Onset of collisionless magnetic reconnection in two-dimensional current sheets and formation of dipolarization fronts

Authors


Abstract

[1] The onset of reconnection in 2-D current sheet equilibria that include an X line separating tail-like regions with magnetized electrons is simulated with a full-particle code. The onset is driven by a finite convection electric field applied outside the current sheet. In the case of tearing stable tails with no accumulated magnetic flux, the convection electric field penetrates the sheet near the X line. In contrast, in multiscale equilibria where the X line is framed by local areas of enhanced flux, the electric field avoids the X line, directly penetrates the areas of increased flux, and ejects them downstream. The ejecta form dipolarization fronts (DFs), sharp magnetic pileups with a thickness on the order of the ion inertial length, much smaller than the mesoscales of the initial flux increase regions. The DFs move with the reconnection outflows in the direction opposite the magnetic field stretching, while behind them new X lines, distinct from the original, form. Simulations with a reduced driving field suggest that DF formation shares properties with the ion tearing instability, which is consistent with its potential destabilization in multiscale equilibria. Weak driving of equilibria with tearing stable tails first forms flux accumulation regions, which then rapidly transform into DFs, making 2-D equilibria inherently metastable. The results are compared with observations of DFs, the statistical visualization of Earth's magnetotail during substorm onset, and the bubble-blob pair formation model.

1. Introduction

[2] Magnetic reconnection is one of the most fundamental processes in space and laboratory plasmas [Angelopoulos et al., 2008; Yamada et al., 2010]. It is usually envisioned as a topological rearrangement of magnetic field that irreversibly converts magnetic energy to plasma energy. To occur, it requires the electric field, which must penetrate into the current-carrying plasma sheet. Magnetic reconnection is particularly difficult to model and understand when plasmas are collisionless, and as a result, both the relevant energy gain and dissipation processes involve complicated multiscale motions of ions and electrons. The reconnection electric field is thought to penetrate into the current sheet near the X line, where it forms the electron diffusion region [Hesse et al., 2001], which organizes the whole energy transformation and transport in the system.

[3] That picture of collisionless reconnection was based mainly on the results of two-fluid and full-particle simulations [e.g., Hesse et al., 2001, and references therein]. It was also based on the linear stability analysis of the tearing mode, which provides spontaneous reconnection in the current sheet. The tearing mode is unstable in the sheet with antiparallel magnetic field lines [Coppi et al., 1966], but it is stabilized in the magnetotail-like configurations, where electrons are magnetized because of the finite magnetic field component Bz normal to the current sheet [Lembege and Pellat, 1982; Pellat et al., 1991]. This suggests, that the electric field penetration into the current sheet is only favorable energetically near the X line, where electrons are unmagnetized.

[4] So far the overwhelming majority of particle simulations of reconnection were based on a simplified setup, the so-called GEM Reconnection Challenge [Birn et al., 2001], where reconnection was initiated by a perturbation of the 1-D Harris equilibrium [Harris, 1962]. The perturbation immediately created an X line, broke the force balance along the original field line direction and caused the electron tearing instability. As a result, studies of sustained fast reconnection had to be postponed to later times, when however, the effect of closed boundary conditions in relatively small simulation boxes might be critically important. A more realistic class of 2-D equilibria was investigated by Pritchett [2005, 2010] with applications to the Earth's magnetotail. However, there the onset of reconnection was provided by the external driving electric field, which was strongly localized along the tail on scales of a few ion inertia lengths, and the mechanism of such a strong localization of the convection electric field in the magnetosphere remained unclear.

[5] In this paper we investigate the onset of reconnection in full-particle simulations with open outflow boundaries, which is driven by broadly distributed electric field imposed outside the current sheet in two equilibria with an already existing X line. Both equilibria belong to the general class of 2-D solutions described by Schindler [1972] and differ in the structure of their magnetotail-like regions framing the X line. In the first case they are similar to the tearing-stable current sheets considered by Lembege and Pellat [1982], with Bz being asymptotically constant along the current sheet and then monotonically decreasing in absolute value to zero toward the X line. In the second case, ∣Bz∣ increases at the tailward end of a thin current sheet before it turns to zero. The increase occurs on a mesoscale, which is intermediate between the current sheet thickness Lz and its original inhomogeneity scale Lx along the magnetic field stretching. Such multiscale magnetotail current sheets were recently shown to have a tendency to tearing destabilization [Sitnov and Schindler, 2010]. Therefore the electric field may penetrate directly into the magnetotail regions where electrons are magnetized. Simulations show that the development of reconnection in these two cases is indeed different. And in the case of 2-D equilibria with multiscale magnetotails the driving electric field avoids the initial X line region, penetrates instead directly into the areas of the initial ∣Bz∣ humps, sharpens and accelerates them in the direction opposite to the initial magnetic field stretching and finally forms structures similar to dipolarization fronts (DFs).

[6] DFs are characterized by a strong and steep increase of the tail magnetic field component Bz normal to the neutral plane. As was shown in recent THEMIS observations [Runov et al., 2009; Sergeev et al., 2009], the increase occurs on a microscale of the order of the ion inertial length d. It is preceded by a sharp but much smaller decrease and followed by a shallower (≳10d) descent from its peak value. The increase of Bz field in DFs is accompanied by a substantial density drop, and as a result, they represent buoyant flux tubes also known as plasma bubbles [Wolf et al., 2009, and references therein]. Another key distinctive feature of DFs is their fast motion in the direction opposite to the original magnetic field stretching with the speed, which is a substantial fraction of the Alfvén speed. That fast motion requires the penetration of the convection electric field into the current sheet (or its spontaneous excitation there) and its amplification and focusing on the corresponding microscale. DFs usually move near the leading edge of flow bursts inside bursty bulk flows (BBFs) and substorm activations, which dominate the transport of plasma and magnetic flux in the magnetotail [Angelopoulos et al., 1992; Slavin et al., 2003; Ohtani et al., 2004].

[7] DFs are well documented in observations, starting from first two-satellite measurements [Russell and McPherron, 1973; Moore et al., 1981] and including both statistical studies [Slavin et al., 2003; Ohtani et al., 2004] and recent multispacecraft investigations based on Cluster and THEMIS data [R. Nakamura et al., 2002, 2005; Eastwood et al., 2005; Runov et al., 2009, 2011; Hwang et al., 2011]. Observations helped distinguish DFs from more global dipolarization processes arising when fast plasma flows are braked in the near-Earth tail region with the higher plasma density and almost dipolar magnetic field [Hesse and Birn, 1991; Shiokawa et al., 1997]. In particular, Ohtani et al. [2004] showed, that DFs are observed throughout the whole magnetotail, from XGSM = −5RE to −31RE (GSM stands for the Geocentric Solar Magnetospheric System with its X axis pointing toward the Sun from the Earth; RE is the Earth radius), while Runov et al. [2009] tracked a DF propagating from XGSM = −20.1RE to −11.0RE using the constellation of 5 THEMIS probes stretched along the tail. DFs have been reproduced in MHD and hybrid simulations of magnetic reconnection [Fujimoto et al., 1996; Hesse et al., 1998; Wiltberger et al., 2000; M. S. Nakamura et al., 2002; Krauss-Varban and Karimabadi, 2003; Ashour-Abdalla et al., 2010; Birn et al., 2011], which used various models of the localized resistivity to trigger the reconnection process in the magnetotail. Observations of DF analogues propagating tailward [Fujimoto et al., 1996; Ohtani et al., 2004] also suggest that DFs are caused by bursts of reconnection in the magnetotail.

