Structures of magnetic null points in reconnection diffusion region: Cluster observations

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Structures of magnetic null points in reconnection diffusion region: Cluster observations

HU YunHui1, DENG XiaoHua1†, ZHOU Meng1, TANG RongXin1, ZHAO Hui2, FU Song1, SU ZhiWen1, WANG JingFang1, YUAN ZhiGang1, R. NAKAMURA3, W. BAUMJOHANN3, H. R`EME4 & C. M. CARR5 1 School of Electronic Information, State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing,

Wuhan University, Wuhan 430079, China; 2 State Key Laboratory of Space Weather, Chinese Academy of Sciences, Beijing 100012, China; 3 Space Research Institute, Austrian Academy of Sciences, 6, A-8042 Graz, Austria; 4 Centre d’Etudes Spatiale des Rayonnements, BP 4346, 31028 Toulouse Cedex 4, France; 5 The Blackett Laboratory, Imperial College London SW7 2BZ, UK

Magnetic reconnection is a very important and fundamental plasma process in transferring energy from magnetic field into plasma. Previous theory, numerical simulations and observations mostly concen-trate on 2-dimensional (2D) model; however, magnetic reconnection is a 3-dimensional (3D) nonlinear process in nature. The properties of reconnection in 3D and its associated singular structure have not been resolved completely. Here we investigate the structures and characteristics of null points inside the reconnection diffusion region by introducing the discretized Poincaré index through Gauss integral and using magnetic field data with high resolution from the four satellites of Cluster mission. We esti-mate the velocity and trajectory of null points by calculating its position in different times, and compare and discuss the observations with different reconnection models with null points based on character-istics of electric current around null points.

Poincaré index, magnetic null point, magnetic reconnection

Magnetic reconnection is a fundamental plasma process for energy conversion and particle acceleration. It plays crucial roles in magnetosphere activities and energy ex-plosive phenomena in experimental plasma. Null point is the place where magnetic field lines can break and reconnect. In 2D model, the null points with zero mag-netic field strength are classified as X points and O points. In 3D, investigations about null point are mainly concentrated on theory and numerical simulation. Lau and Finn[1] defined, classified and illustrated the 3D null points in theory, and Parnell[2] studied the local structure around null point through linear analysis and found the structure could be determined by four parameters. Priest and Titov[3] investigated the reconnection model with 3D neutral point. Wang and Wang[4] proposed a direct and reliable method to detect 2D null. Zhao et al.[5] devel-oped this method to 3-dimensional space. Cai et al.[6]

visualized the topology of magnetotail through 3D elec-tromagnetic Particle-In-Cell simulation. Dorelli et al.[7] studied the separator reconnection on magnetopause by using the method of Greene[8] and tracing null trajectory. As null point is a singular point where magnetic field falls down zero, its detection demands at least four satel-lites. The Cluster mission of four identical satellites make it possible to detect null points by in situ observations. Xiao et al.[9,10] for the first time identified the magnetic null point and null pairs by analyzing magnetic field data from Cluster, and it is very important for the understand-ing of reconnection in 3-D and the plasma process

Received November 6, 2007; accepted January 29, 2008 doi: 10.1007/s11434-008-0173-0 †Corresponding author (email: [email protected]) Supported by the National Natural Science Foundation of China (Grant Nos. 40390153, 40574073, 40574074, 40640420563) and the Outstanding Young Scien-tists Founding of China (Grant No. 40325012)

HU YunHui et al. Chinese Science Bulletin | June 2008 | vol. 53 | no. 12 | 1880-1886 1881

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in astronomy. It is quite important to study the struc-tures and dynamics of null points in diffusion region and its intrinsic relation with reconnection by using the data of fields, waves, and particles. In this paper, we introduce the discretized Poincaré index through Gauss integral and studied the properties and structure of null point in diffusion region using date from Cluster. We estimated the velocity and trajectory by calculating the positions in different times, and compare and discuss the observation with different null point reconnection models based on the characteristics of electric current around null points.

1 Topology of null points and mathe-matical method

1.1 3-D topology of magnetic null point Magnetic field is the point where magnetic field van-ishes. To study the local structure around null, we gen-erally assume that the null point is located at 0=r ( r is the radius vector in Cartesian coordinate) and expands the magnetic field around null point by using Taylor ex-pansions (first order expands by assuming that magnetic field linearly changes around null point)[1]: ( ) ,δ= ⋅B r B r (1)

where δ B is a 3 3× real matrix ,ij i jB B xδ = ∂ ∂

calculated around null point. The trace of δ B equals zero, ( ) 0,tr δ = ∇ ⋅ =B B i.e., 1 2 3 0,λ λ λ+ + = thus the three eigenvalues are either all real or one is real while the two others are conjugate complex. If all the eigenvalues are real with signs of ( , , ),+ − − the null point belongs to type A, while the eigenvalues are real with signs of ( , , ),+ + − it belongs to type B. If the ei-genvalues contain complex, we will call it spiral null point with the subscript of s added[1].

