Magnetic reconnection in quasi-parallel shocks, relevant to the Earth's bow shock, is studied by means of two-dimensional full particle-in-cell simulations. As the Alfvénic Mach number increases, the propagation direction of the waves excited in the transition region changes, and the shock becomes more turbulent with more reconnection sites. In the higher Mach number shock, abundant electron-only reconnection sites are generated with scales on the order of the ion skin depth or less. Non-reconnecting current sheets can also generate electron jets and energy dissipation can occur there as well. However, non-reconnecting current sheets with the magnetic field reversal typically show a smaller energy dissipation rate than reconnecting current sheets. In the shock transition region, two types of waves are responsible for driving reconnection: one has a wavelength around three ion skin depths (di), and the other has a wavelength less than 1 di. Electron and ion distribution functions show that in regions where the former type of waves is excited, there are two ion beams and a single-peaked electron distribution. In contrast, in regions where the latter type of waves is excited, there are multiple electron and ion beams. The waves propagating obliquely to the magnetic field bend the magnetic field lines, and magnetic reconnection occurs where oppositely directed field lines come into contact.

Recent in situ observations of the Earth's bow shock have revealed a number of active magnetic reconnection sites, for instance, studies using Cluster observed many current sheets in the Earth's magnetosheath with magnetic reconnection signatures.1–3 NASA's Magnetospheric Multiscale (MMS) has also been observing the Earth's bow shock, and has identified a number of magnetic reconnection sites in the magnetosheath4–8 and in the shock transition region.9–12 Recently it was discovered, in the magnetosheath7 and the shock transition region,9,10,12 that reconnection can occur due to only electrons' kinetic effects, resulting in the formation of electron reconnection jets, and ions do not participate in the reconnection.7 This electron-only reconnection results from shock turbulence having electron-scale magnetic field structures, and ions cannot respond to such small-scale field gradients.

Shock driven turbulence is a key to understand magnetic reconnection in shocks. It has been demonstrated that quasi-parallel shocks (the shock normal angle 0°<θ<45°) are more turbulent than quasi-perpendicular shocks (45° <θ<90°) (for example, see a simulation result of Ref. 13). Space observations in the Earth's magnetosheath behind the quasi-parallel bow shock show more turbulent current sheets than behind the quasi-perpendicular shock.1,2 A recent statistical study based on MMS measurements in the transition region of the Earth's bow shock also shows the same tendency, and further indicates that reconnection is observed more often in high Mach number, quasi-parallel conditions.12 In this study, we investigate the kinetic physics of waves excited in the shock transition region of quasi-parallel shocks to understand magnetic reconnection driven by shock turbulence, by means of two-dimensional full particle-in-cell (PIC) simulations.

Shock driven reconnection has been addressed in studies using kinetic numerical simulations, both hybrid PIC simulations13,14 and full PIC simulations.15–17 Studies of high Alfvénic Mach number (MA30–40) perpendicular shocks,15,16 with parameters relevant to astrophysical shocks, show that the Weibel instability generates magnetic field fluctuations perpendicular to the initial magnetic field. The result is the generation of turbulence and multiple magnetic reconnection sites. In contrast, studies of much weaker quasi-parallel shocks (MA10), with parameters relevant to the Earth's bow shock,14,17 show that waves different from the Weibel instability are excited. These are responsible for generating reconnecting current sheets. The generation of reconnecting current sheets is correlated with the shock reformation. Our previous study, using a full PIC simulation,17 demonstrated that sub-ion scale magnetic islands generated in the shock turbulence are the sites of electron-only reconnection. In this study, we discuss kinetic waves responsible for creating current sheets, and the field structures, the reconnection rates, and the energy dissipation rates in reconnecting current sheets, as well as those of non-reconnecting current sheets.

It is currently not known which waves are responsible for generating reconnection sites in the Earth's bow shock. For quasi-parallel shocks, waves and instabilities have been studied by means of theory and numerical simulations. Dispersion analyses of ion–ion beam instabilities predict that ultra-low frequency (ULF) waves are excited by the interaction between the incident ion component and the reflected ion component.18,19 One-dimensional and two-dimensional hybrid and full PIC simulations also demonstrated ULF waves whose wavelengths are 10 to several tens of ion skin depths.20–28 Space measurements (Ref. 29 and references therein) as well as laboratory experiments30 have been observing these predicted waves. Also, in a steepened wave structure such as SLAMS (short, large-amplitude magnetic structures),31 high-frequency waves such as whistler waves can be generated27,32 (for space observations, see Ref. 29 and references therein). It is important to elucidate what types of waves are responsible to generate reconnecting current sheets. In this study, we investigate two types of waves: one wave has a wavelength around a few ion skin depths, and the other wave has a wavelength less than one ion skin depth. We examine electron and ion distribution functions in regions of those two types of waves, which generate winding magnetic field lines and cause magnetic reconnection in shocks.

In Sec. II, we describe the simulation method in this study. In Sec. III A, we discuss shock propagation and magnetic reconnection, and we discuss reconnecting and non-reconnecting current sheets. In Sec. III B, we discuss two types of waves excited in the shock transition region, and electron and ion distribution functions that are responsible for instabilities. In Sec. IV, we give discussions and conclusions of this study.

We use 2D PIC simulations to study magnetic reconnection in the transition and downstream regions in the Earth's quasi-parallel bow shock. In the simulation in the xy coordinates, x is the direction of the shock normal. In the simulation domain, there are initially a uniform magnetic field B0=(Bx0,By0,0) and an electric field E0=(0,0,Ez0). Therefore, y is parallel to the shock plane, and the xy plane contains the magnetic field B0. In the z direction, it is assumed that /z=0. The ion and electron densities ni and ne, respectively, are also uniform with ni=ne=n0 initially, at t = 0. The system size is Lx×Ly=375di×51.2di=15000×2048 grids, where di is the ion skin depth (di=c/(4πn0e2/mi)1/2, c is the speed of light, e is the elementary charge, and mi is the ion mass). Initially, 100 particles/cell are loaded for each species.

The y boundary is periodic, and both x boundaries are rigid walls at which particles are specularly reflected. At the two x boundaries, the electric fields are set to be Ey = 0 and Ez=Ez0 as a boundary condition. Particles move in accordance with the E × B drift speed perpendicular to the magnetic field, with an initially imposed bulk parallel velocity V=cEz0Bx0/(B0By0) for both ions and electrons (where B0=(Bx02+By02)1/2), so that the y component of the bulk drift velocity becomes zero, and the bulk drift velocity becomes Vd=(Vxd,0,0) (where Vxd=cEz0/By0 for both ions and electrons). Thus, all the particles initially move in the negative x direction. At t = 0, the velocity distribution functions are drifting Maxwellians with uniform ion and electron temperatures, Ti and Te. At the boundary at x = Lx, we constantly inject particles whose temperature and drift speed are the same as the initial particles.

Near the maximum x boundary, 215di<x<375di, a wave damping region is placed to inhibit artificial wave growth due to particle injection at the boundary x=375di.33 

The simulation parameters are the same as Ref. 17, which are as follows: a mass ratio of mi/me=200, where me is the electron mass, and the initial beta values βi=βe=1.0, where βj=n0Tj/(B02/8π) (j = i or e). The ratio of the electron plasma-to-cyclotron frequency is ωpe/Ωe=4, where ωpe=(4πn0e2/me)1/2,Ωe=eB0/mec.

The shock normal angle θ—the angle between the shock normal (positive x direction) and the magnetic field B0—is set to be 25°; i.e., θ=tan1(By0/Bx0)= 25°. We performed two runs, denoted Run 1 and Run 2. Run 1 uses Vxd=9vA, where vA is the Alfvén speed based on B0 and n0, and Run 2 uses Vxd=5vA. In both Run 1 and Run 2, the shock speed vsh (which will be discussed below) is 11.4vA and 6.5vA, respectively, both of which are smaller than the electron thermal speed, vTe=14.4vA. This condition, vsh < vTe, is relevant to the Earth's bow shock. Further, under this condition, the plasma is stable to Buneman instability.

