GEOPHYSICAL RESEARCH LETTERS, VOL. ???, NO. , PAGES 1–12,

Reconnection onset in the magnetotail:

Particle simulations with open boundary

conditions

A. V. Divin,1

M. I. Sitnov,2

M. Swisdak,3

and J. F. Drake2

A. V. Divin, St. Petersburg University, Universitetskaya nab., 7-9, 199034 St. Petersburg,

Russia. (adivin@umd.edu)

J. F. Drake and M. I. Sitnov, IREAP, University of Maryland, College Park, MD 20742, USA.

(sitnov@umd.edu)

M. Swisdak, Naval Research Laboratory, 4555 Overlook Ave SW, Washington, DC 20375

1St. Petersburg University, St. Petersburg,

Russia

2Institute for Research in Electronics and

Applied Physics, University of Maryland,

College Park, Maryland, USA

3Naval Research Laboratory, Washington,

DC, USA

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2 DIVIN ET AL.: RECONNECTION ONSET SIMULATIONS

Abstract. The mechanism of the onset of magnetic reconnection in col-

lisionless plasmas in the tails of planetary magnetospheres and similar pro-

cesses in the solar corona is one of the most fundamental and yet not fully

solved problems of space plasma physics. Modeling the onset with particle

codes requires either extremely large simulation boxes or open boundary con-

ditions. In this Letter we report on simulations of reconnection onset in the

magnetotail that incorporate open boundaries. It is shown that in a simu-

lation setup with the initial geometry similar to that of the GEM Reconnec-

tion Challenge the onset of reconnection is observed in the outflow regions

that mimic the magnetotails. The onset process strongly resembles the ion

tearing instability predicted by Schindler [1974]. Quenching the onset by re-

placing open boundary conditions for particles with their reintroduction re-

veals the key role of passing particles in the onset mechanism.

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1. Introduction

One of the most universal energy conversion processes in space plasmas, magnetic re-

connection remains insufficiently understood, especially when plasmas are collisionless as

occurs in planetary magnetospheres and the solar corona. Particularly puzzling is the

onset of reconnection, which occurs as a rule in geometries different from the simplest an-

tiparallel magnetic field configuration. An example of such geometry is the tail of Earth’s

magnetosphere, where the reconnection explosions are believed to trigger substorms and

bursty bulk flows [Slavin et al., 2003]. The tail magnetic field lines are strongly bent

rather than antiparallel.

Explosive processes in plasmas are likely caused by instabilities. The tearing instability

responsible for the reconnection onset in the magnetotail was first considered by Coppi et

al. [1966] for the simplest case of antiparallel magnetic fields. Fed by the electron Landau

dissipation near the neutral plane, where the magnetic field turns to zero, it is known as

the electron tearing mode. Yet, electrons in the magnetotail are typically magnetized and

their dissipation is therefore negligible. However, as noted by Schindler [1974], a similar

dissipation due to ions, which can be unmagnetized near the neutral plane where the

magnetic field is minimal, may trigger an even faster instability known as the ion tearing

mode.

The onset problem became more complicated after the discovery that magnetized elec-

trons trapped inside the tail plasma sheet could also change the sign of the tearing mode

energy, making any dissipation useless for destabilization [Galeev and Zelenyi, 1976]. How-

ever, a real crisis appeared in the theory when Lembege and Pellat [1982] and then Pellat

et al. [1991] showed that because of the trapped electrons the ion tearing mode should

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be universally stable, and even the much slower electron tearing would require extremely

small normal magnetic fields to demagnetize electrons and remove their stabilization effect.

Only recently it has been found [Sitnov et al., 2002] that the fast ion tearing instability

can still be possible, because in isotropic self-consistent tail equilibria there is always a

significant population of electrons that are not trapped inside the sheet.

The linear theory of the ion tearing destabilization taking into account passing electrons

is rather cumbersome and difficult to independently verify. Moreover, it does not preclude

the nonlinear stabilization of the ion tearing mode before it reaches an amplitude sufficient

to change the initial tail topology of magnetic field lines and form the X-lines. This is why

the destabilization mechanism must be verified by particle simulations. The main obstacle

is that simulations of magnetic reconnection are usually performed using a combination

of periodic and conducting boundary conditions [e. g. Birn et al., 2001 and refs. therein]

that make all particles artificially trapped. At the same time, simulations with open

boundaries [Pritchett, 2001; Daughton et al., 2006] reveal interesting new effects, such

as the significant stretching of the electron dissipation region [Daughton et al., 2006].