[8] At the same time, DFs are typically not seen in full-particle simulations of magnetic reconnection in electron-proton plasmas, which start from the 1-D Harris current sheet and show rather the formation of the electron-scale current sheets and secondary magnetic islands [Daughton et al., 2006; Klimas et al., 2010, and references therein]. In a few works where signatures of DFs have been reported, they appear either as pileups caused by the interactions of the reconnection outflows with closed/periodic boundaries [Horiuchi and Sato, 1996; Pritchett, 2001a] and secondary plasmoids created by the electron tearing [Hoshino et al., 1998, 2001] or as transient phenomena associated with switching on the global reconnection electric field in the process of its expansion from the initial X line region [Wan and Lapenta, 2008; Sitnov et al., 2009]. In the latter work it was shown, that a sufficiently strong X line perturbation forms a transient 2-D configuration with two tail-like outflow regions where electrons are magnetized by the field Bz and DFs develop in those regions when the force balance is already restored. However, it remained unclear if or how such 2-D equilibria could appear as a result of the quasistatic current sheet evolution. In any case, within the GEM Reconnection Challenge framework [Birn et al., 2001] with its perturbed 1-D Harris setup, the formation of DFs cannot be consistently investigated and their finding can always be ascribed to artificial initial transients caused by the X line perturbation of the 1-D Harris sheet.

[9] 1-D current sheets separating magnetic fields with antiparallel field lines are very rare in self-organized space systems, in which the current sheets have either a substantial guide magnetic field component as, for example, at the magnetopause, or a magnetic field Bz normal to the current sheet plane, characteristic for the terrestrial magnetotail. In this study we consider the onset of reconnection and formation of DFs in 2-D current sheet equilibria with the external driving field, which mimics the convection electric field induced in the magnetotail by the solar wind-magnetosphere interaction. Although 3-D effects, including the ballooning interchange instability, are also important for DFs and plasma bubbles and they determine their structuring along the dawn-dusk direction [e.g., Hurricane et al., 1996; Guzdar et al., 2010; Pritchett and Coroniti, 2010; Birn et al., 2011; Lapenta and Bettarini, 2011], we show, that 2-D simulations with self-consistent initial settings reproduce many key features of DFs, and most importantly, unveil the mechanism of their generation associated with spontaneous reconnection in the magnetotail.

[10] We show in particular that 2-D current sheet equilibria behave as metastable systems and the transition from slow to fast phases of their evolution involves the fast penetration of the convection electric field inside the current sheet, formation and acceleration of DFs. For multiscale equilibria the transition is consistent with the linear stability theory of the ion tearing mode in terms of the stability criteria, growth rate, electric field profile and dominating dissipation species. It should be emphasized that the problem of the ion tearing instability in its classical settings [e.g., Pritchett, 1994], including the zero driving field limit with no topological singularities of the equilibrium magnetic field, is not addressed in this study. At the same time, the new features of reconnection onset in 2-D current sheet equilibria described below open interesting opportunities in a better formulation and further investigation of that problem.

[11] The structure of this paper is as follows. In the next section the two basic equilibria under investigation are described. In section 3 we provide the details on the basic simulation parameters and code features, including the new outflow open boundary conditions suitable for the 2-D case. In sections 4 and 5 we describe the results of simulations of the reconnection onset for strong and weak driving conditions, respectively. The results are further discussed in section 6 and finally summarized in section 7.

2. Two-Dimensional Self-Consistent Equilibria

[12] The 2-D isotropic plasma equilibria considered in our study are described by the following y component of the vector potential [Schindler, 1972]

display math

for any choice of β(x) slowly varying in x: β(x) = exp(ε1g(ξ)), where ξ = x/L and ε1 ≪ 1. The specific choice of the function g(ξ) is made: (i) to allow the formation of a hump in the Bz field at ξ = ξ0 and (ii) to form an X point at ξ = ξ1 > ξ0, which marks also the left/right symmetry line along x, with the mirror Bz hump at ξ = ξ0* = 2ξ1ξ0. Let us choose

display math

where h(ξ) = ξ + (α/ε2)[1 + tanh (ε2(ξξ0))] for ξ < ξ1, while h(ξ) = ξ + (α/ε2)[1 + 2 tanh (ε2(ξ1ξ0)) + tanh (ε2(ξξ0*))] for ξ > ξ1, and h1 = ξ1 + (α/ε2)[1 + tanh (ε2(ξ1ξ0))]. The parameter ξr in (2) determines the scale of the ∣Bz∣ field reduction near the X line. In the limit α = 0 this function describes a 2-D system with an X line separating magnetotails with the constant normal component Bz discussed by Lembege and Pellat [1982], which is the initial current sheet equilibrium in runs 1 and 5. In the case α > 0 the X line separates magnetotails with the Bz humps investigated by Sitnov and Schindler [2010] and such an equilibrium is used to start simulations in runs 2–4. Distributions of the Bz component of the magnetic field and the plasma pressure at the neutral plane z = 0 as well as the current sheet half thickness and field lines for these equilibria are shown in Figure 1. As is seen from Figure 1, a distinctive feature of the equilibrium used in runs 2–4 are two “seeds” of accumulated magnetic flux loaded into the magnetotails. The extension of those seeds along the X direction introduces an additional scale factor into the system, in addition to the current sheet thickness L and its basic inhomogeneity scale L/ε1 making such magnetotail equilibria inherently multiscale [Sitnov and Schindler, 2010]. The 2-D equilibria described by equation (1) contain no By component (guide field).

Figure 1.

Two basic types of current sheet equilibria used in simulations. Run 1: (a) normal magnetic field Bz at the neutral plane z = 0, (b) dimensionless plasma pressure parameter p = 1/(2β2), (c) current sheet half thickness Lz/L = β(x), (d) magnetic field lines for the equilibrium with the magnetotails similar to the Lembege and Pellat [1982] model, and (e) the driving electric field Ey(dr) at top and bottom boundaries z = ±10. Run 5 differs from run 1 by the reduced value of the driving field E0(dr) = 0.05. Run 2: (f–j) parameters similar to those of run 1 for the multiscale equilibrium investigated by Sitnov and Schindler [2010] with the same driving field E0(dr) = 0.2 as in run 1. The strength of the driving field is reduced to E0(dr) = 0.05 in runs 3 and 4. The latter run differs from runs 2 and 3 by the increased size of the simulation box along the X direction: −25 < x < 25.

3. Basic Parameters and Boundary Conditions

[13] Simulations have been performed using an open-boundary modification of the explicit massively parallelized full-particle code P3D [Zeiler et al., 2002]. The magnetic field in the code is normalized by its asymptotic value B0 entering (1), and the time is normalized by the inverse ion gyrofrequency Ω−1 based on the field B0. Consistent with (1), the initial plasma density in the original physics units is given by the expression n(z) = n0β−2cosh−2(z/) + nb, where nb is the background density. In runs 1–3 and 5 nb = 0.2n0, and in run 4 the background density is reduced to nb ≈ 0.12n0 to achieve the proper scaling of the plasma density in the code units in the extended simulation box, and in particular, to provide the same ratio between the background plasma density and its value at the X boundaries of the original (not extended) simulation box nb/n(∣x∣ = 20, z = 0) = 0.2. The plasma density and spatial scales are normalized by n0 and the ion inertial length d = c/ωpi with the ion plasma frequency ωpi based on the density n0.

