated, at Ω_i t = 19.69 (panels (j)–(l)), the radius of the distribution function does not significantly change, suggesting that the electron temperature also does not significantly change after Ω_i t = 19.38, but the distributions show a notable change from the previous time. Panels (k) and (l) show a strong electron beam moving in the positive v_z -direction. Panels (m)–(o) are for the distributions at position A', which is close to position A. There are still multiple electron beams, but the thermal spread in panels (m)–(o) for position A' (${T}_{e}=2.7{m}_{e}{v}_{{\rm{A}}e0}^{2}$) is more significant than that in panels (j)–(l) for position A (${T}_{e}=3.0{m}_{e}{v}_{{\rm{A}}e0}^{2}$). At position A', the z-directional beam component in panel (o) can be seen, but the red part spread in the region −20v_A0 < v_z < 45v_A0, which is more spread than that in panel (l) for position A, where the red region is more localized in a region 20v_A0 < v_z < 45v_A0. This indicates that the z-directional beam is dissipating in a lower v_z region, and electron heating is occurring. We conclude that some high-temperature regions contain multiple electron beams, and those high-speed beams play important roles for electron heating. Detailed mechanisms about how electron beams are scattered to heat electrons are beyond the scope of this paper.

Zoom In Zoom Out Reset image size

Figure 8 shows the time evolution of the electron distribution function at position B (magnetic island). At time Ω_i t = 18.75 (panels (a)–(c)), the distribution function is nongyrotropic, but compared with the distribution at position A (panels (a)–(c) in Figure 7), the distribution at point B is more gyrotropic. As time evolves, the radius of the distribution function increases. Panels (d)–(f) at Ω_i t = 19.06 show that the radius of the distribution is slightly larger than at the previous time. The increase of the radius of the distribution function continues until Ω_i t = 19.69. At the same time of the increase of the radius, a strong z-directional electron beam is generated at Ω_i t = 19.38 (panels (g)–(i)), and the beam remains until Ω_i t = 19.69 (panels (j)–(l)). Again, the beam formation is one of the causes of the electron temperature enhancement in the magnetic island. Compared with the distribution at the X line (Figure 7), the distribution in the island has a persistent strong electron beam that lasts longer than the beam at the X line. Note that the beam direction is not parallel to the magnetic field. This z-directional electron beam can be understood as the beam generated by the E × B drift due to the in-plane electric fields E_x and E_y . As we have seen in Figure 3, in each ion-scale island, there is a strong Hall electric field that points toward the center of the island. Magnetized electrons are drifting in the positive z-direction because of the E × B drift due to the in-plane electric fields in each island, and even unmagnetized energetic electrons can drift in the same direction as the E × B drift; therefore, we observe the strong z-directional beam in panels (h), (i), (k), and (l) in Figure 8.

Zoom In Zoom Out Reset image size

We investigate individual electrons' trajectories to understand where and how they are accelerated in the shock transition region. We choose electrons (actual particles in the PIC simulation, not test particles) in regions where reconnection occurs, and trace their trajectories and the time evolution of their momenta, energies, and the work done by each component of electric field, which is given by −e∫E_j v_j dt, where j represents a component of the electric field, either j = x, y, or z, or j = ∥ (parallel to B ), and the perpendicular work is defined as the total work subtracted by the parallel work.

