ORIGINAL RESEARCH article
Sec. Space Physics
On the Presence and Thermalization of Cold Ions in the Exhaust of Antiparallel Symmetric Reconnection
- 1Space Plasma Physics Group, Department of Physics and Technology, University of Bergen, Bergen, Norway
- 2NASA Ames Research Center, Moffet Field, CA, United States
- 3Department of Electromagnetism and Electronics, University of Murcia, Murcia, Spain
- 4State Key Laboratory of Space Weather, National Space Science Center, Chinese Academy of Sciences, Beijing, China
- 5School of Space and Environment, Beihang University, Beijing, China
Using fully kinetic 2.5 dimensional particle-in-cell simulations of anti-parallel symmetric magnetic reconnection, we investigate how initially cold ions are captured by the reconnection process, and how they evolve and behave in the exhaust. We find that initially cold ions can remain cold deep inside the exhaust. Cold ions that enter the exhaust downstream of active separatrices, closer to the dipolarization front, appear as cold counter-streaming beams behind the front. In the off-equatorial region, these cold ions generate ion-acoustic waves that aid in the thermalization both of the incoming and outgoing populations. Closest to the front, due to the stronger magnetization, the ions can remain relatively cold during the neutral plane crossing. In the intermediate exhaust, the weaker magnetization leads to enhanced pitch angle scattering and reflection. Cold ions that enter the exhaust closer to the X line, at active separatrices, evolve into a thermalized exhaust. Here, the cold populations are heated through a combination of thermalization at the separatrices and pitch angle scattering in the curved magnetic field around the neutral plane. Depending on where the ions enter the exhaust, and how long time they have spent there, they are accelerated to different energies. The superposition of separately thermalized ion populations that have been accelerated to different energies form the hot exhaust population.
Magnetic reconnection is a fundamental plasma process that converts energy stored in the magnetic fields to plasma energy by enabling the reconfiguration of the magnetic field topology. Cold plasma can be abundant in regions of magnetic reconnection, and impact the reconnection process in several ways (for a recent review, see Toledo-Redondo et al., 2021). Examples include, but are not limited to, mass loading (Fuselier et al., 2017; Dargent et al., 2020; Tenfjord et al., 2020), introduction of an extra cold ion diffusion region (Toledo-Redondo et al., 2016a; Divin et al., 2016), modifications to the Hall physics (Toledo-Redondo et al., 2015; André et al., 2016; Toledo-Redondo et al., 2018), and plasma heating (Toledo-Redondo et al., 2016b; Toledo-Redondo et al., 2017). In the magnetotail, cold lobe plasma below a few hundreds of eV (e.g., Engwall et al., 2009) can be heated to several keV (e.g., Ergun et al., 2018), and at the dayside cold magnetospheric plasma of a few tens of eV (e.g., Borovsky and Denton, 2008) can be heated to hundreds of eV (e.g., Toledo-Redondo et al., 2016b) as a consequence of the reconnection process. However, where, how, and to what extent cold plasma is heated (or not) by reconnection is still not fully understood.
Cold ions have been observed deep within the exhaust, both at the dayside and nightside. In the nightside magnetotail, in the vicinity of dipolarization fronts, both Nagai et al. (2002) and Xu et al. (2019) found, in addition to a hot component, two cold beams counter-streaming parallel to the magnetic field. They proposed that the cold ions moved along the reconnected field lines directly from the lobes due to the topological change enabled by reconnection. The counter-streaming beams are attributed to Fermi acceleration of ions initially dwelling on flux tubes downstream of the front that are being swept up as the front expands past them. This process is also described by Eastwood et al. (2015), although the authors attributed the source population to pre-existing plasma sheet, not the lobes. Closer to the X line, Alm et al. (2018) also found that cold ions could dominate the density well inside the separatrices, and account for a significant fraction even when the magnetic field amplitude approached zero. Counter-streaming beams observed during nightside reconnection has also been attributed to acceleration by the Hall electric field (Wygant et al., 2005). This is also shown in numerical simulations (e.g., Divin et al., 2016). While some studies suggest that the Fermi and Hall mechanisms may be dominant in the far and near exhaust, respectively, their relation and effectiveness are not fully understood. For example, Drake et al. (2009) noted that the energization by the Hall electric field arose from a potential, and that the oscillatory motion along the normal direction of the current sheet did not lead to a significant net energization. Drake et al. (2009) also presented a model of the ion heating based on a scenario of counter-streaming ion populations, originating from opposite sides of the current sheet and being subject to Fermi acceleration in the exhaust. Haggerty et al. (2015) developed the model further by taking into account a parallel electric field supported by the heated electrons moving downstream, away from the X line. The corresponding electric field potential slows down ions in the frame of the outflow field lines, reducing the acceleration, and thereby the ion heating.