[14] Other parameters are taken to ensure the separation between ion and electrons scales and magnetization of electrons in the tails. This includes ion-to-electron mass and temperature ratios mi/me = 128 and Ti/Te = 3, the speed of light in the code units c/vA = 15, where vA is the effective Alfvén speed math formula, the initial current sheet thickness constant L = 0.75d, and the following values of the constant parameters in the configurations (1): ξ0 = 14, ξ1 = 0.5Lx/L, ξr = 3, ε1 = 0.03, ε2 = 0.2. With these parameters and the specific value of the parameter α = 3 in runs 2–4, the tearing destabilization factor [Sitnov and Schindler, 2010] for those runs is Cd ≈ 2.1, whereas in runs 1 and 5 with α = 0 the destabilization factor Cd < 1 (more details on the destabilization factor Cd are provided in section 5.2). In runs 1–3 and 5 the box dimensions Lx × Lz = 40d × 20d (with 2304 × 1152 grid points). In the special run 4, which is performed to check the effect of the boundary conditions in the X direction and investigate the details of the ion tearing mode growth, Lx × Lz = 50d × 20d (with 2880 × 1152 grid points). Particles are loaded in the code with shifted Maxwellian distributions corresponding to the locally Harris structure of the 2-D solution (1) [see, e.g., Lembege and Pellat, 1982, equation (1)] using the rejection method [Press et al., 1999] with the number of accepted particles per grid Nppg ≈ 100 outside the current sheet (background) and Nppg ≈ 250 inside it (the maximum current density region). The total number of particles followed in the simulation for each species in runs 1–5 and 5 is Ntot ≈ 3 × 108, while in run 4 Ntot ≈ 4 × 108.

[15] The finite size of the simulation box along the current sheet (X) direction is particularly challenging for simulations of unsteady reconnection. It may artificially limit the length of the diffusion region and block the formation of secondary magnetic islands [Daughton et al., 2006]. Also, it artificially cuts the magnetic flux tube volume, which plays a key role in the tearing stability problem [Sitnov and Schindler, 2010]. 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; Klimas et al., 2010] is inconsistent with the nonzero gradients in 2-D magnetotail equilibria, which are necessary to balance the magnetic tension. To solve this problem we combined the earlier set of open boundary conditions for particle moments [Divin et al., 2007] ∂n(α)/∂x = 0, ∂V(α)/∂x = 0, and Tα = Tα(t = 0), where α = e, i, while n(α) and V(α) are the density and bulk velocity of the species α, with the additional injection of a part of the initial Maxwellian distribution with density δn(α) ∝ (∂/∂x)n(α)(t = 0). The field conditions at the X boundaries ∂Ex,y/∂x = 0, Ez = 0, ∂By/∂x = 0, ∂Bx/∂x = −∂Bz/∂z, and Bz = Bz(t = 0) are taken to provide free propagation of magnetic flux [Pritchett, 2001b].

[16] The top and bottom boundary conditions provide squeezing of the selected 2-D equilibria toward the neutral plane. Particles there are specularly reflected, while the field components satisfy the relations: Ex = 0, Ey = Ey(dr), ∂Ez/∂z = 0, ∂Bx,y/∂z = 0, and Bz = Bz(t = 0), where the driving electric field is taken to be Ey(dr) = −E0(dr)tanh2(t/τ)(tanh2((xδ)/λ) + tanh2((Lxδx)/λ) − 1) and zero outside of the interval (δ, Lzδ) with λ = δ = 0.1Lx and Ωτ = 0.5. This field peaks at the center of the Z boundary and decreases slowly toward its edges (Figures 1c and 1f). Its peak value E0(dr) = 0.2 in runs 1 and 2, and it is reduced to E0(dr) = 0.05 in runs 3–5.

[17] At first sight, a combination of the Z boundaries closed for particles and the finite driving electric field may result in the formation of the vacuum areas and significant electromagnetic noise near those boundaries. However, closer examination shows that this is not an issue in the runs considered below for the following reasons. First, we consider the problem of reconnection onset, which is naturally limited in time. Moreover, plasma is free to penetrate inside the box near top and bottom boundaries through open X boundaries (the flaring tail magnetic field structure favors such an inflow under the action of the same driving field). As a result, at the time of interest the vacuum areas are not formed. Besides, unlike fluid plasma models, full-particle codes, such as P3D, may work normally without background plasmas, and a special investigation revealed no substantial electromagnetic noise at the time of interest.

[18] Note, that in the coordinate system adopted in our simulations the X axis is directed along the magnetic field line stretching in the left part of the simulation box, which is opposite to the GSM coordinate system. As a result, the positive Y direction of the right-handed coordinate system corresponds to the “dusk-dawn” direction for the “magnetotail” in the left part of the box (where initially Bz(z = 0) > 0 consistent with its positive value in the GSM system). To simplify the notations and avoid confusion in the comparison with the magnetospheric parameters, the sign of the Y axis is reversed in the following. Also, for the sake of symmetry in the plots discussed below we shift the X coordinate xxLx/2. The characteristic parameters of five runs discussed below are summarized in Table 1.

Table 1. Some Simulation Parameters
RunEquilibriumaDriving Field E0(dr)Lxnb
1LP19820.2400.2
2SS20100.2400.2
3SS20100.05400.2
4SS20100.05500.12
5LP19820.05400.2

4. Two Different Reconnection Stories: Strong Driving Field Case

4.1. Two-Dimensional Equilibrium With Lembege-Pellat Magnetotails

[19] The results of simulations reveal two drastically different reconnection onset stories. In run 1 (Figures 24) thinning of the current sheet by the field Ey(dr) first results in the current bifurcation (Figures 2a and 2b; note the stronger current on the separatrices at, for example, x = 5d), consistent with the earlier results [Pritchett, 2005; Schindler and Hesse, 2008]. At later time (Figure 2d) a current layer with the thickness of the order of the electron inertial length starts forming near the X line. Note that the initial current thickness in that region exceeds the ion inertial length Lz(t = 0) = β(ξ1)L ≈ 1.8d. As seen from Figures 3b and 4a, the reconnection electric field penetrates the neutral plane z = 0 through the X line vicinity and then expands over the whole current sheet in conjunction with sharp Bz pileup regions (Figure 3a) similar to fronts found in particle simulations of the perturbed 1-D Harris equilibrium [Sitnov et al., 2009].

Figure 2.

Reconnection onset in run 1: (a–e) distributions of the y component of the total current density (color-coded) and magnetic field lines, showing the formation, elongation, and disruption of the electron-scale current layer.

Figure 3.

Reconnection onset in run 1: evolution of the (a) magnetic field Bz and (b) electric field Ey. The dashed line in Figure 3b corresponds to Ey = 0.

Figure 4.

Run 1 at the moment Ωt = 26: (a) color-coded electric field Ey and magnetic field lines and energy dissipation rate j · E for (b) electron and (c) ion species.

