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. ([email protected])
J. F. Drake and M. I. Sitnov, IREAP, University of Maryland, College Park, MD 20742, USA.
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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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.
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Harris, E. G. (1962), On a plasma sheath separating regions of oppositely directed mag-
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system, J. Geophys. Res., 106 (A11), 25,961.
Schindler, K. (1974), A theory of the substorm mechanism, J. Geophys. Res., 79, 2803.
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Geophys. Res., 107 (A9), 1256, doi:10.1029/2001JA009148.
Slavin, J. A., et al. (2003), Geotail observations of magnetic flux ropes in the plasma
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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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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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