1. Shock propagation, reformation, and reconnection

In Fig. 1, panels (a)–(e) show a time sequence of the profile of By along y=25.6di in Run 1 (Vdx=9vA), at five different times, Ωit=12.5, 14.06, 15.63, 17.19, and 18.75, where Ωi is the ion cyclotron frequency based on B0. Each black dashed line represents the position of the shock front, where By becomes the maximum, which is 5–8 times B0. At Ωit=15.63, the shock reformation starts and the By in the foreshock region (around x=50di) starts to grow.17 At Ωit=18.75, there is a new shock front around x=50di, which is indicated by the blue dashed line, and the magnitude of By at the new shock front becomes comparable to that at the old shock front (red dashed line) around x=30di. Based on the distance between the shock front at Ωit=3.125 (not shown), at x=10.5di, and the new shock front at Ωit=18.75, at x=48.6di, the shock speed is vsh=11.4vA (the Alfvén Mach number MA=11.4), considering the upstream drift speed of the bulk plasma in the simulation frame, Vdx=9vA.

FIG. 1.

1D profiles of the magnetic field By at y=25.6di for Run 1 (a)–(e) and Run 2 (f)–(j). The black dashed line represents the shock front at each time. The red dashed line and the blue dashed line in the bottom panel at Ωit=18.75 are the old shock front and the new shock front generated by the shock reformation, respectively.

FIG. 1.

1D profiles of the magnetic field By at y=25.6di for Run 1 (a)–(e) and Run 2 (f)–(j). The black dashed line represents the shock front at each time. The red dashed line and the blue dashed line in the bottom panel at Ωit=18.75 are the old shock front and the new shock front generated by the shock reformation, respectively.

Close modal

Panels (f)–(j) display the result of Run 2 (Vdx=5vA). During the interval between Ωit=12.50 and 15.63, the shock front (black dashed line) is around x=15di to 18di, where the maximum of By3B0. The foreshock region around x30di gradually grows, and the shock reformation proceeds between Ωit=14.06 and 17.19. The magnetic field By in the new shock front becomes comparable to that at the old shock front, 3B0. Based on the shock front at Ωit=3.125 (not shown), at x=5.9di, and the new shock front at Ωit=18.75, at x=29.5di, the Alfvén Mach number is MA=6.5, considering the upstream fluid speed Vd=5vA, and MA in Run 2 is almost twice smaller than MA in Run 1.

Figure 2 shows the time evolution of the magnetic field lines (the contour of vector potential Az) in Run 1 and Run 2, projected on the xy plane. The left-hand column panels (a)–(e) are the plots for Run 1, and the right-hand column panels (f)–(j) are the plots for Run 2. Each panel (a)–(j) in Fig. 2 corresponds to the time in each (a)–(j) panel in Fig. 1, and the red dashed line (old shock front) and the blue dashed line (new shock front) in the bottom panel show the same positions of the old/new shock fronts in Fig. 1. In Run 1 (for MA=11.4), panels (a) and (b) show that the shock front (red dashed line) is around x=18di22di, where the field lines become denser than the upstream region. In the shock transition region (25di<x<40di), where the magnetic field strength is weaker than in the shock front and the downstream region, there are winding magnetic field lines. This undulation in the field lines is due to instabilities excited by interactions between reflected ions and incident ions, which will be discussed later. The wavelength of those winding field lines ranges from di to several di, and the size of the unstable region in the x direction becomes larger and larger as time progresses, comparing panels (a) and (b). The shock plane (red dashed line) is not planar, but features wavy structures. Along the shock plane, the sign of Bx alternates, and there are compression and rarefaction of field lines, with a wavelength around 6di in the y direction.

FIG. 2.

Magnetic field lines projected on the xy plane for Run 1 (a)–(e) and Run 2 (f)–(j). The time in each panel corresponds to the time in each panel in Fig. 1. The red dashed line represents the shock front at each time, and the blue dashed line in the bottom panel at Ωit=18.75 is the new shock front generated by the shock reformation.

FIG. 2.

Magnetic field lines projected on the xy plane for Run 1 (a)–(e) and Run 2 (f)–(j). The time in each panel corresponds to the time in each panel in Fig. 1. The red dashed line represents the shock front at each time, and the blue dashed line in the bottom panel at Ωit=18.75 is the new shock front generated by the shock reformation.

Close modal

As time elapses, the waves in the transition region grow and the winding magnetic field lines start to reconnect in some regions. At Ωit=15.63 [panel (c)], there are several magnetic islands in the transition region between x=27di and 40di, as well as in the shock downstream between x=20di and the shock front at x=27di. The shock reformation started before Ωit=15.63, and in panel (c), the By component of the magnetic field becomes larger in the region between x=40di and 50di, compared with By in previous times [panels (a) and (b)]. During the interval between Ωit=15.63 and 18.75 [panels (c)–(e)], the shock reformation continues, and the steepening of By in the region between x=40di and 50di becomes much more significant. At Ωit=17.19 [panel (d)], there is a strong concentration of the magnetic field lines in front of the magnetic islands around x=40di, and in front of those concentrated field lines, a strong instability that generates winding magnetic fields grows around x=50di. At Ωit=18.75 [panel (e)], the new shock front forms, and the transition region around x=50di contains a number of small magnetic islands due to reconnection that occurs in the instability region. These are sites of electron-only reconnection,7,9,10 in which ions are just passing through the current sheet and only electrons participate in the reconnection. The detailed structure of one of those regions with electron-only reconnection has already been discussed in our previous paper.17 The region between the red dashed line (the old shock front) and the blue dashed line (the new shock front) becomes the shock downstream region, which contains turbulence generated by waves excited before the new shock front is formed, and there are several magnetic islands.

The shock in Run 2 (MA=6.5) in panels (f)–(j) is much less turbulent than the shock in Run 1. In panels (f)–(h), the shock transition region (between x=15di to x=30di) has winding magnetic field lines due to the instability. The wavelength of the waves is of the order of di, shorter than the wavelength in Run 1. As time elapses, By becomes larger and larger in the transition region, and the shock reformation occurs. At Ωit=17.19 [panel (i)], a new shock front is formed at x=27di, and the peak of By in the shock front keeps growing. However, even after the formation of the new shock front, there is no magnetic island formation in the shock transition region and the downstream region [panel (j)], which is contrasted with Run 1, panel (e). Although we observed several magnetic islands during the interval from Ωit=12.5 to 18.75, the number of magnetic islands observed in Run 2 is much smaller than that in Run 1, and in the smaller Alfvén Mach number shock (Run 2), reconnection does not play a significant role in the shock, compared with the higher MA run (Run 1).

Figure 3 displays the out-of-plane current density Jz for Run 1 and Run 2. The left panels (a)–(d) show the results for Run 1 (MA=11.4). At Ωit=14.06, there are many current sheets (red and black stripes) around the shock front (x=22di), while the foreshock region (22di<x) has small amplitude Jz structures, as yellow and blue stripes. Consistent with these Jz stripes, magnetic field lines (black curves) are winding. At Ωit=15.63, around which the shock reformation starts, there are a few long current sheet structures extending from the shock front (x=25di) toward the foreshock. These extended structures contain a few magnetic islands in the region 30di<x<45di. In the region 45di<x, there are coherent waves propagating from the upstream toward the shock, seen as small amplitude (light green/light blue) diagonal stripes. At Ωit=17.19, a large island remains around x=40di, and the magnetic fields pile up in front of the island. The waves in the foreshock region 45di<x grow to much larger amplitudes, rendering the magnetic field lines distorted more significantly than at the previous time. Along a few Jz stripes, short-wavelength waves are excited (see the region 45di<x<50di and y<38di). At Ωit=18.75, the new shock front is formed around x = 45–50di, and there are multiple almost-vertical stripes of Jz around the new shock front. The region between the old shock front (x=30di) and the new shock front (x=45di) becomes the shock downstream region, and there are also complex turbulent structures of Jz in that region. The By pileup in front of the large magnetic island around x=40di becomes almost uniform in the y direction around x=45di, and the magnetic field points almost in the y direction in the new shock front.