[20] Fronts found in run 1 differ from the DFs detected in the magnetotail [Nakamura et al., 2005; Runov et al., 2009, 2011] because the electric field behind the front remains quite substantial (Ey ∼ 0.15) and comparable in strength to its peak value near the Bz front. This persistent trailing electric field is consistent with other simulations of 1-D Harris sheets [see, e.g., Karimabadi et al., 2007, Figure 4; Shay et al., 2007, Figure 1]. However, it is not seen in observations where the characteristic scales of the electric field trailing region often do not exceed few tens of ion inertial lengths [see, e.g., Runov et al., 2011, Figure 5]. In run 1 this residual electric field Ey dominates the current sheet at later time in spite of the elongation of the electron-scale thick layer (Figure 2d) and even the formation of a secondary island (Figure 2e), which just locally modulates Ey. This rather global and persistent reconnection electric field is supported by a very localized electron dissipation (Figure 4b) and a broader, but still well localized and regular in space, ion dissipation region (Figure 4c).

Figure 5.

Reconnection onset in run 2: (a–e) distributions of the y component of the total current density (color-coded) and magnetic field lines, showing the ejection of the enhanced magnetic flux regions, followed by the formation of new X lines behind dipolarization fronts and the current sheet expansion.

4.2. Two-Dimensional Equilibrium With Multiscale Magnetotails

[21] In run 2 (Figures 57) the electric field squeezing the current sheet avoids the central X line region. It penetrates directly into the seed regions with the enhanced ∣Bz∣ (Figures 6b and 7a) and rapidly ejects their plasma and magnetic flux from the system (Figures 5a–5c). The effect resembles the ejection of a melon seed squeezed between the thumb and fingers. The resulting DFs (Figures 6a and 7a) are different from secondary magnetic islands because of monopolar changes of the Bz field. They are similar to the DFs [Nakamura et al., 2005; Runov et al., 2009, and references therein] observed inside flow bursts that dominate plasma and magnetic flux transport in the Earth's magnetotail [Angelopoulos et al., 1992]. Their observed scales (forward and trailing extensions on the order of d and 10d, respectively, according to [Runov et al., 2009, 2011]) are consistent with our simulations showing the Bz ramp-up scale at the front of DFs Δx(f) ≈ 2d and its reduction in the DF trail on the scale Δx(t) ≈ 10d. It is important to note here that the microscales of fronts found in simulations are in drastic contrast with the initial multiscale equilibrium whose Bz humps have the scale ε2−1L, which is intermediate between the current sheet thickness (Ld) and its main inhomogeneity scale along the X direction (∼ε1−1LL) because ε1ε2 ≪ 1. It is the penetration of the electric field inside the Bz humps and its self-focusing in the process of penetration that sharpens the initial multiscale equilibrium, transforming the mesoscale humps into microscale DFs. Another critical distinction of DFs from their seeds in the initial Bz humps is their fast motion in the direction opposite to the original magnetic field stretching with the speed, which is a substantial fraction of the Alfvén speed (∼0.5vA), which is also consistent with observations [Nakamura et al., 2005; Runov et al., 2009, 2011]. DFs are moving with the reconnection outflows just as the observed DFs move at the leading edge of BBFs. However, as we will show below, ions in DFs are not frozen in the magnetic field.

Figure 6.

Evolution of the (a) magnetic field Bz and (b) electric field Ey in run 2.

Figure 7.

Run 2 at the moment Ωt = 26: (a) color-coded electric field Ey and magnetic field lines and energy dissipation rate j · E for (b) electron and (c) ion species.

[22] In agreement with observations, the simulated DFs leave no residual electric field in their wakes (Figure 6b). After their escape from the simulation box, a global drop of the electric field over the whole current sheet is observed with the average Ey < 0.05. The reconnection electric field is building up again when two new X lines form instead of the original one (Figure 5e). At the same time, reconnection does not develop near the initial central X line, which transforms eventually into an O line. Note that in spite of some distortion of the field Bz near the outflow boundaries, caused by our field boundary conditions, fast plasma flows with DFs freely penetrate through those boundaries. This is possible because of the open boundary conditions for particles, which dynamically respond to changes in the plasma density and velocity. The impact of the boundary conditions on the formation of DFs is additionally investigated below in subsection 5.3.

[23] The fact that the electric field avoids the X line vicinity might be explained by the larger thickness of the current sheet in that region, which is almost two times as large as the thickness near Bz humps (Figure 1h). However, such an explanation would be fundamentally incomplete. First, in run 1 the ratio between the current sheet thickness near the X line and its minimum value near X boundaries is also close to two, but in that case the electric field penetrates the sheet near the X line. Second, as will be shown in detail in section 5.2, in case of multiscale equilibria the electric field penetrates the current sheet before any substantial changes of the current sheet thickness at the fixed location in x. In other words, its penetration is not caused by the fact that somewhere outside the X line the current sheet critically thins. The electric field penetrates the current sheet through the regions with a thickness, which is intermediate between its values near the X line and the minimal value Lz = L = 0.75d near the X boundaries. And the most intriguing feature is that it penetrates the current sheet through the regions where Bz is finite and strong enough to magnetize electrons.

[24] The penetration of the electric field directly into the areas with the enhanced values of ∣Bz∣ in the magnetotail-like parts of the current sheet is consistent with the recent tearing stability analysis [Sitnov and Schindler, 2010], which suggests a strong tendency to their destabilization in case of run 2. Moreover, since the ion tearing instability is faster than the electron tearing [Schindler, 1974; Brittnacher et al., 1995; Kuznetsova et al., 1996], which one can expect near the X line, the convection electric field should first arise in the magnetotail parts of the multiscale equilibrium. Note that the distinct regions of localized ion dissipation seen in Figure 7c, in the absence of the electron dissipation region in Figure 7b, agree with the dissipation mechanism proposed for the ion tearing instability [Schindler, 1974]. However, the detailed investigation of tearing stability issues requires the consideration of the reconnection onset under a weaker driving field, well below the sustained reconnection limit Ey ∼ 0.15 [Shay et al., 2007] and will be performed in the next section.

5. Two Different Reconnection Stories: Weak Driving Field Case

5.1. Two-Dimensional Equilibrium With Multiscale Magnetotails

[25] Run 3, which differs from run 2 by the reduced driving field magnitude Ey ∼ 0.05, was performed to investigate the impact of the external driving field on the process of the formation of dipolarization fronts. Its results are summarized in Figures 812. As seen from Figures 8 and 9, the reduction of the driving field substantially delays the excitation of the electric field Ey inside the current sheet and the formation of sharp DF pileup profiles in Bz by Δ(Ωt) ∼ 10. However the comparison of Figures 6 and 9 shows that both the pileup magnetic field Bz and the reconnection field Ey in run 3 reach magnitudes comparable to those in run 2 before DFs escape from the simulation box.

Figure 8.

Reconnection onset in run 3 in a format similar to Figure 5, showing the formation of dipolarization fronts in the case of a relatively weak driving electric field Ey ∼ 0.05.

Figure 9.

Slow (Figures 9a–9d) and fast (Figures 9e–9h) phases of the current sheet evolution in run 3. (a and e) The magnetic field Bz, (b and f) electric field Ey, (c and g) x component of the electron bulk flow velocity vex, and (d and h) x component of the ion velocity vix.

Figure 10.

Evolution of the peak electric field value 〈Eymax(x > 0) in cases of weak (run 3, solid line) and strong (run 2, dotted line) driving. Dashed and dash-dotted lines show the corresponding exponential growth approximations with linear growth rates γ/Ω = 0.06 and 0.1, respectively.

Figure 11.

Preonset evolution in run 3 in the right half of the simulation box (0 < x < 20): (a) current density profiles Jy(x) at z = 0 (dotted lines) and the parameter ∣Bz∣ at z = 0 and (b) current density profiles Jy(z) at the X location where ∣Bz∣ reaches its maximum value in the right half of the simulation box.

