Journal of Geophysical Research: Space Physics

Dipolarization fronts as a signature of transient reconnection in the magnetotail



[1] Dipolarization fronts (DFs), characterized by a strong and steep increase of the tail magnetic field component Bz normal to the neutral plane and preceded by a much less negative dip of Bz, are reported in many observations of bursty bulk flows and substorm activations throughout the whole Earth's magnetotail. It is shown that similar structures appear in full-particle simulations with open boundaries in a transient regime before the steady reconnection in the original Harris current sheet driven out of the equilibrium by the initial X-line perturbation is established. Being secondary reconnection structures propagating with the Alfvén speed, DFs are different from the magnetic field pileup regions reported in earlier simulations with closed boundaries. They also differ from the secondary plasmoids with bipolar Bz changes reported in earlier fluid simulations and particle simulations with open boundaries. In spite of their transient nature, DFs are found to form when the force balance is already restored in the system, which justifies their interpretation as a nonlinear stage of the tearing instability developing in two magnetotail-like structures on the left and on the right of the initial central X-line. Both electrons and ions are magnetized at the front of the dipolarization wave. In contrast, in its trail, ions are unmagnetized and move slower compared to the E × B drift, whereas electrons either follow that drift being completely magnetized or move faster, forming super-Alfvénic jets. In spite of the different motions of electrons and ions, the growth of the front is not accompanied by the corresponding growth of the electrostatic field and the energy dissipation in fronts is dominated by ions.

1. Introduction

[2] It is already well established that the transport of the plasma and the magnetic flux in the tail of Earth's magnetosphere occurs largely in the form of discrete activations, such as substorms and bursty bulk flows [Baumjohann et al., 1990; Angelopoulos et al., 1992; Shiokawa et al., 1997, 1998; Fairfield et al., 1998, 1999]. The distinctive features of any such activation are a high-speed plasma flow burst and a strong and steep increase of the tail magnetic field component Bz normal to the neutral plane. The latter effect, known as the magnetic field dipolarization is usually interpreted as the magnetic field pileup effect, arising when the fast plasma flow ejected from the reconnection region in the mid tail is braked in the near-Earth tail region because of the higher plasma density and stronger more dipolar magnetic field there [Hesse and Birn, 1991; Shiokawa et al., 1997]. The problem however is that the dipolarization phenomena are observed throughout the whole magnetotail, from X = −5 RE to −31 RE [Slavin et al., 1997, 2003; Fairfield et al., 1999; Tu et al., 2000; Nakamura et al., 2002, 2005; Ohtani et al., 2004; Eastwood et al., 2005; Angelopoulos et al., 2008]. As is demonstrated in Figure 1 reproduced from [Ohtani et al., 2004], they appear everywhere in the tail as dipolarization fronts, that is, intervals with a steep increase of the tail magnetic field component Bz normal to the neutral plane, preceded by a much less negative dip of Bz and followed by a more gradual decrease of Bz.

Figure 1.

Dipolarization fronts in the superposed epoch analysis by Ohtani et al. [2004]: (a) ion bulk flow speed, (b) Bz magnetic field component, (c) plasma density, (d) ion temperature, (e) ion pressure, (f) total pressure.

[3] In some cases [Ohtani et al., 2004] dipolarization fronts (hereafter DFs) are accompanied by similar bursts of the plasma flow. In others they were observed near the leading edges of the more extended earthward flows and were interpreted as BBF-type flux ropes [Slavin et al., 2003]. However, most if not all of the DFs have the same common puzzling feature, namely, the profound asymmetry between their southward and northward Bz perturbations. This effect is observed everywhere in the magnetotail, and hence its interpretation cannot be limited by the pileup mechanism within the framework of the conventional flow-braking theory. To extend the latter theory Slavin et al. [2003] proposed the idea that the dipolarization phenomena may arise due to the secondary magnetic islands in the closed field line regions created by the dominating X-line. The idea was based on an earlier argument by Schindler [1974] who noticed that in the process of the tearing instability in the magnetotail one of the X-lines will inevitably outpace the others. Slavin et al. [2003] also suggested that the asymmetry appears because the southward magnetic flux dissipates in the process, which they called “re-reconnection”, when the flux rope is pushed earthward against the main northward geomagnetic field. Ohtani et al. [2004] compared their DF observations with isothermal two-fluid simulations, when the current sheet with antiparallel magnetic fields is perturbed to excite the resistive tearing mode, and the secondary magnetic islands start forming in the exhausts of the main X-line reconnection pattern. They found that many details of the plasma and magnetic field evolution in such a simulation setup, which was based in fact on the above Schindler-Slavin scenario, were consistent with their DF observations. These include the increase of the plasma bulk flow speed measured by virtual satellites placed in the outflow region of the central X-line, negative and then positive variations of the Bz field and a decrease of the plasma density preceded by its significant overshoot. However, they have also found that the north-south asymmetry of the Bz perturbations was not reproduced in the fluid modeling, which showed almost equal positive and negative Bz variations in the secondary plasmoids.

[4] In this paper we demonstrate that the north-south Bz asymmetry and many other distinctive features of DFs can be reproduced in the full-particle simulations with the initial setup similar to the GEM Magnetic Reconnection Challenge (GEM MRC) simulations [Birn et al., 2001] and a new set of open boundary conditions [Divin et al., 2007] similar to other recent models [Daughton et al., 2006; Klimas et al., 2008]. In contrast to the setup used by Ohtani et al. [2004] and similar particle simulations [Hoshino et al., 1998, 2001], where the reconnection is initiated by a very small perturbation of the current sheet with zero Bz and where the growth of magnetic islands on the kinetic level is provided by the electron tearing instability [Coppi et al., 1966], in the GEM MRC setup the initial perturbation is relatively strong, with the maximum value of the Bz field being up to one tenth of the lobe field and the width of an initial magnetic island being comparable to the width of the current layer (with an original rationale to skip the stage of the electron tearing growth). Since initially the j × B force is not balanced by the appropriate pressure gradient, the focus of the original GEM reconnection studies was the long timescale evolution. We found however that the pressure gradient necessary for the MHD force balance is restored in our simulations within one ion gyroperiod. At later time, and before the stage which was considered earlier as a steady state reconnection period, we discovered a new regime of the transient reconnection, which strongly resembles the DF phenomena. We also show that in spite of the fact that some features of this DF reconnection regime resemble earlier results obtained in simulations with periodic/closed boundaries [Horiuchi and Sato, 1997; Hoshino et al., 1998, 2001; Pritchett, 2001a], it has practically no analogs because it is distinguished by the formation and fast propagation of the secondary plasmoids with almost monopolar (northward type) Bz perturbations, where the dissipation of the electromagnetic energy is dominated by ions independent of the initial ion-electron temperature ratio in the system. Interestingly, these new kinetic simulations support the concept of “re-reconnection” by Slavin et al. [2003].

