Observational Features of Charge distribution in Earth’s Inner Magnetosphere

Understanding the motion of charged particles in the electromagnetic �eld in the inner magnetosphere is essential for space science and space weather. Charge accumulation can occur due to the dipole magnetic and convective electric �elds in this region. However, until the recent MMS mission, there were few means to detect charge distribution in situ. We report unambiguous in situ observation of the spatial distribution of the excess charge density in the inner magnetosphere by the magnetospheric multiscale (MMS) mission. We �nd that a positive (negative) charge accumulates in the dusk (dawn) side inner magnetospheres, which is contrary to the long assumed overall quasi-neutrality of space plasma. These observations and results offer insight into magnetosphere–ionosphere coupling.


Introduction
The inner magnetosphere is a signi cant and intricate area of magnetospheric research.The dipole region of the magnetosphere close to Earth is called the inner magnetosphere 1.It spans from approximately 1 to 10 Earth radii and includes a dipole magnetic eld, a co-rotating electric eld resulting from Earth's rotation, and a convection electric eld generated by the solar wind 2. These elds interact with the plasma present in space and give rise to three large-scale structures: the plasmasphere 3,4, the ring current 5, and the radiation belts 6, 7, each corresponding to a population of particles having energies within a certain range.Low-energy particles that exit the ionosphere are in uenced by the convection and co-rotating electric elds, which form the plasmasphere and can produce waves 8, 9 that affect signal transmission between Earth and spacecraft 10, 8. Medium-energy particles are in uenced by the magnetic eld and travel around Earth, forming the ring current 11, which weakens the Earth's magnetic eld 12. High-energy particles in the radiation belts could 'kill' the instrument of spacecraft 13.The inner magnetosphere plays a crucial role in the interactions between elds and plasma in space, energy conversion and matter transport, and serves as a key region in the research of the solar-magnetosphereionosphere coupling.
Medium-and high-energy particles that are consistently injected into the inner magnetosphere from the magnetotail are in uenced by the convection electric eld and dipole magnetic elds, resulting in two distinct types of paths within the inner magnetosphere 14.One is open paths that extend from the magnetotail towards the dayside magnetopause, while the other is closed paths that orbit around the Earth.The boundary between the two types of paths is known as the Alfvén layer.To simulate the Alfvén layer, we used a dipole magnetic eld with a uniform dawn-dusk electric eld and a corotation electric eld and set the initial kinetic energies of protons and electrons to 6 keV and 2 keV, respectively, as shown in Fig. 1.We found that due to the opposite magnetic drift directions of electrons and ions, the Alfvén layer for electrons and ions is asymmetric and does not overlap completely, resulting in net charge accumulation near the separatrix path for particles of each sign.The phenomenon of charge accumulation in the inner magnetosphere was rst proposed by Shield15, but direct observational studies of the accumulation and distribution of charges have been limited due to experimental constraints.
The separation of charge generates a shielding electric eld that protects the Earth's surface from harmful effects of the particles and radiation 16.The charge discharge along magnetic eld lines generate eld-aligned currents (FACs) 17 that ow into and out of the ionosphere on the dusk and dawn sides, respectively 18.The Region-2 FACs play an important role in space weather phenomena such as magnetic storms and substorms 19.Understanding the charge distribution and the relationship between the electric eld, charge, and geomagnetic activity is important in predicting and mitigating the impact of space weather.
The recent Magnetospheric Multiscale (MMS) mission 20, consisting of four identical spacecraft, has enabled highly accurate in situ four-point electric-eld measurements and thus the acquisition of space charge density.The data obtained by simultaneous multi-spacecraft probes have been used to explore various space physics phenomena such as electron holes 21, magnetic reconnection 22, and geomagnetopause 23.This paper utilizes MMS data to investigate the distribution of electric charge density in the inner magnetosphere, its relationship with the electric eld, and how it varies during different geomagnetic conditions.The study reveals the presence of charge accumulation in the inner magnetosphere.The ndings provide valuable insights into the plasma dynamics in the inner magnetosphere and can help clarify the magnetosphere-ionosphere energy exchange mechanism.