Figure 12.

Evolution of the current sheet half thickness LH prior to the dipolarization fronts (DF) onset, estimated using the Harris model at the peak value of the ∣Bz∣ field in the right half of the simulation box (0 < x < 20) in cases of weak (run 3, solid line) and strong (run 2, dotted line) driving. The dashed line shows the critical value LH = LH(T) ≈ 1.7, which corresponds to the current sheet thickness providing the observed initial growth rate γ = 0.06Ω in the single-species linear tearing stability theory [Brittnacher et al., 1995].

[26] A more detailed comparison of the DF formation in runs 2 and 3 using conventional spectral methods, such as the fast Fourier transform, is rather ineffective because of the strong localization of DFs along the X axis (Figure 7a). We consider instead the evolution of the maximum 〈Eymax over the x coordinate (either x < 0 or x > 0) of the parameter 〈Ey〉(x), which represents the local value of the electric field Ey(x, z) averaged over the region ∣z∣ < 0.5d. The results made for the right half of the simulation box x > 0 (Figure 10) show, that while initially (for 〈Eymax ≲ 0.1) the peak value of the reconnection electric field grows rather slowly (〈Eymax ∝ exp γt with γ/Ω ≈ 0.06), compared to the strong driving case (run 2), eventually its growth rate becomes very similar to that in case of the strong driving with γ/Ω ≈ 0.1. The occurrence of an enhanced nonlinear growth stage seems to be a common feature in reconnection simulations [e.g., Karimabadi et al., 2005].

[27] An interesting new feature of weakly driven reconnection in run 3 is the evolution of X lines. In contrast to the strongly driven case in run 2, where the initial X line quickly transforms into the O line (Figure 5e), in run 3 the formation of two new X lines first creates a two-island and three–X line configuration (Figures 8d and 8e), which then slowly evolves into the single-island configuration similar to Figure 5e. The latter configuration appears because of coalescence of the new islands and its formation is finished only at about Ωt = 65 (not shown).

[28] Figures 9 and 10 reveal one of the most impressive new features of reconnection onset in 2-D equilibria, namely the metastable evolution of the current sheet system. In the slow phase (0 < Ωt ≲ 20) the electric field inside the current sheet does not exceed the noise level (Figures 9b and 10), while the magnetic field Bz humps are slowly drifting from the center toward the X boundaries of the simulation box (Figures 9a). According to Figures 9c and 9d, the outflow velocities for both ions and electrons are much less than the Alfvén speed vA and they are at least an order of magnitude less than in the corresponding simulations starting from 1-D current sheets [see, e.g., Sitnov et al., 2009, Figures 5c and 5d].

[29] The fast phase (Ωt ≳ 20) is marked by the exponential growth of the electric field (Figure 10, solid line; see also Figure 9f), which provides the fast acceleration, steepening and growth in magnitude of the initial mesoscale Bz humps and their transformation into microscale DFs (Figure 9e). Both ion and electron outflow velocities become comparable to or exceed the Alfvén speed vA (Figures 9g and 9h). According to Figure 10 (dotted line), the metastable evolution can already be discerned in case of the strong driving (run 2), although in that case the slow phase is shorter (Ωt ≲ 10). Note that the transition from slow to fast phase is not related to any topological changes in the system. In particular, in run 3 secondary X lines form only at Ωt ≈ 39 (red line in Figure 9e). At the same time, as will be shown below in subsection 5.3, the formation of DFs in the fast phase is a non-MHD process, which violates the frozen-in condition. Therefore it must be treated as a part of the general reconnection process.

5.2. Tearing Stability Implications

[30] The main focus of our study is modeling onset of reconnection and formation of DFs in 2-D equilibria driven by a finite convection electric field in a metastable system. Nevertheless, the analysis of our simulations, especially in cases of multiscale equilibria and a weak driving, provides interesting insights into the mechanism of spontaneous reconnection in the magnetotail parts of our system. This includes both the sufficient stability criterion of the ion tearing mode and other conditions, which control the actual transition from stability to fast DF formation and reconnection onset.

[31] The ion tearing instability proposed by Schindler [1974] to explain the onset of reconnection in the magnetotail with electrons magnetized because of the finite Bz component, has long been considered a myth. The problem was that its sufficient stability condition was shown by Lembege and Pellat [1982] to coincide with the WKB approximation kLzB0/(πBz) ≳ 1, where k is the wave number and Lz is the local current sheet half thickness. The WKB approximation allows for the stability analysis of wavy perturbations of the electromagnetic fields ∝ exp(ikx), neglecting variations of the current sheet parameters along the tail. That original conclusion was confirmed by subsequent theoretical studies (see Pritchett [2007] and Schindler [2007] for reviews) and many dedicated full-particle simulations [Pritchett, 1994; Dreher et al., 1996; Hesse and Birn, 2000; Hesse and Schindler, 2001].

[32] Further studies revealed that the stability condition was obtained by Lembege and Pellat [1982] using an approximation of their exact stability condition

display math

in which the flux tube volume V = ∫ dl/B was estimated as 2Lz/Bz, and as a result the potential destabilization factor Cd = VBz/(πLz) was reduced to ∼1. However, the latter flux tube volume estimate is oversimplified for the whole class of 2-D magnetotail equilibria. In particular, Sitnov and Schindler [2010] performed calculations of the flux tube volume parameter for several classes of 2-D equilibria. And they found that the equilibria with ∣Bz∣ nonincreasing in the direction of the magnetic field stretching, including the original equilibrium with Bz = const considered by Lembege and Pellat [1982] and the magnetotails framing the X line in run 1, are indeed tearing stable (Cd ≤ 1) as long as electrons are magnetized. However, they also found a strong tendency for tearing destabilization (Cd > 1) for 2-D equilibria with a hump of Bz at the tailward end of a thin current sheet, such as the magnetotails considered above in runs 2 and 3, for which the peak value of the destabilization parameter Cd exceeds 2.

[33] The analysis of runs 2 and 3, and in particular the results shown in Figure 10, suggest that the growth of the electric field is exponential with two distinct growth rate values γ/Ω ≈ 0.06 and 0.1, characteristic for relatively small and large amplitudes. And as discussed in subsection 4.2 the corresponding dissipation j · E is provided entirely by ions (Figures 7b and 7c), consistent with the ion tearing theory. The aforementioned characteristic values of the growth rate are consistent with the ion tearing mode regime both in time and in space scales. Ions are unmagnetized because their gyrofrequency in the normal field Ωn = (Bz/B0)Ω < γ upstream of DFs (Ωn/Ω ≈ 0.03), while the electron gyrofrequency even in that upstream region Ωen/Ω = (mi/me)(Bz/B0) ∼ 4. The excited DFs resemble rather solitons than monochromatic waves. Therefore, direct assessment of the particle magnetization in space in terms of the parameters αn, where ραn is the gyroradius of the particle species α (α = e, i) in the field Bz, which was extensively used in the past analysis [see, e.g., Pritchett, 2010, and references therein], is questionable in case of DFs. However, the characteristic upstream values ρen ≈ 1.5d and ρin ≈ 30d suggest that electrons are magnetized on the considered spatial scales whereas ions are unmagnetized. Note, that this is consistent with the dissipation picture in Figures 7b and 7c.