At the dayside magnetopause, Li et al. (2017) observed parallel jets, carried by cold magnetospheric ions, on the magnetosheath boundary of the exhaust, showing that initially cold ions could remain cold while traversing the exhaust. Toledo-Redondo et al. (2016b) showed that cold ions could be found inside the reconnection exhaust far away from the X line. Closer to the X line, they found that the cold ions were heated by waves and large electric field gradients inside the separatrix region. Graham et al. (2017) also found that cold magnetospheric ions could interact with magnetosheath ions that enter the magnetosphere through the finite gyroradius effect. The resulting ion-ion streaming instability lead to the formation of lower-hybrid waves that can heat the cold ions. Schriver and Ashour-Abdalla (1990) also suggested that cold inflow plasma in the plasma sheet boundary layer could be heated through an ion-ion streaming instability due to the cold inbound plasma and the hotter parallel ion beams commonly observed there (e.g., Eastman et al., 1986; Nagai et al., 1998). These findings suggest that the separatrices, and the proximity to the X line, may play a role in how cold ions are heated. Dayside observations have also indicated that cold ions may become more efficiently heated if their relative density is lower (Toledo-Redondo et al., 2016b; Toledo-Redondo et al., 2017). These findings are also consistent with those by Xu et al. (2019), who presented one case where the density of the cold populations were large enough such that the density across the front remained approximately constant, in contrast to the more commonly observed density decrease (e.g., Runov et al., 2011).
The thermal population inside the exhaust of reconnection has been studied by several authors. Using hybrid simulations, Nakamura et al. (1998) showed that ion orbit characteristics vary as a function of the distance from the reconnection line due to the change in magnetic field curvature. Based on the magnetic field curvature parameter κ2 = rB/ρi (Büchner and Zelenyi, 1989), where rB is the radius of magnetic field curvature, and ρi is the ion gyroradius, they defined three classes of orbits. Closer to the jet front, in the magnetic pile-up region, the radius of magnetic field curvature typically exceeded the ion gyroradii, and the inflow population could remain magnetized and formed the cold counter-streaming beams mentioned above. In an intermediate region, where the radius of curvature was comparable to the ion gyroradii, the motion became stochastic. Closer to the X line, where the ion gyroradii greatly exceeded the curvature radius, the ions followed typical Speiser orbits (Speiser, 1965), meandering in the field reversal and slowly turning towards the outflow direction over the course of several bounces. The Speiser orbits typically form distributions in the shape of half circles (e.g., Nakamura et al., 1998; Zenitani et al., 2013; Vines et al., 2017) that rotates around the current sheet normal direction with distance from the X line (Arzner and Scholer, 2001; Drake et al., 2015; Nagai et al., 2015). The relative location in the circle is related to the amount of gyroturning around the normal magnetic field, which in turn is related to the number of bounces around the neutral plane the corresponding particles has performed (e.g., Zenitani et al., 2013).
In this work, we take a new integrated look at the acceleration and formation of a thermalized exhaust from initially cold inflow ions. We focus on the distinction between the cold ions being swept up by the exhaust as it expands (e.g., Eastwood et al., 2015) and the cold ions entering the exhaust closer to the X line (e.g., Zenitani et al., 2013). Using a fully kinetic 2.5D particle-in-cell simulation, where a cold inflow is captured by the reconnection process, we address the following questions:
1) When can cold ions potentially appear deep within the exhaust?
2) When do the initially cold ions remain cold, and when are they heated?
3) How are the cold ions heated?
2 Simulation Set-Up
To investigate how the cold ions are accelerated and thermalized, we perform a fully kinetic 2.5D particle-in-cell simulation (Hesse et al., 1999) with symmetric inflow conditions and no guide field. The simulation is initialized with a Harris sheet equilibrium of hot ions and electrons. In addition, we add cold ions and electrons to the inflow regions:
Densities are normalized to n0, times to the inverse ion cyclotron frequency
3 Formation of Dipolarization Front due to Decrease in Inflow Density
The formation of dipolarization fronts have been attributed to three main mechanisms: jet braking (Birn et al., 2011), transient reconnection (Sitnov et al., 2009; Birn et al., 2011; Fu et al., 2013), and spontaneous formation (Sitnov et al., 2013). In this section, we show that the dipolarization fronts in our simulation form due to transient reconnection associated with the transition from (hot) dense to (cold) tenuous inflow plasma.