Event Analysis
On 17 December 2015, during a quiet period with a maximum auroral electrojet index of 309 nT, we used data from the MMS mission as it moved from 8.8 Re away from Earth to 4 Re on the dusk side of the inner magnetosphere.As shown in Fig. 2b, the charge was always positive, the charge density (see the Methods section, calculation of charge density calculation) increased with decreasing L-shell, and the maximum charge density was 49 e/m 3 at approximately L = 5.91.Additionally, the magnetic eld intensity gradually increased from 65 nT to 487 nT, as shown in Fig. 2a.
Figure 3 shows the eld and charge distributions on the dawn side on 23 February 2016 during a quiet period (maximum AE index was 174 nT).The satellites moved from 4 away from Earth to 10 .As shown in Fig. 3a, the magnetic eld intensity decreased gradually from 428 nT to 43 nT.As shown in Fig. 3b, the charge was always negative, and within L = 7, the charge density decreased sharply with increasing L-shell; the maximum negative charge density was − 21 e/m 3 at L = 4, and between L = 7 and L = 9.8, the charge density remained at approximately − 3.47 e/m 3 .As shown in Fig. 3c and d, the electric eld displayed a high-frequency disturbance, and the overall trend was that the electric eld intensity decreased with increasing L-value of the drift shell.
The above comparison between the two cases reveals that positive (negative) charges accumulate at dusk (dawn), and the density decreases with increasing L-shell.To verify this case study and investigate the mechanism of charge accumulation, a statistical study of the net charge distribution is necessary.

Observations of Electric Field and Electric Charge Density
We project , and the charge density directly onto the equatorial plane in SM coordinates along the magnetic eld lines, assuming that the electric eld is constant along a magnetic eld line.To carry out a statistical study, rst, we separate the data into three groups according to geomagnetic conditions as described by the AE index: (i) quiet times for , (ii) weakly disturbed periods for , and (iii) strongly disturbed periods for .Then, we separate the space according to L-shell and MLT with steps of within and in the entire MLT region.Thus, in total, there are 12×24 bins in space.Finally, we calculate the average values for the electric eld components and , as well as the electric charge density in each bin.During the quiet periods, the number of paths in each bin varied in the range of 4-69, whereas during the weakly and strongly disturbed periods, the corresponding ranges were much lower (1-32 and 0-16, respectively).The ratio of path numbers for the quiet, weakly disturbed, and strongly disturbed periods occupied 47.40%, 34.79%, and 17.79% of the whole observational duration, respectively.
Figure 4 shows the calculated results for the electric eld ( , ) and the charge density after being projected and binned onto the equatorial plane in SM coordinates.During strongly disturbed periods, E y is enhanced overall.
Figure 4d-f show the polarization charge distribution, where red corresponds to positive charges and blue corresponds to negative charges.Some positive charges gathered in 6-22 MLT in a limited L-shell range of 4 < L < 10, and their density increased with increasing geomagnetic disturbance.An azimuthally con ned region of 0-6 MLT in quiet periods was occupied mainly by negative charges whose density was much smaller than that of the positive charges on the dusk side.During disturbed periods, the negative charges in this region seemed to fade away.The charge distributions on the dawn and dusk sides were generally consistent with the theoretical charge separation resulting from the grad-B drift and curvature drift of charged particles 24 Next, we analyse the parallel electric eld (Fig. 4g-i) that is closely related to the eld-aligned current.
Because eld-aligned electric elds are generated near the equatorial plane and are symmetrical with respect to it 25, the parallel components of the electric elds measured on the MMS away from the equator (> 5°) in the magnetosphere can be divided into two categories: equatorward (blue) and poleward (red), with the poleward representing the point from the equator to the South and North poles.. the equatorward parallel electric eld.This accordance is consistent with the intuition that an electric eld points away from positive charges and towards negative ones.However, at 6-10 MLT at dayside, the accordance was contravened, perhaps because of some unknown physical processes in this local time sector require more-detailed studies in event analyses.

