CHIN.PHYS.LETT. Vol. 25,No. 8 (2008) 2934

Generation of Electric Field and Net Charge in Hall Reconnection ∗

MA Zhi-Wei(ê��)1,2∗∗ FENG Shu-Ling(¾Ô )11Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031

2Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou 310027

(Received 23 April 2008)Generation of Hall electric field and net charge associated with magnetic reconnection is studied under differentinitial conditions of plasma density and magnetic field. With inclusion of the Hall effects, decoupling of theelectron and ion motions leads to the formation of a narrow layer with strong electric field and large net chargedensity along the separatrix. The asymmetry of the plasma density or magnetic field or both across the currentsheet will largely increase the magnitude of the electric field and net charge. The results indicate that theasymmetry of the magnetic field is more effective in producing larger electric field and charge density. Theelectric field and net charge are always much larger in the low density or/and high magnetic field side than thosein the high density or/and low magnetic field side. Both the electric field and net charge density are linearlydependent on the ratios of the plasma density or the square of the magnetic field across the current sheet. Forthe case with both initial asymmetries of the magnetic field and density, rather large Hall electric field and chargedensity are generated.

PACS: 52. 35.Vd, 52. 65.Kj

Under the Hall MHD description, some importantfeatures of magnetic reconnection can be understood.The ion diffusion region in the spatial scale of the ioninertial length is mainly controlled by the Hall effect.[1]

Hall dominated structures such as bipolar electric fieldand quadrupole magnetic field were observed in bothmeasurements[2] and simulations.[3] However, in theelectron diffusion region, where the frozen-in condi-tion is broken, it still remains a mystery. There isno physical model from the first principles to treatthis electron diffusion region. From the observationalpoint of view, it is difficult to locate the region dueto limitations in the time and space resolutions. Inorder to improve time resolution in the measurementsof the electric and magnetic fields, in late January2004 Mozer et al.[4] modified the telemetry format forthe Polar satellite and found that during the magne-topause crossing there was a new electric field struc-ture of large amplitude and extremely fast fluctuationon the magnetospheric side. These electrostatic fieldsare predominantly perpendicular to the magnetic fieldand usually located inside the local minima of theplasma density. The spatial scale of the structure isof the order of the local Debye length and is believedto be a common feature of the dayside magnetopause.

In two-dimensional particle-in-cell (PIC) simula-tions of collisionless magnetic reconnection with aninitial guide field, Pritchett[5] found that large elec-trostatic field structures perpendicular to the currentsheet appeared on both sides of the current sheet sur-rounding the island. The half-width of this structureis about 3.2λde, where λde is the local electron Debyelength, which is further confirmed by Fu et al.[6] Re-cently, Huang et al.[7] assumed that the initial profile

of the plasma density is asymmetric across the currentsheet in the PIC simulations of magnetic reconnec-tion and found that a strong narrow electrostatic fieldstructure near the separatrice can appear only on thelow/hot plasma density (magnetosphere) side, whichis in good agreement with the observation results. Abimodal structure of electron distribution in the elec-tron diffusion region is also found in the Hall MHDsimulation of the magnetic reconnection from the sym-metric initial condition.[8] The magnetic reconnectionrates under the initially asymmetric conditions werestudied using MHD and Hybrid simulations.[9,10]

In this Letter, we study the structure of the elec-tric field and electron distribution using Hall MHDsimulations of the magnetic reconnection. The plasmadensity or the magnetic field or both are assumed to beinitially asymmetric across the Harris current sheet. Itis found that the narrow layers of the strong electricfield and the net charge exist only on the low plasmadensity or the strong magnetic field (the magneto-spheric) side along the separatrice, which is in goodagreement with the observation results.[4,11]

The Hall MHD equations can be found in the pa-per of Ma.[12] The variables in the equations are nondi-mensionalized by Bs, a, τA = a/vAs, ψ0 = aBs, whereτA is the Alfven time and vAs = Bs/(µ0ρs)1/2 is theAlfven speed. Note that the pressure p is the electronpressure because ions are assumed to be cold.