We plot in Figure 9 the trajectory of an electron that is accelerated at an X line denoted by X1 (at (x, y) = (46d_i , 7.5d_i )) in panel (a) at Ω_i t = 18.91. The black curves are magnetic field lines, and the particle trajectory starting at Ω_i t = 18.75 is shown by the red curve. In panel (a), among several X lines in this plot, two reconnection X lines, X1 and X2, are marked. X1 is the same X line as position A in Figure 6. The electron is interacting with X1, and it is ejected to the outflow region left of the X line. Panel (d) is the time evolution of the electron's Lorentz factor γ, and γ − 1 represents the kinetic energy normalized by m_e c². The electron's γ is increasing at Ω_i t = 18.75 up to 1.8, and after the ejection, it decreases a little bit to 1.5 at Ω_t = 18.91. Panel (b) displays the trajectory until Ω_i t = 19.31. The X line is moving upward during the time interval between Ω_i t = 18.91 and 19.31. After the electron was ejected from X1 at Ω_i t = 18.91, it moves downstream to the left of X1, but the electron is reflected in the downstream region and returns back to X1 again at Ω_i t = 19.31. During the second interaction with X1 (after Ω_i t = 19.2), the electron gets accelerated and gains energy. Panels (e) and (f) are the time evolution of the work done by the perpendicular and the parallel electric fields, and the perpendicular and the parallel momenta of this particle, respectively. (The vertical axis scale in panels (d), (e), and (f) is chosen as the same scale as in Figure 10 for another electron, which shows more dramatic changes of these quantities than this electron in Figure 9.) In panel (d), the Lorentz factor γ becomes almost constant until Ω_i t = 19.2, and after that time, it continuously increases with time until Ω_i t = 19.6. Panel (e) displays the work done by the perpendicular electric field (red) and the parallel electric field (blue). During the interval between Ω_i t = 19.31 and Ω_i t = 19.6, the energy increase is due to the perpendicular electric field. Panel (c) shows the particle trajectory until Ω_i t = 19.69, and this particle is moving back and forth in the y-direction after Ω_i t = 19.31, during the interaction with X1. The magnetic island between X1 and X2 is gradually dissipated between Ω_i t = 19.31 and 19.69, and X1 has already disappeared at Ω_i t = 19.69 (panel (c)). Panel (f) shows the momentum components: the perpendicular momentum (red) and the parallel momentum (blue). During the time interval between Ω_i t = 19.31 and Ω_i t = 19.6, the parallel momentum p_∥ changes its sign five times, during which the perpendicular momentum p_⊥ is slightly increasing. Both ∣p_⊥∣ and ∣p_∥∣ are increasing during the bounce motion. This acceleration resembles Fermi acceleration, but the parallel momentum increase is less significant than the increase of p_⊥, except for at the time of the final reversal of p_∥ around Ω_i t = 19.6. After Ω_i t = 19.6, the electron is ejected to the downstream region with negative p_∥.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figure 10 shows the trajectory of another electron. In panel (a), we see that this electron is moving back and forth between two X lines, X3 and X4. At Ω_i t = 19.06, the Lorentz factor γ is around 1.75 (see panel (d)). After Ω_i t = 19.06, the energy starts to increase, and γ reaches 2.5. Panel (e) indicates that the energy increase is due to the perpendicular work, and both γ and W_⊥ have step-function-like enhancements, consistent with the physical picture of Fermi acceleration, where the energy enhancement occurs due to each bounce motion, as we see in the following. Panel (b) illustrates that during the bounce motion between X3 and X4, another X line (X5) forms between X3 and X4. Then this electron is trapped between X3 and X5, and starts to bounce between X3 and X5. The Lorentz factor γ continues to increase during the bounce motion between X3 and X5, between Ω_i t = 19.35 and 19.58. Panel (f) demonstrates that the parallel momentum p_∥ changes its sign during the bounce motion, and the magnitude of p_∥ increases at each bounce. This step-function-like increase of γ, energization due to the perpendicular electric field, and the increase of ∣p_∥∣ are signatures of Fermi acceleration in contracting islands (Drake et al. 2006).

To see the details of this Fermi acceleration, we plot the dependence of the Lorentz factor γ on the y-position in Figure 10(g), during the multibounce motion between X3 and X5 between Ω_i t = 19.35 and 19.58. The locations of the two X lines, X3 and X 5 are denoted by the vertical dashed lines. Each color in the curve corresponds to the segment during a certain time interval. For example, the magenta curve is the segment during 19.35 < Ω_i t < 19.40, when the particle is moving from right to left (denoted as R → L). During this time interval, the particle is moving from X5 to X3 (see also panel (b)). When the electron reaches the region of X3, around Ω_i t = 19.40, the Lorentz factor γ increases rapidly from 2.0–2.4 during the reflection at X3 (see the rapid rise of the magenta curve near the magenta arrow in panel (g)). After this reflection at X3, the orange segment starts, which represents the electron motion during 19.40 < Ω_i t < 19.47, from left to right (X3 to X5). After the reflection at X3 at Ω_i t = 19.40, γ oscillates between 2.4 and 1.8, and when the electron reaches X5, γ increases again to 2.4. After this second reflection at X5, the green segment starts. The electron moves from X5 to X3, and it is reflected again at X3, where there is also a large increase of γ to 2.6 (see the region near the green arrow). The blue segment for the interval 19.50 < Ω_i t < 19.53 shows the largest γ around 2.7, and the electron encounters the fourth reflection near X5, after which the red segment follows, and the electron moves from X5 to X3. Finally the electron exits from this multibounce motion, moving out toward the left. Panel (h) shows the Lorentz factor γ as a function of time, and each color corresponds to the interval with the same color as in panel (g). In panel (h), there are two major increases of γ, indicated by the magenta and green arrows, which correspond to the same times as indicated by the arrows in panel (g).