[5] The structure of the paper is as follows. In section 2 we overview the new simulation setup with open boundaries. In section 3 we show the results of our basic run, which most clearly exposes the new transient reconnection features. Other runs with different parameters and DF features are described in section 4. In section 5 the results are further discussed, including their detailed comparison with earlier simulations, tearing stability theory and spacecraft observations. The main findings are summarized in section 6.

2. Open Boundary Conditions and Basic Simulation Setup

[6] In our simulations we model the conditions in the magnetotail by imposing a GEM MRC-type perturbation of the magnetic field [Birn et al., 2001] on the Harris current sheet [Harris, 1962]. As is shown in the next section, the MHD force balance, broken by the perturbation, is restored within one ion gyroperiod. Therefore at later time the central X-line created by that perturbation can be interpreted as the distant X-line, separating two tail-like regions with the finite Bz field and with the reconnection at that X-line providing an analog of the magnetospheric convection in those tail-like regions. It can also be interpreted as the major near-Earth X-line where the reconnection has already reached the lobe field region as suggested by Slavin et al. [2003]. The main difference between our force-balanced X-line pattern and earlier simulations with the reconnection initiated by the electron tearing mode [Hoshino et al., 1998, 2001] is that the stronger Bz field in our case allows one to investigate new reconnection regimes with magnetized electrons.

[7] To model the evolution of our system we employ an open-boundary modification [Divin et al., 2007] of the explicit massively parallelized full-particle code P3D [Zeiler et al., 2002]. The Boris algorithm is used in the code to advance in time the Lorentz equation of motion for each particle, and an explicit trapezoidal-leapfrog method employing second-order spatial derivatives in Faraday's and Ampere's Laws is used to advance the fields in time. While the equation ∇ · B = 0 is automatically satisfied by the computational algorithm, this is not the case for the Poisson's equation, so that the latter equation for the discretization error is explicitly solved using the multigrid technique [Press et al., 1999]. Particles are loaded in the code using the rejection method [Press et al., 1999] with the initial number of particles per grid Nppg = 50.

[8] In the following the magnetic field and plasma density are normalized by their maximum values in the Harris equilibrium B0 and n0. The time and space scales are normalized by the inverse ion gyrofrequency Ωi−1 in the field B0 outside the sheet and the inertial length di = c/ωpi based on the plasma density n0. The unperturbed plasma density in the original physics units is given by the expression n(z) = n0 cosh−2(z/λ) + nb, where nb = 0.2n0 is the background density. The Harris magnetic field is given by the formula Bx = B0 tanh(z/λ), and its perturbation is determined by the following flux function ψ(x, z) = ψ0 cos(2πx/Lx) cos(πz/Lz), where Lx = Lz are the box dimensions.

[9] The box dimensions in our simulations do not exceed 40di. Therefore in view of the typical values of the ion inertial length diρ0i ∼ 500 km, we only could simulate a relatively small fraction of the actual magnetotail with its typical length Lx(tail) ∼ 25 RE ∼ 250 di. To overcome this limitation, and in particular, to model the effects of passing particles in the outflow regions of the primary X-line pattern we employ open boundary conditions. Specifically, we impose the conditions of continuity across the X boundary on the first two moments of the distribution functions

equation image

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 equation (1) and having the original temperatures Tα = Tα(t = 0). These conditions combine earlier open setups of Pritchett [2001b], who injected the initial Maxwellian distributions, and Daughton et al. [2006], who required, in addition to equation (1), the continuity of the pressure tensor components. Our choice of the boundary conditions dates back to Divin et al. [2007] who investigated the impact of passing particles on the stability of the tearing mode. It is based on the results of the linear tearing stability analysis [Sitnov et al., 2002, sections 2.1–2.2], which show that passing particles retain their adiabatic response to the tearing perturbation. Specifically, their contribution to the density variation in response to the tearing electrostatic potential ϕ1 is given by the following linear perturbation of the distribution function

equation image

where Hα is the total particle energy invariant in the Harris distribution function. Similarly, the perturbation of the distribution function that contributes to the y component of the bulk flow velocity variation in response to the variation of the vector potential component Ay1 is given by the expression

equation image

where P is the y component of the generalized momentum, another integral of motion in the Harris distribution function, a function of the equilibrium vector potential component Ay0. This means that for the proper description of passing particles, open boundary conditions must include the conditions of continuity for the density and the y component of the bulk flow speed. We further discuss different types of open boundaries for particles in section 5.

[10] The problem of finding a realistic and numerically stable set of open boundary conditions on the electric and magnetic fields is far from being solved, especially for collisionless plasmas [Lindman, 1975; Engquist and Majda, 1977; Higdon, 1986; Renaut, 1992; Horiuchi et al., 2001; Pritchett, 2001b; Daughton et al., 2006]. 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 [Divin et al., 2007], the following set was found to yield the most interesting and numerically stable results with no artificial wave or magnetic flux 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 [2001b] to provide free propagation of the flux through the boundary. Our conditions appear to mimic the removal of the magnetic flux through the dayside magnetopause. However, in contrast to the real magnetopause, we have no parameter such as the interplanetary magnetic field clock angle to control the rate of effective reconnection at the boundaries, and the “double-tail magnetosphere” in our simulation box remains always “open”.

[11] 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 [2001b]) 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 timescales considered it does not result in any noticeable loss in the total flux or a reduction in the total plasma and magnetic pressure. Besides, the closed Z boundaries help model the effect of the northward turning of the IMF Bz component, which is often considered as a prerequisite of the substorm onset [Caan et al., 1977; Rostoker, 1983].