Figure 1A shows the magnetic field and density profiles along a vertical cut (z) through the X line, at tωci = 5. We note that the deviation of the originally hot plasma ni,hot in Figure 1A from a Harris sheet density profile is due to the initial perturbation. Figure 1B shows the temporal evolution of the density and magnetic field in the inflow, 1 di above the X line. As the denser Harris sheet plasma is processed by the reconnection process, and moved downstream of the separatrices and X line, the inflow adjacent to the X line is gradually replaced by the more tenuous and cold population, resulting in a density decrease. The magnetic field also decreases, albeit slower than
FIGURE 1. (A) Magnetic field |B| and densities ni,hot, and ni,cold in the inflow along a vertical cut through the X line at tωci = 5. (B) |B|, ni,hot, ni,cold, ntot, and vA in the inflow 1di above the X line. (C) Reconnection rate, here defined as the out-of-plane electric field Ey at the X line. (D) Magnetic field Bz around the equatorial plane as function of time and x. The thicker black line shows the location of the X line, while the thinner black lines show equipotential contours of Ay. Newly reconnected field lines appear at the X line and propagate outwards. (E) Magnetic field |B| and densities ni,hot, and ni,cold in the outflow along a horizontal cut through the X line at tωci = 120. Note that the colors of the lines are consistent throughout the figure, and we therefore only define the legend the first time a quantity appear.
The increase in reconnection rate leads to the formation of two flux pile-up regions propagating away from the X line. This is seen in Figure 1D where we plot Bz(x, t) at z = 0. The thicker black line shows the location of the main X line. The thinner black lines are equipotential contours of the magnetic vector potential Ay, defined by the in-plane magnetic field (Bx, Bz) through B = ∇ ×A, and show the motion of field lines in the outflow. Newly reconnected field lines appear at the main X line and adjacent to islands, and propagate outwards. The islands are seen as localized regions of Bz reversals, numbering five in total, with the first one appearing at tωci ∼ 85. The slopes of the lines indicate the propagation speeds of the field lines. When the reconnection rate increases, we can see how newly reconnected field lines appear at a greater rate and propagate outwards at a higher speed, such that they stack up and form two regions of stronger magnetic field Bz. In magnetotail nomenclature, these flux pile-up regions are commonly referred to as dipolarizing flux bundles (DFB), and the leading edges of the two DFBs as dipolarization fronts (DFs) (e.g., Liu et al., 2013). In the magnetotail, the tailward propagating front is also referred to as an anti-dipolarization front (e.g., Li et al., 2014). At tωci = 120, these fronts have reached x ≈ 70 and x ≈ 135. In Figure 1E, we can see how the fronts roughly separate the initially hot and cold plasmas. From this, we conclude that the dipolarization fronts can be considered as compressed versions of the initial current sheet edge. The field lines that reconnected faster, and went on to form these flux pile-up regions, were also the ones that separated the hot from the cold plasma. This is why the initially cold plasma end up deep within the exhaust, all the way to the dipolarization fronts. Whether or not this initially cold plasma can remain cold depends on the heating mechanisms.
In the following, we will mainly focus on the time tωci = 120, where the two flux pile-up regions have expanded to x = 70 and x = 135. In Figure 2, we show the density, pressure, and temperature of the initially hot and cold ions, respectively. The gray contour lines show the in-plane magnetic field. The main X line is located at x = 109, with a second and third X line at x = 100, and x = 91. These three X lines straddle two magnetic islands that are centered at x = 95, and x = 102. In order to more easily compare the hot and cold ion contributions to density and pressure in the exhaust, we have saturated the color scales. In the equatorial plane, the cold ion density exceeds the hot ion density for x ≳ 71, while the cold ion pressure exceeds the hot ion pressure for x ≳ 72 in the left exhaust. While the hot temperature exceeds the cold temperature in large parts of the exhaust, their effect on the dynamics inside the exhaust is negligible due to their low density. The initially cold ions are heated to temperatures comparable to the initial temperature of the hot Harris sheet ions, where the initial Harris sheet temperature is given by
FIGURE 2. Density, pressure, and temperature of initially hot and cold ions at tωci = 120: (A)nih, (B)nic, (C)Pih, (D)Pic(E)Tih, (F)Tic. The initially cold plasma dominate the density and pressure at x ≳ 71 and x ≳ 71, respectively. While the hot temperature exceeds the cold temperature in large parts of the exhaust, their effect on the dynamics inside the exhaust is negligible due to their low density. The initially cold ions are heated to temperatures comparable to the initial temperature of the Harris sheet ions.
