Magnetotail dipolarization front and associated ion reflection: Particle-in-cell simulations



[1] For the Earth magnetotail, the Svenes et al. (2008) statistical study infers the lobe density to be highly variable, in the range of 0.007–0.092 cm−3. Such lobe density variation modifies reconnection diffusion region physical processes and reconnection rate drastically. This letter addresses observable reconnection signatures in the vicinity of the X-line that are to be affected by the dynamic changes of reconnection. Using a 2.5D particle-in-cell (PIC) code, we find the dipolarization front (DF) a moving ram that pushes the initial equilibrium plasma sheet in front of it. The DF propagation velocity scales with the upstream Alfvén speed, leading to faster, steeper, and more compressed DFs with strongerBz at low lobe densities. These DFs reflect plasma sheet ions in a streaming manner. The streaming ions create a bipolar magnetic field straddling the central plasma sheet and can excite various instabilities.

1. Introduction

[2] Although magnetotail reconnection has been the focus of intense scrutiny, no previous computational work has analyzed the effect of lobe density on the reconnection downstream region signatures. Recently, Wu et al. [2011] reported that a low lobe density drastically enhances the reconnection rate, affects diffusion region physical processes and leads to faster outflows. It is then natural to expect that such violent reconnections will also significantly modify the downstream signatures, particularly the observables such as the dipolarization front [e.g., Angelopoulos et al., 1992] and the suprathermal particles [e.g., Vaivads et al., 2011].

[3] Historically, the dipolarization front (DF) refers to the leading edge [e.g., Nakamura et al., 2002; Runov et al., 2009] of the earthward propagating magnetic flux pileup region (FPR) [e.g., Fu et al., 2011]. Earlier, magnetohydradynamics (MHD) simulations [e.g., Hesse and Birn, 1994] and hybrid simulations [e.g., Hesse et al., 1998] found an increase of the normal magnetic field Bzthat propagates away from the X-line as a result of reconnection. Recently, particle-in-cell (PIC) simulations with open boundary conditions [Sitnov et al., 2009] measured the approximate propagation of the DFs that are in agreement with the observation by Runov et al. [2009]. The above works establish that the DF can be a generic and transient signature associated with reconnection, rather than a consequence of the compression of the Earth's dipole field. A new feature is the recent observation of a plasma sheet ion population reflecting off the DF [Zhou et al., 2010] that has not been studied in previous simulations.

[4] The DF propagates away from the x-line which originally generated it. In this letter, we define “behind” the DF to be on the side nearest this x-line, and “in front” of the DF to be the opposite side. We perform large scale kinetic PIC simulations, systematically studying the DF at various lobe densities. A long history of the DF propagation is followed, allowing us to trace the DF propagation before the computational boundary can have an effect on the DF. We find that the DF propagates away from the reconnection site with a small initial speed (near zero). The speed soon increases and reaches a steady rate that scales, almost linearly, with the upstream Alfvén speed. At low lobe densities, the DF magnetic fieldBz is more compressed and in front of it, there is a sharp rise of plasma sheet density followed by a gradual decrease along X, corresponding to a rise of ion flow velocity in front of the DF that has been observed by Runov et al. [2009]. In addition, there is an X directional preferential heating for the ions in the region of flow velocity rise, as a consequence of their streaming away from the DF upon reflection, due to the lack of magnetic confinement in the central plasma sheet in the vicinity of the X-line. This streaming is different from the behavior of ions reflected off of a perpendicular shock, which gyrate and drift across a sizeable magnetic field. Further, we examine the ion phase space density to illustrate the streaming reflection processes and to re-affirm the localized feature of the DF. The DF is identified as a moving ram that pushes the initial equilibrium plasma sheet, as well as a separator between the sub-Alfvénic reconnection outflow and the super-Alfvénic reflected ion flow. Finally, we predict that the streaming of the reflected ions generates a bipolar magnetic field.

