Reconnection onset in the magnetotail: Particle simulations with open boundary conditions



[1] The mechanism of the onset of magnetic reconnection in collisionless plasmas in the tails of planetary magnetospheres and similar processes 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 conditions. In this Letter we report on simulations of reconnection in the magnetotail that incorporate open boundaries. In a simulation setup with the initial geometry similar to that of the GEM Reconnection Challenge bursts of spontaneous reconnection are detected in outflow regions that mimic magnetotails. They strongly resemble the ion tearing instability predicted by Schindler [1974] as a mechanism of magnetospheric substorms. Quenching the onset by replacing open boundary conditions for particles with their reintroduction reveals the key role of passing particles in the tearing destabilization.

1. Introduction

[2] One of the most universal energy conversion processes in space plasmas, magnetic reconnection remains insufficiently understood, especially when plasmas are collisionless as is the case 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 antiparallel 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.

[3] Explosive processes in plasmas are likely caused by instabilities. The tearing instability responsible for 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 instability. Yet, electrons in the tail 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 instability.

[4] The onset problem became more complicated after the discovery that magnetized electrons 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]. However, 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 should 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 tail equilibria there is always a significant population of electrons that are not trapped inside the sheet.

[5] 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 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 references therein] that artificially trap particles. At the same time, simulations with open boundaries 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 to the outflow regions that may be good models of the magnetotail.

2. Basic Simulation Setup

[6] The explicit particle-in-cell 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 Poisson's equation, which makes it especially attractive for implementing non-periodic and, in particular, open boundary conditions.

[7] 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 a GEM-type perturbation 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 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 normalized 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 Ωi−1, based on the unperturbed magnetic field B0 outside the sheet. The initial plasma density is given by n(z) = n0 cosh−2(z/λ) + nb, where nb = 0.2 n0 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(2 πx/Lx)cos(πz/Lz), where Lx = Lz = 19.2 di are the box dimensions. The magnitude of the initial perturbation is taken to be relatively large ψ0 = 0.3 B0di to provide a Bz component of the magnetic field in the outflow regions that is strong enough to magnetize electrons.

3. Open Boundary Conditions

[8] 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

display math

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.

[9] 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 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 tested 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.

[10] At the z-boundaries conducting boundary conditions are retained and particles there 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

[11] 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 = 0.387 di = 0.5 ρ0i, 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/equation image taken to be c/vA = 15. The basic simulation grid had 768 × 768 cells with 60 particles per cell on average. The time step was δt = 0.0025 Ωi−1. As can be seen in Figures 1 and 2a, 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 ΩiΔt ∼ 1–3. On these time scales ions are unmagnetized near the neutral plane because (Bz/B0iΔt ≪ 1. At the same time, this Bz field is strong enough to magnetize electrons, because (Bz/B0iΔt(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. In Figure 2b we show the evolution along the line z = 0 of the perturbation to the magnetic field δBz due to the instability. A comparison of the logarithmic slopes of δBz and the electric field Ey (Figure 2c, solid line) for the ion tearing mode with the similar parameter based on the electric field Ey in an electron tearing run (Figure 2d, note different abscissa) shows that the ion tearing instability indeed develops at least two times faster.

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 corresponds to the maximum current density; numbers in the color bar are omitted because each plot is separately normalized in order to show the full range of current variations.

Figure 2.

Run 1: (a) Normal magnetic field Bz (x) at the neutral plane z = 0 at times Ωit = 5 (blue line), 6 (red), 7 (green), and 8 (black). (b) Plot of δBz, the difference between the maximum and the minimum values of Bz to the left of the primary neutral line versus time. (c) Time history of the maximum absolute value of the out-of-plane electric field Ey near the plasmoid region (solid line) and the primary neutral line (dashed line), with the maxima taken over the boxes marked by the white frames in Figure 1. To reduce noise, the original field Ey is averaged over a time interval Ωi−1. (d) Similar Ey time history for the electron tearing instability simulated in Run 2 (ψ0 = 0, periodic boundary conditions along the x-direction; max∣Ey∣ is taken over the box ∣x/di∣ < 9.6 and ∣z/di∣ < 0.5).

[12] Yet because of the relatively large amplitude ψ0 of the initial perturbation in Run 1, there remains a possibility that the plasmoid formation 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 (Figure 2c, solid line) that is noticeably shorter than that of the primary reconnection (Figure 2c, 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.

[13] 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 differed from Run 1 only in 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 for the reconnection onset is the effect of passing electrons.

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.

[14] 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). This result may have important implications for present and future multiprobe missions aimed at the investigation of bursty reconnection in the tail.

Figure 4.

Run 1: Out-of-plane magnetic field By/B0 at Ωit = 8. Global pattern reflects the quadrupolar structure typical for collisionless reconnection in an antiparallel magnetic field geometry. White frame highlights a different out-of-plane magnetic field pattern, arising due to the reconnection onset in one of the tail-like regions.

[15] 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

[16] 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 number 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 must exceed several wavelengths of the tearing mode. This prediction is consistent with recent Geotail observations on plasmoid statistics. As shown by Ieda et al. [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.


[17] The authors acknowledge useful discussions with P. Pritchett, W. Daughton, D. Swift, K. Schindler, and V. Semenov. This work was supported by NASA grants NAG513047 and NNG06G196G and NSF/DOE grant ATM0317253. 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.