Magnetic Reconnection by a Self-Retreating X-Line
M. Oka, M. Fujimoto, T. K. M. Nakamura, I. Shinohara, K.-I. Nishikawa
aa r X i v : . [ a s t r o - ph ] O c t Magnetic Reconnection by a Self-Retreating X-Line
M. Oka, ∗ M. Fujimoto, T. K. M. Nakamura, I. Shinohara, and K.-I. Nishikawa Center for Space Plasma and Aeronomic Research,University of Alabama in Huntsville, Alabama, 35805, USA Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Sagamihara, Japan (Dated: December 7, 2018)Particle-in-cell (PIC) simulations of collisionless magnetic reconnection are performed to studyasymmetric reconnection in which an outflow is blocked by a hard wall while leaving sufficientlylarge room for the outflow of the opposite direction. This condition leads to a slow, roughly con-stant motion of the diffusion region away from the wall, the so-called ‘X-line retreat’. The typicalretreat speed is ∼ PACS numbers: 52.35.Vd, 52.65.Rr, 94.30.cp, 96.60.qe
Magnetic reconnection triggers many explosive phe-nomena in laboratory and astrophysical plasma. Impor-tance lies in the diffusion region where the MHD breaksdown and the kinetic scale comes in. It is a scientific chal-lenge to understand the structure of the diffusion regionand its role in controlling the reconnection rate. Owingto the growing power of computer capabilities, full parti-cle simulations has become popular these days. It is nowclear that the out-of-plane current remains localized andthe reconnection rate remains fast while electrons form ahigh-velocity jet that extends large distances downstreamfrom the X-line [1, 2, 3, 4]. The region with the local-ized current is termed ‘inner diffusion region’ whereas therest of the diffusion region including the elongated elec-tron current layer is termed ‘outer diffusion region’. Ingeneral, simulations generate magnetic reconnection thatis symmetric in both inflow and outflow directions.Actual magnetic reconnection, however, occurs in amore complicated situation. During a magnetic recon-nection in the Earth’s magneto-tail, an earthward out-flow soon collides with the dipole field while the tailwardoutflow is directed to the interplanetary space. As forthe solar flares that take place above the area of emerg-ing fluxrope, the upward outflow eventually merge intothe solar wind while the downward outflow collides withthe magnetic obstacle in the lower part of the corona. Inthese cases, magnetic reconnection may not be symmet-ric about the X-line.In this letter, we investigate the consequences of block-ing one side of the outflow. A hard wall is set up justahead of an outflow while leaving a sufficiently largeroom for the outflow of the opposite direction. As a re-sult, reconnected magnetic fields are piled-up against thewall. By the time the pile-up region reaches the diffu-sion region, the X-line starts to move away from the wall(‘X-line retreat’). Under such a strong influence of theboundary condition, the structure of the diffusion regionis modulated. In the past, the X-line motion was consid-ered by an analytical treatment [5], MHD simulations [6], and a full particle simulation [7], although these work didnot shed light on the modulated structure of the diffusionregion. This paper will be the first to report the struc-ture of the diffusion region in a self-consistent simulationof asymmetric outflow reconnection.We utilized the two dimensional, particle-in-cell code[8, 9]. The initial condition consists of a Harris cur-rent sheet. The anti-parallel magnetic field and thedensity are given by B y = B tanh(( x − L x / /D ) and N cs = N / cosh (( x − L x / /D ), respectively, where B is the magnetic field at the inflow boundary, D isthe half-thickness of the current sheet, N is the den-sity at the current sheet center and L x and L y are thedomain size in ˆx and ˆy direction, respectively. D ischosen to be 1.5 λ i where λ i is the ion inertial length.The inflow, background plasma is represented by N B = N B (1 − / cosh (( x − L x / /D )) where N B =0.2 N . Thepressure imbalance by the non-uniform density is not im-portant to the results presented below. The electron toion temperature ratio is set to be T e /T i =1/5 for both thecurrent sheet and the background. The frequency ratio ω pe / Ω ce =1.5 where ω pe and Ω ce are the electron plasmafrequency and the electron cyclotron frequency, respec-tively. The conducting walls are used at x =0 and x = L x and symmetric boundary conditions were used at y =0and y = L y . At the symmetric boundary, particles arespecularly reflected whereas fields are given by ∂B n /∂n = 0, B t = 0, ∂ E t /∂n = 0, and E n = 0, where E and B are the electric field and the magnetic field vectorsand the subscripts n and t denote the normal and tan-gential components, respectively. Reconnection is ini-tiated at the distance H from the bottom ( y =0) wallwith a small magnetic island given by the vector poten-tial A z = − A exp[ −{ ( x − L x / + ( y − H ) } / (2 λ i )]where A = 0 . B λ i . L y is chosen to be large enoughto simulate effectively the free boundary condition of thereality.Five simulation runs are performed to study the depen-dence on the initial distance H . The values of H along TABLE I: Initial parameters and results of each run. Lengthsare normalized by the ion inertial length λ i , times are by theinverse ion cyclotron frequency Ω − ci and speeds are by the ionAlfv´en speed v A . ∆ is the