4 Kinetic Structure of Ion Distributions Within the Exhaust
In this section we investigate the structure of the ion distributions within the exhaust. Figure 3 shows the reduced distributions of initially cold ions fic(x, vx), fic(x, vy), fic(x, vz) at z = [0, 2, 4] between the two dipolarization fronts at time tωci = 120. The reduced distributions are integrated over the remaining velocity dimensions, e.g. f(vx) = ∫f(vx, vy, vz)dvydvz. At this time, the initially cold ions dominate the density and pressure within the exhaust (see Figure 1E and Figure 2), which is why we do not include the initially hot population in the further analysis. The distributions are sampled over boxes spanning a spatial domain of Δx ×Δz = 0.5 × 0.5, i.e. x = 75 → x = 75 ± 0.25, and z = 4 → z = 4 ± 0.25. The dotted vertical lines show the location of the separatrices, the dashed vertical lines at z = 0 show the locations of the DFs, and the black solid lines show the corresponding components of vE×B. The still cold populations are identified by higher phase space densities that occupy a relatively small velocity interval. The initially cold ions that have become heated are seen as (relatively) lower phase space densities that occupy larger velocity intervals.
FIGURE 3. Reduced distributions of initially cold ions at tωci = 120, at (A–C)z = 4, (D–F)z = 2, and (G–I)z = 0. The vertical dashed lines show the location of the separatrix, which at z = 0 maps to the X line. The solid line shows vE×B. We do not show these in panels (H) and (I) because they contain Bx, which is small there. While the cold ions are most apparent in the inflow, they are also present in the outflow. In an intermediate region, just inside the separatrices, the cold incoming population is affected by the dynamics at the separatrices, and becomes partly thermalized. This is perhaps most prominently seen in f(x, vz).
Heated ions are observed throughout the exhaust. Cold ions are most prominent in the inflow region at z = 2 and z = 4, but are also found throughout the exhaust, at all z. In an intermediate region, inside the separatrices, the inbound ions (vz < 0) are affected by the convergent electric field associated with the separatrix density cavities. At z = 4, this region is clearly seen between x = 118 and x = 127 for the right exhaust (Figure 3C). At z = 2, this region has extended over a larger range along x (x = 113–124), and the inbound ions have become more thermalized (Figure 3F). At z = 0, this region maps to the intermediate exhaust (Figure 3I). In f(vz, z = 0), we see counter-streaming populations throughout the exhaust. The coldest populations are found in the vicinity of the X lines and the DFs (where they are superposed on a hot population). Outbound cold ions (vz > 0) in f(vz, z = 2) are only observed in the immediate vicinity of the DFs (70 < x < 75 in left exhaust). This means that only a portion of the cold beams observed at z = 0 are actually leaving the equatorial region with their temperature relatively intact. In the rest of the exhaust (closer to the separatrices and X lines), the outbound plasma at z ≥ 2 is strictly thermal.
In f(vy, z = 0), we see a distinct transition in the speed of the cold ion population at around x = 82 in the left exhaust (with a corresponding transition in the right exhaust). At x < 82, the cold ions are observed at vy > 0, while at x > 82 the cold ions are observed at vy < 0 (with gradually increasing vy towards the X line). As explained by Divin et al. (2016), the initially vy < 0 is acquired through a process where the inbound ions are first accelerated by the Hall electric field Ez leading to an enhanced flow vz towards the equatorial plane. This vz is turned towards
4.1 The Role of Separatrices in Ion Thermalization
Depending on where the ions cross into the exhaust the inbound ion motion vz, mentioned in the previous section, can either be on average magnetized (v⊥ ≈ vE×B) or demagnetized (v⊥ ≠ vE×B). This is illustrated in Figure 4, where we plot the reduced distributions fic(vz) as a function of z at x = 75 and 85. The dotted line (now horizontal) again shows the separatrix location, while the solid lines show vE×B,z. As expected, the ion edge (e.g., Gosling et al., 1990; Lindstedt et al., 2009), i.e. the location at where the first reconnected ions appear (here represented by the thermal population centered on vz > 0) inside of the separatrix, is further inside the separatrix deeper within the exhaust. At x = 75 and x = 85, Bx dominates the magnetic field in the range |z| > 1.5 and |z| > 0.5, respectively. Where Bx dominates, vz constitutes a perpendicular component of the ion flow, and a deviation of the cold ions from vE×B,z signifies demagnetization. At x = 75, the cold ion population follows vE×B,z until z ≈ 2 (see also Toledo-Redondo et al., 2018), while at x = 85, they deviate shortly after crossing the separatrix. At x = 85, the Hall electric field Ez is quasi-stationary over the time it takes for an ion to cross from the separatrix to the neutral plane. For this location, we have also plotted the speed
FIGURE 4. Reduced ion distributions f(z, vz) at (A)x = 75 and (B) 85 at tωci = 120. The solid, dashed, and dotted lines show vE×B,z,
To further illustrate, and to better understand the differences between the populations that remain cold deep within the exhaust, and the population inside the separatrices that are slightly more thermalized, we make use of test particles integrated in the dynamically changing electric and magnetic fields. The initial velocities of the test ions are chosen from the cold component of