Note that the PIC simulations by Matsumoto et al. (2015) and Bohdan et al. (2020) in astrophysical perpendicular shocks (where M_A is of the order of 50) also show electron energization by Fermi acceleration due to interactions with multiple reconnection regions (in particular in merging islands). In our simulation with the parameters in the Earth's bow shock, M_A (=10.5) is much smaller than in those studies, the shock is quasi-parallel, and the shock speed (10.5v_A0) is smaller than the upstream electron thermal temperature v_Te0 = 14.4v_A0. Even in such a lower-energy regime than Matsumoto et al. (2015) and Bohdan et al. (2020), clear signatures of Fermi acceleration are observed. The particle in Figure 9 shows a similar acceleration history as shown in Matsumoto et al. (2015) and Bohdan et al. (2020), where the electron is accelerated as it encounters X lines (in other words, head-on collisions with reconnection jets) as discussed in Hoshino (2012). In contrast, the particle in Figure 10 shows a different history, where the particle is trapped between two X lines, and accelerated in the contracting islands, showing stepwise increases of its energy (Drake et al. 2006).

Figure 11 displays the trajectory of an electron that is accelerated in an ion-scale magnetic island generated by ion-coupled regular reconnection. Panel (a) shows that this electron is moving around the ion-scale island at Ω_i t = 19.69. This island is the same island denoted as island 2 in Figures 1, 2, and 5. This electron enters the ion-scale island at Ω_i t = 19.34, and goes around the island once until Ω_i t = 19.84. After then, it is ejected from the island, and moves around the region below this ion-scale island where smaller (electron-scale) islands are generated (see panel (b)). Panel (c) shows that the electron is ejected from this region, and it enters the downstream region of the shock. Panel (d) is the x–z trajectory of the electron, and after this electron is trapped in the island at Ω_i t = 19.34, it is accelerated in the positive z-direction in the ion-scale island and the small-scale islands until Ω_i t = 21.34. Note that the simulation is 2D in the x–y plane, but z in panel (d) is computed as the time integral of v_z .

Zoom In Zoom Out Reset image size

Panel (e) shows the time evolution of the Lorentz factor γ. This electron has γ ∼ 6 at Ω_i t = 18.75, and after it enters the ion-scale magnetic island at Ω_i t = 19.34, γ increases with time. The time interval in which the electron is going around this island is denoted by the blue arrow between the blue vertical lines. When it is ejected from the island, at Ω_i t = 19.84, γ reaches ~10. After the ejection, γ is still gradually increasing while the electron is moving around in the region below the ion-scale island, and γ becomes close to 12. Panel (f) shows the work done by the parallel electric field (blue) and the perpendicular electric field (red). Most of the energy increase is due to the perpendicular electric field.

The time evolution of the parallel and perpendicular momenta is shown in panel (g). While the electron is in the ion-scale island, first, the parallel momentum is almost zero between Ω_i t = 19.3 and 19.6. During that interval, the perpendicular momentum increases with time. In panel (a), we see that the electron is moving along the left side of the ion-scale magnetic island between Ω_i t = 19.3 and 19.6. During this time interval, the electron is magnetized, and the zigzag pattern along the left side of the island in panel (a) represents the gyro-motion. After Ω_i t = 19.6, panel (g) shows that the parallel momentum increases rapidly, and the perpendicular momentum decreases. After the electron is ejected from the ion-scale island, the parallel momentum p_∥ becomes negative and the magnitude ∣p_∥∣ increases with time until Ω_i t = 20.5. After that, p_∥ oscillates a few times, while the perpendicular momentum p_⊥ increases with time until Ω_i t = 21.25, around when γ reaches the maximum value, γ ∼ 12.

Panel (h) presents the decomposition of the work into three components of electric fields, E_x (red), E_y (blue), and E_z (green). During the time interval in the ion-scale magnetic island, the green curve for W_z , which represents the work done by E_z , the same direction as the reconnection electric field, is decreasing with time. The dominant contributions to energize this electron are the in-plane electric fields, E_x and E_y . During the time interval in the ion-scale island, first, the red curve for W_x increases until Ω_i t = 19.6. After that, the blue curve for W_y increases rapidly, and the work done by E_y becomes the largest contribution before the electron is ejected from the ion-scale island. The blue curve continues to rise in the region below the ion-scale island until Ω_i t = 20.25. After that, W_x starts to increase, and it becomes the highest contribution at the end.

Let us see more details of the energization by the in-plane electric fields while the electron is moving around the ion-scale magnetic island. Figure 12 illustrates the time evolution of the spatial distributions of E_x and E_y in the ion-scale island (grayscale), and the electron trajectory (red). The light blue curves are magnetic field lines, and we see that the island is moving from the lower-right region to the upper-central region in the 2D domain. The yellow dots in each plot show the position of the electron at each time denoted in the yellow box. At Ω_i t = 19.41 (panels (a) and (b)), the electron is at the upper-left part of the island, and it has passed through the upper-right part of the island. In the island, there are strong in-plane electric fields E_x and E_y , pointing toward the center of the island. These electric fields cause the formation of the electric potential ϕ that has a dip at the center of the island. Since the electron has a negative charge, the electron feels a potential −e ϕ that has a positive peak at the center of the island.