FIG. 3.

Current density Jz (color) in the xy plane for Run 1 (a)–(d) and Run 2 (e)–(h). The black curves show the magnetic field lines projected on the xy plane.

FIG. 3.

Current density Jz (color) in the xy plane for Run 1 (a)–(d) and Run 2 (e)–(h). The black curves show the magnetic field lines projected on the xy plane.

Close modal

The right panels of Figs. 3(e)–3(h) show Jz for Run 2 (MA=6.5). Since the Alfvén Mach number in Run 2 is much smaller than that in Run 1, we see less turbulent structures in the shock, even though waves with a significant amplitude are excited. At Ωit=14.06, the shock front is around x=15di, and there are structures of alternating Jz in the foreshock region 20di<x. The wave planes seen in 20di<x<30di are almost perpendicular to the magnetic field. In the simulation frame, these waves are propagating toward the upstream. Field lines are winding because of those waves. At Ωit=15.63, the amplitude of waves in the foreshock region 25di<x<35di becomes larger than that in the previous time, and some modulations appear in the direction of the wave plane. At Ωit=17.19, the new shock front forms around x=27di, and the front of the waves in the foreshock (30di<x) becomes more non-uniform than at the previous time. At Ωit=18.75, the new shock front has a larger By, and the wave amplitude in the foreshock 30di<x becomes smaller than that in the previous time. During the shown interval from Ωit=14.06 to 18.75, only coherent waves are generated in the shock transition region, and reconnection does not play a significant role.

In the following, to discuss detailed structures of current sheets, we will show the results of Run 1 (Vdx=9vA). Figure 4 plots the blowups of several small regions that contain current sheets at Ωit=18.75. Figure 4(a) is the current density Jz in the shock including the foreshock, transition, and downstream regions, and we marked four areas, from #1 to #4 in the shock transition region, by yellow boxes, and magenta Xs represent the positions of reconnection X-lines. There are two types of current sheets: reconnecting current sheets, which have reconnection X-lines with a field-line topology change, and non-reconnecting current sheets, which do not have reconnection X-lines. In areas #1,#2, and #3, there are multiple X-lines; in contrast, area #4 does not contain X-lines. Figure 4(b) shows Jz in area #1, which has already been discussed in the previous paper.17 This area contains two X-lines, and the right X-line, at (x,y)=(48.18di,27.05di), is on the positive (red) current sheet, and readers can refer to Ref. 17 for more details about the field structures in this current sheet.

FIG. 4.

(a) Current density Jz (color) for Run 1 at Ωit=18.75. The black curves are magnetic field lines projected on the xy plane. Magenta Xs are the positions of reconnection X-lines. Four yellow boxes, #1 to #4, are areas where the details of current sheets are discussed. Area #1 is the same region discussed in Ref. 17, and an example of the plot Jz is given in panel (b). Panels (c)–(e) are for area #2, panels (f)–(h) are for area #3, and panels (i)–(k) are for area #4. In each area, the current density Jz, the electron fluid velocity Vey or Vex, and the energy dissipation rate J·E, where E=E+Ve×B/c, are plotted. Magenta lines are magnetic field lines projected on the xy plane, magenta Xs represent the positions of X-lines, and white arrows are vectors of electron fluid velocity.

FIG. 4.

(a) Current density Jz (color) for Run 1 at Ωit=18.75. The black curves are magnetic field lines projected on the xy plane. Magenta Xs are the positions of reconnection X-lines. Four yellow boxes, #1 to #4, are areas where the details of current sheets are discussed. Area #1 is the same region discussed in Ref. 17, and an example of the plot Jz is given in panel (b). Panels (c)–(e) are for area #2, panels (f)–(h) are for area #3, and panels (i)–(k) are for area #4. In each area, the current density Jz, the electron fluid velocity Vey or Vex, and the energy dissipation rate J·E, where E=E+Ve×B/c, are plotted. Magenta lines are magnetic field lines projected on the xy plane, magenta Xs represent the positions of X-lines, and white arrows are vectors of electron fluid velocity.

Close modal

Figures 4(c)–4(e) show physical quantities in area #2: the current density Jz, the electron fluid velocity Vey, and the energy conversion rate J·E, where E=E+Ve×B/c is the electric field in the electron rest frame, respectively. In this area, there are three X-lines, and let us focus on the top two X-lines, at (x,y)=(50.35di,39.225di) and (x,y)=(50.4di,38.7di). The upper X-line is located on the negative (black) Jz current sheet [see panel (c)], while the lower X-line is located between the positive (red) current sheets. In the vicinity of the lower X-line, Jz is negative (green to light blue colors). The time evolution of the field lines (not shown) indicates that the reconnection around the lower X-line occurs between the field lines in the upper right quadrant and the field lines in the lower left quadrant, which means there is a negative current sheet Jz extending diagonally from the upper left to the lower right regions across the X-line. Panel (d) shows that Vey becomes positive around those X-lines, and electron jets exist in the upper sides of the X-lines. Panel (e) demonstrates that around these X-lines and electron jets, J·E becomes positive, although some regions along the jets have negative values.

Figures 4(f)–4(h) are the current density Jz, the electron fluid velocity Vex, and the energy conversion rate J·E for area #3, in which four X-lines are formed. Panel (f) shows that there are two magnetic islands whose centers are around (x,y)=(49.75di,30.6di) and (x,y)=(49.7di,29.6di), and Jz becomes positive in those islands. The two islands are separated due to the negative (black) current sheet, and there is an X-line at (x,z)=(49.675di,30.1di) on the negative Jz. Panel (g) shows that along this negative current sheet with the X-line, there is a negative Vex flow. Above the upper island and below the lower island, there are other X-lines (the X-line below the lower island is outside of the box), around which there are also negative electron Vex flows. Panel (h) demonstrates that along these negative Jz sheets and negative Vex flows, J·E features large positive values.

The above two areas (#2 and #3) are examples of reconnecting current sheets, where strong electron jets and positive J·E are generated. In contrast, there are many non-reconnecting current sheets, too, and Figs. 4(i)–4(k) show such an example. Panel (i) plots Jz in non-reconnecting current sheets. There are several positive and negative Jz stripes in the vertical direction, and the magnetic field By reverses across the right-most negative Jz around x=47di (see the field lines in that current sheet). Even though By reverses along this negative current sheet, there is no reconnection X-line. Nonetheless, panel (j) exhibits a strong positive electron jet Vey along the rightmost vertical negative current sheet. Panel (k) shows that J·E becomes positive along that current sheet, although some regions along the current sheet have a negative J·E.

The energy conversion rate J·E in non-reconnecting current sheets with the magnetic field reversal tends to be smaller than that in reconnecting current sheets. The magnitude of the positive J·E in the non-reconnecting current sheet in area #4 [panel (k)] is around 5cB02/di throughout the negative Jz sheet, except for a local region around (x,y)=(47.1di,41.9di) where it reaches around 10cB02/di. In contrast, the reconnecting current sheet in area #3 [panel (h)] shows that J·E in these negative Vex jets around the X-lines reaches 2530cB02/di. In addition, the reconnecting current sheet in area #1 (see Fig. 2 in Ref. 17) shows that J·E exceeds 15cB02/di. The reconnecting current sheet in area #2 [panel (e)] exhibits a little bit smaller J·E around 7.5cB02/di, except for small numbers of local regions where J·E10cB02/di. This is because the reconnection in area #2 is not fully developed at this time Ωit=18.75, indicated by the small magnitude of the negative Jz around the X-line at (x,z)=(50.4di,38.7di). If we consider the area average, using a square 0.5di×0.5di around the X-line, the reconnecting current sheets in area #2 and #3 show J·E=0.60cB02/di and 1.63 cB02/di, respectively. In contrast, the area average around the non-reconnecting current sheet in area #4 is J·E=0.31cB02/di.

