Collisionless Magnetic Reconnection in an Asymmetric Oxygen Density Configuration

Combined with the magnetic field, the distribution of charged particles in the inflow region is expected to control the rate of magnetic reconnection. This paper investigates how the reconnection process is altered by a cold, asymmetrically distributed, oxygen population, which is initially located away from the current layer in the inflow regions. A particle‐in‐cell simulation is used to gain further insight into the dynamics of the system. The time evolution of the reconnection process proceeds rapidly compared to the cyclotron period of O + . Therefore, the oxygen remains, to a good approximation, demagnetized. Thus, Alfvén scaling is not an adequate description of the reconnection rate. A scaling relation for the reconnection rate for an asymmetrically distributed, demagnetized species has been developed. Additionally, we find that an asymmetric density configuration leads to a distinct motion of the reconnection site and generates an asymmetry of the diffusion region and the Hall electric field.


Introduction
Magnetic reconnection is one of the most important energy release and transport processes in plasmas. In the Earth's magnetosphere, magnetic reconnection at the magnetopause is the primary mechanism responsible for the transport of energy, mass, momentum, and magnetic flux into Earth's magnetic cavity. Magnetic reconnection involves a violation of the ideal frozen-in condition (Vasyliunas, 1975), where the bulk motion of the particle species separates from the transport of magnetic flux.
In addition to the more abundant plasma species, that is, protons and electrons, heavier species, such as oxygen, may also be present in the plasma sheet as a result of ionospheric outflow (e.g., Baker et al., 1982;Chappell et al., 1987;Moore et al., 2001). The cold magnetotail oxygen originates from high-latitude ionosphere, and the ionospheric outflow rate is proportional to geomagnetic activity (Baker et al., 1982). Presence of oxygen of ionospheric origin in the Earth's magnetotail is supported by various observations (e.g., Frank et al., 1977;Grande et al., 2013;Moore et al., 2001;Mouikis et al., 2018;Wilken et al., 1995;Zong et al., 1998). Spacecraft observations reveal that oxygen may be the dominating ion species during storm time conditions (Kistler et al., 2005;Wygant et al., 2005a). At certain times, north-south asymmetries of ionospheric outflow may generate asymmetric oxygen inflow densities, and such asymmetries are also quite generically expected at the Earth's magnetopause (Li et al., 2013).
The inclusion of additional ion populations introduces distinctive kinetic behavior and different spatial and time scales for the system (Divin et al., 2016;Shay & Swisdak, 2004;Toledo-Redondo et al., 2015). The presence of heavy magnetized ions may result in a reduction of the Alfvén velocity, thus indicating a reduction of the reconnection rate (Hesse & Birn, 2004;Shay & Swisdak, 2004). Different mechanisms to slow down the reconnection rate have previously been proposed such as the effect of the tearing growth rate in the presence of O + (Karimabadi et al., 2011) and induced charge separation (i.e., ambipolar electric fields) (Liang et al., 2016(Liang et al., , 2017. Previous studies (Tenfjord et al., 2018(Tenfjord et al., , 2019, investigated the effects of oxygen being symmetrically distributed in the inflow regions and proposed a scaling relation for the inclusion of a demagnetized species. Cluster reported density composition of n p = 0.03 cm −3 and n O + = 0.07 cm −3 for protons and oxygen in the current sheet, respectively, (Wygant et al., 2005b) leading to a reduction of the reconnection rate of a factor of 6.3 through mass-loading effects and a factor of 3.3 for the proposed scaling relation (Tenfjord et al., 2019). During some phase of the evolution, this might imply that the dynamics of geomagnetic storms and substorms evolves twice as fast as expected when oxygen is present. This paper examines the effects on magnetotail reconnection imposed by an asymmetrically distributed oxygen population. The outline of this paper is as follows: Section 2 will give an overview of the simulation setup employed for this study. Section 3 examines the evolution and dynamics of the reconnection site, for example, motion of x-point and the arising asymmetry of the diffusion region. Finally, in section 4 a scaling relation for an asymmetrically distributed, demagnetized, species is developed.