Zoom In Zoom Out Reset image size

After the electron enters the island, the electron starts to move in the positive y-direction. While the electron is moving in positive y, the relative position of the electron in the island does not change much. Panels (a)–(d) indicate that the electron stays in the upper-left region of the island between Ω_i t = 19.41 and 19.63. If the island is stationary, the electron does not gain energy when it stays at the same position. However, the island is moving upward, and the electron gains energy with the amount, ΔW_y = − e∫E_y v_y dt = e∣E_y−av∣l_y , where ∣E_y−av∣ is the average of ∣E_y ∣ in the path of the electron, and l_y is the electron displacement in the y-direction. Therefore, as we see in panel (h) in Figure 11, the electron shows a significant increase of W_y during the time interval in the ion-scale island. In addition, the electron gains energy from E_x when the electron moves left during the gyration, since there is strong E_x pointing right within the left side of the island. The island is moving in the negative x-direction, and the electron is also slightly moving in the negative x-direction between Ω_i t = 19.41 and 19.63 (see panel (c)). Therefore, the electron also gains energy from E_x while the island and the electron move in the negative x-direction together, and Figure 11(h) shows an enhancement of W_x in the island. The sum of W_x and W_y increases with time while the electron is in the island. Panels (e) and (f) in Figure 12 show the electron trajectory after Ω_i t = 19.63. After the electron reaches the upper edge of the island, it moves downward, passing through the region where E_y > 0 until Ω_i t = 19.84. The electron gains further energy while the electron is moving gradually to a position farther from the center of the island, since the electric potential, -e ϕ, becomes smaller there. As we see in panels (e) and (h) in Figure 11, the Lorentz factor γ increases from 6 to 10 while the electron is in the island (see panel (e) in Figure 11), and the contribution from the in-plane electric field E_y reaches 5m_e c² (see the blue curve in panel (h) in Figure 11). Note that this energy gain, Δγ ∼ 4, is exaggerated because of the artificial simulation parameters such as m_i /m_e = 200 and ω_pe /Ω_e = 4, which make the electron thermal speed and electron Alfvén speed close to the speed of light. In Section 3.4, we will discuss realistic electron energization by translating the simulation result into a real result based on realistic parameters.

Figure 13 displays another example of electron energization in an ion-scale magnetic island. Panels (a)–(c) demonstrate that the electron is once accelerated in an X line (X6), and it is later trapped in a magnetic island associated with X7. After this electron is trapped in the island, the island is moving in the y-direction, and the island passes through the top boundary at y = 25.6d_i and reappears from the bottom boundary at y = 0. The electron is kept trapped until Ω_i t = 21.88. During the trapping, panel (d) shows that the electron is moving in the z-direction. The energy γ continues to increase until Ω_i t = 21.88 (panel (e)), and the work done by the perpendicular electric field is the major source of the energization (panel (f)). Panel (g) shows that after the electron was trapped in the magnetic island around Ω_i t = 20.63, the parallel momentum is almost constant, but the perpendicular momentum is gradually increasing. Panel (h) indicates that after Ω_i t = 20.63, the most significant increase is W_z , the work done by E_z , and it rapidly increases after Ω_i t = 21.0. The work done by the in-plane electric field (the sum of W_x and W_y ) is almost constant between Ω_i t = 20.63 and Ω_i t = 21.0, and the sum starts to decrease after Ω_i t = 21.0. This result shows that, in contrast to the particle in Figure 11, the particle in Figure 13 is energized in the ion-scale island mainly due to the out-of-plane electric field E_z . This example resembles the island surfing acceleration discussed by Oka et al. (2010), in which electrons trapped in a magnetic island generated in laminar standard reconnection are accelerated in the island mainly due to the out-of-plane electric field.