[34] What causes the transition to instability? And what is the role of the driving electric field if the process is spontaneous? To answer these questions we note that the sufficient stability condition (3), which is relaxed in case of multiscale magnetotail current sheets, does not determine the stability threshold itself, which may be determined by other factors. One key factor is the current sheet thickness (and the current density related to it) in the regions where (3) is relaxed, because it determines the access to the ion Landau dissipation. It determines in particular the tearing growth rate γ, whose theoretical estimate is inversely proportional to the current sheet half thickness Lz [Schindler, 1974; Pritchett et al., 1991; Brittnacher et al., 1995]. The growth rate must become comparable to the ion gyrofrequency inside the current sheet γ ≳ Ωn to make the instability real [Schindler, 1974; Pritchett et al., 1991; Brittnacher et al., 1995]. Interestingly, further analysis shows that the driving electric field barely thins the current sheet globally, to substantially change the current density at the given point along the current sheet (for run 3 this is seen from the dotted lines in Figure 11a). Instead, it shifts the area of the maximum relaxation of the condition (3), corresponding to the Bz maximum, toward the higher current density region as shown by solid lines in Figure 11a. Figure 11b demonstrates the evolution of the current density Jy near those Bz maxima. In Figure 12 we show the resulting evolution of the current half thickness LH, which is inferred from the approximation of the current density profiles in Figure 11b and similar profiles for run 2 by the Harris model with Jy ∝ cosh−2(z/LH). The dashed line in Figure 12 shows the critical value of LH = LH(T) ≈ 1.7ρ0i, which provides the observed initial growth rate γ = 0.06Ω in the linear theory of the single-species ion tearing (see Brittnacher et al. [1995], particularly their Figure 3a). Figure 12 reveals an impressive consistency with the linear theory of the ion tearing mode. It shows in particular that, consistent with the electric field evolution plots in Figure 10, the strong driving in run 2 brings potentially unstable Bz humps toward the threshold of the actual linear instability ∼10 units of the dimensionless time Ωt earlier than it occurs in the case of run 3 with the reduced driving field. Thus, the role of the driving field in both cases is similar, and it brings the potentially unstable system to its actual instability threshold.

5.3. Onset of Reconnection in Multiscale Current Sheets: Further Details

[35] To further investigate the transition from slow to fast evolution in multiscale equilibria we performed the additional run 4 with the extended X dimension −25 < x/d < 25. Such an extension helps us also to evaluate the possible distortion of the simulated reconnection onset picture caused by the outflow boundary conditions. Note here that our study is focused on the DF formation process, which begins deeply inside the simulation box (∣x∣ ∼ 10d). Figure 13 shows that even the transition to the enhanced nonlinear growth (when γ/Ω changes from ≈0.06 to 0.1) occurs in our simulations at the moment Ωt ∼ 30–33, when DFs are far from the X boundary (see green lines in Figures 9e and 9f; in case of run 4 this is also clearly seen from Figure 15, showing the electric field distribution at Ωt = 40). Moreover, both fast plasma flows and magnetic flux associated with DFs freely penetrate through the outflow boundaries. This is possible, in particular, because the electric field Ey is not fixed at the boundary and because our boundary conditions for particles dynamically respond to changes in the plasma density and velocity. At the same time, the propagation of the plasma with the frozen-in magnetic flux through the X boundaries may result in an artificial distortion of the field Ey in response to the reduction of the Bz peak in DFs dictated by the corresponding boundary condition. Run 4 helps us to check the possible distortions in the electric field growth caused by the DF interaction with the X boundary, because at the moment of such an interaction in run 3, the distance between DFs and X boundaries in run 4 strongly exceeds the front extension. The comparison of runs 3 and 4 in Figure 13 prior to the moment Ωt = 40, when the DF reaches the right simulation boundary in run 3, shows that the distortion of the DF electric field Ey by our boundary conditions is insignificant: in run 3 the parameter 〈Eymax follows the same exponential growth as in run 4 (γ/Ω ≈ 0.1) and it sharply drops down only at Ωt > 40 when the DF escapes the simulation box.

Figure 13.

Evolution of the peak electric field value 〈Eymax(x > 0) in the case of weak driving for the original (run 3, dotted line) and extended (run 4, solid line) simulation boxes. Dashed and dash-dotted lines show the corresponding exponential growth approximations with growth rates γ/Ω = 0.06 and 0.1, respectively.

[36] A key feature revealed by Figure 13 is the large value of the parameter 〈Eymax ≈ 0.4 achieved in run 4. It exceeds the original driving electric field by an order of magnitude and strongly suggests that the formation of DFs is a spontaneous process caused by an instability. Its tearing nature is further supported by Figure 14, where the z profile of the field Ey, averaged over the interval δx = d near its peak value Ey = 〈Eymax at Ωt = 40 and denoted as 〈Ey*, is compared with the tearing eigenmode structure in the WKB approximation (see equation (37) of Pritchett et al. [1991]).

Figure 14.

Z profile of the electric field Ey averaged over the interval δx = d near its peak value Ey = 〈Eymax at Ωt = 40 (solid line) in run 4 compared to the analytical scaling of the ion tearing eigenmode Eyimage [Pritchett et al., 1991] for Lz = d and kLz = 0.5 (dotted line).

[37] Another important feature of the DF formation process, which is clarified by run 4, is the formation of secondary X lines in DF wakes. They form at Ωt ≃ 39, well after the transition from slow to fast phase of the current sheet evolution (Ωt ≃ 15), and as seen from Figure 13, their appearance does not affect the peak values of the electric field associated with the DF growth and acceleration. According to Figure 15, their formation involves violation of the frozen-in condition and it is driven by differential plasma motions, fast DF regions and relatively slow-moving plasma in their wakes. Figure 15a shows that the DF formation is a non-MHD process, because the ion fluid is not frozen in the magnetic field and it does not follow the E × B drift. The secondary X line formation looks similar to the bubble-blob pair formation process, which has recently been described on the basis of MHD simulations by Hu et al. [2011] (compare, in particular, the interpretation of their simulations given in Figure 7 of that work with the ion outflow pattern in Figure 15e). Similar to the latter work, in our simulations the DFs have properties of plasma bubbles with the stronger magnetic field and smaller plasma density and they move indeed much faster than their wake regions. The latter effect is also seen in run 3 (Figures 9g and 9h), where plasma outflow velocities inside the region ∣x∣ < 10d remain strongly sub-Alfvénic even in the fast phase. Figures 15b–15d further support that the DF formation is a rather spontaneous process, weakly dependent on the external driving field and dominated by the ion dissipation. It is not caused by any changes of magnetic topology but rather promotes these changes being a core part of the reconnection onset mechanism.

Figure 15.

Run 4 at the moment Ωt = 40: (a) electric field Ey at the neutral plane (z = 0) compared to the corresponding −v × B terms for ions (red) and electrons (blue), (b) color-coded electric field Ey and magnetic field lines, energy dissipation rates j · E for (c) electron and (d) ion species, and (e) x component of the ion bulk flow velocity vix.