3. Dipolarization Front in Simulations With Open Boundaries

[12] Recent particle simulations with open boundaries [Daughton et al., 2006] revealed a number of new effects, such as the elongation of the electron dissipation region and the formation of the secondary magnetic islands that were not detected before for the case of antiparallel undriven reconnection in the simulations with closed boundaries [Hesse et al., 2001; Pritchett, 2001a; Shay et al., 2001; Ricci et al., 2002; Karimabadi et al., 2005]. Later, Divin et al. [2007], using a similar set of open boundaries, demonstrated that in a simulation setup with the initial geometry corresponding to the GEM Reconnection Challenge [Birn et al., 2001] bursts of spontaneous reconnection occur in the tail-like outflow regions of the central X-line. Bursts developed on the timescales when the electrons were magnetized by the field Bz whereas ions remained unmagnetized, and the corresponding reconnection was faster than the electron tearing instability of a current sheet with the same thickness. That strongly suggested that they dealt with the ion tearing instability. Moreover, quenching the formation of plasmoids by replacing open boundary conditions for particles with their reintroduction confirmed the theoretical predictions [Sitnov et al., 2002] that the onset of reconnection in the magnetotail is controlled by the availability of passing particles.

[13] The present study was originally intended to refine the results of Divin et al. [2007], and in particular, to clarify the role of the initial GEM perturbation and the response of ions and electrons with the aim to understand the mechanism of reconnection in the magnetotail and its relation to the ion tearing instability. In simulations by Divin et al. [2007] the initial GEM-type perturbation was rather strong. In particular, in their basic run with ψ0 = 0.3 B0di in the box with Lx = Lz = 19.2 di the maximum value of the Bz component at z = 0 was equal to 0.1, and the unbalanced force jyBz might be an additional trigger of reconnection in the tail-like outflow regions of the central X-line reconnection pattern. To reduce that effect, in the basic run of the present study (Run 1) we used a weaker initial perturbation ψ0 = 0.15 B0di. At the same time, we increased the mass ratio mi/me = 128 to keep electrons magnetized on the timescales Δt of the expected instability (Bz/B0iΔt(mi/me) ≫ 1. Other parameters of our basic run are the following: half thickness of the Harris sheet λ = 0.433 di = 0.5 ρ0i, temperature ratio Ti/Te = 3, and the speed of light in the code units c/vA = 15, where vA is the effective Alfvén speed vA = B0/equation image. Run 1 employed 6.6 · 107 particles per species and that number was increased by a factor of 4 in Runs 3 and 4 discussed in the next section.

[14] Common wisdom suggests that the reduction of the field Bz should result in the formation of conventional plasmoids, similar to the magnetic islands that appear in the unperturbed Harris sheet due to the electron tearing instability. However, Figures 2 and 3 show a completely different picture. According to Figure 2, one island is indeed formed in each of the two outflow regions. However, the width of these islands in the Z direction is very small. At the same time, on the right (left) of the left (right) island one can observe regions of very strong pileup of the Bz field. Figure 3 shows that those pileup regions look like sharp fronts propagating with the Alfvén speed from the central X-line to the boundaries of the simulation box. The front has a strong and steep increase of the original tail field Bz, up to a half of the lobe field, in contrast to the much shallower trail region of this dipolarization wave, and in contrast to a relatively small and shallow negative dip of Bz in the front precursor, comparable in amplitude to the field Bz at t = 0. It is interesting that the formation of the secondary X-line (at Ωit ≈ 10 in the left tail and ≈12 in the right in Figure 3) does not result in any dramatic changes in the evolution of the system (it is also seen in the evolution of other parameters discussed below), and in particular, in any further increase of the Bz variation opposite in sign to the original tail field (negative at x < 0 and positive at x > 0).

Figure 2.

Magnetic field lines and the color-coded current density component −Jy for Run 1 at the moment Ωit = 12.

Figure 3.

Evolution of the normal magnetic field Bz at the neutral plane z = 0 in Run 1.

[15] These characteristic changes of the Bz component are consistent with observations of similar DF stuctures in the Earth's magnetotail. The examples include observations of substorm dipolarizations using Wind and Geotail [Slavin et al., 1997, Figure 3; Fairfield et al., 1999, Figure 7] as well as recent Themis data [Angelopoulos et al., 2008, Figure 4h] and similar BBF observations using Geotail and Cluster data [Tu et al., 2000, Figure 2; Nakamura et al., 2002, Figure 1; Slavin et al., 2003, Figure 3; Nakamura et al., 2005, Figure 3; Eastwood et al., 2005, Figures 1 and 2]. These case studies are substantiated by the statistical analysis of 818 fast earthward flow events in Geotail data [Ohtani et al., 2004, Figures 4 and 5; the latter figure is reproduced as Figure 1 in the present paper]. It is important that Ohtani et al. [2004] compared their observations with two-fluid simulations and found that the key DF feature, namely, the small decrease of the Bz field prior to the front, could not be reproduced by the fluid modeling, which shows almost equal positive and negative Bz variations in the secondary plasmoids.

[16] The crucial question is whether the new DF-like structures may indeed be interpreted as a sustained regime of reconnection in the presence of such a relatively strong GEM MRC perturbation. As shown by Pritchett and Coroniti [2004], for the typical GEM MRC parameters the plasma pressure drop along the X direction from the boundaries to the center of the box, which is necessary to balance the corresponding jyBz force, is comparable to the total pressure B02/8π. In other words, such a strong perturbation creates an order of unity MHD stress imbalance along the current sheet. No real current sheet could ever be so far out of pressure balance. However, as is shown below, the imbalance in our case is compensated rather quickly. The evolution of the MHD force balance terms in Run 1 is shown in Figure 4 (since the ion temperature is much in excess of the electron temperature and the electrostatic field Ex is strongly attenuated due to our open boundary conditions, we show in Figure 4 only the ion contributions). According to Figure 4b, the ion pressure gradient necessary to balance the jiyBz force is built up in the main part of the simulation box already by the moment Ωit = 4, before the onset of wavy excitations leading to the formation of DFs. Figure 4 also shows that the ion pressure gradient follows the jiyBz force up to Ωit = 10, and it is again strongly violated only when the DF wave is transformed into a steep front. Moreover, the force imbalance in a fully developed DF (Figure 4f) is an order of magnitude stronger than even the initial imbalance, which therefore cannot drive the DF formation process. (Note here, that the latter imbalances are unlikely to be caused by boundary conditions because they are also seen deep inside the simulation domain in a similar Run 3 made in a bigger simulation box.) Thus the formation of a DF has nothing to do with any current sheet collapse in response to the broken MHD force balance in the system, and it rather resembles an instability of an already force-balanced system. The latter force balance in the period Ωit = 4–10 is transient and incomplete, because the reconnection has already started in the system and plasma particles are getting accelerated on their way from the central X-line toward the box boundaries. However, similar particle acceleration takes place in any steady reconnection regime and it should also occur near any X-line in the magnetotail, including the distant and near-Earth X-lines.