grid size. See text for details. L x L y H µ λ i / ∆ t col t ret v p Run 1 76.8 204.8 12.8 25 20 80 83 0.3Run 2 76.8 204.8 25.6 25 20 90 98 0.5Run 3 102.4 204.8 51.2 25 20 110 140 1.0Run 4 102.4 204.8 102.4 25 20 120 - -Run 5 170.7 341.3 25.6 9 12 90 98 0.5 with the other initial values are compiled in Table I. Theinitial X-line is at the center of the domain for Run 4,and thus, it is a non-retreating case. Run 5 is an addi-tional case intended to study the dependence on the ionto electron mass ratio µ ≡ m i /m e . In all runs, we usedaverage of 64 particles in each grid cell. 276 particles percell represents the unit density. electron stagnation point ion stagnation point X-line 6040200 X -3-2-10123 X Y X W ci t = 75 W ci t = 115 W ci t = 225 J ze (a) (b) (c)(d) i on f l o w s peed no r m a li z ed b y V A ci t 76543210-1-2-3-4 e l e c t r on f l o w s peed no r m a li z ed b y V A upward ion flow maxdownward ion flow x-line upward ele flow maxdownward ele flow max X-point retreat speed at the X-line ion flow max ele flowat the (e)
FIG. 1: Overview of the retreating X-line for Run 1: (a-c)normal electron current J ez . (d) positions of the X-line andthe stagnation points. (e) the X-line retreat speed (blackcurve), upward and downward maximum flow speeds for ions(blue solid curves) and electrons (red solid curves), and theflow speeds at the X-line for ions (blue dashed curve) andelectrons (red dashed curve). A stagnation point is definedby the flow reversal point. Upper panels of Figure 1 show snapshots of electronout-of-plane current for Run 1. It is evident that as mag-netic reconnection proceeds, a downward flow is blockedby the bottom wall and creates the pile-up region whichexpands in time and eventually pushes the diffusion re-gion upward. The upward flow, on the other hand, con-tinues to grow in length until its leading edge reachesthe upper wall. The choice of L y is not sensitive to theresults presented in this paper and thus any effect fromthe upper wall is not important. The diffusion region isalso elongated and the final length of the electron currentsheet becomes as long as ∼ λ i .The black curve in Figure 1d shows the X-line posi-tion defined by the B x reversal point, B x =0. A distinctretreat process starts at Ω ci t ∼
85. The X-line movesaway from the wall monotonically. Although a jump ap-pears at Ω ci t ∼
145 due to a magnetic island generatedby a secondary tearing instability, the X-line motion con-tinues even after the upward outflow reaches the upperwall. The approximate time of the outflow collision withthe wall is Ω ci t ∼ ∼ v A at Ω ci t ∼
80 when it is blocked by thebottom wall (marked by the blue arrow). The downwardflow speed soon decreases and is followed by the upwardmotion of the X-line. In contrast, the retreat speed andthe ion flow speed at the X-line are roughly constant atthe same value ∼ v A , although disturbances appear atΩ ci t ∼
145 due to a secondary tearing.A striking feature can be found around the diffusionregion. Figure 2 shows an enlarged view of the diffusionregion for Run 1 at Ω ci t ∼
115 when the upward electronspeed is saturated. In this figure, data are accumulatedover one gyro period. Coordinates are shifted so thatthe X-line comes to the origin. This simulation, alongwith others with different H , reveals that the ion andelectron stagnation points are not collocated with theX-line. As the pile-up region expands, the lower halfof the outer diffusion region is modulated and ion flowsare deflected. As a result, the ion stagnation point isshifted downward about the distance of λ i from the X-line(marked by the blue horizontal arrow). This separationbetween the stagnation point and the X-line is almostconstant in time as shown in Figure 1d. The shift of theion stagnation point leads to an upward ion flow at theX-line, resulting in a slow, rising motion of the X-line.While there is a high speed ion flow in the upper half ofthe diffusion region (the curve labeled ‘I’ in the bottompanel), the typical value of the upward ion flow at the X-line was ∼ v A (‘II’). The deflection can be recognizedas the double peak feature in the cut of the downwardoutflow (‘III’).In contrast to ions, electrons do not show strong deflec-tion and exhibit well developed outflow jets both upward V iy / V Ai V ey / V Ae B x / B -2 0 2 A B observed E z inertia term Lorentz term pressure term sum of the terms I II III V iy / V Ai V ey / V Ae ou t e r DR i nne r DR FIG. 2: Enlarged view of the retreating X-line for Run 1 obtained at Ω ci t=115. The center panel shows the electron flow in ˆy direction V ey (color code) as well as ion flow directions in the x − y plane (vectors). The left panel shows quantities B x , V ey / √ µ , and V iy along x =0 whereas the right panel shows the electric field terms of the generalized Ohm’s law along x =0.The bottom panel shows V iy cut along the horizontal lines in the color code. The purple and yellow arrows in the right panelshow the outer and inner diffusion regions (DR), respectively. and downward, although the downward jet is limited inlength. The typical length of the downward electron jetis ∼ λ i and remains constant throughout the run. Themaximum value of the electron flow speed, however, isquite fast and reaches 0.65 v Ae , as can be seen in the leftpanel. This is almost the same as the maximum speed ofthe upward electron jet. We have found that