Magnetic field lines have complex structures when multiple nulls exist in space[6]. Figure 1 depicts the con-dition when an A null and a B null coexist. A∑ plane

(also called fan plane) intersects with B∑ plane. As

every magnetic field line on A∑ ( B∑ ) connects null A (B), the intersect line between A and B must be the line connecting A and B, which implies that there must be a null-null line between A and B. Therefore, B∑

infinitely approaches Aγ (also called spine line) but cannot intersect with A.γ The only possibility is that

A∑ is a half infinite plane bounded by Aγ on one

direction, magnetic field lines on B∑ diverge around

A and then radiate along A.γ It is the same that Bγ is

the boundary of [1]A .∑

Figure 1 Structure of two null points (adapted from Lau and Finn[1], 1990).

1.2 Analysis method

The geometry characteristics around null points can be illustrated by the Poincaré index[9―11]. Wang and Wang[4] introduced and defined the 2D Poincaré index, they also showed the topology structures of field lines around so-lar flare in the solar activity region. Zhao et al.[5] defined the 3D Poincaré index through geometry of manifold and analyzed the magnetic field of the sun. Here we de-scribe the Poincaré index in another different way.

By assuming that there is a simple smooth closed curved surface S which contains volume V. ( , , )ξ η ζ is a point on S and n is the outer normal of S on ( , , ),ξ η ζ r is a vector connecting ( , , )ξ η ζ and ( , , ),x y z that is,

2 2 2( ) ( ) ( ) .r x y zξ η ζ= − + − + −

We can define the Poincaré index of point ( , , )x y z as integral

[12]2

1 cos( , )( , , ) d .4 S

I x y z Srπ

= ∫∫r n

The integral is proved to be 0 or ±1 by Gauss integral formula. The closed surface will not encompass ( , , )x y z when the integral equals zero. The closed curved surface encompasses ( , , )x y z when the integral equals ±1, and the signs will depend on the direction between normal and radius vector.

To facilitate our calculation, we confine the curved

1882 HU YunHui et al. Chinese Science Bulletin | June 2008 | vol. 53 | no. 12 | 1880-1886

surface S to a unit sphere by Gauss projection [12]

3 2: \ {0} , .R Sγ →vvv

We need to check

whether the projected curved surface encompasses the center of sphere.

According to above information, the method demands a closed curved surface formed by known physical quantities. Four identical spacecrafts of Cluster mission form a tetrahedron in space and the magnetic fields re-corded by them will form a closed curved surface. Thus we can calculate and decide whether a null point exists inside the tetrahedron formed by Cluster. Figure 2 shows the projection of tetrahedron to unit sphere[12].

On unit sphere, the recurrent number surround center of sphere (Poincaré index ) is

4

1

1 1cos( , )d sign( ) ,4π 4π i i

iS

I S S S=

= = ⋅∑∫∫ r n

where Si is the area of spherical triangle formed by pro-jecting three points from tetrahedron to sphere. n is the outer normal of each surface of the tetrahedron after projecting. Obviously, the normal is parallel to r, so sign( ) cos( , ) 1.iS = = ±r n

The method and the geometry properties around null points[2] can be summarized in Table 1.

We could use this method to determine whether there is a null point or not, and then reconstruct the local

magnetic field structure by eigenvalues and eigenvectors of the gradient matrix. In addition, eq. (1) can be ex-panded linearly as follows[4]:

0 00

0 0 0

0 00

0 0 0

0 0 0

0 0 0

( ) ( ) ( )

( ) ( ) ( ).

( ) ( ) ( )

x x xx

r rr

y y yy

r r r

z z zz

r rr

B B BB x x y y z z

x y z

B B BB x x y y z z

x y z

B B BB x x y y z zx y z

⎧ ∂ ∂ ∂⎪ = − + − + −

∂ ∂ ∂⎪⎪

∂ ∂ ∂⎪= − + − + −⎨

∂ ∂ ∂⎪⎪

∂ ∂ ∂⎪ = − + − + −⎪ ∂ ∂ ∂⎩

(2) The equation set consists of 12 unknown variables in-

cluding 9 variables of δ B matrix and 3 position vari-ables 0 0 0( , , ).x y z We could transform this equation set into a linear equation set of these variables if the gradi-ent matrix is full rank. Magnetic field and position re-corded on each satellite of Cluster will form 3 equations, so the data from the four satellites can be used to exactly form 12 equations. Then we could obtain the position as well as velocity of null point by solving the linear equa-tion set.

2 Analysis of one typical event with 3-D null point in diffusion region

We have statistically analyzed the null points inside dif-

Figure 2 Projection from tetrahedron to unit sphere.