Figure 4.

Distribution of the main parameters responsible for the force balance in the system, jiyBz (blue lines) and ∂Pixx/∂x (red lines) along the neutral plane z = 0 at different moments in Run 1. Both parameters are averaged in space over the interval δx ≈ 0.8di to reduce noise in the data.

[17] The existence of a relatively steady current sheet with a significant plasma pressure buildup toward the boundaries prior to the formation of DFs is also seen in Figures 5a and 5b in the form of the bunching of the density and temperature profiles for the moments Ωit = 2,4, and 6, compared to the later profiles. The results of the force balance analysis shown in Figure 4 are further supported by the evolution of the bulk ion and electron velocity components along the X direction (Figures 5c and 5d) that do not reveal any signatures of the catastrophic plasma acceleration at early times.

Figure 5.

Evolution of (a) plasma density ni, (b) ion temperature Ti, (c) ion bulk flow speed Vix, (d) electron speed Vex, (e) electron current density jey, and (f) total current density jy at the neutral plane z = 0 in Run 1. All parameters are averaged in space over the interval δx ≈ 0.4di to reduce noise in the data.

[18] It is interesting, that the strongest (an order of magnitude stronger than the initial) force imbalance in the system (Figure 4f) is very localized in space and it occurs in the regions of practically zero jiyBz force. The corresponding unbalanced pressure gradient might be balanced in case of a conventional plasmoid structure with a negative dip of Bz comparable in amplitude to the positive perturbation, which is surprisingly missing in DFs. At the same time, such a pressure gradient is indeed observed near the minimum of Bz field in Geotail data (as is shown in Figure 1 [Ohtani et al., 2004]). As follows from Figure 5a, the corresponding changes of the plasma pressure are mainly provided by the plasma density overshoot followed by its drop below the initial values. According to Figure 5b, the ion temperature, which is estimated as Ti = (Pixx + Piyy + Pizz)/3ni, where Piαβ and ni are the components of the pressure tensor and the plasma (ion) density, increases rather monotonically and more gradually than the plasma density. These density and temperature variations are consistent with Geotail observations [Ohtani et al., 2004] shown in Figure 1.

[19] Another overshoot is seen in the ion bulk flow velocity component Vix (Figure 5c) and it shows that the DF is also a burst of the plasma flow. However, further reduction of the ion bulk flow speed as a function of time at the given distance from the X-line, which is seen in DF observations [Ohtani et al., 2004], is not well seen in this run (there is only a mild decrease in the region −3 di < x < 4.5 di for Ωit > 12). This occurs because of the effects of the global reconnection in the system associated with the central X-line. Its rate does not depend on the amplitude of the initial perturbation ψ0 and can only be reduced using a more sophisticated simulation setup that would allow a gradual transition from open to closed boundary conditions. Note that the persistent increase of the bulk flow speed well after the magnetic field front is nevertheless fully consistent with another group of DF observations by Slavin et al. [2003], who found that in most cases their magnetic flux ropes were observed near the leading edge of the earthward flow event. We further discuss this issue in section 5.3.

[20] The effect of the central X-line reconnection is also seen in the profile of the electron bulk flow speed (Figure 5d). It reveals even stronger overshoot signatures that have however no direct relation to DFs as is shown below. As one can see from Figure 5d, near the electron dissipation region associated with the central X-line electrons form two jets moving with the speed that strongly exceeds the Alfvén speed vA. Although such super-Alfvénic jets were reported earlier [see, for instance, Pritchett, 2001a, Figures 2 and 5] those narrow (∣δz∣ < di) high-speed jets attracted special attention after publishing the first results on simulations with open boundaries [Daughton et al., 2006] and were extensively discussed later in [Karimabadi et al., 2007; Shay et al., 2007; Drake et al., 2008]. As is concluded by Drake et al. [2008], being an interesting distinctive feature of the collisionless reconnection, those jets appear to play a minimal role in convecting electrons away from the X-line.

[21] Figure 5e shows the evolution of the electron current density jey in the plane z = 0. It is interesting that by the time Ωit = 8 the maximum electron current density saturates (having presumably formed a fully fledged electron dissipation region), and further its spatial profile only slightly spreads not to exceed ∼10 di, consistent with the results by Shay et al. [2007]. On the other hand, as follows from Figure 5f, the most prominent changes in the total current density, namely the formation of strong depression regions and sharp peaks off the central X-line are clearly associated with the formation and propagation of DFs (note, that in view of evident symmetry properties of the parameters shown in Figure 5 they are displayed only for x > 0).

[22] Figure 6 shows two-dimensional distributions of some field components and the plasma density at the moment Ωit = 12. Its most distinctive feature, compared to the GEM MRC picture provided by the simulations with closed boundaries [e.g., Pritchett, 2001a], is the bifurcation of the reconnection electric field. Note also, that the comparison of the By pattern in Figure 6b with the classical patterns, such as, for instance, Drake et al.'s [2008] Figure 5e, suggests that the DF provides a By perturbation opposite in sign compared to that associated with the central X-line. This picture is more similar to the formation of the secondary plasmoids in the process of the electron tearing instability [Hoshino et al., 1998, 2001]. However, in contrast to the latter works, the formation of the secondary plasmoids in our case starts when electrons are magnetized by the normal field Bz. This results in new drift and dissipation features discussed below and other important distinctions that will be further analyzed in section 5.

Figure 6.

Spatial distribution of some key parameters in Run 1 at the moment Ωit = 12: (a) reconnection electric field Ey, (b) out-of-plane magnetic field component By, (c) electric field component Ez, (d) ion density ni.