In the following, we will investigate further each reconnecting/non-reconnecting current sheet: reconnecting current sheets with a regular bipolar jet structure (area #2), reconnecting current sheets with a one-sided jet structure (area #3), and non-reconnecting current sheets (area #4).

2. Reconnecting current sheets, area #2

In Ref. 17, a reconnection region in area #1 is analyzed, and a bipolar jet structure is shown. Area #2 contains another example of a reconnection region with a bipolar jet. Figure 5 displays the contour plots and profiles for several quantities for the reconnecting current sheet in area #2. Panels (a)–(f) show: (a) the out-of-plane magnetic field Bz, (b) electron fluid velocity Vex and (c) Vey, (d) electron density ne, (e) ion fluid velocity Vix, and (f) Viy. Let us focus on the second top X-line at (x,y)=(50.4di,38.7di). This X-line is moving, with a velocity [3.00,5.11]vA in the xy plane. These six panels plot the quantities in the reference frame where the second top X-line is stationary. In the following, we call this frame the X-line stationary frame.

FIG. 5.

Details of field structures in area #2, where there are reconnecting current sheets. (a) magnetic field Bz, (b) and (c) electron fluid velocity Vex and Vey, (d) electron density ne, and (e) and (f) ion fluid velocity Vix and Viy. In each plot, the magenta lines are magnetic field lines projected on the xy plane, and magenta Xs are the positions of X-lines. White arrows are velocity vectors of electron fluid velocity [panels (a)–(d)] and ion fluid velocity [panels (e) and (f)]. In (a), the green/red ovals indicate the electron inflows/outflows, respectively. In each panel, the yellow solid line on the second top X-line is the cut line across the X-line, along which the field profiles are plotted in panels (g)–(l). (g) Each component of the magnetic field, (h) electric field, (i) current density, (j) electron fluid velocity, (k) ion fluid velocity, (l) electron density (red), and the energy dissipation rate J·E (blue).

FIG. 5.

Details of field structures in area #2, where there are reconnecting current sheets. (a) magnetic field Bz, (b) and (c) electron fluid velocity Vex and Vey, (d) electron density ne, and (e) and (f) ion fluid velocity Vix and Viy. In each plot, the magenta lines are magnetic field lines projected on the xy plane, and magenta Xs are the positions of X-lines. White arrows are velocity vectors of electron fluid velocity [panels (a)–(d)] and ion fluid velocity [panels (e) and (f)]. In (a), the green/red ovals indicate the electron inflows/outflows, respectively. In each panel, the yellow solid line on the second top X-line is the cut line across the X-line, along which the field profiles are plotted in panels (g)–(l). (g) Each component of the magnetic field, (h) electric field, (i) current density, (j) electron fluid velocity, (k) ion fluid velocity, (l) electron density (red), and the energy dissipation rate J·E (blue).

Close modal

Panel (a) demonstrates that there is a guide field Bz in the reconnection region (the second top X-line). The value around the X-line is Bz3B0, which is almost twice larger than the reconnecting component of the magnetic field. In this X-line stationary frame, panels (b) and (c) show that the electron flow has a bipolar jet structure around the X-line. The positive Vex and Vey flows around the X-line are easily recognized, especially in the Vey plot [panel (c)], but there is a small localized region where Vex and Vey are negative [see panel (c), below the X-line], around (x,y)=(50.2di,38.5di). The arrows represent the velocity vector for the electron fluid in the X-line stationary frame. They show that there are the electron inflows from the negative x side and the positive x side toward the vicinity of the X-line, and the upward electron jet flows out from the X-line. The downward jet is weak, except for the localized region around (x,y)=(50.2di,38.5di). In panel (a), the green ovals and the red ovals indicate the electron inflows and the electron outflows around the X-line, respectively.

The electron density [panel (d)] shows that a layer with a large density is elongated diagonally in the direction from the upper right to the lower left. Panels (e) and (f) are the ion fluid velocities Vix and Viy, and there are only almost uniform ion flows around the X-line, from the upper right to the lower left regions. There are no ion jets from the X-line. Therefore, this is a region of electron-only reconnection,7 where only electrons participate in reconnection, and ions just pass through the region, similar to area #1 discussed in Ref. 17. This electron-only reconnection is also consistent with previous theoretical and numerical studies34–36 with a laminar electron-scale current sheet where ions are unmagnetized, and ions are not involved in reconnection physics inside the thin layer.

Panels (g)–(l) are the profiles of quantities along the yellow line in the contour panels (a)–(f), crossing the second top X-line diagonally. This yellow line is almost perpendicular to the reconnecting component of the magnetic field. Note that the magnetic fields in the first quadrant (positive x and positive y sides) around the X-line, and those in the third quadrant (negative x and negative y sides) around the X-line, are reconnecting. In the panels (g)–(l), the horizontal axes represent the x position of the yellow line. The magnetic field By [see panel (g)] changes from negative to positive as we move from the right to the left. The vertical dashed line in panels (g)–(l) is the position of the X-line, at x=50.4di. The guide field Bz is around 3B0 at the X-line, and its magnitude is almost twice larger than the reconnecting component of the magnetic field (1.5B0) in the right side of the reconnection region, around x=50.6di. The electric field [panel (h)] shows some fluctuations, and Ez at the X-line becomes a small negative value. The magnitude of the reconnection electric field (|Ez| at the X-line) is around 0.026B0, from which the normalized reconnection rate is 0.15BdmVout/c, where Bdm=1.35B0 is the magnitude of the reconnecting magnetic field in the upstream region (the mean value at the two positions, at x=50.2di and at x=50.6di), and Vout=7.27vA is the maximum outflow speed, (Vex2+Vey2)1/2, in the upward directing electron jet from the X-line. Note that we here assumed symmetric reconnection, since the magnetic field and the density profiles are almost symmetric within the region up to 0.2di away from the X-line. The reconnection electric field averaged around the X-line (using a square with its side 0.5di) is Ez=0.020B0±0.010B0, where the uncertainty is due to the time fluctuation during 90 time steps. Using this value, the reconnection rate is 0.12±0.06BdmVout/c. Panel (i) plots the current density, and Jz is close to zero or negative around the X-line, but |Jz| is not large there. This indicates that the negative current sheet Jz around this X-line has just started to form. The time history of the reconnection in this current sheet (not shown) shows that the magnitude of the negative Jz grows as time evolves, and a clearer reconnecting current sheet appears in a later time. This reconnection looks like driven reconnection.

The electron fluid velocities [panel (j)] show that Vey is non-zero at the X-line. This is because the electron jet (Vey > 0) is generated slightly away from the X-line, as shown in panel (c), in which the jet starts slightly below the X-line. In the negative x side of the X-line, there is a negative Vey flow, because the electron flow has a bipolar jet structure around the X-line. The ion flow velocities [panel (k)] show that ion flows are almost uniform background flows. Panel (l) demonstrates that the density is almost uniform along the yellow line in the contour plots. The energy conversion rate, J·E, fluctuates along this yellow line, and it becomes positive around the X-line, although the value at the X-line is negative. The negative J·E results partially from a small positive Jz on the X-line, instead of a negative value. Due to fluctuating Jz and the fact that reconnection is in its developing stage, no fully developed negative Jz exists around the X-line yet. However, if we average out such a fluctuation, J·E around the X-line gives a positive value. In a later time (not shown), the negative Jz grows and the value of J·E becomes larger.