Two Distinguished Regions
The observations presented above reveal that there were two distinguished MLT sectors: (i) the local time sector limited by 13-16 MLT (sector A), where positive charges were accumulated, and (ii) the sector limited by 2-5 MLT (sector B), where negative charges were accumulated.The observations in these two sectors are consistent with the theoretical predictions for the dusk-and dawn-side parts of the Alfvén layer 15 26.Then, we perform further analysis on the two sectors by averaging the charge density over every (horizontal axis) in Fig. 5a-c (sector A) and Fig. 5d-f (sector B) during different geomagnetic conditions (shown by the columns in the gures).
As shown in Fig. 5a, the charge density in sector A was positive.During quiet periods, the charge density diminished gradually towards the outer shells; for instance, it was 36.65 e/m 3 at L = 4 and 15.39 e/m 3 at L = 10; the same trend occurred during disturbed periods, as shown in Fig. 5b and c.By comparing Fig. 5a-c, it was evident that the positive charge accumulations grew with the geomagnetic disturbances.
The maximum charge density for all L-shells was 47.97 e/m 3 in weakly disturbed periods and 72.02 e/m 3 in strongly disturbed periods.
As shown in Fig. 5d, the charge density in sector B during quiet periods was always negative, with an average value of − 8.59 e/m 3 ; during disturbed periods, as shown in Fig. 5e and f, its magnitude diminished slightly.
By comparing the charge density distributions in Fig. 5a-f, in sector B on the dawn side, the amount of charge was signi cantly less than that in sector A on the dusk side.For L = 4-10, the average charge density in sector A was 22.92 e/m 3 during quiet periods (approximately three times that in sector B) and 27.17 e/m 3 during strongly disturbed periods [six times that in sector B (-4.93 e/m 3 )].This nding implied a signi cant dawn-dusk asymmetry in the net charge distribution of the Alfvén layer in the inner magnetosphere, and this asymmetry become more severe with enhanced geomagnetic disturbances.

Conclusion
Based on the electric eld data from the EDP onboard the MMS satellites, we revealed and analysed the changes in the charge and electric eld in the Earth's inner magnetosphere under different distribution conditions and exhibited, for the rst time, the observational features of the Alfven layer.The new ndings are summarized as follows.1) The inner magnetosphere accumulates a positive charge at dusk and a negative charge at dawn.The charge decreases with increasing L-shell value and varies with magnetic activity.
2) The distributions of charge and electric eld con rm the existence of charge separation in the Alfvén layer observed by multipoint satellites.
3) The charge distribution in the Alfvén layer has a dawn-dusk asymmetry, with the positive charge density at dusk being much greater than the negative charge density at dawn.
The data used in this study were measured during a solar minimum, and few samples were collected for events with an AE index greater than 1000 nT.Future work is needed to study the charge distribution during periods of stronger geomagnetic activity.

Discussion
Here, we discuss possible mechanisms responsible for the asymmetry of the Alfvén layer shown in Fig. 4.
The current carriers for the region-2 eld-aligned current should be protons and electrons.Because protons have a larger mass and inertia, a strong eld-aligned electric eld at the dusk side is needed to drive them to ow into the ionosphere along the magnetic eld lines; moreover, electrons have a smaller mass and inertia and thus ow more easily into the ionosphere along the magnetic eld lines with a rather weak eld-aligned electric eld.This would lead to the observed asymmetry of the charge distribution shown in Fig. 4.
Figure 4 also shows that the electric eld distribution and charge distribution have a westward rotation of 45°.It is commonly assumed that the electric current distribution is symmetrical at approximately 0-12 MLT in the magnetosphere and ionosphere 27, 28,29, and that the convection electric eld in the ionosphere is perpendicular to the line of 0-12 MLT 30, 31.However, as shown in Fig. 4, the axis of symmetry of the E || distribution is roughly at 3-15 MLT, and the equatorial electric eld in the magnetosphere is mainly a tailward electric eld.The phenomenon of westward rotation was also found in previous studies 32, 33.
The region we observed, which is between 4-10 RE from Earth, partly overlaps with the plasmasphere that contains dense cold plasma with a density ranging from over 10 4 cm − 3 down to about 10 cm − 3 , which is much greater than the charge density.Although charge compensation may occur, our statistical ndings and individual event observations suggest that the charge compensation process in the plasmasphere cannot entirely offset the excess charge resulting from charge separation.It is unfortunate that the plasma data are severely missing when MMS is in the inner magnetosphere, despite being equipped with FPI instruments.With better and more complete data in the future, we can further analyze the process of accumulation and release of excess charge.