Below we show and discuss the results of simulations with the code P3D [Zeiler et al.,

2002], which has been modified to investigate various types of open boundaries. Except

for these new boundary conditions, our simulation setup is similar to that of the GEM

Reconnection Challenge [Birn et al., 2001]. However, in contrast to the latter group of

works, we shift the focus of the study from the X-line vicinity with its electron dissipation

region to the outflow regions that may be good models of the magnetotail.

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2. Basic Simulation Setup

The explicit code P3D [Zeiler et al., 2002], which is used in our 2D simulations, retains

the full dynamics of both ions and electrons. It is massively parallelized using MPI routines

with 3D domain decomposition. An important distinctive feature of the code is the use of

the multigrid technique [Press et al., 1999] instead of the fast Fourier transform to solve

the Poisson’s equation, which makes it especially attractive for implementing non-periodic

and, in particular, open boundary conditions.

The initial state in our simulations is the Harris current sheet [Harris, 1962] with a

background plasma, which is driven out of the equilibrium by the GEM-type perturba-

tion of the magnetic field [Birn et al., 2001] to form the initial X-line, which mimics

the distant neutral line in the magnetotail. The perturbation also ignites the global

reconnection in the system, similar to the process, which drives the steady nightside

convection in the magnetosphere. The magnetic field and plasma density are normal-

ized by the maximum values of these parameters in the equilibrium, while the space and

time scales are normalized by the ion inertial length di = c/ωpi, based on the maximum

equilibrium plasma density n0, and the inverse ion gyrofrequency Ω−1i , based on the un-

perturbed magnetic field B0 outside the sheet. The initial plasma density is given by

n(z) = n0 cosh−2(z/λ) + nb, where nb = 0.2n0 is the background density. We use here the

system of reference with the X-axis directed to the right, Z-axis directed upward, and the

unit vector in Y-direction ey = ez × ex. The magnetic field is determined by the Harris

field Bx = B0 tanh(z/λ) with the perturbation specified by the magnetic flux function

ψ(x, z) = ψ0 cos(2πx/Lx) cos(πz/Lz), where Lx = Lz = 19.2di are the box dimensions.

The magnitude of the initial perturbation is taken to be relatively large ψ0 = 0.3B0di to

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provide a Bz component of the magnetic field in the outlow regions that is strong enough

to magnetize electrons.

3. Open Boundary Conditions

To model passing particles in the outflow regions of the primary X-line pattern we

impose the conditions of continuity across the x-boundary on the first two moments of

the distribution functions

∂n(α)

∂x= 0,

∂V(α)

∂x= 0, α = e, i (1)

where n(α) and V(α) are the density and bulk velocity of the species α. Particles that

cross the x-boundaries are excluded from the simulations and new particles are injected

into the system with shifted Maxwellian distributions obeying (1) and having the original

temperatures Tα = Tα(t = 0). These conditions combine earlier open setups of Pritchett

[2001], who injected the initial Maxwellian distributions, and Daughton et al. [2006], who

required, in addition to (1), the continuity of the pressure tensor components. Our choice

is based on the results of the linear tearing stability analysis [Sitnov et al., 2002], where

passing electrons adiabatically respond to the changes of the electrostatic and vector

potentials that control density and bulk flow velocity.

The problem of finding a realistic and numerically stable set of open boundary conditions

on the electric and magnetic fields is far from a final solution, especially for collisionless

plasmas. The ideal set should prevent charging of the simulation box and allow free

escape of the flux and all types of waves from the system. After a series of test runs

we have found the following set to yield the most interesting and numerically stable

results with no artificial wave accumulation: ∂Ex,y/∂x = 0, Ez = 0, ∂Bx,y/∂x = 0, and

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Bz = 0. A condition similar to the latter was used earlier by Pritchett [2001] to provide

free propagation of the flux through the boundary. We also tried to replace it by the

so-called radiation condition (∂/∂x± c−1∂/∂t)Bz = 0 used by Daughton et al. [2006] and

found, consistent with the latter work, that significant current sheet stretching occurred.

However, further discussion of that very interesting effect goes beyond the scope of the

present brief communication and will be discussed in more detail elsewhere.

At the z-boundaries the set of conducting boundary conditions is retained and particles

there (they largely belong to the background population) are specularly reflected. Note,

that the use of these conditions (also employed by Pritchett [2001]) does not preclude the

openness of outflow regions, the main objects of our study, because they are located far

from the z-boundaries. Also, at the time scales considered in our work it does not result

in any noticeable loss in the total flux or a reduction in the total plasma and magnetic

pressure.