3. Reconnecting current sheets, area #3

Reconnection sites in the shock-driven turbulence can have jet structures distinct from the regular bipolar jets (such as area #1 and #2). Area #3 contains such an example, characterized by one-sided jets. Figure 6 shows the quantities in area #3. In this area, there are two magnetic islands and four X-lines. Panel (a) plots Bz, and let us focus on the second from the bottom X-line at (x,y)=(49.675di,30.1di). The magnetic field at the X-line is Bz4B0, which is almost twice larger than the reconnecting magnetic field Bx in this region. This X-line is moving with a velocity [0.545,5.02]vA in the simulation frame. All the quantities in Fig. 6 are shown as the values in the X-line stationary frame. Panels (b) and (c) are the electron fluid velocities Vex and Vey, and in the vicinity of the X-line, there is a large electron jet with negative Vex and positive Vey. This jet is almost one directional around the X-line, and the opposite directional jet with a positive Vex and a negative Vey is located far away from the X-line [see the green/red ovals in (a), which indicate the electron inflows/outflows, respectively]. The electron flow patterns around the two magnetic islands show a vortex-like structure, in which the electrons moving clockwise around those two islands (Vey > 0 in the left side of those islands, while Vey < 0 in the right side of the islands). The electron density [panel (d)] is almost flat but there is a slight asymmetry between the top and the bottom sides of the X-line, and the density above the X-line is slightly higher than below the X-line. Ion flows [panels (e) and (f)] show that there are almost uniform Vix and Viy flows around the X-line.

FIG. 6.

2D fields and 1D cuts for area #3, where there are reconnecting current sheets, the same format as Fig. 5.

FIG. 6.

2D fields and 1D cuts for area #3, where there are reconnecting current sheets, the same format as Fig. 5.

Close modal

These field structures around the second bottom X-line can also be applied to the X-line above the upper island. On the top of the upper island, there is an X line and a negative Vex layer, similar to the negative Vex layer above the lower island. Also, below this box, there is another X-line (not shown), and there is another negative Vex layer below the lower island. In addition, the Bz structure [panel (a)] and the density structure [panel (d)] are similar in the upper and the lower islands: the lower right quadrant of each island has a positive Bz, while the upper left quadrant of each island shows a negative Bz. The density becomes high in the lower left and the bottom regions of each island.

Panels (g)–(l) are the profiles of field quantities along the yellow line across the second bottom X-line [see the yellow vertical line in panels (a)–(f)]. Panel (g) shows the profiles of magnetic field, where the reconnecting magnetic field is almost in the x direction, reversing its sign at the X-line at y=30.1di (the vertical dashed line). The upper side of the X-line (30.1di<y) has a larger magnetic field Bx than the lower side (y<30.1di). Since both the magnetic field and the density exhibit asymmetry across the current sheet, let us apply the asymmetric reconnection model.37 The average magnitude of the reconnecting magnetic field Bdm=2B1B2/(B1+B2)=1.49B0, where B1 and B2 are the reconnecting field in the upper side and the lower side, respectively, and we used the absolute values of the maximum and the minimum Bx in panel (g), i.e., B1=Bx,max and B2=Bx,min. The guide field Bz4B0, whose magnitude is almost three times larger than the reconnecting field. The electric fields [panel (h)] fluctuate a lot, and the reconnection electric field Ez at the X-line is Ez=0.047B0, which corresponds to the reconnection rate 0.17BdmVout/c, where Vout=10.4vA is the maximum outflow speed (Vex2+Vey2)1/2 in the left side of the X-line. Using the spatially averaged Ez around the X-line (using a square with its side 0.5di), Ez=0.0045±0.017B0, where the uncertainty is due to the time fluctuation during 90 time steps. This area #3 contains a larger fluctuation than area #2, and the averaged Ez around the X-line in area #3 is about five times smaller than in area #2. Therefore, the high value of the reconnection rate 0.17BdmVout/c is an instantaneous local value, and the averaged reconnection rate is 0.017±0.063BdmVout/c.

Panel (i) displays the current densities, and Jz has a negative value at the X-line, and the negative Jz peak is in the upper side of the X-line, y=30.175di. In addition, Jz becomes positive in both sides across the X-line. Panel (j) plots the electron flow velocities. The Vey plot exhibits no signature of the electron inflows toward the X-line, and the Vex plot shows a negative flow at the X-line. This suggests that the structure of the electron flow is different from the regular reconnection picture, where the inflow Vey is generated toward the X-line, and the outflow Vex comes out from the X-line. Instead, the negative Vex coming from the positive x side of the X-line [see panel (b) and the electron flow with negative Vex covering the region around the X-line] may play a role as the inflow toward the X-line, and the one sided outflow comes out from the X-line. Panel (k) exhibits almost uniform ion flows. Panel (l) demonstrates that the electron density has small asymmetry across the current sheet, and the densities at the upper side and the lower side are n1=5.5n0 and 3.2n0, respectively, where we took the maximum and the minimum values of ne in panel (l). Using these densities and the magnetic fields, the asymmetric model37 predicts that the outflow becomes Vout=[B1B2(B1+B2)/(n1B2+n2B1)]1/2(1/4πme)1/2=12vA, and the observed outflow is Vout=10.4vA, slightly smaller than the prediction. The energy conversion rate J·E is a positive value around the X-line. Note that Fig. 4(h) shows that J·E in the left side (outflow side) of the X-line becomes much larger than at the X-line, and the energy conversion to electron is more significant in the outflow region than that at the X-line.

4. Non-reconnecting current sheet, area #4

Figure 7 illustrates quantities in the non-reconnecting current sheets in area #4. Let us focus on the negative Jz around x=47di. Since there are no X-lines in this region, the quantities are shown in the simulation frame. Panel (a) demonstrates that along the non-reconnecting current sheet, Bz is negative in the left side and positive in the right side. The reversal of Bz is because there is an electron jet Vey, which is seen in pane (c). Panels (b) and (c) are the electron fluid velocities, and there is an electron jet in the y direction along the current sheet. The maximum Vey13vA in the jet. Electron density [panel (d)] shows an enhancement along the current sheet. Panels (e) and (f) are the ion fluid velocities. There is y directional ion flows, but no ion jet is formed associated with the current sheet around x=47di.

FIG. 7.

2D fields and 1D cuts for area #4, where there are non-reconnecting current sheets, the same format as Fig. 5.

FIG. 7.

2D fields and 1D cuts for area #4, where there are non-reconnecting current sheets, the same format as Fig. 5.

Close modal

Panels (g)–(l) display the profiles of field quantities across the yellow line [see panels (a)–(f)]. Panel (g) shows that By reverses at x = 47.11 (the dashed line), and Bz also changes its sign around the dashed line. Panel (h) plots the electric fields. The signs of the electric fields are consistent with the signs of Ve×B/c (not shown), although the electrons are unmagnetized in this area and EVe×B/c: the electric field Ex reverses its sign across the current sheet, and Ez becomes almost uniformly negative because of the positive Vey flow and the negative Bx across the current sheet.

Panel (i) shows that Jz becomes negative, and at the same time Jy becomes negative due to the strong Vey flow. Panel (j) displays the electron fluid velocities, and both Vey and Vez become positive, and Vex0. Panel (k) shows that the ions have almost uniform flows. Panel (l) shows that the electron density has an enhancement around the current sheet. The energy conversion rate J·E becomes positive in the current sheet, and the peak of J·E is near the By reversal (dashed line). Note that as seen in Fig. 4(k), the magnitude of J·E is almost uniformly 5cB02/di along the current sheet, and this value is much smaller than J·E in area #3, which reaches around 30cB02/di in the downstream jet region of the second bottom X-line in area #3.