[23] The most striking and distinctive feature of DFs is their dissipation source. As follows from Figure 7, while the reconnection at the central X-line is provided by the dissipation of electrons in the central diffusion region (top panel), the similar dissipation in DFs is provided by ions. The latter source of dissipation would not be surprising at the initial phase of the front formation, which, as argued by Divin et al. [2007], strongly resembles the ion tearing instability. However, the domination of the ion dissipation at the nonlinear stage of the ion tearing evolution, especially after the formation of the secondary X-lines (Figure 2) and the corresponding electron diffusion regions, is really surprising. This domination suggests that the mechanism of the ion tearing instability that was originally proposed by Schindler [1974] to describe the initial stage of the reconnection in the tail is more fundamental and is also responsible for a special regime of the fully developed reconnection in the form of DFs. Note, that similar sharp ion energization fronts were observed in the current disruption events [Lui et al., 1988], in substorm injections [Baker et al., 2002], and in solitary electromagnetic pulses accompanying bursty bulk flows [Parks et al., 2007].

Figure 7.

(a) Electron and (b) ion contributions to the energy dissipation j · E in Run 1 at the moment Ωit = 12.

[24] Different bulk flow motions of electrons and ions as well as their specific contributions to the energy dissipation at different stages of the DF evolution are presented in Figure 8. Panels b, f, j, and n in Figure 8 show that ions mostly drop behind the E × B drift. This is not surprising, because the current sheet is thin and the initial Bz field is small. However, that “sub-E × B” drift of ions further supports one of the main points of the present study, namely the absence of any catastrophic collapse of the current sheet on the DF formation timescales, where the initially unbalanced j × B force would quickly expel ions out of the central X-line forcing them to move faster than magnetized electrons that follow the E × B drift. Electrons move slower than the E × B drift only inside the central diffusion region. In the rest of the simulation domain they either follow the E × B drift or move even faster, forming super-Alfvénic jets. The formation of separate ion dissipation regions in DFs, different from the central electron dissipation region, is clearly seen in panels c, g, k, and o. Comparison of the latter panels with the upper panels b, f, j, and n shows also that, in contrast to the original expectations [Daughton et al., 2006], super-Alfvénic jets do not extend the central electron dissipation region as their peaks rather correlate with the dynamo regions where je · E < 0. Finally, panels d, h, l, and p in Figure 8 clearly demonstrate that the growth of the electromagnetic components of the DF perturbation Bz and Ey is not accompanied by the corresponding growth of the electrostatic component Ex. The latter effect was predicted in the kinetic theory of the tearing instability [Sitnov et al., 2002], which shows that the electrostatic field must be shortcircuited by the population of passing electrons, whose availability is provided due to the open boundary conditions (equation (1)). Thus the results of Run 1 clearly identify DF as a new region of reconnection, complementary to the electron diffusion region, and possibly even a new regime of magnetic reconnection, distinct from more conventional antiparallel and guide-field cases [Rogers et al., 2003; Drake and Shay, 2007] due to its characteristic fingerprints in magnetic and electric fields as well as the location and sources of the energy dissipation.

Figure 8.

(a, e, i, m) Normal magnetic field, (b, f, j, n) out-of-plane electric field Ey (black line) compared with VixBz (red line) and VexBz (blue line) parameters, (c, g, k, o) total energy dissipation jtot · E (black line) and its ion (red line) and electron (blue line) components, and (d, h, l, p) electric field component Ex at the neutral plane z = 0 in Run 1 for different moments in time.

4. Other DF Examples

[25] To confirm our original findings described above we performed several additional runs with different parameters. Run 2, which differs from Run 1 by a stronger perturbation ψ0 = 0.3B0di and a smaller mass ratio mi/me = 64, is an important reference run with the regime of magnetized electrons similar to Run 1 (due to the same product Bz(max)mi/me) and to the basic run described in the earlier work by Divin et al. [2007]. It differs from the latter run only by the more realistic temperature ratio Ti/Te = 3. Compared to our basic Run 1, it shows less prominent signatures of DFs (Figure 9) that now arise together with a conventional plasmoid (the strongest bipolar Bz perturbation at the central X-line in Figure 9).

Figure 9.

Evolution of the normal magnetic field Bz at the neutral plane z = 0 in Run 2 with the twice stronger GEM perturbation compared to Run 1.

[26] DFs are better contrasted with a plasmoid in Run 3 (Figures 1012 ), an analog of Run 1, which is performed in a bigger box with Lx = Lz = 38.4 di, and where to keep the same maximum value of the Bz field in the tail-like regions Bz(max) = 0.05B0 the perturbed vector potential amplitude ψ0 is doubled ψ0 = 0.3B0di. The formation and propagation of DF is especially well seen in the left (x < 0) part of Figure 10. It shows, in particular, that DFs propagate much faster than plasmoids. The bigger box allows us to trace the evolution of DFs much longer, and Figure 11 shows that both the DF perturbations occupy a wider area in the Z direction at Ωit = 18, compared to the perturbations at the end of Run 1 (Figure 2). According to Run 3 together with Run 1 (Figures 2, 3, 4, 7b, and 8), DFs strongly resemble solitary waves, such as, for instance, the solitary electromagnetic pulses found in Cluster observations by Parks et al. [2007].

Figure 10.

Evolution of the normal magnetic field Bz at the neutral plane z = 0 in Run 3 performed with the parameters similar to those of Run 1 in a bigger simulation box.

Figure 11.

Magnetic field lines and the color-coded current density component −Jy for Run 3 at the moment Ωit = 18.

Figure 12.

Some details of Run 3 at the moment Ωit = 18 at the neutral plane z = 0: (a) normal magnetic field; (b) drift parameters VixBz for ions (red line) and VexBz for electrons (blue line) compared with the out-of-plane electric field Ey (black line); (c) total energy dissipation jtot · E (black line) and its ion (red line) and electron (blue line) components; (d) electrostatic (jixEx, blue line) and electromagnetic (jiyEy, red line) contributions to the ion dissipation shown in Figure 12c; (e) electric field component Ex. Red line in Figure 12a provides an additional profile of Bz field at the moment Ωit = 3.

