We present full-particle simulations of 2-D magnetotail current sheet equilibria with open boundaries and zero driving. The simulations show that spontaneous formation of dipolarization fronts and subsequent formation of magnetic islands are possible in equilibria with an accumulation of magnetic flux at the tailward end of a sufficiently thin current sheet. These results confirm recent findings in the linear stability of the ion tearing mode, including the predicted dependence of the tail current sheet stability on the amount of accumulated magnetic flux expressed in terms of the specific destabilization parameter. The initial phase of reconnection onset associated with the front formation represents a process of slippage of magnetic field lines with frozen-in electrons relative to the ion plasma species. This non-MHD process characterized by different motions of ion and electron species generates a substantial charge separation electric field normal to the front.
Earth's magnetotail is a global collisionless plasma system where energy is gradually stored and then suddenly released due to kinetic processes, the main of which is magnetic reconnection. The tail magnetic field lines are sharply curved rather than anti-parallel, and according to one of the main substorm scenarios [McPherron et al., 1973], magnetic reconnection results in the formation of an X-line, accompanied by the severance of a magnetic island (plasmoid or flux rope) and the reduction of the magnetic field line stretching earthward of the X-line, termed dipolarization. Recent observations have confirmed that reconnection plays a key role in magnetotail activity, including substorms [Angelopoulos et al., 2008; Sergeev et al., 2012]. At the same time, elementary dipolarizations do not fit the standard picture based on the plasmoid formation in a 1-D current sheet with anti-parallel magnetic field lines. They rather represent sharp dipolarization fronts (DFs) at the leading edge of fast earthward plasma flows with a strong and steep increase of the original northward magnetic field component Bz, preceded by a much smaller dip and followed by a shallower decline [Moore et al., 1981; Nakamura et al., 2002; Ohtani et al., 2004]. Thus, they are different from conventional plasmoids and flux ropes, which have bipolar Bz variations of comparable amplitudes. The north-south Bz asymmetry cannot be explained by near-Earth flow braking [Shiokawa et al., 1997] as it is observed throughout the whole tail [Ohtani et al., 2004].
The evolution of the empirical picture of magnetotail reconnection onset went along with progress in its theoretical understanding and numerical modeling. Since electrons are magnetized prior to onset (because of the finite Bz field), the original kinetic theory based on the models of electron [Coppi et al., 1966] and ion [Schindler, 1974] tearing instabilities encountered problems. In particular, Lembege and Pellat  found that a sufficient stability condition of the ion tearing mode coincides with the WKB condition, which allows for the stability analysis of wave-like perturbations of the electromagnetic fields, neglecting variations of the current sheet parameters along the tail. However, recently Sitnov and Schindler  conjectured that the sufficient tearing stability criterion of Lembege and Pellat  is not so universally prohibitive. In particular, it may be relaxed, relative to the WKB condition, for 2-D magnetotail equilibria with an accumulation of magnetic flux at the tailward end of a thin current sheet due to a hump in the Bz field on a mesoscale, intermediate between the current sheet thickness and the main inhomogeneity scale along the tail. Such a hump is clearly seen in the magnetotail prior to sub-storms in statistical visualizations based on Geotail data [Machida et al. 2009]. It reflects a natural process of energy accumulation in the tail prior to its explosive release. Later Sitnov and Swisdak , using a 2-D equilibrium with open boundaries and weak external driving, showed that onset of reconnection indeed occurs in tails with accumulated magnetic flux, consistent with the linear stability theory predictions. The onset in such 2-D equilibria starts from the formation and acceleration of DFs.
In past simulations, DFs appeared as initial transients that could be caused by the external perturbation of a 1-D current sheet [e.g., Sitnov et al., 2009; Wu et al., 2011] or might be a result of external driving [Sitnov and Swisdak, 2011]. However, to understand the instability responsible for spontaneous DF formation in the tail, simulations of 2-D current sheet equilibria without external driving are required. The simulation boundaries along the tail must be open so as not to suppress reconnection-driven dynamics. Such simulations are described in the present paper.