As this example reveals, non-reconnecting current sheets are also the region where the field energy is converted to particles' energy, because J·E>0. There can be a magnetic field reversal across a non-reconnecting current sheet, and also an electron jet can be formed, which is not the result of magnetic reconnection. Note that panels (g) to (l) exhibit a similarity between this non-reconnecting current sheet and a rotational discontinuity, in which the direction of the magnetic field rotates across the current sheet. There is an electron jet between the two sides of the current sheet, and the rotational discontinuity-like structure in this current sheet region dissipates energy with a smaller energy conversion rate than reconnecting current sheets.

In the shock transition region, waves are excited due to interactions between incident particles and reflected particles, as well as multiple components of beams, which eventually make magnetic field lines winding and many current sheets form.

Figure 8 exhibits the field lines (gray curves) and the current density Jz (color) at two different times. At Ωit=15.63, the shock front is located at x=27di, and panel (a) shows the foreshock region. There are wave structures seen as stripes of positive (red–yellow) and negative (blue–black) Jz, in the region 40di<x<65di. The wavelength is around 3di, indicated by the two magenta lines drawn along the two crests of the wave near x=57di. From the time sequence of Jz (not shown) in the simulation frame, in which the shock front is moving toward the positive x direction with a speed around 2–3 vA, we found that the waves are propagating toward the shock (in the negative x direction), obliquely to the magnetic field. The propagation angle between the wave vector (perpendicular to the wave plane) and the quasi-static magnetic field in the xy plane is between 100° and 110°. For example, when we measure the propagation angle at x=60di, the wave vector points almost 110° from the direction parallel to the magnetic field, and the propagation angle is 103° when we measure at x=55di. In the simulation frame, the phase speed is around 6vA during the time interval from Ωit=15.63 to 17.19. Based on the three components of the magnetic field pattern (not shown), the polarization of the wave is right-handed in the simulation frame. The wave frequency in the simulation frame is around 13Ωi. As time evolves, the wave amplitude grows, and at Ωit=17.19 [panel (b)], field lines (gray curves) have many winding structures. In panels (c)–(e), the diagrams illustrate the relationship between positive and negative Jz values and crests and troughs of winding magnetic field lines. Since the wave propagates obliquely to the magnetic field, magnetic field lines are bent where there are positive and negative Jz. Eventually, as the wave grows, the bent magnetic field lines can reconnect, as in panel (e), after two oppositely directed magnetic field lines come into contact.

FIG. 8.

(a) Current density Jz (color) at Ωit=15.63 and (b) Jz at Ωit=17.19. Gray curves are magnetic field lines projected on the xy plane. Panel (a) shows waves whose wavelength is 3di. In panel (b), white oval areas are regions where shorter wavelength waves whose wavelength is 0.7di are observed, and two crests of waves are shown by the white solid lines in the middle and the bottom oval regions. (c)–(e) diagrams illustrate how magnetic field lines are bent as the wave grows. Black arrows are magnetic field lines, and the blue and red lines show negative and positive Jz, respectively. In panel (e), the magenta X is the position where two opposite directional magnetic field lines contact and magnetic reconnection occurs.

FIG. 8.

(a) Current density Jz (color) at Ωit=15.63 and (b) Jz at Ωit=17.19. Gray curves are magnetic field lines projected on the xy plane. Panel (a) shows waves whose wavelength is 3di. In panel (b), white oval areas are regions where shorter wavelength waves whose wavelength is 0.7di are observed, and two crests of waves are shown by the white solid lines in the middle and the bottom oval regions. (c)–(e) diagrams illustrate how magnetic field lines are bent as the wave grows. Black arrows are magnetic field lines, and the blue and red lines show negative and positive Jz, respectively. In panel (e), the magenta X is the position where two opposite directional magnetic field lines contact and magnetic reconnection occurs.

Close modal

In Fig. 8(b) at Ωit=17.19, there is another type of waves, whose wavelength is much shorter than 3di. Examples of such waves are seen in the regions marked by the white ovals in panel (b). For example, in the bottom white oval, along the right red stripe, there are many sub-di wavelength phase fronts, and the white two lines represent one wavelength in those short-wavelength waves. Also, in the middle white oval in panel (b), along the red stripe, similar short-wavelength structures are seen. The wavelength in those regions is around 0.7di, and this short-wavelength mode is generated almost along the wave plane of the long-wavelength mode whose wavelength λ3di.

Figure 9 plots the current density Jz, showing the time evolution of the long-wavelength mode (λ3di, in the following, we denote it as the LW mode) and the short-wavelength mode (λ0.7di, in the following, we denote it as the SW mode) in a local region, 45di<x<55di and 27di<y<37di. In panel (a), LW modes are seen as diagonal yellow and blue stripes, and one crest (Jz>0) is marked by the while line. Around this white line, the black curves are magnetic field lines projected on the xy plane, covering one wavelength of the LW mode. Note that in this region at this time Ωit=16.66, there is a large non-uniform |Bz| up to 5B0 (not shown), because of the shock reformation. This makes the foreshock region eventually the new shock front. Then, the wave propagation angle changes to closer to 90° because the magnetic field BzBx and By, when the wave vector is in the xy plane. Along this marked crest of the LW wave (indicated with the white line), there are SW wave patterns shown as red/yellow and blue/dark-blue stripes. Two crests of the SW mode are marked by the short magenta lines. The whole wave structure of the SW mode is moving together with the motion of the crest of the LW mode, suggesting that the SW mode is propagating along the wave plane of the LW mode in the reference frame where the LW mode is stationary. Let us call this frame the LW wave frame. However, the wave plane of the SW mode (magenta line) is not perpendicular to the wave plane of the LW mode (white line), and the angle between the wave vectors (perpendicular to the wave planes) of those two modes is 143°.

FIG. 9.

(a) Current density Jz (color) at Ωit=16.66 and (b) at Ωit=17.16. In panel (a), the white solid line represents one crest of the LW (long-wavelength) wave, and black curves are magnetic field lines projected on the xy plane for the region of one wavelength of the LW mode including one crest and one trough of the wave. Along the white line, the SW (short-wavelength) mode is excited, and the short magenta lines represent the positions of crests of the SW mode. In panel (b), the white solid line is the crest of the LW mode at Ωit=17.16 propagated from the original position at Ωit=16.66, denoted by the white dashed line. The short magenta lines on the white solid line are the crest positions of the SW mode at Ωit=17.16 propagated from the original position of the short magenta line on the white dashed line at Ωit=16.66. The magenta arrow illustrates the phase velocity of the SW mode. This magenta vector is considered the sum of the solid yellow vector, which is the phase velocity of the LW mode, and the dashed yellow vector, which is the phase velocity of the SW mode in the LW wave frame. The blue X shows the reconnection X-line generated due to the LW mode, while magenta Xs are the reconnection X-lines due to the SW mode.

FIG. 9.

(a) Current density Jz (color) at Ωit=16.66 and (b) at Ωit=17.16. In panel (a), the white solid line represents one crest of the LW (long-wavelength) wave, and black curves are magnetic field lines projected on the xy plane for the region of one wavelength of the LW mode including one crest and one trough of the wave. Along the white line, the SW (short-wavelength) mode is excited, and the short magenta lines represent the positions of crests of the SW mode. In panel (b), the white solid line is the crest of the LW mode at Ωit=17.16 propagated from the original position at Ωit=16.66, denoted by the white dashed line. The short magenta lines on the white solid line are the crest positions of the SW mode at Ωit=17.16 propagated from the original position of the short magenta line on the white dashed line at Ωit=16.66. The magenta arrow illustrates the phase velocity of the SW mode. This magenta vector is considered the sum of the solid yellow vector, which is the phase velocity of the LW mode, and the dashed yellow vector, which is the phase velocity of the SW mode in the LW wave frame. The blue X shows the reconnection X-line generated due to the LW mode, while magenta Xs are the reconnection X-lines due to the SW mode.