The initial equilibrium plasma density and mag-netic field are given by

ρ(z) = (ρs + ρm)/2 + (ρs − ρm) × tanh(z/a)/2,

Bx(z) = (Bs − Bm)/2 + (Bs + Bm) × tanh(z/a)/2,

By =0, Bz = 0, (1)

∗Supported by the National Natural Science Foundation of China under Grant Nos 40536030 and 40474058.∗∗Email: zwma@zju.edu.cnc© 2008 Chinese Physical Society and IOP Publishing Ltd

No. 8 MA Zhi-Wei et al. 2935

where ρm, ρs, Bm, and Bs are the plasma densitiesand magnetic fields at magnetosphere (hot and lowdensity) and the magnetosheath (cold and high den-sity) side of the current sheet, respectively. With theassumption of the force balance, we have the plasmapressure

pn = pBs(1 + βs) − pB , (2)

where pB = B2/2µ0 is the magnetic pressure, andβs = pns/pBs is the plasma beta in the magne-tosheath. When the pressure and density are given,the temperature is automatically adjusted to satisfyp = nT . In our simulation, periodic boundary con-ditions are applied to the two boundaries in the xdirection, and free boundary conditions are applied inthe z direction.

The Hall MHD simulations reported here are car-ried out in the whole simulation domain: −Lx/2 ≤x ≤ Lx/2, −Lz/2 ≤ x ≤ Lz/2. The Hall MHDequations are solved by a fourth-order Runge–Kuttascheme in time and space. The uniform mesh is usedin the x direction, but in the z direction we employa dynamic nonuniform mesh which means that thegrid size will decrease when the scale length of a cur-rent sheet is reduced. In the present study, the high-est resolution reaches dzmin = 4 × 10−3 in the latestage of the simulation. The other parameters cho-sen are di = 0.1, η = 10−3, γ = 5/3, and Lx = 64,Lz = 32. Note that the reconnection process is mainlycontrolled by the Hall effects because di = 0.1 is muchlarger than ∆ ∼ η1/2 = 0.032.

Table 1. Parameters (normalized) for all conducted cases.

di ρm ρs Bm Bs

Case 1 0.1 2/11 1 1 1Case 2 0.1 3/11 1 1 1Case 3 0.1 4/11 1 1 1Case 4 0.1 1 1 2 1Case 5 0.1 1 1 3 1Case 6 0.1 1 1 4 1Case 7 0.1 2/11 1 2 1Case 8 0.1 1 1 1 1

We considered totally eight cases in three cat-egories: (1) plasma density asymmetries and mag-netic field symmetry; (2) plasma density symmetryand magnetic field asymmetries; and (3) asymmetryin both plasma density and magnetic field, as listedin Table 1. At the dayside magnetopause, typical val-ues of the parameters are a = 106 m, di = 105 m,Bs = 40nT, n = 15 × 106/m3.

The magnetic reconnection process is triggered bya small initial perturbation in the magnetic flux:

ψ = −ψ0 cos(2πx/Lx) cos(πz/Lz), (3)

where ψ0 = 0.25. This small initial perturbation leadsto formation of the X-point at the centre of the sim-ulation domain x = 0, z = 0.