FIGURE 5. Test particle positions at five different times. The initial conditions for each particle are taken at t = 120, from the locations in phase space shown in (A, B). (C–G), and (H–L) shows the particle position for five different times, overlaid Ez and
The main difference between the yellow/black and the white/green ions is where they cross the separatrices, or rather, what sort of separatrices they cross. While the separatrices, by definition, span the entire range of x in the simulation, they are highly active closer to the X line, and comparatively inactive, further away. Here, we define active separatrices as characterized by large amplitude parallel electron flows toward the X line, associated density cavities, and electric fields, electrostatic waves propagating along the magnetic field in the direction of the electron flow, and in general stronger Hall fields. While a Hall electric field can still be observed inside the inactive separatrices, it is typically weaker, and is generally characterized by the absence of deeper density cavities, small to none parallel electron flow, and less wave activity. The white/green ions enter the exhaust at active separatrices, are slightly thermalized at the separatrices, substantially accelerated by Ez, and eventually form the hot thermal population. The green ones first enter an island, while the white ones directly enter the open exhaust. The yellow/black populations enter the exhaust at inactive separatrices, or rather, the separatrices expand to encompass them. When they are caught up by the expanding exhaust region, they are picked up by the motional electric field Ey, and are accelerated while remaining relatively magnetized.
In addition, the black and white ions also show distinct differences in their z distribution. In Figure 5B, we can see that the white ions overall have larger |vz| at z = 0. The white ions, affected by the Hall electric field, gain higher vz, and are able to penetrate deeper into the southern side of the exhaust. This is reflected in the density
The subdivision into the yellow and black populations is mainly based on their behaviour around, and after having crossed, the neutral plane. While the time period of the test particles is not large enough to properly display their different behaviours, the density structure is. At all times, there is a clear cutoff in
4.2 The Role of Magnetic Field Curvature in Ion Thermalization
To illustrate how the ion magnetization changes from directly adjacent the front, to a distance away, we take a closer look at the magnetic curvature
FIGURE 6. (A) Density of initially cold ions originating from the north,
Figures 6D–H show 2D reduced distributions f(vx, vy) at z = 0, and x = [72, 74, 76, 78, 80] (white squares in Figures 6A,B). At z = 0, the magnetic field is dominated by Bz, and f(vx, vy) is therefore the distribution in the plane perpendicular to B, such that
To investigate the evolution of the ion distributions around the neutral plane, we plot the 1D reduced distributions
We start by looking at the field line located closer to the front (Figures 6I–K). At about s = −10, the cold populations slowly drift with the magnetic field. In
We now look at the field line located further away from the front, in the intermediate exhaust (Figures 6L–N). With exception for the ion trapping seen closer to the front, the behaviour is approximately similar until s = −3. One important difference between the two field lines is the behaviour in
4.3 The Role of Streaming Instabilities Deep Within the Exhaust
Inside the exhaust, we observe two off-equatorial regions (one in the north and one in the south) where electrostatic solitary waves (ESW) form (Figure 7). The ESWs form at around z = ±4 and grow both in amplitude and perpendicular (to B) extent as they propagate downstream with the magnetic field at low parallel speeds. To investigate how they form we examine a timestep (tωci = 115) before they have grown to their full strength. Figures 8A–D shows a few quantities plotted as a function of the distance s along a magnetic field line marked in Figure 8E (the same field line is also marked by the thicker black line in Figure 7). s = 0 marks the neutral plane z = 0, s > 0 is in the region z < 0 (since the magnetic field direction is from north to south), and s < 0 is in the region z > 0. Figure 8A shows the parallel electric field E‖ in which the ESWs are seen as bipolar spikes. Figure 8B shows the reduced distribution function of the initially cold ions as a function of v‖. The inbound ions move at low parallel speeds further out, and are gradually accelerated as they approach and cross the neutral plane. This acceleration is due to Fermi acceleration, where the initial vy gained by Ey is first turned towards z by the magnetic force − vyBx, and thereafter towards − x by vyBz. We note that the final speed gain is consistent with what we expect due to Fermi acceleration: vafter = −vbefore + 2vDF, where the speed of the front vDF ≈ 0.5. Oscillations in the cold component indicate that they are affected by the parallel electric field. The results of the wave-ion interaction, for a closely located field line, are also seen in Figures 6I,K where we only plot the ions originating from the north.