[27] The larger size of the simulation box in Run 3 not only allows one to see more perturbation modes, compared to Run 1, but it also better reveals the distinctive features of DFs, because it increases their separation from the central electron diffusion region. Run 3 confirms that electrons are magnetized in DF and largely follow the E × B drift (Figure 12b), while ions are not and they fall behind the electrons. In spite of that as well as the strong perturbation of the Bz field in DFs (highlighted in Figure 12a due to the additional red line showing the Bz profile at the beginning of the run), the perturbations of the electrostatic field Ex shown in Figure 12e are small everywhere on the X axis except the region of closed field lines in the central plasmoid. This difference in Ex perturbations between regions with open and closed particle orbits further supports the earlier results [Sitnov et al., 1998, 2002] that the electron compressibility effect being strong for trapped electrons is attenuated in open systems. Figure 12c confirms that the dissipation in DFs is provided by ions. It shows that ions also dominate the changes of j · E in plasmoids, although the net change of the latter parameter integrated over the plasmoid length is much less than that in DFs.

[28] Figure 12 clearly shows that the DFs are certainly something other than just the interaction between a secondary X-line and the main X-line. It shows, in particular, that the DFs are different from the conventional secondary plasmoid seen on the right of the original central X-line. The central X-line is only necessary insofar as it provides an initial condition for our simulations with two magnetotail-like regions. As is seen from Figure 12, both the reconnection electric field (panel b) and the dissipation rate (panel c) are much stronger in the DF regions, compared to the central X-line dissipation region. More importantly, the dissipation is of the different type, because it is fully dominated by ions. The dissipation rate pattern in DFs is also drastically different from that in the secondary plasmoid with comparable true dissipation (j · E > 0) and dynamo (j · E < 0) regions. Besides, the domination of the electromagnetic part of the ion dissipation jiyEy over its electrostatic part jixEx, which is far more pronounced in DFs compared to the conventional plasmoid, also shows that ions are accelerated and heated largely by the reconnection electric field Ey rather than by the drag force from the side of electrons that move faster.

[29] Run 4 differs from Run 3 due to the increased current sheet half thickness λ = ρ0i. Its comparison with Run 3 shows similar DFs that grow slower (Figure 13). The latter result is consistent with the tearing instability theory, which suggests that the tearing growth rate γ is inversely proportional to the current sheet thickness γλp, where p = 5/2 [Schindler, 1974] or 3 [Pritchett et al., 1991]; in Divin et al. [2007], the formation of DF and plasmoids in the case λ = ρ0i was missed presumably because of a smaller simulation box and a shorter duration of the run. Figure 14 shows that though the formation of DF in Run 4 is slower, compared to the cases with thinner current sheets, it eventually results in the magnetic field and current perturbations in a broader area in the Z direction. According to Figure 15, thicker sheets also provide larger areas for the ion dissipation, while the electron dissipation region remains approximately of the same size. This difference between the size of regions of the ion and electron dissipation is expected to further increase for the actual mass ratio.

Figure 13.

Evolution of the normal magnetic field Bz at the neutral plane z = 0 in Run 4, having larger compared to Run 3 current sheet half thickness λ = ρ0i.

Figure 14.

Magnetic field lines and the color-coded current density component −Jy for Run 4 at the moment Ωit = 27.

Figure 15.

(a) Electron and (b) ion contributions to the energy dissipation j · E in Run 4 at the moment Ωit = 27.

5. Discussion

5.1. Comparison to Previous Simulation Results

[30] In view of the long history of the particle simulations of reconnection with various initial and boundary conditions, including works related to the dipolarization phenomena or describing similar effects, it is important to compare the present results with the preceding studies. In particular, the bifurcation of the dissipation pattern j · E for ions was reported by Horiuchi and Sato [1997] (see their Figure 4). However, in their simulations with periodic boundaries the bifurcation effect was found only for the electrostatic part jixEx + jizEz and it was interpreted as the energy conversion from electrons overtaking the slower ions in the reconnection exhaust regions and pulling them downstream. In contrast, in our simulations with open boundaries (Figure 12d) the dissipation of ions in DFs is dominated by the electromagnetic component jiyEy, and according to Figures 8 and 12, it is well separated in space from the electron dynamo regions with j · E < 0. It is interesting to note here, that the j · E pattern, most similar to our Figure 7b, has been found in the magnetic pileup regions in simulations of magnetic island coalescence [Pritchett, 2008], which started from the multiisland equilibrium [Fadeev et al., 1965].

[31] Bifurcation of the reconnection electric field has recently been reported in simulations of forced reconnection by Wan and Lapenta [2008] who used an implicit full-particle code with the open boundary conditions most similar to ours and Divin et al. [2007]. Strong front-like magnetic pileup regions and bifurcation of the reconnection electric field Ey similar to our Figure 6a were found earlier by Hoshino et al. [2001] in their particle simulations of the electron tearing instability. Moreover, Hoshino et al. [1998, 2001] emphasized and investigated in detail the role of those DF-like structures as a new source of electron and ion energization. A similar pileup effect was shown by Pritchett [2001a] in his Figure 6, although no secondary plasmoids were associated with the pileup regions. In contrast to those works, where fronts appear either because of the collision of the reconnection flow with the preexisting plasmas near the O-line of a secondary plasmoid [Hoshino et al., 1998, 2001], or because of a deceleration of a relatively slow (Vx < ∼0.25 vA) reconnection outflow caused by its interaction with the periodic boundary [Pritchett, 2001a], in our simulations DFs appear in the absence of any obstacle for the reconnecting flow, as an instability of the force-balanced tail-like equilibrium with the magnetized electron species. In spite of the fact that our boundary conditions are not explicitly open for the magnetic flux, they provide its removal in the process similar to the magnetopause reconnection (due to the openness of the system for the plasma flow and our field boundary conditions, including zero perturbations of the Bz field) and our outflow boundaries neither decelerate DFs nor result in any additional pileup effects.

[32] DFs are also different from the more conventional secondary plasmoids, such as the central plasmoid in Run 3 (Figures 1012). Such plasmoids are usually observed on longer times scales in simulations with open boundaries [Daughton et al., 2006; Klimas et al., 2008], and they are also observed in our simulations at later times (for instance at Ωit = 45 in Run 4). In contrast to DFs and similar to fluid simulations in [Ohtani et al., 2004], they have comparable positive and negative perturbations of the field Bz, and they appear either inside an extended electron dissipation region or very close to it. According to Figures 12d and 12e, the electrostatic effects are more significant in those closed-field line structures, compared to DFs. Their formation was also shown by Daughton et al. [2006] to significantly change the reconnection rate in the system, although recent studies of a similar reconnection regime [Klimas et al., 2008] suggest a more complex relationship between the formation of plasmoids and the reconnection rate.