Next, we study the asymmetries in plasma den-sity and symmetry in the magnetic field. Figure 1

shows the z component Ez of the electric field, thenet charge density, and their profiles at x = 26 att = 100τA for ρm/ρs = 2/11 (case 1). The time ischosen because at this time the magnitudes of theHall electric field and the charge density are nearly attheir maximum values and remain nearly unchangedfor a quite long period. The black solid lines in Fig-ures 1(a) and 1(b) represent the magnetic field lines.One can clearly see that a large perpendicular electricfield Ez with small spatial scale is built up in the ini-tially low-density region at the edge of the magneticisland. Both the electric field and charge density arenearly four times larger in the low density (magneto-sphere) side than the higher density (magnetosheath)side. The magnitude of this Hall electric field reachesa peak value of about Ez ≈ 1mV/m, which is muchsmaller than Ez ≈ 60mV/m observed by the Polarsatellite.[2] The x component of the electric field ismuch smaller than its z component. If this electricfield is assumed to be electrostatic and is mainly as-sociated with the charge separation, the net chargedistribution can be obtained from the Coulomb lawρ = −ε0∇ · E. The resulting distribution of the netcharge is shown in Fig. 1(b). It is found that the mag-nitude of the net charge can be in order of 10−19 C/m3,which is much larger than that obtained by Huang etal.[8] However, the value of the charge density is alsoone order smaller than that observed.

Fig. 1. Contour plots of (a) Hall electric field and (b) netcharge density and the profiles of (c) Hall electric field and(d) net charge density along the line x = 26 at t = 100 forthe case 1.

As is well known, the inclusion of the Hall term in

2936 MA Zhi-Wei et al. Vol. 25

the generalized Ohm’s law mainly leads to decouplingof the electron and ion motion. The Hall effects playan important role near the reconnection region andalong the separatrix, where the spatial scales are usu-ally smaller than the local ion inertial length. That is,in these regions, ions are unmagnetized while electronsare still magnetized. Therefore, during the process ofthe magnetic reconnection, ions are left behind whenelectrons move across the separatrix, which leads tothe formation of a positive charge layer in the inflowregion and a negative charge layer in the outflow re-gion near the separatrix. The Hall term (J × B)z/ρindicates that decoupling of the ion and electron mo-tions is severer in the low density region than that inthe high density region. The separation of the ionsand elections forms a narrow net positive and nega-tive charge layer at the edge of the magnetic islandand generate a spiked electric field, which is clearlyshown in Figs. 1(c) and 1(d).

Fig. 2. Dependence of the maximum Hall electric fieldand charge density on ratios of the plasma density acrossthe current sheet.

Figure 2 shows the dependence of the magnitudesof the Hall electric fields and the charge densitieson the ratios of the plasma densities in the magne-tosheath and magnetosphere. The left (right) labelis for the magnitude of the net charge density (theHall electric field). The results indicate that both themagnitudes of the Hall electric field and charge den-sity increase roughly linearly with the ratios of the(asymmetric) plasma densities cross the initial cur-rent sheet. The slope between the charge density (theelectric field) and the ratio of the plasma densities is0.449 (0.171), which means that the generated Hallelectric field and net charge density are weakly de-pendent on the asymmetry of the plasma density. Inorder to reach the observed magnitude of the elec-tric field, we should have the difference between theplasma densities on the two sides of the dayside mag-netopause should be about a thousand times if themagnetic field is symmetric. This suggests that fromthe Hall MHD model, it is impossible to achieve theobserved electric field of about 60mV/m if only theplasma density asymmetry is considered.

Fig. 3. Contour plots of (a) Hall electric field and (b) netcharge density and the profiles of (c) Hall electric field and(d) net charge density along the line x = 26 at t = 100 forthe case 4.

Fig. 4. Dependence of the maximum Hall electric fieldand charge density on ratios of the magnetic field squareacross the current sheet.