FIGURE 7. Electrostatic solitary waves inside the southern exhaust. The arrow recurring in every panel identifies one given structure.
FIGURE 8. Thermalization due to ion-acoustic waves within the exhaust. Panels (A–E) show quantities sampled along a magnetic field line marked by the black line in panel (E). They are plotted as a function of the distance along the field line s, where s = 0 is the equatorial plane and s > 0 is in the region z < 0 (since the magnetic field direction is from north to south). (A) Parallel electric field E‖. (B) Reduced ion distribution of initially cold ions, log 10fic. (C) Mixing level of ions originating from the north and south, respectively,
The level of plasma mixing, defined by the phase space density ratio of cold ions originating from the top to all cold ions:
In Figure 7, we can see that the ESWs convect downstream with the magnetic field. In the frame of the magnetic field, their motion is approximately field-aligned. To investigate what kind of instability may generate the waves that grow into ESWs we therefore solve the one-dimensional electrostatic plasma dispersion equation:
where Z is the plasma dispersion function (Fried and Conte, 1961), ωps again is the plasma frequency,
5 Formation of the Thermalized Exhaust
In this section we investigate the formation of the thermal population within the exhaust. The distribution fic(x, vy) in Figure 9 (see also Figure 3) exhibits discrete structures in phase space. One of the most distinct structures is a cold beam ranging between vy ≈ −0.5 at x ≈ 82 and vy ≈ 0.5 at x ≈ 92 (with corresponding structures in the right exhaust). As explained in Section 4, this beam is formed by a combination of forces Ey > 0 and vzBx < 0 acting on the ions during their first inbound leg towards the neutral plane from the inflow regions. Where vy < 0, vzBx is dominating, while where vy > 0, Ey has been dominating. Divin et al. (2016) showed that this striation in phase space followed contours of constant canonical momentum py = mivy + qAy. Echoes of this initial striation are seen in both exhausts. While the first echo is towards higher vy, the striations successively rotate counter-clockwise (clockwise) in the left (right) exhaust, such that the range of vy eventually covers a large range of positive and negative values. The first echo is comprised of ions that have crossed the neutral plane once, have been turned around by Bx, and returned to z = 0 again. During this motion, they have been accelerated to slightly higher vy, by the reconnection electric field Ey. Once they have acquired a vy > 0, they will start to become turned downstream by vyBz < 0. These two forces will jointly move the initial striation to higher vy and lower x. By plotting contours of constant py in Figure 9, we show that also the higher order (in terms of bounces) striations approximately follow contours of constant py. This kind of trajectory was first described by Speiser (1965). That is, the thermal exhaust is formed by a superposition of Speiser orbits with varying initial conditions, related to where they enter the exhaust. As explained by Divin et al. (2016), the change in vy of the zero order striation is due to spatially different acceleration by the Hall electric field Ez. Closest to the X line, the integrated electric field is smaller than further downstream, and therefore the initial vz leading to vy through vzBx is larger further away from the X line. We also note that each time the ions cross the neutral sheet, they are scattered in the highly curved magnetic field, smearing out and further blending the superposed structures.
FIGURE 9. Reduced ion distribution f(x, vy) at z = 0, and t = 120. The lines show contours of constant canonical momentum py = mivy + qAy, which well explains the overall phase space structure (x, vy) of the thermalized exhaust. Since vy(t = 0) = 0, ions with the same py originated from the same flux tube, defined by a given Ay. Ions with vy > 0 (< 0) are currently downstream (upstream) of their original flux tube.
By integrating test particles in the dynamically changing fields, we find that py is constant to a good approximation. Initially at t = 0, all of these ions had vy = 0, such that
We can also think of ΔAy and Δvy in terms of diffusion. As described above, an ensemble of ions that dwells in the vicinity of the X line, being accelerated by Ey, sees the magnetic field lines move past them. Or, in other words, the magnetic field diffuses across the ensemble of ions. Further downstream, when vy is turned to vx, the ions are catching up to their original field lines, and the diffusion is reversed. The primary diffusion is done by the electric field, which performs work on the ions. Since the reversed diffusion is done by magnetic forces, no work is done, and a net energy transfer between magnetic field and plasma is realized.