Charge density calculation
We use the electric-and magnetic-eld data from all four MMS spacecraft 20 that orbit Earth in nearequatorial space at low latitude with a perigee of 1.2 and an apogee of 12 .The data were collected three years from September 2015 to September 2018, during which the constellation swept the full range of magnetic local time (MLT) more than three times.The data from the eld instrument suite 34, 35-including electric eld measurements-were provided by two sets of biased double-probe sensors (EDP) 36 with a temporal resolution of 8 − 1 s to 32 − 1 s.
Based on the reliability of the data product, we have retained both ok data and good data.In the inner magnetosphere, the magnitude of the electric eld is typically less than 100 mV/m37, 38, so to ensure statistical quality, we have removed any outliers whose values exceed 1000 mV/m.
Let the position vectors of the four MMS spacecraft be in barycentric coordinates 39, with .The electric elds measured by the four spacecrafts can be expressed as , and the linear gradient of component E i of the electric eld can be derived as follows 40: 1 , where the volumetric tensor is de ned as , which characterizes the constellation geometry.The truncation error for the gradient of the electric eld is on the order of S/D, where S is the characteristic size of the constellation, and D is the spatial scale of the electric eld.The characteristic scale of MMS is generally S ≈ 20 km, while the spatial scale of the electric eld is assumed at the scale of one Earth radius ( = 6371 km), as indicated by the empirical model of large-scale electric eld in the inner magnetosphere 41.So the truncation error for the gradient of the electric eld is S/D ≈ 0.3%, which implies the high accuracy of the calculation with formula (1).
Based on the linear gradient of the electric eld measured by each spacecraft, the charge density at the barycentre of the spacecraft constellation is calculated using Gauss's law 21, 22, 23 as follows: To simulate the actual charge density measurement, we randomly placed a tetrahedron inside a nonuniformly charged sphere.The error in the charge calculation, denoted as , is de ned as the difference between the charge density( ) calculated from the electric eld divergence and the actual charge density .After running the model 1000 times, we obtained a maximum error of 0.2 and an average error of 0.1%.These results suggest that the method used for the charge density calculation is reliable.
As shown previously 23, the observational error in the charge density calculated from MMS electric measurements is , where the measurement accuracy of the axial electric eld , and the spin axial electric eld 34.So that , where e is the charge of electron, and e = 1.602 176 634×10 − 19 Coulombs.Here, we remove outliers in the charge density (which accounted for less than 0.001% of all the data) whose values exceed 1000 e/m 3 .Notably, the MMS constellation, at which the electric eld is measured, is moving at an orbital velocity of approximately 1 ~ 10 km/sec relative to Earth.The charge density calculated with Eq. ( 2) needs to be transformed under a Lorentz transformation 42 to obtain the corresponding charge density as viewed in the Earth centre frame of reference.However, because the orbital speed of MMS is much less than the speed of light in a vacuum, the change in the charge density under the Lorentz transformation is small and well below the observational error of the charge density ( ) given above.Therefore, we neglect the effect of relativity here and assume that the charge density of the plasmas observed in the Earth centre frame of reference is the same as that measured in the MMS constellation frame of reference. Illustration

Figure
Figure4a-cshow the distribution of E y , where red corresponds to a duskward electric eld (E y > 0) and blue corresponds to a dawnward electric eld (E y < 0).During the quiet periods (i.e., in the early morning sector of 0-4 MLT with L = 6-10 and in the local time sector of 10-22 MLT on the dusk side with L = 5-10) are dawnward electric elds that contain the polarization eld produced by the accumulated polarization charges in the Alfvén layer 15,16.In the 5-8 MLT sector and the region limited by 22-24 − 4 MLT and L = 4-6 are duskward electric elds that are enhanced in disturbed periods and spread to cover 5-24 MLT, whereas the early morning sector of 0-6 MLT is dominated by a dawnward electric eld.During strongly disturbed periods, E y is enhanced overall.
Figure 4d-i clearly show a distributional accordance in the local time sector of 10-19 MLT between positive charges and the poleward parallel electric eld and in 0-6 MLT between negative charges and

4 ∇
E(r α ), α = 1, 2, ⋅ ⋅ ⋅, of the formation of the Alfvén layer viewed from the Northern Hemisphere with the sun at the top.The protons and electrons are injected into the inner magnetosphere from the plasma sheet on the night side under the action of a dawn-dusk electric eld.The red and blue thin lines represent the drift paths of protons and electrons, respectively, in a dipole magnetic eld with a uniform dawn-dusk electric eld and a corotation electric eld.The initial kinetic energies of protons and electrons are set to 6 keV and 2 keV, respectively.The red and blue thick lines denote the proton and electron Alfvén Layer, respectively.The pink area and sky-blue area indicate the regions where the positive and negative charges accumulate due to charge separation.

Figure 2 Overview
Figure 2

Figure 3 Overview
Figure 3