4. Results

The formation of plasmoids in the outflow regions of the primary X-line reconnection

pattern was first detected in a run with the parameters λ = 0.387di = 0.5ρ0i, where ρ0i

is the thermal ion gyroradius in the field B0, mi/me = 64, and Ti/Te = 3/2, where mα

and Tα are the mass and temperature for the species α. Another parameter, common for

all other runs, was the ratio between the speed of light c and the effective Alfven speed

vA = B0/√

4πn0mi taken to be c/vA = 15. As is seen from Figures 1 and 2 (top panel),

the plasmoid (we focus our attention on the biggest one on the left side) appears in the

region of the substantial normal magnetic field Bz = (0.05 − 0.1)B0 in a few gyrotimes

Ωi∆t ∼ 1−3. On these time scales ions are unmagnetized near the neutral plane because

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(Bz/B0)Ωi∆t≪ 1. At the same time, this Bz field is strong enough to magnetize electrons,

because (Bz/B0)Ωi∆t(mi/me) ≫ 1. The reconnection instability developing under such

conditions, the ion tearing mode [Schindler, 1974] is known to develop much faster than

the electron tearing with the growth rate ratio γi/γe = (miTi/meTe)1/4 ≈ 3. To check this

theoretical prediction we performed another run with the same parameters, but without

the initial perturbation, ψ0 = 0, and with periodic boundary conditions along the x-

direction, and compared the evolution of the electric field Ey (Figure 2, bottom panel)

with the similar evolution of the secondary reconnection electric field in Run 1 (Figure 2,

central panel, solid line). The comparison shows that the latter instability indeed develops

at least two times faster than the electron tearing, and it has therefore another signature

of the ion tearing mode.

Yet, taking into account the relatively large amplitude ψ0 of the initial perturbation

in Run 1, there remains a possibility that the formation of plasmoids was driven by the

primary reconnection associated with the central X-line. To clarify this issue an estimation

of the electric field was performed in the two different areas marked by white rectangles

in Figure 1. The comparison of the temporal evolution of the maximum absolute value

of Ey in these two regions shows that the onset of the secondary reconnection, starting

in the left box and then gradually shifting to the left boundary, has a time scale that is

much shorter than that of the primary reconnection (Figure 2, central panel, dashed line).

Moreover, its peak amplitude is several times larger than the primary reconnection electric

field. We conclude that the most plausible interpretation of the plasmoid formation in

Run 1 is a burst of spontaneous reconnection in the form of the ion tearing instability.

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What causes the destabilization? According to the linear theory [Sitnov et al., 2002], the

stabilizing effect of trapped electrons [Lembege and Pellat, 1982], which appears when the

Bz field is strong enough to magnetize electrons, can be eliminated by passing electrons.

One of the reasons for the strong destabilizing role of passing electrons is their significant

relative number in the plasma sheet [Sitnov et al., 2002, Appendix D]. To clarify the role

of passing particles in our simulations we performed another test run, which was different

from Run 1 only by the particle boundary conditions: instead of the reinjection (1),

particles crossing the x-boundaries were re-introduced on the same field line with z → −z

and vx → −vx. Hence all particles in Run 3 were effectively trapped inside the sheet.

As is seen from Figure 3, this single modification of the Run 1 setup drastically changes

the evolution of the system: instead of the secondary reconnection bursts in the tail-like

regions and the subsequent current sheet stretching we observe only the classical GEM-

type signatures of the primary reconnection approaching its quasi-steady regime. Thus,

our simulations confirm the linear stability results in that the most likely mechanism of

the reconnection onset is the effect of passing electrons.

Note however, that on the basis of this simulation one cannot exclude the influence of

other factors on the onset mechanism. Apart from the finite rate of the primary/global

reconnection in the system, these factors may include the finite By component of the

magnetic field associated with that global reconnection process and shown in Figure 4. On

the other hand, Figure 4 shows that the Hall-MHD pattern associated with the formation

of plasmoids in the magnetotail (the region inside the white frame in Figure 4) may

strongly differ from its classical GEM analog arising from the Harris equilibrium with

antiparallel field lines [e. g., Shay et al., 2001; see also the global pattern in Figure 4].

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This result may have important implications for present and future multiprobe missions

aimed at the investigation of bursty reconnection in the tail.

The onset of secondary reconnection detected in Run 1 does not seem to strongly depend

on the mass and temperature ratios as it has also been detected for mi/me = 25 and 128

as well as for Ti/Te = 3. At the same time, it disappears with the doubling of the current

sheet thickness (which then becomes equal to one ion gyroradius).