6 Discussion and Summary
In this study, we have investigated the occurrence and heating of initially cold ions during symmetric magnetic reconnection. We showed that ions that enter the exhaust at active separatrices with large gradients and electric fields become heated and thermalized more efficiently than ions that enter the exhaust at inactive separatrices lacking large gradients and electric fields. These findings are consistent with observations at the dayside magnetopause (Toledo-Redondo et al., 2016b). The ions that enter the exhaust at inactive separatrices are typically picked up as the exhaust expands into them, both in the outflow (x) and normal (z) directions. While Eastwood et al. (2015) suggested that ion beams observed close to the dipolarization front originated from the pre-existing plasma sheet, Birn et al. (2017) concluded that they originated from the lobes. We find that the cold beams can originate from an extended range of z’s that may include, but also extends well outside, the pre-existing hot current sheet population. The cold ion beams closer to the front originate on lower z’s, while those further away originate at higher z’s. This is illustrated both in Figure 5 (c.f. original locations of yellow and black ions) and Figure 9 where the cold beam (here seen in fic(vy)) extend over a range of py. Recall that py indicated the original z location of the field line/particle, and that lower py corresponded to a field line initially located further out in the inflow. How far this range of z’s extends likely depend on the stage of reconnection. The further the front has propagated, the larger range of z’s can be picked up without being significantly thermalized at the separatrices. In addition, while cold counter-streaming beams could be observed close to the equatorial plane throughout the exhaust, the beams that entered the exhaust at inactive separatrices could remain colder in all three components. Also, closer to the X line, the vz was gained mainly through acceleration by Ez, while further downstream, the vz gain was through Fermi acceleration (mediated by Ey).
The separatrix environment, and therefore also the ion thermalization at the separatrices may be sensitive to the reduced mass ratio we employ in this simulation (e.g., Lapenta et al., 2010). However, we note that observations also show divergent electric field at the separatrix electron flow channels. The integrated potential associated with these fields in observations could be comparable or higher than the inflow energy of the cold ions (Norgren et al., 2020). We also note that out-of-plane instabilities active at the separatrices, such as the lower-hybrid-drift instability, or ion-ion cross-field drift instabilities do not develop in our 2D simulation. Based on previous observations at the magnetopause, such instabilities would also act to enhance the ion thermalization at the separatrices (e.g., Toledo-Redondo et al., 2017; Graham et al., 2017).
In our simulation, electrostatic field-aligned waves were also found to impact the ion thermalization in the off-equatorial region deep within the exhaust. The waves were produced by an ion-acoustic instability due to the cold inbound ion population and the drifting electrons. Field-aligned electrostatic waves has also been found in observations in the vicinity of dipolarization fronts relatively close to the neutral plane (Liu et al., 2019). These waves were likely generated by cold proton, and potentially oxygen, beams. Field-aligned electrostatic waves have also been identified in regions closer to the separatrix in the magnetopause boundary layer at the dayside (Steinvall et al., 2021). Similar to our results, they found the waves to be driven by an ion acoustic instability. Altogether, these results shows that waves may play a role in the thermalization of cold ion populations in many different subregions during reconnection.
We showed that the cold ions that entered the exhaust downstream of active separatrices could remain cold all the way to the neutral plane, and beyond. After having crossed the neutral plane, ions in the intermediate exhaust, reflected back at about a distance of 2di from the neutral plane after having crossed it. Closer to the dipolarization fronts, due to the higher level of magnetization, the cold ions could also continue outward beyond z = ±2di. At the neutral plane, the incoming cold populations also experienced pitch angle scattering due to magnetic field curvature. As such, at a given location, the inbound beam (approaching the neutral plane) is colder than its outgoing counterpart. This is consistent with observations from the vicinity of dipolarization fronts in the magnetotail where Xu et al. (2019) observed cold beams propagating along the magnetic field. The beam approaching the neutral plane was colder than the beam leaving it. Overall, the region along
Since the majority of the outflow is dominated by the species that were initially cold, the division into a hot and cold ion diffusion region such as explored by Divin et al. (2016), and typically invoked at the magnetopause (Toledo-Redondo et al., 2015), is not applicable here. At the dayside, the multiple ion diffusion regions are typically a result of the multiple inflow populations, e.g. cold magnetospheric plasma and warm magnetosheath plasma (Toledo-Redondo et al., 2015). Although the heated ions seen inside the separatrix in Figure 3 do not follow vE×B, this is not an indication of multiple ion diffusion regions, it is merely a superposition of inbound and outbound populations of the same origin. However, multiple ion diffusion regions could exist in the tail if there are multiple species in the inflow (e.g., oxygen ions (Wygant et al., 2005; Tenfjord et al., 2018), if there is an extended region of overlap of hot and cold plasma in the transition from the plasma sheet to the lobes, if there are overlapping regions with ions of different origins (for example the ionosphere and solar wind), or if there are significant north-south plasma asymmetries, for example due to mantle plasma (e.g., Artemyev et al., 2017) or asymmetric ionospheric outflow (e.g., Glocer et al., 2020).