Close modal

Figure 9(b) displays the plot at Ωit=17.16. At this time, the crest of the LW mode with the white solid line is shifted leftward. The white dashed line in the right is the original position of the crest at Ωit=16.66. The yellow arrow pointing from the white dashed line to the white solid line is the direction of the LW mode propagation. This arrow is parallel to the wave vector of the LW mode, which is perpendicular to the wave plane. According to the propagation of the wave plane between Ωit=16.66 and 17.16, the phase speed of the LW wave is 5.9vA. The short magenta lines on the white dashed line are the original positions of the SW mode, the same as in panel (a). These wave planes are propagating downward along the crest of the LW mode, and the short magenta lines on the solid white line are the positions of the same crests of the SW mode at Ωit=17.16. The magenta arrow represents the direction of the propagation of these wave planes of the SW mode in the simulation frame. If we assume that the SW mode is propagating along the wave plane of the LW mode in the LW wave frame, we can consider that the wave front motion of the SW mode in the simulation frame is the combination of motion of the crest of the LW mode (yellow arrow, from the white dashed line to the white solid line) and the wave propagation of the SW mode along the wave plane of the LW mode (dashed yellow arrow). In other words, the dashed yellow arrow represents the phase velocity (along the wave plane of the LW mode) of the SW mode in the LW wave frame, and the phase speed in that reference frame is 4vA.

Due to those waves, reconnection X-lines are generated. Focusing on the region around the white solid line in panel (b), there are two types of X-lines: one is due to the LW mode, which is denoted by the blue X-line, located at (x,y)=(47.475di,32.875di), above which a di-scale magnetic island is seen. The other is due to the SW mode, denoted by the red X-lines right to the white line. The SW mode generates multiple sub-di to di-scale magnetic islands.

To understand the wave excitation mechanisms, we investigate electron and ion distribution functions in the region where those wave modes are observed. Figure 10 exhibits the distribution functions at locations from #1 to #4 at Ωit=15.63, at which time the LW mode is observed in the shock transition region. Location #1, at (x,y)=(70di,40di) is in the region where no LW waves are observed, while location #2, at (x,y)=(60di,40di), is in the region where LW waves are seen as stripes in the current density Jz [panel (a)] and the electron density ne [panel (b)]. Location #3, at (x,y)=(55di,40di), and location #4, at (x,y)=(55di,30di), are also where LW waves are observed, and larger amplitude waves are seen at location #4. At each of the four locations, we collected particles in a square region around the location, with its side length 0.1di, for ten time steps (2.5ωpe1), and we generated the distribution functions of electrons and ions. Panels (c1)–(c3), (d1)–(d3), (e1)–(e3), and (f1)–(f3) are for electron distribution functions for locations #1, #2, #3, and #4, respectively, and panels (c4)–(c6), (d4)–(d6), (e4)–(e6), and (f4)–(f6) are for ion distribution functions at each location. Each panel shows a reduced distribution function, in which the particle count is integrated along the velocity component perpendicular to the 2D velocity components in each panel. The white dashed arrow represents the projection of the magnetic field direction in each 2D velocity plane.

FIG. 10.

(a) Current density Jz (color) and (b) electron density (color) at Ωit=15.63. Black curves are magnetic field lines projected on the xy plane. The white dots, #1 to #4, are the positions where the electron and ion distribution functions are measured. Location #1 is where no LW mode is observed. Locations #2 to #4 are where LW waves are generated. Location #4 is where the largest LW waves are observed. (c1)–(c3) are the electron distribution functions and (c4)–(c6) are the ion distribution functions in location #1. Each panel shows the reduced distribution function in which particle counts are integrated in the velocity component perpendicular to each 2D velocity plane. White arrows are the projection of the magnetic field direction in each velocity plane. The same format for (d1)–(d6) for location #2, (e1)–(e6) for location #3, and (f1)–(f6) for location #4.

FIG. 10.

(a) Current density Jz (color) and (b) electron density (color) at Ωit=15.63. Black curves are magnetic field lines projected on the xy plane. The white dots, #1 to #4, are the positions where the electron and ion distribution functions are measured. Location #1 is where no LW mode is observed. Locations #2 to #4 are where LW waves are generated. Location #4 is where the largest LW waves are observed. (c1)–(c3) are the electron distribution functions and (c4)–(c6) are the ion distribution functions in location #1. Each panel shows the reduced distribution function in which particle counts are integrated in the velocity component perpendicular to each 2D velocity plane. White arrows are the projection of the magnetic field direction in each velocity plane. The same format for (d1)–(d6) for location #2, (e1)–(e6) for location #3, and (f1)–(f6) for location #4.

Close modal

Panels for locations #2 and #3, where LW waves are observed, show the following characteristics of the electron and ion distribution functions: Electron distribution functions in (d1) and (e1), in the vzvy plane, exhibit an almost circular shape, and panels (d2) and (e2), in the vzvx plane, and panels (d3) and (e3), in the vyvx plane, show that the distribution function has a peak around vx=17.5vA. In contrast, ion distributions in (d4)–(d6) for location #2 and in (e4)–(e6) for location #3 have cold incident ions (lower red dot) and reflected ions (upper diffuse component). The incident component has a peak of the distribution function at (vx,vy,vz)=(8.5vA,0.5vA,1.5vA) in location #2.

Compared with locations #2 and #3, where LW waves are observed, the ion distribution function at location #1, where LW waves are not observed, has less numbers of particles in the reflected component, and these ions are separated from the cold incident component farther in the vx direction than those in locations #2 and #3. The growth of the LW mode is likely due to the interactions between the incident ions, the reflected ions, and the single electron beam. In location #1, the LW mode is not growing because the number of reflected ions is small.

Location #4 is in the region where the strongest LW mode is observed [see panels (a) and (b)]. In this location, larger numbers of reflected ions are seen in the ion distribution functions in (f4)–(f6), although the density at location #4 is less than that in location #3 [see panel (b)]. The large fraction of the number of reflected ions compared with the incident ions may cause the increase in the growth rate of the LW mode, resulting in the larger amplitude of the LW mode in location #4 than in location #3.

Figure 11 shows the same format of plots of electron and ion distribution functions for later time, at Ωit=16.66. We selected four locations: Location #1 is where LW waves are observed, but no SW waves are seen. Location #2 is where SW waves are observed on top of a crest of the LW wave. Location #3 is where a significant amplitude of SW waves and distorted field lines is observed. Location #4 is next to the SW wave turbulence, but it is where SW wave activity is quiet and only LW waves are seen. Panels (c1)–(c3) and (c4)–(c6) are for location #1, and they exhibit similar distribution functions to those in Figs. 10(d1)–10(d3) and 10(d4)–10(d6), respectively, where only LW waves are observed.

FIG. 11.

Electron and ion distribution functions at Ωit=16.66, the same format as Fig. 10. Location #1 is where only LW waves are observed. Location #2 is where both LW waves and SW waves are observed. Location #3 is where strong SW waves are observed. Location #4 is where LW waves are observed but no significant SW waves are observed.

FIG. 11.

Electron and ion distribution functions at Ωit=16.66, the same format as Fig. 10. Location #1 is where only LW waves are observed. Location #2 is where both LW waves and SW waves are observed. Location #3 is where strong SW waves are observed. Location #4 is where LW waves are observed but no significant SW waves are observed.

Close modal

Panels (d1)–(d3) in Fig. 11 are the electron distribution functions in location #2, where small-amplitude SW waves are observed along the crest of the LW wave. In these panels, there are multiple components of electron beams. Ion distribution functions in panels (d4)–(d6) show distributions similar to those in location #1, with a cold core and the reflected ions. Therefore, SW waves are attributed to multiple beam components in both electrons and ions, i.e., electron beams seen in panels (d1)–(d3) and ion beams in panels (d4)–(d6).

Location #3, where the strongest SW wave turbulence is observed, has more outstanding electron beam structures in panels (e1)–(e3). The contour lines in panels (e1)–(e3) show outstanding two electron beam components. The strong SW turbulence is likely driven by these electron beams. Ion distributions (e4)–(e6) have similar structures to those in locations #1 and #2.