5. Conclusion

An important new result of full-particle simulations with open boundary conditions

is the onset of the secondary bursty reconnection in the outflow tail-like regions of the

primary X-line reconnection pattern. The onset process strongly resembles spontaneous

reconnection resulting from the ion tearing instability predicted by Schindler [1974] as

a mechanism of magnetospheric substorms. Simulations also confirm the destabilizing

effect of passing particles as a key mechanism of the reconnection onset in the magnetotail

[Sitnov et al., 2002]. Our results suggest that (1) explosive reconnection may be possible

in the tail notwithstanding the stabilizing role of the normal magnetic field; (2) the key

parameters that determine the onset conditions are the current sheet thickness and the

length of the tail, which controls the relative amount of passing particles. Moreover,

simulations complement the linear theory by showing that the tearing amplitude can be

large enough to change the initial tail topology and to form an X-line. On the other

hand, the linear theory justifies the extrapolation of the simulation results to a broader

range of parameters, including the real mass ratio and the magnetotail spatial scales. It

also complements simulations by providing an estimate for the critical length of the tail.

In particular, according to Sitnov et al. [2002], the length of the unstable plasma sheet

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must exceed several wavelengths of the tearing mode. This prediction is consistent with

recent Geotail observations on plasmoid statistics. As shown by Ieda [1998], the minimum

distance at which the plasmoids start to form in the tail is 24 RE , while their average

size at that distance is only 4 RE . It is also consistent with recent results of Nagai et

al. [2005], who showed that the spontaneous (not driven by the immediate solar wind

trigger) onset of reconnection in the tail is only possible beyond 25 RE .

Acknowledgments.

The authors acknowledge useful discussions with P. Pritchett, W. Daughton, D.

Swift, and K. Schindler. This work was supported by NASA grants NAG513047 and

NNG06G196G. Simulations were performed at the National Energy Research Scientific

Computing Center at the Lawrence Berkeley National Laboratory. The work of A. V. D.

was performed during his stay as a J-1 Trainee Visitor at the University of Maryland.

References

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Daughton, W., J. Scudder, and H. Karimabadi (2006), Fully kinetic simulations of un-

driven magnetic reconnection with open boundary conditions, Phys. Plasm., 13, 072101.

Galeev, A. A., and L. M. Zelenyi (1976), Tearing instability in plasma configurations, Sov.

Phys. JETP, 43, 1113.

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Harris, E. G. (1962), On a plasma sheath separating regions of oppositely directed mag-

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Press, W. H. et al. (1999), Numerical Recipes in Fortran 77, Cambridge Univ. Press, New

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Pritchett, P.L. (2001), Collisionless magnetic reconnection in a three-dimensional open

system, J. Geophys. Res., 106 (A11), 25,961.

Schindler, K. (1974), A theory of the substorm mechanism, J. Geophys. Res., 79, 2803.

Sitnov, M.I., et al. (2002), Reconnection onset in the tail of Earth’s magnetosphere, J.

Geophys. Res., 107 (A9), 1256, doi:10.1029/2001JA009148.

Slavin, J. A., et al. (2003), Geotail observations of magnetic flux ropes in the plasma

sheet, J. Geophys., 108 (A1), 1015, doi:10.1029/2002JA009557.

Zeiler, A., D. Biskamp, J. F. Drake et al. (2002), Three-dimensional particle sim-

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DIVIN ET AL.: RECONNECTION ONSET SIMULATIONS 13

Figure 1. Magnetic field lines and color-coded current density Jy for Run 1 with open

boundary conditions at times Ωit = 6, 7, 8, and 9 (black color corresponds to the maximum

current density).

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Figure 2. Run 1: Normal magnetic field Bz(x) at the neutral plane z = 0 (top panel) at times

Ωit = 5 (blue line), 6 (red), 7 (green),and 8 (black), and time history of the maximum absolute

value of out-of-plane electric field Ey (central panel) near the plasmoid region (solid line) and

the primary neutral line (dashed line) calculated for left and central white frames in Figure 1.

To reduce noise, the original field Ey is averaged over a time interval Ω−1i . Bottom panel shows

similar Ey time history for the electron tearing instability simulated in Run 2 (ψ0 = 0, periodic

boundary conditions along the x-direction). Note the different timescales on the bottom two

panels.

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DIVIN ET AL.: RECONNECTION ONSET SIMULATIONS 15

Figure 3. Magnetic field lines and color-coded current density Jy for Run 3 with particle

reintroduction at times Ωit = 6, 7, 8, and 9.

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Figure 4. Run 1: Out-of-plane magnetic field By at Ωit = 8. Global pattern reflects the

quadrupolar structure typical for collisionless reconnection in an antiparallel magnetic field ge-

ometry. White frame highlights a different out-of-plane magnetic field pattern, arising due to

the reconnection onset in one of the tail-like regions.

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