Zoom In Zoom Out Reset image size

Let us compare acceleration by magnetic reconnection and that by other mechanisms in the shock. First, let us consider an electron that is accelerated by the shock potential (Woods 1969; Goodrich & Scudder 1984; Scudder 1995). Since the shock potential e ϕ (= − e∫E_x dx) is positive in the shock transition region, the electron gains energy from E_x as it passes through the shock. The magnitude of the shock potential is of the order of $2{m}_{i}{v}_{{\rm{A}}0}^{2}{M}_{{\rm{A}}}$ when M_A ≫ 1 (see Ohsawa (1985)). From Figure 4(a), using the value of 〈E_x 〉 at Ω_i t = 20.31, if we numerically integrate 〈E_x 〉 from x = 55d_i to x = 46d_i , which is the region including the first peak ∼0.4B₀ around x = 48d_i with its thickness ∼3d_i , the magnitude of the electric potential is $e\phi =54{m}_{i}{v}_{{\rm{A}}0}^{2}$, which is more than twice as large as $e\phi \sim 2{m}_{i}{v}_{{\rm{A}}0}^{2}{M}_{{\rm{A}}}$ with M_A = 10.5. Let us consider a situation where electrons are accelerated by the electrostatic potential to form an electron beam, and heating occurs subsequently due to another heating mechanism such as scattering by fluctuating fields, which can produce a flat-top distribution function (Schwartz et al. 1988; Chen et al. 2018). In this case, the electron temperature can increase to a value close to the potential energy. This potential value $e\phi =54{m}_{i}{v}_{{\rm{A}}0}^{2}\,=54{m}_{e}{v}_{{\rm{A}}e0}^{2}$ is much greater than the actual temperature increase, $2{m}_{e}{v}_{{\rm{A}}e0}^{2}$, seen in Figure 5, because electrons that are energized by the shock potential are also decelerated by E_z during their drift motion to the positive z-direction, mainly due to the E_x × B_y drift and the parallel motion along the magnetic field line that has negative B_z in the shock transition region (see Figure 4(d)). The total energy gain is the sum of the positive shock potential and the negative work done by E_z while electrons are drifting in the positive z-direction. To see this point, let us follow Bessho & Ohsawa (1999) to derive the energization in an oblique shock. We consider the energy equation in the shock rest frame for an electron in a laminar, 1D, stationary shock, given as

$$\begin{eqnarray}&&{m}_{e}{c}^{2}\displaystyle \frac{d\gamma }{{dt}}=-{{eE}}_{x}{v}_{x}-{{eE}}_{z0}{v}_{z},\end{eqnarray} \tag{ 1 }$$

where we only need to consider the electric field E_x and E_z . In the 1D stationary shock, E_y in the shock is zero because ∂B_z /∂t = − c∂E_y /∂x = 0 and E_y0 (E_y in the far-upstream region) is zero. The value E_z0 is a constant value ${E}_{z0}={v}_{\mathrm{sh}}{B}_{0}\sin \theta /c$, where v_sh is the shock speed (v_sh = M_A v_A0). If we integrate the first term in the right-hand side with respect to time (note that the shock is assumed to be stationary and E_x is not time dependent), we obtain the shock potential e ϕ. The integration of the second term on the right-hand side is the work done by E_z0 during the motion in the z-direction, which gives the deceleration. To compute the integration, we consider the y-component of the equation of motion,

$$\begin{eqnarray}&&{m}_{e}\displaystyle \frac{d\gamma {v}_{y}}{{dt}}=-\displaystyle \frac{e}{c}{v}_{z}{B}_{x0}+\displaystyle \frac{e}{c}{v}_{x}{B}_{z},\end{eqnarray} \tag{ 2 }$$

where B_x0 in the 1D shock is a constant. Substituting v_z from this equation into Equation (1), and integrating with respect to time, we obtain

$$\begin{eqnarray}&&{m}_{e}{c}^{2}{\rm{\Delta }}\gamma =e{\phi }_{p}+{m}_{e}c\displaystyle \frac{{E}_{z0}}{{B}_{x0}}(\gamma {v}_{y}-{\gamma }_{0}{v}_{y0}),\end{eqnarray} \tag{ 3 }$$