5.2. Implications for Tearing Stability Theory

[33] Unlike recent GEM MRC studies with open boundaries [Daughton et al., 2006; Karimabadi et al., 2007; Klimas et al., 2008], motivated by the necessity to avoid the electron recirculation effect in periodic systems on long timescales, the original motivation of the present studies was to model the system evolution on a shorter timescale with the aim, in particular, to describe the effect of passing particles on the stability of the tearing mode in a system with magnetized electrons [Divin et al., 2007]. The ion tearing mode proposed by Schindler [1974] to replace the electron tearing [Coppi et al., 1966], which is stabilized because of the missing Landau dissipation for electrons magnetized by the normal magnetic field Bz, was also found stable [Lembege and Pellat, 1982; Pellat et al., 1991]. The stabilization effect arises because in the case of the finite initial field Bz, electrons perturbed by the tearing electric field E1y drift in the crossed fields E1y and Bz and drag heavy unmagnetized ions along the X direction. Lembege and Pellat [1982] showed that because of that effect, the sufficient condition for the tearing stability coincides with that of the WKB approximation kλB0/Bn > 4/π (here k is the mode wave number, λ is the half thickness of the current sheet, and B0 is the lobe magnetic field), which allows for a local stability analysis in the X direction, neglecting variations of the current sheet parameters along the tail. Thus the onset of reconnection in the tail in the form of the ion tearing was found to be forbidden. That result was confirmed by subsequent theoretical studies [Brittnacher et al., 1994; Quest et al., 1996] and full-particle simulations [Pritchett, 1994; Dreher et al., 1996; Hesse and Birn, 2000; Hesse and Schindler, 2001].

[34] However, the theoretical studies treated electrons as a single fluid of particles trapped inside the current sheet, while the corresponding particle simulations were performed with closed boundary conditions where all particles were effectively trapped (reflected from the boundaries or reintroduced in a similar way). In the meantime, it was found that the tearing mode in the magnetotail could be destabilized due to different response to tearing perturbations of trapped and passing electrons, whose excursion along the magnetotail strongly exceeds the tearing wavelength [Sitnov et al., 1998, 2002]. It was predicted that a population of passing electrons could effectively shortcircuit the electrostatic potential created by the trapped population due to the mechanism of Lembege and Pellat [1982], and thus eliminate its stabilization effect. It was also predicted, that the reconnection onset should start when the tail current sheet becomes not only thin enough to demagnetize ions but also long enough to provide a kinetic response of electrons with a significant contribution of their passing population. Divin et al. [2007] confirmed the destabilizing effect of passing particles by direct comparison of simulations with the outflow boundaries open and closed for particles.

[35] The present study further substantiates the results by Divin et al. [2007] by showing that the formation of the secondary islands discussed in that work may indeed occur when the MHD force balance, broken by the GEM MRC perturbation of the Harris equilibrium, is already restored in the system (Figure 4). Therefore that island formation can indeed be treated as an instability and compared with the corresponding linear tearing instability theory. Domination of the ion dissipation in DFs and more conventional secondary plasmoids in Figures 8 and 12 further suggests that they may form in the process of the ion tearing instability [Schindler, 1974] fed by the ion Landau dissipation. The latter is consistent with the fact that on the timescales Δt of the instability (Ωit = 4–8 in Run 1) electrons are magnetized by the normal field Bz ((Bz/B0)(mi/meiΔt > ∼1) outside the region ∣Bz/B0∣ < ∼0.002. At the same time, we cannot exclude that the electron dissipation near the central X-line in our simulations also affects the development of the instability. Whether our front formation instability may exist independent of the central X-line reconnection and the corresponding electron dissipation (as is suggested by the comparison of our virtual satellite data with observations discussed in the next subsection) remains an interesting open question.

[36] Our simulations further suggest that the ion Landau dissipation dominates the nonlinear evolution of the instability in the form of DFs in spite of the formation of the pileup regions with very strong Bz field that should magnetize ions and the formation of the secondary magnetic islands where one should expect demagnetization of electrons. The ion dissipation domination in DFs could have a more prosaic reason, namely, could arise because DFs represented fragments of the original Harris current sheet moving from the central X-line because of the unbalanced j × B force. In that case the ion dissipation domination could be explained by the domination of the ion current jiy in the original Harris equilibrium with hotter ions (Ti = 3 Te). To check that hypothesis we performed an additional Run 5 with equal ion and electron temperatures. Its results are shown in Figure 16. They do not support the hypothesis, that the ion dissipation in the simulated DFs is caused by a disruption of the force-unbalanced Harris sheet, because ions keep dominating the dissipation in DFs notwithstanding their equal initial temperature with electrons (Figure 16c).

Figure 16.

Similar to Figure 8 showing Bz and Ey fields, ion, and electron drifts, as well as dissipation components for Run 5, with equal ion and electron temperatures at the moment Ωit = 13.

[37] In spite of these test results, closer examination of the dissipation patterns in Figures 7, 8, 12, 15, and 16 forces us to conclude that the dissipation in DFs is not limited to ions. Smaller spikes and areas of the electron dissipation in DFs are also clearly seen in Figures 7, 8, 12, 15, and 16. It is interesting however, that they do not coincide with the secondary X-lines as is seen, in particular, from the comparison of Figures 2 and 7a. This finding is consistent with the results of Hoshino et al. [2001] and Pritchett [2008], who emphasized the role of the magnetic pileup regions as important areas of the electron energization.

[38] Another possible interpretation of the strong ion domination in DF might be the effect of collision of the fast reconnection jet from the central X-line with the preexisting plasma near the magnetic pileup regions. The latter appears to be the case in simulations by Hoshino et al. [1998, 2001]. Since ion and electron bulk flow speeds for ions and electrons in those outflow jets are comparable, one could expect stronger heating of ions because of their higher dynamic pressure. However, in our simulations (compare, for example, Figures 3 and 5) the front is not static. It propagates with the speed close to that of the plasma bulk flow. Thus the amount of bulk flow energy available for conversion to thermal energy is relatively small.