In contrast, in the electron distributions in location #4, panels (f1)–(f3), where only strong LW waves are generated, there are no clear electron beam components, and the distributions are more similar to panels (c1)–(c3) for location #1, than to distributions for location #2 and #3, although the distribution (f1) shows slight anisotropy compared to (c1). Again, ion distribution functions (f4)–(f6) are similar to those in locations #1 to #4.

We have studied magnetic reconnection in quasi-parallel shocks whose shock angle is 25°, by means of two-dimensional full PIC simulations, using parameters relevant to the Earth's bow shock. We have compared a low Mach number shock and a high Mach number shock, and have found that the number of reconnection sites increases as the Mach number increases. In the low Mach number shock (MA6), waves whose wavelength is 1di are excited in the shock transition region. No strong turbulence is generated in the shock transition region, and a small number of magnetic reconnection sites are observed in the shock transition and downstream regions. In contrast, in the high Mach number shock (MA11), waves whose wavelengths range from 3di to sub-di are excited in the shock transition region, and strong turbulence is generated as those waves interact with the shocked particles and fields.

We have investigated the properties of shock-driven magnetic reconnection in the high Mach number shock. In the shock transition region, there are several sub-di to di-scale magnetic reconnection sites in which only electrons participate in the reconnection, and ions are just passing through the X-line. There are many such small-scale reconnection sites where only a one-sided electron jet is generated, which lacks the electron jet in the opposite direction across the X-line. The reconnection rates in reconnecting current sheets have been measured, and they are around 0.1 to 0.2 based on the local values of Ez, and 0.02 to 0.1 based on the spatial averages of Ez. Note that around the reconnection X-lines, the reconnection electric field gives the dominant force to accelerate particles in the z direction, compared with the magnetic force (eVe×B/c), and the electric force in area #2 is twice larger than the magnetic force, and the electric force in area #3 is an order of magnitude larger than the magnetic force, at Ωit=18.75 in the simulation.

There are also non-reconnecting current sheets, some of which show the reversal of magnetic field across the current sheet without magnetic reconnection. Both reconnecting current sheets and non-reconnecting current sheets can dissipate magnetic energy into thermal and kinetic energies of plasma; therefore, even non-reconnecting current sheets can have electron jets, and the magnetic field reversal with an electron jet and a positive energy dissipation, i.e., J·E>0, does not necessarily mean that reconnection is occurring. Comparing the magnitude of J·E between reconnecting current sheets and non-reconnecting current sheets, we have found a tendency that non-reconnecting current sheets with the magnetic field reversal show smaller J·E than reconnecting current sheets.

We have studied the kinetic physics of waves excited in the high Mach number shock. In the shock transition region, waves whose wavelengths 3di (LW modes) are first excited, propagating obliquely to the magnetic field, generating current sheets. As these waves grow, magnetic field lines are bent, and magnetic reconnection sites are generated after the oppositely directed magnetic field lines come into contact. Short-wavelength (λ0.5di to 1di) waves (SW modes) are excited along the wave plane of the LW mode, and sub-di to di-scale small reconnection regions are generated as these SW modes grow. We have investigated electron and ion distribution functions in the shock transition region where LW modes and SW modes are excited. In regions with LW waves, electron distribution functions exhibit a single-peaked shape, while ion distribution functions show a cold incident ion component and a reflected ion component. It is expected that these two ion beams and a single-peaked electron beam destabilize LW modes. In regions with LW waves and SW waves, electron distribution functions have multiple electron beams. Ion distributions show the ion core component and the reflected ion component. It is expected that these multiple electron beams are interacting with multiple ion beams to excite SW waves.

Further studies of kinetic instabilities for LW modes and SW modes, such as analyses of the growth rates and the dispersion relations of those modes, remain to be conducted to understand the properties of excited waves, which are responsible to generate reconnecting/non-reconnecting current sheets. The LW wave we have observed in the simulation may be consistent with ion–ion beam modes observed in the Earth's bow shock,38–40 which are due to the incident ions and the reflected ions. Since we use an artificial mass ratio (200) in the PIC simulation, the wavelength (3di) of the LW wave observed in the simulation may vary in a simulation with a more realistic mass ratio.

In the simulation, we have observed that the LW waves are right-handed and propagating toward the shock in the simulation frame. To identify the type of the wave, the wave phase velocity in the plasma rest frame needs to be considered. According to the plasma bulk velocity in the region 50di<x<55di at Ωit=15.63, the LW waves are propagating in the negative x direction in the plasma rest frame, opposite to the reflected ion component. If we analyze the phase speed 55di<x, the LW waves propagate with almost the same velocity as the plasma bulk velocity, and it is hard to determine whether the LW waves propagate in the negative x direction in the plasma rest frame, because of uncertainties in the wave phase velocity and the plasma bulk velocity in the simulation frame. Considering the analysis in the region 50di<x<55di, it is likely that the LW waves excited in this simulation are propagating toward the shock in the plasma rest frame, and we consider that the LW modes are due to non-resonant ion–ion beam instability,18 which are right-handed waves propagating away from the reflected ion component.

The SW modes we have observed in the simulation are more complicated, but one of the features of the electron distribution function in the SW mode region is outstanding two electron beams. We analyzed the wave phase velocity and the polarization in the plasma rest frame in the SW mode region in Fig. 11, around location # 3. In the plasma rest frame, the SW waves are propagating in the negative y direction along the LW wave plane, and the polarity is right-handed. The wave frequency in the plasma rest frame is estimated to be around 25–40 Ωi (0.13 − 0.2 Ωe in the simulation where the mass ratio is 200). These results indicate that the SW waves can be whistler waves, excited under multiple electron and ion beams. Since the LW mode contains the longitudinal electric field, which generates the density fluctuations seen in Fig. 10, there exists an electric field component parallel to the magnetic field, which accelerates electrons. The electron beams accelerated within the LW waves can be the source of the secondary instability to generate whistler waves.41,42 The electron distributions observed in the SW modes also resemble electron distributions in modified two stream instabilities (MTSIs),43,44 which can generate oblique whistler waves with frequencies near the lower hybrid frequency, 0.07 Ωe in the simulation with the mass ratio of 200, although the waves by MTSIs have been discussed in quasi-perpendicular shock waves.45,46 As shown in Fig. 3, as time evolves, the magnetic field lines in the shock become close to the quasi-perpendicular condition, even though the upstream field lines are fixed to be a quasi-parallel condition. Therefore, waves similar to the ones excited in quasi-perpendicular shocks might operate in the turbulent quasi-parallel shock region. More investigations of wave properties and instability mechanisms are necessary, by performing linear dispersion analyses using the ion and electron distribution functions obtained in the simulations.

The above results are useful to guide the search of reconnecting current sheets and to understand the properties of magnetic reconnection driven by shocks in the Earth's bow shock region by space measurements. An abundance of current sheets has been observed in the Earth's bow shock by NASA's MMS measurements, and many of these are potential reconnection sites. Magnetic field reconstruction methods47–49 are available to see the magnetic field topology to determine whether observed current sheets are reconnecting or non-reconnecting. Further studies of magnetic reconnection in the Earth's bow shock are needed to understand the roles of reconnection in contributing to shock heating and particle acceleration.

This work was supported by DOE Grant No. DESC0016278, NSF Grant Nos. AGS-1619584 and AGS-1552142, NASA Grant No. 80NSSC18K1369, and the NASA MMS project. This work was partially supported by the International Space Science Institute's (ISSI) International Teams programme. Work at the University of Bergen was supported by the Research Council of Norway/CoE under Contract No. 223252/F50.

PIC simulations were performed on Pleiades at the NASA Advanced Supercomputing Division, and the simulation data are available upon request from the authors.

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