Magnetic Null-Pairs within Magnetic Reconnection Ion Diffusion Region in the Magnetotail: A Case Study

The 3-dimentional structure of magnetic reconnection ion diffusion region has been studied in this paper. Steady magnetic null-pair structure is found among the Cluster tetrahedron within a thin current sheet when magnetic reconnection takes place in the near-Earth magnetotail. Two magnetic null points in the null-pair are well coupled, with an angle of about 3~7˚ between the spin line of one and the fan surface of the other. The magnetic null-pair detected in the ion diffusion region, is quasi-stable in spatial structure but fast evolved in time, consistent with the fast reconnection scenario. The spatially steady magnetic null-pair within the diffusion region of the collision less fast magnetic reconnection presents an advanced understanding of the magnetic reconnection process.

The magnetic null structure in the magnetic reconnection diffusion region (hereafter called "MRDR") has been studied in detail. Firstly, Xiao et al. [12] applied the null concept and arithmetic to space physics to interpret the 3-D magnetic reconnection in the near-earth central current sheet. They went even further to investigate the geometry of the magnetic null in detail and found a null-pair around which lower-hybrid wave was found [13]. Secondly, He et al. [14] developed a new method to construct the magnetic topology around a magnetic null in the magnetotail. The method is developed for reconstructing the local magnetic field based on the four-point measurement from Cluster tetrahedron. The method makes use of a fitting function with 10 fitting parameters in 10 spherical harmonic functions and another two in the Harris current sheet model, thus matching the 12 observed field components. Thirdly, He et al. [15] studied the electron dynamics in the region close to the null point, observing electron beams in two directions as seen in the pitch angle distributions (PAD) of electron differential energy flux measured by SC2/PEACE, and Deng et al. [16] have studied the structure of magnetic null and the micro process of wave turbulence that are very close to the coupled magnetic null in the diffusion region.
However, some issues remain ambiguous. Poincare index [17] and the magnetic null fitting method [14] to the study the geometry of magnetic null-pairs, which are observed within the MRDR on 10 September 2001. This event has been studied by Wang et al. [18] [19] and Li et al. [20] extensively which mainly focused on the energetic electrons within the reconnection diffusion region [18] and electron pitch angle distributions in the vicinity of the X line and the outflow region [19]. We have also investigated the waves and particle dynamics around the magnetic null point in our previous work [20]. We will go further on this event to extend our understanding on the magnetic structure of the magnetic reconnection diffusion region.
Data used in this paper include Spin-resolution (4 seconds) data from the FluxGate Magnetometer (FGM) instrument [21] and the Cluster Ion Spectrometry (CIS) [22]. The full resolution (22.46 Hz) data is from the FGM. This paper is arranged as follows. In Section 2, the magnetic reconnection process is reviewed briefly. In Section 3, geometry of the magnetic null-pair in the diffusion region is studied in detail. Summary and discussion will be addressed in the final section.

Magnetic Reconnection Overview
The magnetic reconnection event occurred on 10 September 2001, when Cluster is located in the near-earth tail at (−19.29, 2.19, 0.46) R E in the Geocentric Solar Magnetospheric (GSM) coordinate system (the same coordinate system will be used throughout this paper), has been studied extensively by Wang et al. [18] [19]. The crossing of all the spacecrafts to the ion diffusion region during the time interval of 07:50 -08:05 UT has been confirmed by the key observation characteristics in Wang et al. [18] [19] which include 1) A reversal of high-speed flow x V (from negative to positive) coincides with a reversal of z B (from negative to positive); 2) Out-of-plane Hall magnetic field y B is observed by the four satellites of the Cluster.
In this study, we will focus on the structure of the ion diffusion region. We will brief review this event by plotting the X-and Z-component of the magnetic current density ( xyz J ) and the half thickness of the current sheet (   Figure 1(f)) is estimated [24] by pressure balance condition [25] providing that the spacecraft is in the plasma sheet boundary layer (PSBL). Figure 1  Forming a tetrahedron in space, the Cluster mission provides data from four similarly instrumented spacecraft and thus provides unique opportunities to study the 3-dimentional structures within the ion diffusion region of magnetic reconnection. The Poincar'e-index method, which was originally introduced by Greene [17] and developed by Zhao et al. [28], has been successfully employed to infer the presence of a true magnetic null point [12] [13] [14] [15]. In bottom part of Figure 1, the Poincare index and the corresponding B ∇ ⋅ from 07:55:00 UT to 07:59:00 UT by applying the Poincar'e-index method are shown. With poincare index of +1 or −1, magnetic null points are included by the Cluster tetrahedron during this time (Figure 1(g)). The corresponding B ∇ ⋅ ( Figure   1(h)) is very small (from −0.005 to 0.005, the values indicated by the two horizontal dotted lines), confirming that the singular points are physical magnetic nulls. In this paper, we will mainly study the magnetic structure within reconnection diffusion region at around 07:57 UT.