2 Equilibrium, Stability, and Simulation Setup
A distinctive feature of this study is the use of 2-D current sheet equilibria characteristic of the terrestrial magnetotail. 1-D current sheets, including the well-known Harris equilibrium [Harris, 1962], reveal many interesting dynamical effects, such as the formation of secondary plasmoids and interacting flux ropes [Daughton et al., 2009, 2011]. However, many of these effects are blocked in 2-D equilibria by the tearing stabilization [Lembege and Pellat, 1982]. In this study, following Sitnov and Schindler , we consider a subset of the general class of approximate 2-D equilibria [Schindler, 1972] described by the flux function (y component of the vector potential) ψ = LB0ln[β(x)cosh(z/(Lβ(x)))], where β(x) is slowly varying: β(x) = exp(ε1g(ξ)), with ξ = x/L and ε1 ≪ 1. A feature of the subset is the presence of a flux accumulation region at the end of a thin current sheet, which can be modeled by the following choice of the function g: g(ξ) = ξ + (α/ε2)[1 + tanh(ε2(ξ − ξ0))] and corresponds to a hump in the Bz magnetic field component at the neutral plane:
on a scale intermediate between the current sheet thickness L and its length L/ε1: ε1 ≪ ε2 ≪ 1. Thus, the parameter α measures the amount of flux accumulated in the mesoscale region around ξ = ξ0. As was shown by Sitnov and Schindler , the sufficient stability criterion derived by Lembege and Pellat 
where k is the tearing wave number, while Lz and V = ∫ dl/B are the local values of the current sheet half-thickness and the flux-tube volume, differs from the WKB condition kLzB0/(πBz) > 1 by the factor . If the parameter , the criterion (2) is satisfied within the WKB condition and hence the tail is stable. In the opposite case, there is a potential for instability. In particular, for a class of equilibria with the Bz profile (1) and α = 0, the destabilization parameter Cd = 1, which agrees with the result from Lembege and Pellat  for the specific case Bz(x, z = 0) = const considered in their work. However, for α > 1, according to Sitnov and Schindler , one can expect destabilization because Cd(α > 0) > 1. In particular, in the limit α ≫ 1, the destabilization factor can be estimated as Cd ≈ α.
To investigate possible tearing destabilization, we performed simulations using an open-boundary modification of the explicit massively parallel full-particle code P3D [Zeiler et al., 2002]. The simulation setup is described in detail in the previous study by Sitnov and Swisdak , where an additional driving electric field was added to nudge the current sheet toward the instability threshold. The problem is that condition (2) is a sufficient criterion of stability and its deviation from the WKB condition does not guarantee instability. It was found that the transition to instability also requires a sufficiently small current sheet half-thickness Lz near the Bz hump. In simulations by Sitnov and Swisdak , the external driving shifted the initial Bz hump toward the region of higher current density and smaller thickness. Explosive DF formation occurred when Lz reached its critical value providing the single-species tearing growth rate [Pritchett et al., 1991; Brittnacher et al., 1995] consistent with its simulated value γ > (Bz/B0)ω0i, where ω0i is the ion gyrofrequency in the field B0. To circumvent that problem, in our simulations with zero driving field, we used a smaller value of the current sheet thickness constant L = 0.5d, where d is the ion inertial length based on the maximum plasma density parameter n0 (n(z) = n0β− 2cosh− 2(z/Lβ) + nb, where nb = 0.2n0 is the background density).
The magnetic field in the code is normalized by its value B0, and the time is normalized by the inverse ion gyrofrequency . The specific values of the simulation parameters correspond to Runs 1–3 and 5 in [Sitnov and Swisdak, 2011], except for the parameters α and ε2. Values of the former parameter discussed below are α = 0, 1, 2, and 3, whereas the latter parameter value ε2 = 0.133 was changed to have a similar x scale of the Bz hump in the code units after the thickness change. Note also that the specific value of the ion-to-electron mass ratio mi/me = 128 was taken to ensure sufficient separation between ion and electron scales and magnetization of electrons in the tail, while the temperature ratio Ti/Te = 3 is consistent with the most recent statistical data [Artemyev et al., 2011].
The formation of DFs and onset of reconnection in full-particle simulations can be blocked in the case of small simulation boxes and closed/periodic boundary conditions [Pritchett, 1994; Daughton et al., 2006, and references therein]. This problem can be mitigated by the use of either very big simulation boxes [e.g., Daughton et al., 2009] or open boundaries [Daughton et al., 2006]. The use of 2-D equilibria favors the latter approach because in contrast to 1-D current sheets, the increase of the simulation box along the x direction requires an additional reduction of the normalized magnetic field Bz/B0 to keep the same current sheet thickness near the hump region as the most likely instability location. This, in turn, would require a substantial further increase of the ion-to-electron mass ratio to avoid artificial demagnetization of the electron species leading to electron tearing near the x boundaries.
At the same time, the use of recent types of open boundaries, which provide zero density gradients across the boundary [Daughton et al., 2006; Divin et al., 2007] is inconsistent with the nonzero gradients in 2-D magnetotail equilibria, which are necessary to balance the magnetic tension. To solve this problem at the “earthward” end of the box, we combined the earlier set of open-boundary conditions for particle moments [Divin et al., 2007]: ∂ n(ν)/∂ x = 0, ∂ V(ν)/∂ x = 0, and Tν = Tν(t = 0), where ν = e, i, while n(ν) and V(ν) are the density and bulk velocity of the species ν, with the additional injection of a part of the initial Maxwellian distribution with the density δn(ν) ∝ (∂/∂ x)n(ν)(t = 0). Since balancing the magnetic tension at the “tailward” end of the box cannot be done using injection of particles from the outside buffer regions, the simulation box was doubled to add a mirror-looking magnetotail separated from the original one by the equilibrium X-line as described by Sitnov and Swisdak . The field conditions at the x boundaries are taken to provide free propagation of magnetic flux [Pritchett, 2001].