2. Simulations

[5] We analyzed the set of relativistic and electromagnetic 2.5D particle-in-cell (PIC) simulations as described in our other publication [Wu et al., 2011]. For completeness, we re-state the normalization scheme and the essential parameters. Normalization is based on the initial double Harris sheet maximum densityn0 and the asymptotic magnetic field B = 1. Therefore, lengths (x, y, z) are normalized to an ion skin depth calculated from di0 = c/ωpi0, where ωpi0 is the ion plasma frequency calculated from n0 = 1. Time t is normalized to an inverse of ion cyclotron frequency Ωi0−1 calculated from B. The electric fields are normalized to vA0B/c. The velocities are normalized to a referenced Alfvén speed vA0 = B/(4πn0mi)1/2. The electron mass is me = 0.04mi. The ion and electron temperatures are initially Ti = 0.4167 and Te = 0.0833, where temperature T is normalized to T0 = mivA02. The speed of light is chosen to be c = 15vA0. The simulations are 102.4di0 × 102.4di0in the X-Z plane. Double Harris plasma sheets are initiated with the widthw0 = di0, Bx(z) = B[tanh((z − Lz/4)/w0) − tanh((z − 3Lz/4)/w0) − 1] and nx(z) = n0(B2 − Bx(z)2)/(2(Ti + Te)). To this Harris equilibrium is added a second non-drifting plasma population with densitynb. Note that since the Harris equilibrium has a zero density in the lobes, the lobe density becomes nb. The six simulations analyzed have nb = 0.01, 0.015, 0.03, 0.1, 0.2, and 1.0, respectively. There is no guide field. Small magnetic perturbations are included for reconnections to develop (through the tearing mode). The double Harris sheet with periodic boundary condition reproduces large values of the tearing mode stability parameter and does not exhibit artificial saturation due to conducting boundaries [e.g., Wu et al., 2011].

3. The Dipolarization Front and Ion Reflection

[6] Using the PIC simulations described above, we analyze each dipolarization front (DF) at a time when the DF magnetic field BDF reach a maximum. The DF is fully developed and the periodic boundary condition has not affected the DF yet. The general properties of the DF are illustrated in Figure 1. The two extreme density simulations shown both have X-lines nearx∼ [25,35]. Focusing on the region to the right of these X-lines, the low lobe density simulation has a steeper and larger DF magnetic field magnitude |BDF| relative to the higher density case. In front of the DF for the low density case, the reflected plasma sheet ions cause a density pileup (Figure 1b). Such a density pileup is not seen for the high density case which has too small a BDF to reflect any ions visibly. Figure 1c traces the propagation of a DF by following the maximum Bz along X. During reconnection onset, the DF is forming in the diffusion region (DR) with an initial BDF smaller than the DR fluctuating Bz, as seen in the discontinuities at t = 53, 55. After reconnection onset, the DF is gradually accelerated to a maximum and constant propagation speed vDF (t = [56–77]). After t = 77, the periodic boundary condition starts to affect the DF propagation, resulting in an unphysical velocity reversal. Figure 1d compares vDF to the upstream Alfvén speed vA,up. The linear scaling vDF ∼ 0.3 × vA,up is similar to the maximum ion outflow scaling vio ∼ 0.4 × vA,up discussed by Wu et al. [2011].

Figure 1.

DFs overview. (a) Cuts of Bzalong X through the X-line. (b) Cuts of ion densitynialong X through the X-line. Both Figures 1a and 1b only show two extreme background density simulations:nb = 0.01 (red) and nb = 1.0 (black). The other four simulation results (not shown) gradually transition from one extreme to the other. Dashed lines mark the DFs. (c) The propagation of the nb = 0.01 DF. Note that the DF is undefined before reconnection starts and thus the discontinuities at t = 53, 55. At t = 77, the periodic boundary conditions begin to play a role and result in an unphysical velocity reversal. (d) DF propagation as a function of the upstream Alfvén speed. (e) Maximum dipolarization front magnetic field strength BDF as a function of the upstream magnetic field Bup, the upstream density nup, and the plasma sheet density ncs.