Magnetic Null-Pairs in the Reconnection Diffusion Region
The magnetic structure around the reconnection diffusion region is much more complex than that in the ideal 2-dimensional model. In the case presented here, as indicated in Figure 1(f), the half thickness of the Current Sheet decreases rapidly to about 400 -500 km when the flow reverses at about 07:57 UT. The thickness of the thin CS is as small as the scale of the ion inertial length i d ( i d -360 km). Magnetic reconnection which takes place in a thin current sheet seems more likely to be three-dimensional [17] [28]. In this section, we will study the 3-D reconnection structure by employing the FGM data with a resolution of 22.46 Hz [21].
There are six types of null in space, i.e., the X-type and O-type in 2-D; A-type and B-type in 3-D and As-type and Bs-type in 3-D. The type of null is determined by the eigenvalues of the B ∇  matrix [5]. One can refer to Dorelli et al. [11] for details on the determination for these types of magnetic null structure. We classify the null types into A-, B-, As-and Bs-types according to the eigenvalues of the B ∇  matrix [5]. The field lines along the spine line of an A-type null direct out from the null, while field lines in the fan plane direct toward the null. And for the B-type null, the field line direction is opposite to the A-type null. The field lines around the As-and Bs-types are more complex than those in the A-and B-type nulls, with a spiral structure. Therefore, the type of contains types B-As-B (Bs)-As. The types of the null points are also marked in Figure 1(g) and shown in Table 1. Note that in Table 1, "j" is the current density calculated by the Curlometer technique [23] and the "spine" and "fan" denote the γ-line and the Σ-surface of the magnetic null geometry, respectively. The results are shown in Table 1, in which columns 3 to 6 are the type of null points, angle between j and Σ-surface, angle between j and γ-line, and the averaged value of the corresponding B ∇ ⋅ .
It is very interesting that there are 4 null points (they are hereafter named as N2a, N2b, N2c and N2d as in Table 1) in N2, separated by small time intervals.
This suggests a very complex magnetic structure in the magnetic reconnection diffusion region. To study the relationship between 2 adjacent magnetic nulls, we take N2a-N2b, N2b-N2c, N2c-N2d as candidate magnetic null-pairs [13] [14] [15] [16]. For each candidate null-pair, we calculate the angle between the current density to the γ-line and the Σ-surface, and the angles between the γ-line and the Σ-surface. The results are summarized in Table 2 (N1 and N3 are analyzed with the same procedure, but we only want to illustrate the characteristics for these two magnetic nulls).
The angles are calculated under the assumption that there are two different magnetic nulls adjacent to each other, and the "1" and "2" denote the first and the second ones, respectively. "j" is the current density; "spine" and "fan" represent the γ-line and the Σ-surface for the corresponding magnetic null. Taking the N2a-N2b pair for an example, the angle between the γ-line of N2a (B-type null) and the Σ-surface of N2b (As-type null) is as small as 3.36˚, indicating that the γ-line of the N2a null is almost in the Σ-surface of N2b; the angle between the As-type null γ-line and the Σ-surface of the B-type null is 34.72˚.
This angle is not very small. It is suggested to be caused by the deviation brought about from the spiral characteristic of the γ-line of the As-type null. Thus, this magnetic null-pair is confirmed to be a "B-As" coupled null-pair. Figure 2 illustrates the magnetic null-pair of N2a-N2b. The rough dashed lines represent the γ-line.
The same analysis is performed for the magnetic null-pair of N2c-N2d, but with the result slightly different from that of the N2c-N2d pair with regard to the angles. For the N2b-N2c null-pair, the situation is almost the same as for the N2c-N2d pair, with the angle of the γ-line of the Bs-type null (N2c) and the Σ-surface of the As-type null (N2b) as small as 3.09˚. Though the angle between the γ-line of the B (Bs) type null and the Σ-surface of the As-type null is not  International Journal of Geosciences exactly the same as the former null-pair addressed above, their analogy can still be reliable. The relationship between the two groups of the magnetic null-pairs will be dressed in the following.