In this study, we have investigated the acceleration, thermalization, and occurrence of cold ions in the exhaust of symmetric antiparallel magnetic reconnection. The overall temperature increase was comparable to the predictions based on the counter-streaming ion model by Drake et al. (2009). However, we found that several processes conspire to form the eventually hot thermalized exhaust (Figure 10). The initially cold populations are thermalized due to interactions at the separatrices, around the neutral plane, and due to waves inside the exhaust. At the same time, the individual populations are accelerated (mainly) by the out-of-plane electric field Ey. At a given position, we can find populations that have been accelerated to different energies, depending on where they entered the exhaust. Together, the thermalization and superposition of ions that have been accelerated to different energies form the hot exhaust population. The acceleration by Ey lead to super-Alfvénic speeds. If we assume an inflow magnetic field of 20 nT, and a plasma sheet density of 0.3 cm−3 (recall that the Alfvén speed in our simulation is based on the inflow magnetic field and central Harris sheet density), the Alfvén speed is vA ≈ 800 km/s. An ion accelerated to 1vA and 2vA thus has an energy of about 3300 eV and 13400 eV, respectively. As for the overall temperature seen in Figure 2F, Ti = 0.3 corresponds to about 2000 eV. These energies are comparable to average ion temperatures in the central plasma sheet (e.g., Baumjohann et al., 1989). This suggests that magnetic reconnection can heat cold lobe plasma to large enough energies to replenish the hot plasma sheet.
FIGURE 10. Illustration of thermalization process of cold ions. Individual populations are thermalized by one or several processes, for example separatrix activity, magnetic field curvature scattering, or plasma waves. The hot exhaust is formed by a superposition of these individually thermalized populations that are accelerated to different energies.
Data Availability Statement
The dataset used in this study can be found at UiB Open Research data: https://doi.org/10.18710/0257WR.
CN planned the study and executed the simulation, performed the data analysis, scientific interpretation and lead the manuscript writing. PT, MH, YX shared in the conception of the study. PT, MH shared in the simulation execution and data analysis. PT, MH, ST, WL, YX, NK, SS, HK shared in the scientific interpretation and discussion of the simulation results. All authors shared in the writing of the manuscript.
This study was supported by NOTUR/NORSTOR under project NN9496K. CN and PT received support from the Research Council of Norway under contract 300865. MH's work prior to moving to NASA ARC was supported by NASA contract NNG04EB99C at SwRI. STR, WL, and PT acknowledges support from the International Space Science Institute (ISSI) international team ‘Cold plasma of ionospheric origin in the Earth's magnetosphere’. The open access publishing fee is funded by the University of Bergen.
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Keywords: magnetic reconnection, particle-in-cell (PIC), space physics, cold plasma, cold ion heating, plasma waves
Citation: Norgren C, Tenfjord P, Hesse M, Toledo-Redondo S, Li W-Y, Xu Y, Kwagala NK, Spinnangr S, Kolstø H and Moretto T (2021) On the Presence and Thermalization of Cold Ions in the Exhaust of Antiparallel Symmetric Reconnection. Front. Astron. Space Sci. 8:730061. doi: 10.3389/fspas.2021.730061
Received: 24 June 2021; Accepted: 30 August 2021;
Published: 17 September 2021.
Edited by:Marian Lazar, Ruhr-Universität Bochum, Germany
Reviewed by:Giovanni Lapenta, KU Leuven, Belgium
Jan Egedal, UW Madison International Division, United States
Copyright © 2021 Norgren , Tenfjord , Hesse , Toledo-Redondo , Li , Xu , Kwagala , Spinnangr , Kolstø and Moretto . This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Cecilia Norgren , firstname.lastname@example.org