5.4. Two-Dimensional Equilibrium With Lembege-Pellat Magnetotails: Transition to a Multiscale Configuration

[38] In run 5 we investigated the onset of reconnection in the equilibrium with Lembege-Pellat magnetotails (Figure 1a) under the action of the reduced driving field with E0(dr) = 0.05, below the sustained reconnection limit Ey ≈ 0.15 [Shay et al., 2007]. Figure 16 shows the evolution of the total current density in this run, which appears to be similar to run 1 (Figure 2) and consistent with the conventional picture known from previous simulations of perturbed 1-D current sheets with open boundaries [Daughton et al., 2006; Klimas et al., 2010] and in very large simulation boxes [Fujimoto, 2006; Shay et al., 2007]. This includes the formation and subsequent elongation of the electron-scale layer near the initial central X line (we also observed the formation of secondary islands at Ωt > 62, not shown in Figure 16).

Figure 16.

Reconnection onset in run 5 in a format similar to Figures 5 and 8 showing the reconnection onset in the case of Lembege-Pellat magnetotails and the weak driving electric field Ey ∼ 0.05.

[39] On the other hand, similar to another group of kinetic simulations based on 1-D equilibria [Wan and Lapenta, 2008; Sitnov et al., 2009], the formation and elongation of the electron diffusion region is accompanied by the formation and acceleration of DFs (Figures 17e and 17f). Moreover, similar to previous runs 3 and 4 with the reduced driving field, the current sheet evolution in run 5 can be clearly separated in the slow (Figures 17a–17d) and fast (Figures 17e–17h) phases. Note, that several reconnection stages had also been identified in simulations of 1-D Harris equilibria [Wan and Lapenta, 2008]. However, the slow phase cannot be faithfully reproduced in any simulations starting from perturbed 1-D Harris sheets because of the inherently nonequlibrium nature of the initial state of the system. In contrast, the slow phase in run 5 is relatively long (0 ≲ Ωt ≲ 35), and similar to run 3, it is characterized by strongly sub-Alfvénic ion and electron outflows (Figures 17c and 17d). Interestingly, one of the main changes in the slow phase is the formation of the Bz humps in the original Lembege-Pellat magnetotails (Figure 17a), which makes the original equilibrium at the end of the slow phase more similar to the multiscale equilibria used in runs 2–4.

Figure 17.

(a–d) Slow and (e–h) fast phases of the current sheet evolution in run 5 in a format similar to Figure 9.

[40] The fast phase in run 5 is also similar to that in run 3 and it is characterized by the rapid growth of the electric field (Figure 17f) as well as the accompanied acceleration and steepening of the Bz humps (Figure 17e). Note that the electric field Ey near the central X line does not reach the sustained reconnection limit Ey ∼ 0.15 [Shay et al., 2007] and saturates at the level on the order of the driving field value E0(dr) = 0.05. This is different from the transient picture observed in simulations with open boundaries in case of a perturbed 1-D Harris sheet [Daughton et al., 2006, Figure 6; Wan and Lapenta, 2008, Figure 1b] and in large boxes [Fujimoto, 2006, Figure 1], where the electric field near the X line has a significant overshoot above its asymptotic value. In contrast to the relatively small saturated value of the electric field near the X line, in run 5 the electric field in the outflow regions quickly grows to the level much larger than the driving field (Figure 15d). This is also different from the case of the strong driving in run 1 and from simulations with 1-D equilibria, where the initial electric field peak near the X line is getting bifurcated and propagates away from the X line as a pair of switch-on fronts [Wan and Lapenta, 2008; Sitnov et al., 2009].

[41] Interesting new features of the fast phase in run 5, compared to the similar phase in run 3, are stronger plasma outflows (Figures 17g and 17h). While the ion outflow speed is approximately two times larger than in run 3 (Figure 9h), the electron outflow speed is larger than in run 3 (Figure 9g) by an order of magnitude. Similar fast electron outflows were found in simulations of DFs with 1-D equilibria [see, e.g., Sitnov et al., 2009, Figure 5d]. However, in contrast to those simulations, rapid changes of plasma flows and electromagnetic fields in run 5 appear rather spontaneously, after an extended slow phase without any significant changes, except the buildup of the Bz humps. It is tempting to interpret the transition from slow to fast phases in run 5 in terms of the tearing instability, whose sufficient stability condition (3) is indeed relaxed because of the formation of Bz humps. However, in contrast to multiscale equilibria, in this case such an analysis must include the effect of fast plasma outflows [e.g., Loureiro et al., 2007], which is not done yet for collisionless plasmas with the finite Bz component.

6. Discussion

[42] New features of reconnection onset in 2-D current sheet equilibria discovered in our simulations and associated with the formation of DFs with subsequent changes in magnetic topology are surprisingly consistent with recent models and ideas in substorm theory. In particular, the multiscale magnetotail equilibria and the resulting reconnection onset features revealed in runs 2–4 bear a strong resemblance to recent statistical visualizations of substorm onset in the terrestrial magnetotail based on Geotail data [Machida et al., 2009]. First, prior to substorm onset, it shows the formation of distinct Bz humps at the radial distance 19–25 RE (see the first four panels in Machida et al.'s Figure 2c). Their scale size (∼2 RE) is of the same order of magnitude as the equilibrium considered above, given that with the plasma density in that region n0 ∼ 0.2 cm−3 [Ohtani et al., 2004] the ion inertial scale d ∼ 0.1RE. Note, that the formation of thin current sheets prior to the onset of reconnection at their tailward end was suggested in other studies [Asano et al., 2004; Miyashita et al., 2009]. These effects are also seen in global MHD [Merkin and Goodrich, 2007, Figure 2] and full-particle [Pritchett and Coroniti, 1995, Figure 8] simulations of convection in the magnetotail as well as in the 2-D magnetohydrostatic theory of the magnetotail boundary deformations [Birn and Schindler, 2002, Figure 6]. Earlier the formation of a thin current sheet extended from near geosynchronous orbit to the middle of the magnetotail (∼30 RE) during the growth phase of a substorm was inferred by Pulkkinen et al. [1999] from multisatellite measurements. The formation of the near-Earth minimum in the Bz profile was found as a consequence of the steady state adiabatic convection in the tail plasma sheet [Erickson, 1984; Hau et al., 1989; Hau, 1991]. Recently an extension of the interval of very small (less than 5nT) Bz values up to ∼10 RE along the tail prior to a DF event was registered by THEMIS probes P1-P3 near the neutral plane [Runov et al., 2009].

[43] The Geotail visualization further shows the penetration of the dawn-dusk electric field toward the neutral plane just tailward of those humps and quite far away from the X line (see the first panels in Figure 2f of Machida et al., 2009), which is consistent with the similar effects shown in Figures 7a and 15b. The main feature of the empirical onset picture termed “catapult current sheet relaxation” are fast earthward flows [Machida et al., 2009, Figure 2a] providing the dipolarization of the original stretched magnetotail and preceding the formation of the near-Earth neutral line. In our simulations a similar relaxation has the form of the formation and ejection of DFs with the subsequent formation of new X lines in their wakes. It is also worth noting that, after the DF ejection in run 2 (Figure 5e), the magnetotails not only restore their ion-scale thickness (Figure 3, Ωt = 36), but they become even thicker than in the original equilibrium, though with rather complex internal structure with two, three and even four current peaks. This expansion is consistent with the observation that the plasma sheet becomes thicker in association with reconnection in the magnetotail [Ohtani and Mukai, 2006], but it is quite distinct from the reconnection process in runs 1 and 5, dominated by the current sheet with the electron-scale thickness.