Simulation Setup
The analysis is performed by using a 2.5-D particle-in-cell (PIC) simulation, that is, two spatial components and three-fields and velocity components, designed to model the evolution of a current sheet subjected to asymmetric oxygen inflow. Initially, the configuration of the magnetic field is given as a two-dimensional generalized Harris-type equilibrium (see Hesse and Birn, 2004 for details) defined as B x = B 0 tanh(z∕ ), where = 2d p is the half width of the initial current sheet. The factor d p = c∕ pi (= c √ m p ∕4 n 0 e 2 ) is the proton inertial length, where n 0 is the foreground density at the center of the initial current sheet.
In the upper inflow region (top lobe), oxygen O + is homogeneously distributed with a number density of n O + = 0.2 at a distance of z > 2.5d p , where z = 0 is the initial location of the x-line (see Figure 1b). Density is normalized to n 0 . Initially, the oxygen has zero thermal and bulk velocity. In addition to the oxygen, a uniform proton H + with a density of n b = 0.2 is added to the Harris-sheet density configuration (n p = n 0 ∕cosh 2 (z∕2d p ) + n b ) resulting in a peak density of 1.2 in the current layer.
The following mass ratios are employed: m p ∕m e = 25 for the protons/electrons and m O + ∕m p = 16 for the oxygen/protons. A total of 4.8 · 10 9 macroparticles are used. Boundary conditions are periodic at x = x min and x = x max . At z = z min and z = z max , specular reflection is enabled and the out-of-plane electric field E is set to zero, preserving magnetic flux in the simulation domain.
Lengths are normalized to d p , whereas time is normalized to the inverse of the proton cyclotron frequency Ω −1 p = m p ∕eB. The size of our simulation domain is 200 × 50 d p with a grid size of 3,200 × 1,600. We employ a time step of Ω e t = 0.5. The velocity normalization is the proton Alfvén speed, based on the foreground current sheet density n 0 . The foreground temperatures fulfill with n 0 = 1 and B 0 = 1, and the ratio of proton-electron temperature is chosen to be T p ∕T e = 5. The ratio between the electron plasma frequency and gyrofrequency is pe ∕Ω e = 2.

Evolution and Dynamics of the Reconnection Site
In Figure 1, the O + and H + density are shown at three selected times. Close to the initial configuration, Ω p t = 4, oxygen is present in the inflow region only at a distance of z > 2.5d p . At Ω p t = 64 reconnection has initiated. However, the oxygen is still not present at the reconnection site. The reconnection rate peaks at Ω p t = 80, and an increasing amount oxygen is transported into the reconnection region, thus becoming present in the reconnection region. At Ω p t = 156 we observe horizontal layers of increased oxygen density close to the x-point. These striations form as a result of quasi-steady flow of oxygen from the inflow to the reconnection region (Tenfjord et al., 2018). Inclined oxygen density striations, which are evident at Ω p t = 156, are governed by the formation and expansion of the localized Hall electric field E z . As the Hall electric field expands, oxygen particles at rest are collectively accelerated resulting in an oxygen density wave (Tenfjord et al., 2018).