all retreat-ing cases show nearly the same maximum speed of thedownward electron jet, i.e. v Ae . The length of the jet,on the other hand, becomes slightly larger for larger H .For Run 3, the length reached ∼ λ i .We further examined the out-of-plane component ofthe generalized Ohm’s law along x =0 as shown in theright panel of Figure 2 [4]. It is expressed as E z = − m e v ey e ∂v ez ∂y + 1 c v ey B x − ne ∇ · Γ (1)where v e is the electron bulk velocity, Γ = p exz ˆx + p eyz ˆy is the flux of z -directed electron momentum in the re- connection plane (not including convection of momen-tum) with p e the electron pressure tensor. The observedelectric field (solid black curve) is balanced by the sum(dashed black curve) of the electron inertia (red), theLorentz force (brown), and the divergence of the momen-tum flux (green). Note that the time variation term isnegligible and is not considered here. As is the case fornon-moving, symmetric reconnection reported previously[4], the major contribution to the inner diffusion comesfrom the pressure term. The upward outer diffusion re-gion is also similar to the symmetric case. On the otherhand, the downward outer diffusion region is highly mod-ified. Steeper acceleration of v ey and piled-up B x makethe Lorentz term profile to be sharper. The inertia termwith the opposite sign is also enhanced because of steepergradient of v ez in the short jet. The striking feature isthat both are enhanced in a balanced way such that E z as a whole shows the almost flat profile along the x -axis.The above finding leads to the idea that the downwardouter diffusion region adjusts itself to buffer the effectsof the wall such that the reconnection electric field is de-termined irrespective of the asymmetric boundary con-dition. This idea is inspected in Figure 3 which showsthe reconnection rate, E R , of all runs. E R is determinedby averaging the electric field over a small square regioncentered at the X-line. We have verified that taking thetime derivative of the total magnetic flux between theX-line and the bottom center of the pile-up region yieldsidentical profile as E R . For Runs 1-3 and 5, the upwardoutflow jet reaches the upper wall well after Ω ci t ∼ ci t ∼ E R does not dependon µ [10]. Relatively large fluctuations are due to sec-ondary islands whose appearance times are indicated bythe horizontal arrows. Also illustrated in Figure 3 are thetimings of the jet collision with the wall as well as theX-line retreat. It clearly shows that the rising motion ofthe diffusion region is associated with the expansion ofthe pile-up region rather than the peak E R .To summarize, the retreat motion of the X-line is con-stant ( ∼ v A ). This speed does not depend on the ini- r e c onne c t i on r a t e r e c onne c t i on r a t e ci t FIG. 3: Dependencies of reconnection rate on the initial dis-tance from the wall (upper panel) and the mass ratio µ (lowerpanel). The horizontal arrows show periods of secondary is-land appearances. The straight vertical arrows show the ap-proximate times of the collision of the downward jet t col whichis defined by the peak in the profiles of the downward ion flowspeed (see, for example, the blue arrow in Figure 1). Thedashed vertical arrows show the start times of the X-line re-treat, t ret . The numbers annotated below each dashed arroware the downward ion flow maximum v p /v A . There is no ver-tical arrow for Run 4 because it is the non-retreating case. tial distance of the X-line from the wall, H , except ofcourse the non-retreating case. The downward outflowregion is largely modulated by the asymmetry. The iondeflection pattern depends on H and so is the length ofthe electron downward jet. The outer diffusion regionis not elongated as in the case of symmetric reconnec-tion. In terms of the reconnection rate, however, it staysthe same as the symmetric case thanks to the internalbalance within the downward outer diffusion region.In addition to the initial distance from the wall, a de-pendence on the mass ratio µ should also be addressed.What we have found here is the tendency for more mag-netic islands to emerge with smaller µ . These islands mayhelp maintain fast reconnection [1]. In fact, as shown inFigure 3, the µ =9 case (Run 5) generated two large is-lands, yet keeping the same reconnection rate as the otherruns. Another important aspect of the µ dependence isthe separation between the ion stagnation point and theX-line, but we could not measure δ xs for the µ =9 casebecause of the successive generation of the secondary is-lands. The mass dependence of the size of the separationwill be studied in our future paper. Note the fact, how-ever, that the retreat speed was roughly equal to the ionflow speed so that the X-line and the stagnation will becollocated in the X-line rest-frame of reference.Finally, it should be mentioned that the X-line speedcan be faster than the self-retreating speed of 0.1 v A if,say, it is blown by another dominant X-line. Recent two-fluid simulations show that its reconnection rate is re-duced and reconnection at the moving X-line is even-tually terminated. Particle simulations for this kind ofsituations are also needed.This work was partially supported by the Grant-in-Aid for Creative Scientific Research (17GS0208) from theMEXT, Japan. MO was supported by the Grant-in-Aidfor JSPS Postdoctoral Fellows for Research Abroad. ∗∗