where the subscript 0 represents the quantity in the upstream region, Δγ is the increase of the Lorentz factor γ − γ₀, and e ϕ_p is a pseudo-potential, defined as e[ϕ − (E_z0/B_x0)A_y ] (see also Scudder 1995), and A_y is the vector potential given by A_y = ∫B_z dx. Note that the derivative of −ϕ_p along the field line (−d ϕ_p /ds = − (d ϕ_p /dx)(B_x0/B), where $B={({B}_{x0}^{2}+{B}_{y}^{2}+{B}_{z}^{2})}^{1/2}$ and ds = (B/B_x0)dx) is equal to the parallel electric field E_∥ = (E_x B_x0 + E_z0 B_z )/B. When we consider electrons with v_y ∼ 0 and v_y0 ∼ 0, the energy gain of an electron that passes through the shock is given by the pseudo-potential (e ϕ_p = − e∫E_∥ ds) instead of the shock potential (e ϕ = − e∫E_x dx). The magnitude of the pseudo-potential e ϕ_p is discussed in Ohsawa (2018), and its maximum magnitude in our simulation at Ω_i t = 20.31 is obtained as $e{\phi }_{p}=e\phi -e({E}_{z0}/{B}_{x0}){A}_{y}\,=12{m}_{i}{v}_{{\rm{A}}0}^{2}$, where we used A_y based on the numerical integral of 〈B_z 〉 shown in Figure 4(d). If we neglect the second term in the right-hand side of Equation (3), the increase of the Lorentz factor (e ϕ_p /m_e c²) due to the pseudo-potential $\sim 12{m}_{i}{v}_{{\rm{A}}0}^{2}$ in the shock rest frame is computed to be around 0.75. Since E_∥ > 0 will accelerate electrons along the magnetic field where the dominant component is B_y , most of the electrons are flowing in the negative y-direction with v_y < 0; therefore, the effect of the second term in Equation (3) becomes large, and the increase of the gamma factor is much less than the pseudo-potential. This picture is consistent with the fact that the temperature increase in Figure 5, $\sim 2{m}_{e}{v}_{{\rm{A}}e0}^{2}$ is much smaller than the pseudo-potential $e{\phi }_{p}=12{m}_{i}{v}_{{\rm{A}}0}^{2}\,=\,12{m}_{e}{v}_{{\rm{A}}e0}^{2}$. Note that rigorously speaking, the simulation result is in the simulation frame, which is different from the shock rest frame; however, the shock in the simulation frame is moving with a small positive speed (∼1.5v_A0) in the x-direction, which is much smaller than the speed of light (1.5 v_A0 = 0.027c), and about 14% of the shock speed v_sh = 10.5v_A0. Considering the Lorentz transformation from the shock rest frame to the simulation frame with the relative velocity v_rel = − 1.5v_A0, Δγ in the simulation frame (${\rm{\Delta }}{\gamma }_{\mathrm{sim}}$) is obtained using the quantities in the shock rest frame as

$$\begin{eqnarray}&&{\rm{\Delta }}{\gamma }_{\mathrm{sim}}={\gamma }_{\mathrm{rel}}{[{\rm{\Delta }}\gamma -{v}_{\mathrm{rel}}({v}_{x}\gamma +{v}_{\mathrm{sh}}{\gamma }_{0})/{c}^{2}]}_{\mathrm{shock}},\end{eqnarray} \tag{ 4 }$$

where γ_rel is a Lorentz factor based on v_rel, and the quantities in the right-hand side are the values in the shock rest frame (denoted by the subscript shock). Here, v_x is the velocity in the shock frame after the acceleration, and we assume that v_x0 in the shock rest frame (v_x before the acceleration) is v_x0 = − v_sh in the upstream region. Since the electron is accelerated by E_∥ along the magnetic field where B_y is the most dominant component in the magnetic field, ∣v_x ∣ is much smaller than ∣v_y ∣ after the acceleration. Suppose that the electron becomes relativistic, passing through the shock with the speed of light along the magnetic field after the acceleration by E_∥, and the speed can be approximately expressed as v_x ∼ − cB_y /B_x0. Even in this extreme case, the second term on the right-hand side in Equation (4) is negligibly small as long as γ is of the order of unity. Therefore, under a small relative velocity v_rel, ${\rm{\Delta }}{\gamma }_{\mathrm{sim}}\sim {\rm{\Delta }}{\gamma }_{\mathrm{shock}}$ because γ_rel ∼ 1, and we can apply the estimate obtained by Equation (3) in the shock rest frame to the simulation frame.

Figure 14 shows an electron that is accelerated mainly by the parallel electric field (in other words, by the pseudo-potential). This electron is at x = 55d_i at Ω_i t = 20.31, and it is moving along the magnetic field. After the electron enters the shock transition region, it starts to drift to the positive z-direction (see panel (b)). This is due to the E_x × B_y drift and the parallel motion under B_z < 0. (Note that E_x × B_y drift in the z-direction is determined by E_x B_y . In the region 48d_i < x, B_y < 0, which gives a negative z drift if E_x > 0; however, E_x ( > 0) becomes large only in the region x < 48d_i , where B_y > 0. In fact, at Ω_i t = 21.25, 〈E_x 〉 < 0.18B₀ in 48d_i < x, while 〈E_x 〉 ∼ 0.43B₀ around x = 48, where 〈E_x 〉 is the y-averaged value of E_x (not shown).) In addition, the electron starts to gain energy, and γ increases (see panel (c)). At Ω_i t = 21.13, the energy reaches the maximum, γ = 2.1. Panel (d) shows that the energy increase is due to the work done by the parallel electric field (blue curve). This is because of the pseudo-potential, and the perpendicular work (red curve) gives a negative contribution. The energy increase, Δγ = γ − γ₀ is around 1.0, and this increase is consistent with the increase due to the pseudo-potential estimate $e{\phi }_{p}=12{m}_{i}{v}_{{\rm{A}}0}^{2}$, which gives an increase Δγ ∼ 0.75. Note that the estimate is under an assumption that the shock is stationary and 1D, and the actual increase Δγ ∼ 1.0 is larger than 0.75 because the shock is 2D and nonstationary in the simulation frame, and also many magnetic islands are generated in the shock transition region, across which the electron passes.