Motion of the X-Point
The asymmetric oxygen distribution leads to distinct motion of the x-point across the initial x-line. The blue dotted line in Figure 2 shows the displacement of the x-point in the z direction with respect to time. We observe a shift of the x-point toward positive z, reaching a value of z = 0.7d p . From Ω p t > 120 the x-point moves in the negative z direction, reaching its minimum at z = −1.7d p . Close to the end of the simulation, the x-point moves toward its initial value. The two processes governing the shift toward positive and negative z values are distinct. We now examine them further.
We start by investigating how the shift of the x-point relates to the differences in the flux velocity between the top and bottom lobe. The motion of the magnetic flux in z direction is determined by finding the frame in which dA dt = 0. This leads to the magnetic flux velocity w z Figure 3. Reconnection site before (after) reconnection has initiated on left (right). The oxygen front is illustrated by the wall, hindering the replenishment of magnetic field lines.
In order to investigate the correlation between the magnetic flux velocity and the actual velocity of the x-point, we average w z throughout the simulation domain, excluding the diffusion region. The difference in the magnetic flux velocities between the two inflow regions is attributed to a drag imposed by the higher inertia of the oxygen species. As the far more agile electrons move with the magnetic field, an (ambipolar) electric field is set up to prevent charge separation (Karimabadi et al., 2011;Liang et al., 2016). The resulting electric field impedes the transport of magnetic flux from the top lobe. However, in the lower inflow region, the absence of oxygen lets the electrons and protons move, to a higher extent, unencumbered toward the x-line.
The limited flux mobility in the region covered by cold oxygen results in an upward movement of the current layer. This can be shown analytically by approximating the oxygen region as nearly stationary such that the flux velocity and the overall plasma bulk velocity is much smaller here than elsewhere. The different regions of flux mobility above and below are indicated by z U and z L , see Figure 3. If magnetic reconnection removes the same amount of flux Δ from the upper region and the lower region, force balance requires that the current layer moves upward to keep the magnetic pressure equal on both sides. At a time t 1 , indicated by the subscript, the flux contained in the two regions can be written as where the numbered subscripts refer to the times of evaluation, t 0 and t 1 . The length of the combined regions are to be remained constant, thus Combining equations (1)-(3), the change in length of the upper and lower regions can now be expressed as Equations (4) and (5) imply that the x-point shifts to positive values of z. In order to reach equilibrium, the upper region needs to be compressed and the lower region expanded, leading to an upward motion of the current layer (see Figure 3).
The x-point reaches its highest value at Ω p t = 88 and is thereafter shifted toward negative z due to a different process. The Hall electric field E z extends into the upper lobe and accelerates the demagnetized oxygen toward the x-point, resulting in an increase of the dynamic oxygen pressure. At this point the oxygen no longer imposes a drag on the transport of magnetic flux, but instead contributes to an increase of w z as a result of its enhanced dynamic pressure, see Figure 5. An overall higher w z on the oxygen side, compared to the southern lobe, shifts the x-point to negative z-values. In the next section we will see how the increase of the dynamic pressure affects the diffusion region.

Asymmetry of the Diffusion Region
As the oxygen gets involved in the reconnection process, an asymmetry in the diffusion region arises. Figure 4 shows an overview of the Hall electric field and its potential. Figure 4a shows the structure of the Hall E z , and in Figure 4b a clear asymmetry across the diffusion region is evident.
To investigate the asymmetry of the Hall electric field E z we start by examining the pressure balance along z through the x-point. We obtain Equation (6) only includes the dominant contributions (shown for Ω p t = 132 in Figure 5) where C is a constant, and n avg O + is the average oxygen density. In Figure 5, we see that the contribution to the overall pressure balance of the dynamic and thermal pressures for both oxygen and protons are substantially higher in the northern lobe than in the southern lobe. On the z > 0 side a clear reduction of the gradient of the magnetic pressure is seen. Additionally, asymmetries in the dynamic oxygen pressure and the thermal pressure of both the oxygen and protons are found.
Due to acceleration of oxygen by the Hall electric field, the dynamic oxygen pressure increases and a reduction of the magnetic pressure arises to preserve the pressure balance. The higher contribution of the pressure terms from the plasma species in the northern lobe suppresses the contribution of the magnetic pressure to preserve the pressure balance. The immediate effect of this is the broadening of the B x profile which, through Ampères law 0 ∼ ∕ z B x , leads to a reduced current density. The reduction in the magnetic field strength results in a decrease of E z in the northern lobes through Ohm's law.
In the region shown in Figure 5, the current is, to a good approximation, mainly carried by the far more agile electrons. We may therefore write the Hall electric field in the following manner which may be written, in terms of the magnetic pressure, in the following way From equation (7) we see that the Hall electric field is directly dependent on the gradient of the magnetic pressure. Thus, on the oxygen side, the flatter profile of the magnetic pressure results in a smaller gradient and thereby reducing the Hall electric field.