[7] Let ncs be the initial plasma sheet density. We examine the DF in its rest frame, where the stationary plasma sheet ions appear to be injected into the DF with a velocity vDF ∼ 0.3 × vA,up, corresponding to a ram pressure Pram,csmincs(0.3vA,up)2. In the region where ions are reflected from the DF, this Pram,cs is being converted into the plasma thermal pressure. Immediately in front of the DF, the conversion rate is ∼100%, Pthermal,front ∼ Pram,cs. Integrating the curvature force along X across the DF yields an estimate for an effective “curvature pressure”: δPcurv,DF ∼ BxBzδx/(4πδz) ∼ BxBzδx/(4πδz). We have found in our simulations that the magnetic field lines in the vicinity of the DF are nearly circular, yielding Bx ∼ Bz ∼ BDF, and δz ∼ δx because ∇ ⋅ B = 0. Therefore, δPcurv,DF ∼ PB,DF. Pressure balance requires Pthermal,front ∼ δPcurv,DF + PB,DF, which now can be written as Pram,cs ∼ 2PB,DF. This estimate gives 2BDF2/8πmincs(0.3vA,up)2 ∼ 0.32mincsBup2/(4πminup). Re-arranging the equation, we findBDF/Bup ∼ 0.3(ncs/nup)1/2, which matches Figure 1e's systematically shown relation: BDF ∼ 0.3Bup(ncs/nup)1/2. The scaling indicates that the DF is a moving ram that pushes the initial equilibrium plasma sheet ahead of it. The scaling also provides a way to predict the maximum DF magnetic field in the reconnection vicinity using upstream conditions. It should be noted that our simulations only address the reconnection vicinity where the Earth's pre-existing dipole field and other thermalization processes such as betatron and Fermi accelerations [e.g.,Wu et al., 2006] are not affecting the Pthermal,behind (neglected above) behind the DF. Observational studies of DF events [e.g., Li et al., 2011] are necessary for understanding the DF pressure balance in the near Earth region.

[8] Another piece of evidence of ion reflection is the rise of ion flow in front of the DF, as seen in the Figure 2a(left). The over-plotted Aflvén speed illustrates an interesting feature of the DF. The reconnection outflow behind the DF is sub-Alfvénic relative to the local Alfvén speed, while the flow in front of the DF is super-Alfvénic owing to the small plasma sheet magnetic field. This super-Alfvénic flow is created by a reflected beam of ions streaming away from the DF. This beam coexists with the ambient plasma sheet population, which is manifested as an increase inTixx in front of the DF.

Figure 2.

(a) Cuts of ion flow vix (black solid line), local Aflvén speed vA (red dotted line), and temperatures Tixx(b) along X through the X-line. (left)nb = 0.01, (right) nb = 1.0. The DFs are marked by a vertical dashed line.

[9] Another interesting feature present in the Tixx (Figure 2a) is that behind the DF, there is a region of heating (x = [43,53]) created as the high speed outflowing ions decelerate and compress behind the DF. The high speed outflowing ions are sub-Alfvénic, however, so this left hand edge of the compression region is not a fast shock. Furthermore, it does not match any other known discontinuity criteria (contact, tangential, intermediate shock). This structure steepens with time, reaching a peak density compression ratio of 2 and an associated deceleration of about 1vA0 at t = 69.

[10] The presence of streaming ions in front of the DF is not limited to the lowest density case, and is evident for even the nb = 0.2 case. To further understand these reflected ions, we examine phase space densities (PSDs) of the nb = 0.1 case at t = 93 when the DF magnetic field is maximum (Figure 3). Figure 3c shows the spatial scope of the DF as a reference to the PSD panels. In the PSD panels (Figure 3a), there are two populations of plasma sheet plasmas in front of the DF, one with a small averaged vix (population 1, as highlighted by an orange dashed rectangle) and one with a larger averaged vix (population 2, as highlighted by a green dashed rectangle). Near the DF, the two populations are close enough to mix in velocity space and create a elongated oval. Slightly far away from the DF, the two populations clearly separate in vix. Population 2 fades out as we move away from the DF. Hence, population 1 is the ambient plasma sheet ions. Population 2 is the DF reflected plasma sheet ions. Due to the relatively modest size of these simulations, these reflected ions originate from both the growing and steady phase of the DF. Simulating realistic reflected ion dispersion will require a future study with much larger simulation sizes.

Figure 3.