Zoom In Zoom Out Reset image size

This increase Δγ ∼ 1 due to the parallel electric field in Figure 14 is comparable to the Lorentz factor increase for electrons in Figures 9 and 10, where those electrons are accelerated by multiple reconnection sites associated with small-size islands, but much smaller than the acceleration during the interactions with ion-scale magnetic islands, as we see in Figures 11 (Δγ ∼ 4) and 13 (Δγ ∼ 2). As we see in Figure 5(d), Lorentz factors for the most significantly accelerated electrons reach on the order of 10, and these high-energy electrons cannot be explained by the acceleration due to the pseudo-potential (the acceleration by E_∥), but they should be partially accounted for by the mechanism we observed in ion-scale magnetic islands.

The large Lorentz factor increase (Δγ ≳ 2) in ion-scale magnetic islands can be understood in the following way. As we see in Figures 11 and 13, electrons can be energized by both the in-plane electric field and the out-of-plane electric field in islands. In reconnection in the shock transition region, electrons are accelerated and the fluid speed reaches of the order of the electron Alfvén speed v_Ae0 (see Figure 2 and also Bessho et al. 2022). The fast flow generates a strong Hall electric field (for example, −V_ez × B_y convection field), B_r v_Ae0/c, where B_r is the reconnecting magnetic field. Let us use a typical value of B_y in the shock transition region to represent B_r . According to Figure 1(d), B_y ∼ 4B₀ is a reasonable estimate. Therefore, in an ion-scale magnetic island, the in-plane electric field is E_in ∼ 4B₀ v_Ae0/c ∼ B₀, where v_Ae0/c = 0.25 in the simulation. Figure 3 shows that both E_x and E_y are of the order of B₀. Each magnetic island is moving in a random direction with a speed of the order of v_A0 (Bessho et al. 2022), and electrons trapped in such an island are moving in the same direction as the island motion. Therefore, roughly speaking, if the island is moving in a certain direction with a distance l, a trapped electron can gain energy ∼eE_in l.

Let us discuss first the electron energization shown in Figure 11. According to Figures 11 and 12, the island speed is around 6v_A0, and the electron enters the island around Ω_i t = 19.3 and reaches the top of the island around Ω_i t = 19.7. During this time, the island moves upward with a distance 2.5d_i . The minimum E_y in the upper part of the island is around −B₀, but E_y exerted on the electron fluctuates during the motion. The electron feels $\bar{{E}_{y}}\sim -0.4{B}_{0}$ on average (the time average of E_y exerted on the particle (not shown)). In the following discussion, a quantity Q with a bar in the top, $\bar{Q}$, refers to the averaged field value exerted on the particle. As a result of the acceleration by E_y , the increase of the Lorentz factor ${\rm{\Delta }}\gamma \sim e| \bar{{E}_{y}}| l/{m}_{e}{c}^{2}\sim 3.5$ during 19.3 < Ω_i t < 19.7, where l is the upward displacement 2.5d_i . Note that this obtained energy is larger than the potential energy in the island, which can be estimated, roughly speaking, as $e| \bar{{E}_{y}}| {r}_{\mathrm{island}}/{m}_{e}{c}^{2}\sim 1.4$, where r_island is the radius of the island, r_island ∼ 1d_i . Since the electron travels a longer distance than the island radius, the electron can gain energy lager than the potential energy of the island. From Ω_i t = 19.7 to 19.8, the electron moves downward around the island (see Figures 12(e) and (f)), and it is accelerated by a positive E_y in the island and obtains a similar amount of energy as in the upward motion. However, in this downward displacement in y around 4d_i , the electron passes through a region where $\bar{{E}_{y}}\sim 0.15{B}_{0}$, which is weaker than in the upper part of the island (see Figure 12(f)), and the increase of γ during the downward motion is ${\rm{\Delta }}\gamma \sim e| \bar{{E}_{y}}| l/{m}_{e}{c}^{2}\sim 2.1$, using l ∼ 4d_i . The total contribution of E_y acceleration to the increase of γ is around 5.6 (see Figure 11(e), the blue curve in the interval of the island). However, while the electron gains energy from E_y , Figure 11(h) shows that the electron loses energy due to E_z . The sum of W_y and W_z indicate that the total increase of γ during the motion in the island is around 4. Note that the contribution of W_x during the time interval of the island is almost zero.