3 Main Phases of the Tail Current Sheet Evolution
Prior to the main series of runs of the tails with flux accumulation regions, we made a benchmark run with α = 0 corresponding to the tail model of Lembege and Pellat . Figure 1a and Animation S1, included as Supporting Information, show the persistent stability of that tearing-stable tail in spite of the presence of the equilibrium X-line at x = 0. As follows from Sitnov and Swisdak , the X-line cannot serve as a universal “tailward” boundary condition in case of external driving, because for α = 0, the driving electric field penetrates the X-line and it becomes an active region. In contrast, in cases with zero driving, the tailward equilibrium X-line remains stable in all runs, which justifies its use as an open-boundary condition at the “tailward” end of the simulation box.
Figures 1b–1d and Animation S2 show the evolution of the magnetotail in case of the tail equilibrium with an accumulation of magnetic flux (α = 3). They show that after the relaxation of initial fluctuations of the electric field Ey caused by the approximate nature of these 2-D equilibria (Figure 1b), the electric field in the current sheet explosively grows from the noise level. Its increase is accompanied by the accelerating earthward motion of the Bz hump and its steepening to form a DF (Figure 1c). As is seen from Figure 2b, the propagation speed of the full-fledged DF is ∼ 0.5vA, where is the effective Alfvén speed, while the x scale of the DF Δx ∼ d. These features are consistent with earlier simulations [Sitnov and Swisdak, 2011] and with DF observations [Runov et al, 2009]. After DF absorption by the left boundary and the current sheet stretching in its wake, two magnetic islands are formed one after another. The fast DF formation does not involve a significant change of magnetic topology as only a relatively small magnetic island forms ahead of the growing front (small negative Bz region in Figure 2b). Large plasmoids formed at ω0it > 44 drift tailward (Figure 2c) with a speed visibly less than the earthward propagation speed of the DF.
4 Explosive DF Formation and Magnetic Topology Change
In the study with weak driving, Sitnov and Swisdak  already showed that the plasma instability responsible for the rapid DF formation has the properties of ion tearing, including the dominant ion Landau dissipation and the eigenmode structure across the current sheet. The simulations with zero driving presented here now provide a key element missing in the verification of the linear stability theory predictions: Figure 3a shows that this mode spontaneously and explosively grows for α = 3 and 2 (with maximum growth rates γ = 0.23 and 0.18ω0i, respectively) but there is no such growth for a smaller value α = 1. Together with our benchmark run for α = 0, these results confirm the stability results by Sitnov and Schindler . Moreover, simulations show that the ion tearing mode whose potential destabilization was predicted in the latter work, becomes indeed linearly unstable if the current sheet is sufficiently thin and the accumulation of magnetic flux tailward of the sheet is sufficiently strong. The spatial structure of the key tearing electric field component Ey is shown in Figure 3b. Interestingly, this tearing instability in its DF formation phase does not involve a significant change of magnetic topology. It looks rather like slippage of magnetic lines with frozen-in electrons relative to the ion plasma species [see the different motions of frozen-in electrons and unmagnetized ions in Sitnov and Swisdak, 2011, Figure 15a]. Figure 3c shows that those different motions of ion and electron species generate the appropriate electric field Ex caused by the charge separation, which grows together with the DF and is peaked in the region of the steepest Bz buildup. Such electric fields normal to the DF have recently been reported in Cluster [Fu et al., 2012] and THEMIS [Runov et al., 2011] observations.
Figures 4a–4c show that the DF propagates through the open boundary without any flow reversal or flux accumulation. The proper performance of the open boundary is confirmed by simulations in a larger box with x/d = (−30, 30). In this run (Animation S3), the formation of new X-lines behind a DF occurs before it reaches the “earthward” boundary, whereas Figure 4d shows similar current sheet evolution combining the DF formation and topology change. The latter occurs in the progressively stretched DF's wake where the plasma pressure is anisotropic and even agyrotropic for ions, while the total current is bifurcated (not shown). Its mechanism may therefore involve secondary tearing instabilities of agyrotropic thin current sheets [Zelenyi et al., 2008] and bifurcated sheet collapse [Camporeale and Lapenta, 2005]. Further reduction of the Bz field should eventually activate reconnection mechanisms arising from electron non-gyrotropy [Hesse and Schindler, 2001]. It is clear however that the primary trigger of the X-line formation is the original ion tearing/slippage instability, which creates the entire developing DF structure.
We have provided the first explicit demonstration that spontaneous onset of reconnection is possible in 2-D magnetotail equilibria with accumulation of magnetic flux at the tailward end of a sufficiently thin current sheet, consistent with recent findings in the linear stability analysis of the ion tearing mode [Sitnov and Schindler, 2010]. A key onset element is the formation of a dipolarization front, which triggers the magnetic topology change. The formation of the front proceeds according to the same underlying mechanism of mutual attraction of parallel current filaments of the sheet which provides spontaneous reconnection in 1-D current equilibria. Similar processes may occur in 3-D due to magnetic buoyancy [Pritchett and Coroniti, 2011].
This work was supported by NASA grants NNX09AH98G, NNX09AJ82G, and NNX12AD31G, as well as NSF grant AGS0903890. The contribution of NB and TEM was supported by the NASA Magnetospheric Multiscale Mission. Simulations were made possible by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center.