[44] A mechanism similar to the catapult current sheet relaxation has been recently proposed on the basis of kinetic ring current and MHD simulations [Yang et al., 2011; Hu et al., 2011]. It involves the formation of a bubble and a blob whose different motions along the magnetotail result in the formation of a new X line. The mechanism is based on the ballooning/interchange instability providing different motions of bubbles and blobs because of their different buoyancy properties but it also requires some kinetic effects providing the violation of the frozen-in condition. Such processes have indeed been found in 3-D full-particle simulations of the magnetotail in the form of the plasma sheet disruption by interchange-generated flow intrusions when the normal component of the magnetic field Bz increases in the tailward direction [Pritchett and Coroniti, 2011]. Our simulations (see, in particular, Figure 15) show that similar non-MHD plasma motions, which lead eventually to the X line formation in the magnetotail, can be a part of the general reconnection onset process at its initial phase, dominated by the DF generation and the tearing instability.

[45] The metastability of 2-D current sheet equilibria was predicted already in early kinetic models of reconnection onset based on linear tearing stability theory [Schindler, 1974; Galeev and Zelenyi, 1976]. The metastable regimes found in our study are also shown to be consistent with tearing stability theory including its recent updates. This theory may be useful in further interpretation of simulations of the DF formation process. For example, distinctive features of the ion tearing instability, and in particular, its large growth rate γi, compared to that of the electron tearing mode (γi/γe = (miTi/meTe)1/4 [e.g., Kuznetsova et al., 1996]) help explain one of the key puzzling features of DFs in the magnetotail, namely, the north–south asymmetry of their Bz profile. The tearing instability starts as a bipolar perturbation in the tail current sheet with the finite Bz field, when the electron Landau dissipation is blocked. However, when the negative part of the bipolar perturbation reduces the Bz field value and makes it close enough to zero to demagnetize electrons, the tearing instability there transforms into a much slower electron tearing (γi/γe ∼ 4 in our runs and ∼10 for realistic magnetotail mass and temperature ratios). As a result, further reduction of Bz below the zero level strongly slows down, causing the effect termed “re-reconnection” in the DF phenomenology [Slavin et al., 2003]. The re-reconnection concept is further strengthened in the model of the so-called nightside flux transfer events (NFTE) [Sergeev et al., 1992] and its MHD basis, the model of impulsive reconnection [Semenov et al., 2005, and references therein]. The latter model presupposes conservation of the magnetic field topology in the outflow regions during periods of the enhanced reconnection, and it may serve as a prototype for the MHD description of DFs [Runov et al., 2011].

[46] At the same time, the conventional tearing stability theory for steady state plasmas may be insufficient for the interpretation of the DF formation simulations. This is particularly relevant to their original setups based on perturbed 1-D current sheets where the DF onset is strongly influenced by the reconnection plasma outflow, which is known to affect the tearing instability properties [Loureiro et al., 2007]. A challenging result here is the finding of the DF formation in the low-density electron-positron plasmas [Bessho and Bhattacharjee, 2010]. In the latter case DFs have strong peaks of the electric field with its relatively small values in DF wakes, similar to our results for 2-D multiscale equilibria. The DF enhancement in the electron-positron plasma case is still consistent with the magnetotail plasma theory, which predicts tearing destabilization even in the presence of the finite Bz field for mi = me [Zwingmann et al., 1990; Dreher et al., 1996]. However, the Bz asymmetry in that case cannot be explained by different growth rates for electron and ion tearing, and it requires a special investigation such as a kinetic generalization of the tearing stability analysis for current sheets with shear flows or a self-consistent kinetic model of fully fledged DFs.

[47] Even for multiscale equilibria the interpretation of the DF formation in terms of the conventional tearing stability theory must be made with caution. First, the transition from slow to fast evolution in metastable systems, as is the case in our simulations with weak driving, must involve nonlinear effects [see, e.g., Chen, 1974, chapter 6]. Also as is already mentioned above, the excited DFs resemble rather solitons than monochromatic waves. Therefore their comparison with the linear WKB stability theory is inherently limited and the further interpretation of the DF formation process requires a generalization of the tearing stability theory beyond the WKB approximation.

7. Conclusion

[48] In this work full-particle simulations of magnetic reconnection onset have been performed, starting from 2-D self-consistent equilibria and being triggered by broadly distributed convection electric field, settings that have never been explored before. The main result of our work is the discovery that the formation of dipolarization fronts is a generic feature of the reconnection onset and an important element of its mechanism. This result could not be obtained in conventional simulation settings, when reconnection was triggered by a perturbation of the 1-D Harris current sheet, because in that case the DF formation could be attributed to artificial transients arising from the non-self-consistent nature of the latter simulation setup. Moreover, the use of the new simulation setup revealed new effects in the reconnection process, including its dependence on the structure of the initial 2-D equilibrium, metastable nature of the DF formation and the consistency of the reconnection onset mechanism with the tearing instability theory. Below we list these specific new findings.

[49] 1. In the onset process the driving/convection electric field may penetrate the current sheet aside of the X line. It penetrates instead regions of accumulated magnetic flux and increased ∣Bz∣ field. The initially broadly distributed electric field is focused on the scale of the order of the ion inertial length and strongly amplified in strength whereas the original mesoscale ∣Bz∣ humps sharpen to form microscopic and rapidly moving DFs. Then electron diffusion regions form in the DF wakes rather then near the initial X line.

[50] 2. Simulations with weak driving show that the formation of DFs is a metastable process consisting of slow and fast evolution phases. The transition from slow to fast phase is not caused by topological changes, such as the formation of secondary magnetic islands. The fast phase associated with the formation of DFs rather resembles the formation of bubble-blob pairs, which then promote the formation of secondary islands and the corresponding electron diffusion regions.

[51] 3. In the case of multiscale equilibria with Bz humps and in the slow phase the areas of the relaxed stability in terms of the sufficient stability criterion (3) are shifted to the areas with the stronger current density and smaller current sheet thickness. When the latter parameter reaches a critical value, the fast exponential growth of the electric field inside the current sheet commences. In case of the 2-D equilibrium with tearing-stable magnetotails the slow phase includes also an additional process of the formation of Bz humps that form as a result of slow flux transfer through the initial central X line.

[52] 4. Fast phase of reconnection onset in case of equilibria with multiscale magnetotails has properties of the ion tearing instability: (1) the ion-dominated energy dissipation, (2) exponential growth of the reconnection electric field inside the current sheet with the growth rate and spatial structure of the field across the current sheet being consistent with the linear stability theory, and (3) stability thresholds consistent with the tearing stability theory. At the same time the structure of DFs along the direction of their propagation resembles solitary waves and caution must be exerted in further interpretation of their formation process in terms of the quasi-monochromatic wave stability theory based on the WKB approximation.

[53] 5. Reconnection onset and the formation of dipolarization fronts observed in the present simulations are found to be consistent with the new catapult current sheet relaxation model of magnetospheric substorms [Machida et al., 2009] and with another recent substorm concept of the bubble-blob pair formation associated with the violation of the frozen-in condition [Hu et al., 2011; Yang et al., 2011].

Acknowledgments

[54] The authors acknowledge useful discussions with K. Schindler, P. Pritchett, J. Drake, N. Bessho, A. Divin, B. Anderson, B. Mauk, V. Merkin, D. Mitchell, A. Runov, and S. Ohtani. This work was supported by NASA grants NNX09AH98G and NNX09AJ82G as well as NSF grant AGS0903890. Simulations were performed at the NASA Advanced Supercomputing Division.

[55] Masaki Fujimoto thanks the reviewers for their assistance in evaluating this paper.