Scaling of the reconnection rate
In this section we evaluate how the reconnection rate scales for asymmetrically distributed oxygen. There are several reasons to expect that the aforementioned configuration would affect the reconnection rate. Being under the influence of oxygen, we have seen that the flux velocity is reduced in the top lobe compared to the bottom lobe. Furthermore, the cyclotron period of the oxygen species is much longer than the characteristic time scale of the reconnection process (Ω −1 p ), oxygen is therefore mainly influenced by electric forces. This can also be seen in Tenfjord et al. (2018), where a similar simulation setup is employed.
Mass loading effects predict a reduction in the reconnection rate that scales as the total Alfvén velocity, that is, Tenfjord et al. (2019) investigated how the reconnection rate scales in the presence of demagnetized oxygen and found that the reconnection rate was reduced much less. Performing a scaling analysis similar to that of Tenfjord et al. (2019) leads to a relation for the reconnection rate in the presence of an asymmetrically distributed oxygen density. The Hall electric field for the protons scales in the following manner The factor is a proportionality constant relating the off-diagonal pressure components to the diagonal components. We have here assumed that off-diagonal pressure components shows a linear dependence on Substituting the derived pressure relations into equation (6) gives Writing out the average oxygen density to relate the density of the upper n U O + and lower n L O + lobes as n Neither the right-hand side, the scale lengths, nor the proportionality constants, and , depends on the oxygen density. This leads us to express the scaling relation of the reconnection electric field as a function of the oxygen and proton density In Figure 6, we compare the reconnection rate, which can be expressed as d dt rec = ∫ E d , to numerical simulations, hereafter referred to as runs, of different oxygen densities. The oxygen species are initially a distance of 2.5d p from the x-line for the three oxygen runs. The reconnection rate of the symmetric density configuration of n O + = 0.1 (red) (Tenfjord et al., 2019) displays a good agreement with the asymmetric density configuration of n O + = 0.2 (dash-dotted line). The blue solid line represents a run with absence of oxygen.
The scaling relation in equation (8) coincides with the reconnection rates as seen in Figure 6. All runs follow the same evolution toward the peak of the reconnection rate. This indicates that an increase of the density reduces the maximum value of the reconnection rate and not the rate at which it reaches its peak. This implies that the reconnection rate is determined by the oxygen content and not its initial configuration.

Discussion and Summary
Our results provide insight into the rate reconnection actually proceeds with. It further provides a generalization of the estimated reconnection rate when the plasma system includes heavier species in addition to the protons and electrons. This generalization is valid when heavier ion species are distributed asymmetrically, and such scenarios may occur for ionospheric outflow, for example, when the Northern Hemisphere is illuminated, and the Southern Hemisphere is in darkness. The smaller gyrofrequency of the O + introduces longer coupling time scales. Therefore, the oxygen is largely demagnetized on the time scales of this investigation and Alfvén scaling of the reconnection rate does not apply in our model. It should be noted that steady state reconnection lasting for a significantly longer time would be expected to magnetize the oxygen at some point, and Alfvén scaling may apply for these cases. Through pressure balance arguments, we succeeded in extending previous results (Tenfjord et al., 2018) to derive a scaling relation for the reconnection rate for an asymmetrically distributed, demagnetized, species.
The reconnection rate during the fast phase shows no dependency on the oxygen density. We have seen that reduction of the reconnection rate is dependent on the overall content of the demagnetized species and not its initial configuration. The initial temperature of oxygen is chosen to be zero to isolate oxygen inertia on the reconnection process from other effects, such as streaming and finite temperature. In Tenfjord et al. (2019) it was shown that an initial temperature of the O + has no significant effect on the reconnection rate. The additional effects arising from an asymmetric oxygen population, such as asymmetric E z , bulk z-directed oxygen velocity in the diffusion region and motion of x-point, does not appear to have any significant effect on the reconnection rate.
Regarding observations, signatures as the oxygen striations seen in Figure 1 and asymmetries in the Hall fields should be detectable. The results obtained in this study may apply to magnetotail reconnection in case of north-south asymmetries of ionospheric outflow. In addition, they may play a role in reconnection at the magnetopause, even though many effects may be masked by the typical and often rather large asymmetries across the current layer. They may further apply to laboratory devices and other reconnection geometries, where either history or asymmetric particle injection may lead to similarly asymmetric situations. Regarding space applications, we hope our results provide useful insight for future analysis of space mission data, in particular, the Magnetospheric Multiscale mission.