Magnetic Fitting of Magnetic Null-Pairs
The Bs-As coupled magnetic null-pair. The difference between topologies of these 3 magnetic null-pairs is very small. This small difference suggests that the magnetic geometry can change itself slightly during the reconnection process at this time.
In this subsection, further study of the magnetic null geometry will be done.
To study the relationship among these 3 above-mentioned magnetic null-pairs, the magnetic field fitting method [14] is applied here to reconstruct the field line around the magnetic null. The fitting method is restated below.
The magnetic fitting method was designed by He et al. [14] on the year of 2008. This method is to fit the recorded 4 vectors with 12 magnetic field components simultaneously measured by the Cluster satellites. There are 12 functions in the fitting model, including ten spherical harmonic functions and a function taken from the Harris current sheet model by Harris [29] in 1962, together with a constant background field. They adopted the spherical harmonic functions as part of the fitting model for their convenience of describing a potential field. Considering the special feature of magnetic field configuration in the magnetotail, we add the Harris current sheet function with a constant background field to the fitting function. Such a fitting can be expressed as where ( , ,    (1) is designed specifically to represent the magnetotail environment.   Provided that the 3-D magnetic reconnection takes place in the space at a fixed time point, forming a magnetic null-pair as shown in Figure 2. In the first stage, Cluster moves into the diffusion region, and includes one part of the null-pair, which is presented to be the B-type null during 07:56:33.3 -07:56:33.8 UT as seen in Figure 3(a). During this time span, the magnetic null structure is stable, only with its B-part included in the cluster tetrahedron. After that, the magnetic reconnection pauses temperately. About 20 seconds later, the magnetic reconnection re-occurs, forming the magnetic null-pair again. Firstly, one part of the null-pair (B-type null) is included by the Cluster tetrahedron, ref. to the illustrating drawing in Figure 2; and as the spacecrafts moving forward, the other part of the null-pair (A or As-type) is in turn included (Figure 3(c)). The magnetic null-pair lasts only several seconds. It moves from left to right (in International Journal of Geosciences  Figure 3) and again, after ceasing for about 5 seconds, the reconnection re-occurs and form a new magnetic null-pair, which is detected by the tetrahedron during 07:57:10.7 -07:57:14.6 UT (Figure 3(d)). In the third stage from 07:58:08.0 UT to 07:58:10.8 UT (Figure 3(e), Figure 3(f)), the results in Table 2 tell the same story for the As-type part of the magnetic null-pair of the magnetic reconnection diffusion region as what is addressed in the first stage.

Temporal Evolution of Magnetic Null-Pairs
The analysis above suggests that the magnetic null-pair, which is detected in the magnetic reconnection diffusion region, is a quasi-stable in spatial structure but with fast evolution in time. The temporal evolution and spatial quasi-stability of the magnetic null-pair shed a new light on the magnetic diffusion geometry in the reconnection region.

Conclusion and Discussion
We have studied the magnetic diffusion structure in regimes of 2-D ideal reconnection model and 3-D magnetic null as the reconnection taking place in the neutral current sheet in the near-earth tail on 10 Sep. 2001. Main points are summarized as follows: 1) Many magnetic null points have been detected by the Cluster in the diffusion crossing time. The N1 from 07:56:33.3 -07:56:33.8 UT is a B-type single magnetic null; the N2 from 07:57:05.2 UT to 57.14.6 UT is composed of 4 magnetic nulls, which are regarded to be the coupled magnetic null-pair according to the angle between the γ-line of one and the Σ-surface of the other for each null-pair. N3 is also a single magnetic null (As type).
2) The magnetic field fitting result around the magnetic null or null-pair in the reconnection diffusion region has shown good magnetic structure around International Journal of Geosciences the null in different stages. The small difference in magnetic null-pairs suggests that the magnetic spatial topology in the reconnection diffusion region is very steady. For spatial geometry, the magnetic field forms a steady magnetic null-pair (B-As) in the magnetic diffusion region, suggesting a steady spatial geometry for the 3-D reconnection. The 3-dimensional geometry of the magnetic null-pair, in this case, is different with the case on 15 September 2001 studied by Xiao et al. [12]. There is only one single magnetic null in the case. Though it can be inferred to be one part of the magnetic null-pair from the magnetic lines as in our case (N1), the two are not the same because of the lack of other parts of the magnetic null-pair. A magnetic null-pair is more stable than a single magnetic null in a 3-D magnetic regime. Our case is also different from the A-B-As coupled magnetic null structure in 1 October 2001 introduced by Xiao et al. [13] and studied in detail by He et al. [15] and Deng et al. [16]. The magnetic null-pair in the presented case evolves with time and is detected by Cluster time again, and therefore reinforces the topology modification during the relative stable 3-D magnetic geometry.
In Figure 1(g), when C1 C3 and C4 are located in the outflow region ( Figure   1(c), |Vx|~300 -400 km/s), there are several places where Poincare indexes were 1. They denote that there were also magnetic nulls there. However, this is just one single magnetic null, or at least it is one part of the magnetic null-pairs, which is included by the Cluster tetrahedron. Though the satellites are mainly located in the outflow region, however, the C3 maybe go back to the other side since Vx (c3) decreases rapidly to zero at this time. Thus it is possible for the magnetic nulls to be included by the tetrahedron.
The current sheet during this time is unstable. As addressed in part 2, the current sheet presents a kink-like configuration with a tendency of "thickening, thinning, thickening, and thinning" of the current sheet variation. The thickness of the current sheet is less than 1000 km in the magnetic null observation, i.e., the magnetic null-pair forms a steady 3-D reconnection structure in a thin current sheet. Moreover, SC2 is located northern relative to the other 3 satellites.
Thus the relative location of SC2 to SC3 is further than those of SC1 and SC4 to SC3. However, the X-component of magnetic field ( x B ) detected by SC2 is smaller than what detected by the SC1 and SC4 during the time interval from 07:56:20 UT to 07:57:40 UT. This suggests a tilt of the current sheet X-Y plane.
Thus, the current sheet during this time is completely in the 3-D regime, which might be a response or an exciter of the 3-D magnetic structure in the 3-D magnetic reconnection process. The large scale of the current sheet associated with the 3-D geometry of magnetic diffusion region remains to future work.