5.3. Comparison to Observations

[39] To facilitate the comparison of our findings with observations we presented the results of Run 4 in the form of records of two virtual satellites located at x = 10di and 15di at the neutral plane z = 0. These results shown in Figure 17 can be readily compared with the similar format results by [Ohtani et al., 2004] shown in our Figure 1 and other observations. Important consistency points between these virtual satellite results and observations include similar profiles of Bz field, ion density, temperature and pressure (the time unit Ωi−1 in Figure 17 is ∼0.5 s for B0 = 20 nT). The persistent increase of the ion bulk flow speed similar to the result of fluid simulations in [Ohtani et al., 2004] and different from their observations, where the ion bulk flow speed peaks right after the peak of Bz, is consistent with the results of Slavin et al. [2003] who found DFs near the leading edge of the more extended fast plasma flows. One of the explanations of the inconsistency with observations in [Ohtani et al., 2004] might be that many BBFs are caused by the reconnection on closed field lines only, whereas in our simulations we assume the existence and close proximity to our satellites of either an active distant X-line or a similar near-Earth X-line, which already involves the lobe field reconnection. This hypothesis is confirmed by the comparison of our electric field profile Ey (Figure 17g) showing a reduced but still quite significant residual reconnection electric field after the front passage (Ωit > 40) with a similar profile in BBF observations by Nakamura et al. [2005] (Figure 4), which shows a decrease of Ey to zero after the front. This comparison has important implications for further tearing stability studies because it suggests that the onset of the reconnection, that is, the tearing mode instability may occur in the magnetotail even in the absence of an active X-line region nearby.

Figure 17.

Different parameters of the simulated DFs in Run 4 presented in the form of virtual satellite data: (a) ion bulk flow speed, (b) Bz magnetic field component, (c) plasma density, (d) ion temperature, (e) ion pressure, (f) plasma beta, and (g) dawn-dusk electric field Ey. Location of the two virtual probes is shown in Figure 17a.

[40] Consistent with [Ohtani et al., 2004] and even more with [Slavin et al., 2003], our virtual satellite observations show a significant overshoot in the plasma density, whose importance as a driving source for DFs has been found in section 3 (see, in particular, Figure 4). The subsequent decrease of density down to the background level, which is less in observations, can be explained by the fact that our satellites are located too close to the X-line. It is interesting, that a similar strong reduction of the plasma density was reported by Fujimoto et al. [1996] for a fast tailward flow in the distant tail. The plasma beta profile in our virtual observations is consistent in shape with [Ohtani et al., 2004], while its absolute value (∼102) is more similar to the case shown in Slavin et al.'s [2003] Figure 3, where Geotail was very close to the neutral plane.

6. Conclusion

[41] The main result of this study is the demonstration that many distinctive features of the dipolarization fronts outside the near-Earth vicinity of the magnetotail, where they are naturally explained by the flow-braking theory, can be explained by a relatively unusual type of the transient reconnection, where the dissipation is dominated by the ion species. We have also shown that in spite of the fact that the initial strong GEM MRC perturbation of the Harris equilibrium with zero normal magnetic field creates a strong force imbalance in the system, the formation of dipolarization fronts commences when the force balance is already restored in the main part of the simulation box. This justifies the interpretation of the front formation in terms of the ion tearing instability [Schindler, 1974]. It is noteworthy that ions dominate the dissipation in DFs even when ion and electron species have equal temperatures, and even after the formation of a very strong pileup magnetic field region on the one side of a DF and the secondary X-line on the other.

[42] The importance of DFs as the sources of the particle energization, complementary to the well-known electron dissipation region was strongly advocated by Hoshino et al. [1998, 2001], Hoshino [2005], and Pritchett [2008]. However, the detailed analysis of the ion and electron energization mechanisms in those works was performed either for the coalescence process in a multiplasmoid system [Pritchett, 2008] or for another regime, where the pileup appears because of the collision of the reconnection flow from the dominating X-line with a dense plasma region near the O-line of a plasmoid that is already formed in a pure Harris sheet with Bz = 0 [Hoshino et al., 1998, 2001]. Thus they cannot be directly applied to our DF structures that appear in the absence of any obstacle for the reconnecting flow, as an instability of the force-balanced tail-like equilibrium with the magnetized electron species, and they propagate from the X-line with approximately constant speed close to the Alfvén speed vA. Our simulations suggest that the main dissipation in DFs is dominated by ions. However, the whole picture of the dissipation is more complex having also the distinct regions of the electron dissipation, whose location is different from the secondary X-lines. Also, it remains unclear if the ion dissipation is indeed of the same nature as the ion Landau dissipation suggested by Schindler [1974] for the ion tearing mode. It may also be caused by the nonadiabatic ion motion in thin current sheets [Speiser, 1965] or by direct interaction of ions with the front as is suggested by the fact that the dissipation regions are more localized in the X direction than the Bz and Ey profiles (compare, for instance Figures 12a, 12b, and 12c in the region 10di < x < 16di). A key to understanding the dissipation mechanism may be the test particle simulation results with ad hoc DF-like Bz and Ey profiles [Li et al., 1998, 2003; Fok et al., 1999; Jones et al., 2006] that explain many features of the dispersionless substorm injections [McIlwain, 1974; Moore et al., 1981; Mauk and Meng, 1987; Reeves, 1998].

[43] Furthermore, as follows from Figures 2, 5, 11, and 14, DFs represent also the regions of the enhanced current density. Therefore they may be unstable with respect to current-aligned instabilities [Lui et al., 1991; Daughton, 2003]. Three-dimensional simulations and multiprobe observations must clarify the possible contribution of the current-aligned instabilities to the dissipation at DFs and explain their fine structure along the propagation direction [Slavin et al., 1997; Nakamura et al., 2002] as well as their localization and wave activity along the main current direction [Nakamura et al., 2004].


[44] We thank S. Ohtani, B. Mauk, B. Anderson, J. Slavin, T. Lui, D. Mitchell, J. Drake, K. Schindler, M. Hoshino, G. Lapenta, V. Angelopoulos, D. Sibeck, and Themis Telecon community for useful discussions of our results. We also thank one of the reviewers for the suggestion of a test run with equal ion and electron temperatures. This work was supported by NASA grant NNX08AD85G and NSF grant ATM-0539038. Simulations were performed at the NASA Advanced Supercomputing (NAS) Division and the National Center for Supercomputing Applications (NCSA), a part of the NSF TeraGrid infrastructure.

[45] Wolfgang Baumjohann thanks the reviewers for their assistance in evaluating this paper.