Ion phase space densities (PSD), vix versus viz and vix versus viy, (a) along the X-axis and (b) above the X-axis in front of thenb= 0.1 DF. The orange and green dashed rectangles mark two distinct populations (see text). The physical location (x, z) of each PSD plot is marked on top of the images. (c) For comparison with the PSD panels, the intensity and physical domain of the DF are shown. (top) 1-D cut ofBzalong X through the X-line. (bottom)2-D illustration of the DF domain. (d) (top) A cut ofjix, jex, and jxalong X through the X-line. (bottom) 2-D illustration of the bipolar magnetic field straddling the symmetric line. The solid vertical lines mark the dipolarization front and the dashed lines mark the X-line. Note in Figures 3c and 3d that there is a secondary island between the X-line and the DF. This island does not affect the DF physical processes due to its long distance from the DF: ∼15di.

[11] The scope of reflection is examined in Figure 3b, which shows that slightly off the central plasma sheet, the number of reflected ions is greatly reduced. About 5di0 above and below the central plasma sheet, there is no ion reflection at all. Comparing with Figure 3a, we find that the ion reflection is closely associated with the DF and therefore is rather localized, like the DF itself as shown in Figure 3c. Figure 3d shows a current jixgenerated by the streaming ions. Despite being partially counter-balanced by an electron currentjex, jix leads to the net current jx which induces a bipolar magnetic field straddling the central plasma sheet (Figure 3d, bottom) in front of the DF. Such a feature is not seen in the highest density simulation where the DF is too weak to reflect ions.

4. Summary and Discussions

[12] Particle simulations performed here focus on the specific topic of the dipolarization front (DF) in the vicinity of the X-line. At small lobe density, the dipolarization front propagates faster at a speed that scales with the upstream Alfvén speed. The fast propagating DF has a largerBDF due to compression. We find that BDF can be evaluated using upstream conditions and a scaling relation. A large BDFhas the characteristic of a moving ram that pushes the plasma sheet in front of it, creating a population of reflected ions. Recall that for a perpendicular shock, reflected ions will gyrate and drift around the magnetic field and form a gyro-tropic population [e.g.,Wu et al., 2009]. The DF reflected ions are, on the contrary, initially unmagnetized and streaming strongly in X, resulting in: (1) The DF marks a boundary of the sub-Alfvénic reconnection outflow and the super-Alfvénic reflected ion flow. (2) The reflected ions are preferentially accelerated in X initially. The highly anisotropic ions excite various instabilities inside the primary island, a topic of future investigations. (3) A strong current along X induces a bipolar magnetic field straddling the central plasma sheet, as is evident in a Cluster observation (L.-J. Chen, personal communication, 2010).

[13] In the simulations reported here, for simplicity, there is no guide field. In reality, events with a small guide field By ∼ 0.2 have been observed by Cluster [e.g., Cattell et al., 2005] in the magnetotail when reconnection is already in progress. A guide field leads to the formation of electron holes and bipolar electric fields [e.g., Drake et al., 2003; Lapenta et al., 2011]. In addition, we mean to simulate dynamics close to the X-line. One expects that in the near Earth region, the terrestrial dipole fieldBz becomes important and the plasma sheet is no longer unmagnetized. The Bz modifies the initial streaming nature of the reflected ions [Zhou et al., 2011], allowing the ions to be re-magnetized and re-gain gyrations. In this letter, we perform a systematic study of the basic physics controlling the properties of the DF as well as the ions it reflects. Future work will address the topic in more realistic cases, including a guide field, a backgroundBz, a larger simulation box, and possibly an open boundary. We can also investigate other initial equilibrium configurations [e.g., Yoon and Lui, 2004; Pritchett and Coroniti, 2010; Sitnov and Swisdak, 2011].


[14] This work was supported by NSF grant ATM-0645271 and NASA grants NNX08AM37G and NNX08AO83G. Simulations were performed at the NCAR Computational Information Systems Laboratory, sponsored by NSF. PW thanks Tai Phan, Marit Oieroset, Mitsuo Oka, Dan Winske, S. Peter Gary, William Mattheaus, Mikhail Sitnov, Andrei Runov, Michael Hesse, Thomas Moore, Vassilis Angelopoulos, and Xuzhi Zhou for discussions.

[15] The Editor thanks one anonymous reviewer for assisting with the evaluation of this paper.