In the same way, let us estimate the energization in an ion-scale island due to E_z , as observed in Figure 13. The out-of-plane field E_z in an island is much less than the in-plane electric field, and of the order of 0.01–0.1B₀. An electron trapped in an island is moving in the z-direction with its speed around a few v_Ae0 (see Figures 8(k) and (l), where the red population shows v_z ∼ 40 − 50v_A0 ∼ 2.8 − 3.5v_Ae0, using the mass ratio 200 in the simulation). For the electron shown in Figure 13, the electron is trapped in the ion-scale island around Ω_i t = 20.5 until the end of the simulation at Ω_i t = 21.88, and the trapping time is ${t}_{\mathrm{trap}}\sim 1.5{{\rm{\Omega }}}_{i}^{-1}$. While the electron is trapped in the island, the electron moves in the z-direction with v_z ∼ 2.5v_Ae0 (not shown), and a travel distance l_z ∼ v_z t_trap ∼ 53d_i (see Figure 13(d)). Using $\bar{{E}_{z}}\sim -0.01{B}_{0}$ (based on E_z exerted on the particle (not shown)) and l_z = 53d_i , the increase of the Lorentz factor becomes ${\rm{\Delta }}\gamma \sim e| \bar{{E}_{z}}| {l}_{z}/{m}_{e}{c}^{2}\sim 2$, which is consistent with Figure 13(h) (see the green curve for W_z ).

In both cases (the energization by E_in and by E_z ), Δγ during the trapping in the ion-scale island is much larger than Δγ due to acceleration in sub-ion-scale reconnection regions where electron-only reconnection occurs. This is mainly due to the fact that ion-scale islands last longer than sub-ion-scale islands. Note that this estimate Δγ ∼ 2–4 is for the simulation with v_Ae0/c = 0.25. In the actual bow shock, v_Ae0/c is much smaller than 0.25, and the plasma is nonrelativistic. The energy increase should be estimated as eE_in l ∼ er_B B₀(v_Ae0/c)l, where l is of the order of d_i , and r_B is a ratio of B_y to B₀. When B₀ = 5 nT, r_B = 4, and n = 20cm⁻³ (Gingell et al. 2017, 2019; Wang et al. 2019), we obtain E_in ∼ 20 mV m⁻¹, and Δγ ∼ 1 keV.

Finally, let us compare with the shock drift acceleration (Holman & Pesses 1983; Wu 1984; Ball & Melrose 2001). In the shock rest frame, this acceleration occurs when electrons drift in the direction opposite to the motional electric field E_z ( > 0) due to the grad B drift, which points in the negative z-direction. Ball & Melrose (2001) argued the maximum ratio of the reflected electron's energy E_r to the incident electron's energy E_i , and also the maximum ratio of the passing electron's energy E_p to the incident energy E_i . They demonstrated that for both reflected electrons and passing electrons, the energy increase ratio is large, of the order of 10, when the shock angle is close to 90°. In our simulation, the shock is quasi-parallel, and in Appendix B, we discuss the maximum energies of the reflected electron and the passing electron due to the shock drift acceleration, under an assumption that the shock is 1D. The energy increase, Δγ = (E_r − E_i )/m_e c² for the reflected electron that reaches the maximum energy, and Δγ = (E_r − E_i )/m_e c² for the passing electron that reaches the maximum energy, are given in Equation (B16) and Equation (B27), respectively (note that the maximum value of Δγ among reflected electrons is given in Equation (B22), while the maximum of Δγ among passing electrons is given as Equation (B27)), and Δγ for both cases under the simulation parameters is less than 0.2 (see the discussion in Appendix B), which is much smaller than Δγ for electrons that are accelerated by reconnection in the shock.

Figure 15 shows an example of an electron accelerated by the shock drift acceleration. Panel (a) illustrates that the electron enters the shock around Ω_i t = 21.28, and panel (b) shows that a drift starts in the negative z-direction, which is opposite to the direction of the E_z ( > 0) field. After Ω_i t = 21.28, the energy starts to increase (panel (c)), and the Lorentz factor γ reaches its maximum value around 1.8 at Ω_i t = 21.64. Panel (d) shows that the most dominant contribution is the perpendicular electric field (see the red curve for W_⊥), which indicates that the electron gains energy because it drifts to the direction opposite to the direction of the perpendicular electric field, the same mechanism as the shock drift acceleration. The increase of Lorentz factor, Δγ, due to this negative z drift motion is around 0.6, which is greater than the prediction (Δγ ∼ 0.2) based on the 1D shock model discussed in Appendix B. This may be because the shock in this study is 2D, and the effect of nonuniform fluctuations (which generate magnetic islands) in the y-direction could be important, but studying such effects is beyond the scope of this paper. As this example shows, Δγ due to the shock drift acceleration is less than 1.0, and this is much smaller than Δγ due to magnetic reconnection.

Zoom In Zoom Out Reset image size

Since we have only investigated a limited number of particles, there might be electrons accelerated by other different mechanisms. Further studies, including statistical studies, are necessary to understand how much the contribution of magnetic reconnection is on the production of nonthermal electrons and the total electron heating in the shock.