Further, we study the symmetry in the plasmadensity and asymmetries in the magnetic field. Fig-ure 3 shows the z component Ez of the electric field,the net charge density, and their profiles at x = 26 att = 100τA for Bm/Bs = 2/1 (case 4). It is evident thatthe Hall electric field and charge density with smallspatial scale are larger in the higher magnetic field(magnetosphere) side than that in the low-magneticfield side, with a difference of roughly four times. Thegenerated electric field and charge density are nearlyof the same order as in case 1 of the density asym-metry. We also carried out simulation for different

No. 8 MA Zhi-Wei et al. 2937

asymmetries of the magnetic fields across the initialcurrent sheet. Figure 4 shows the dependences of theHall electric fields and charge densities on the ratiosof the square of the magnetic field between the mag-netosphere and the magnetosheath. The linear depen-dence of the electric field (or charge density) and theasymmetry ratio of the square of the magnetic fieldsbetween the two sides is easy to understand from theHall term (J ×B)z/ρ ∼ B2

x/ρδz, where δz is the per-pendicular spatial scale of the magnetic field. Theslope between the charge density (the electric field)and the ratio of the magnetic field asymmetry is 2.85(0.718), which means that the generated Hall elec-tric field and net charge density are much stronglydependent on the asymmetries of the magnetic fieldas compared with the dependency of plasma densityasymmetries. However, we still need a larger asymme-try of the magnetic field, or a difference of an order,across the magnetopause in order to obtain the ob-served magnitudes of the charge density and electricfield. That is, the Hall electric field and charge densityis too small to explain the observed electric field if weuse the typical values of the magnetic field asymmetryat the dayside magnetopause.

Fig. 5. Contour plots of (a) Hall electric field and (b) netcharge density and the profiles of (c) Hall electric field and(d) net charge density along the line x = 26 at t = 100 forthe case 7.

Finally, we investigate the asymmetry in both the

plasma density and the magnetic field. Figure 5 showsthe result for the case (case 7) with both the magneticfield and plasma density asymmetric. In this case,both the Hall electric field and net charge density inthe low density/high magnetic field (magnetosphere)side are almost an order larger than that in the highdensity/low magnetic field (magnetosheath) side. Al-though we obtain a rather larger Hall electric fieldand charge density, the electric field and charge den-sity are still not large enough to be comparable withthe observed ones.

In summary, large Hall electric field and chargedensity with small perpendicular spatial scales are ob-served in our Hall MHD simulation. The electric fieldand charge density in the low density/high magneticfield (magnetosphere) side are much larger than inthe high density/low magnetic field (magnetosheath)side, which is in good agreement with the satelliteobservations.[2,11] However, their magnitudes are stilltoo small for explaining the observations. From ourHall MHD simulations, it is found that the only possi-bility for achieving the observed magnitudes of theelectric field is to reduce the perpendicular spatialscale of the charge density if the typical observed val-ues of the magnetic fields and plasma densities in themagnetosphere and magnetosheath are used in theHall MHD simulations. We shall address the prob-lem of how to realize the reduction of the spatial scalein a future work. It may be of interest to note thatfrom particle-in cell simulations, Huang et al.[7] foundthat the obtained electric field in the low density sidecan be larger that 100mV/m with reasonable ratio ofthe plasma density across the current sheet, which in-dicates that electron kinetic effects missed in the HallMHD model may play crucial roles in the magneticreconnection.

References

[1] Birn J et al 2001 J. Geophys. Res. 106 3715[2] Mozer F S et al 2002 Phys. Rev. Lett. 89 015002[3] Ma Z W and Bhattacharjee A 2001 J. Geophys. Res. 106

3773[4] Mozer F S et al 2004 Geophys. Res. Lett. 31 L15802[5] Pritchett P L 2005 Phys. Plasmas 12 062301[6] Fu X R, Lu Q M and Wang S 2006 Phys. Plasmas 13

012309[7] Huang J, Ma Z W and Li D 2008 Geophys. Res. Lett. 35

in press.[8] Huang F C, Sheng Z M, and Ma Z W 2007 Phys. Plasmas

14 062901.[9] Borovsky J and Hesse M 2007 Phys. Plasmas 14 102309

[10] Nakamura M and Scholer M 2000 J. Geophys. Res. 10523179.

[11] Mozer F S et al 2003 Phys. Rev. Lett. 91 245002.

[12] Ma Z W 2008 Phys. Plasmas 15 032906.