Table 1 Eigenvalues Eigenvectors γ line (spine) Σ plane (fan surface) normal Null type and Poincaré index

λ1, λ 2<0; λ3>0 v1; v2; v3 v3 v1×v2 A, index = 1 λ1, λ2>0; λ3<0 v1; v2; v3 v3 v1×v2 B, index = −1 λ1, λ 2 conjugate; λ3>0 v1; v2; v3 v3 u1×u2 u1=v1+ v2; u2 = −i(v1−v2) As, index = 1 λ 1, λ 2 conjugate; λ3<0 v1; v2; v3 v3 u1×u2 u1 = v1 + v2; u2 = −i(v1−v2) Bs, index = −1 The table does not consider the condition when one or more eigenvalues equals zero. When gradient matrix has the eigenvalue equal to zero, the null

will decay into 2D or 1D. The detailed description can be found in the paper of Parnell et al.[2], 1996.


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fusion region on both magnetopause and in magnetotail region. Here we concentrate on the event of Oct. 8, 2001.

2.1 Event overview

Cluster was located in the magnetotail near X = −15.4Re, Y = 9.6Re, Z = 0.087Re (GSM), the distance between each satellites is about 2000―2250 km. Figure 3 shows the three components of magnetic field from FGM with the resolution of 44 ms and x component of proton bulk velocity from CIS with the resolution of 4s. We can see

that during 12:52UT to 12:59UT, Cluster crossed the neutral sheet several times and detected the reverse of bulk flow from tailward to earthward[13].

Figure 4 illustrates the x and y components of the magnetic field on SC1 and SC4 during the interval when satellites stayed around the flow reverse region[14―16]. Bx and By shows the Hall quadrupolar structure, which is consistent with the simulations.

The Poincaré index shown in Figure 3 indicated that there are several null points in the diffusion region. Considering that | | | |∇ ⋅ ∇ ×B B must be small enough

Figure 3 The data of magnetic field and bulk flow of Cluster during 12:49:00UT to 12:59:00UT on Oct. 8, 2001. From up to bottom are: magnetic field, x component of proton bulk velocity, Poincaré index. At the bottom, it shows the x component of magnetic field and the Poincaré index in the extended period indicated by vertical line. The rectangle marks the interested null point here. —— , C1; — —, C2; ------, C3; −-−-−, C4.


Figure 4 Hall quadrupolar structure observed by SC1 and SC4 when crossing the neutral sheet. Left part shows the x and y components of magnetic field on C1 during 12:50:30 to 12:52:00. Right part shows the x and y component of magnetic field on C4 during 12:51:00 to 12:53:00.

in physics, here we focus on the null pair during 12:54:01 to 12:54:04 UT. The Poincaré index is 1 during the period of 12:54:01.832―12:54:03.616 UT while it becomes −1 during the period of 12:54:04.106―12:54: 04.686 UT. These two nulls belong to different types in theory and we discuss their characteristics in detail.

(1) 12:54:01.832 to 12:54:03.616. Calculating the gradient matrix by using the least square method[7]

0.001402 0.0019746 0.0054480.000728 0.001562 0.000874 .0.001677 0.003509 0.0001584

−⎛ ⎞⎜ ⎟∇ = − −⎜ ⎟⎜ ⎟− −⎝ ⎠

B

The divergence is 61.81 10−− × and ∇ ⋅ ∇ × =B B 0.02%, thus the eigenvalues and eigenvectors of null point is reliable[9,18,19]. The eigenvalues are 1λ = 0.000693, 2 0.000348 0.00299 ,iλ = − + 3 0.000348λ = − − 0.00299 ,i and it belongs to As according to Table 1. The direction of the spine is ( )0.887 0.385 0.255 ,− − the

normal direction of fan plane is (0.1848 0.8709−

0.4555), and the angle between them is 22.5 .θ = The null trajectory deduced from eq. (2) is shown in Figure

5, and the estimated velocity is about 1089 km/s. (2) 12:54:04.106 to 12:54:04.686 UT. Similarly, we

calculated its gradient matrix: 0.0015775 0.0033236 0.0050593

0.00026455 0.0013912 0.00058304 .0.0010005 0.0023571 0.0000179

−⎛ ⎞⎜ ⎟∇ = −⎜ ⎟⎜ ⎟− − −⎝ ⎠

B

Divergence is 41.69 10 ,−∇ ⋅ = ×B relative error of cal-culating gradient matrix is 2.2∇ ⋅ ∇ × =B B %, the

eigenvalues are 1 0.000974,λ = − 2 0.000571λ = +

0.00243 ,i 3 0.000571 0.00243iλ = − it is a Bs type null

point. The direction of spine is ( )0.59 0.506 0.63 ,−

normal direction of fan plane is (0.1108 0.9936−

0.0197), and the angle between them is 33.7 .θ = Figure 6 shows the position and trajectory of null point. Its estimated velocity is about 1502 km/s.

It is shown that the angle between the spine line of As and fan plane of Bs is about 28°, while the angle be-tween spine of Bs and fan of As is 15°. In theory, the angle between spine of one null and fan of the other null should approach zero[1] if the two null points are related

Figure 5 Position and trajectory of null point. Left part shows the x, y, z coordinate of null point during the interval. Right part shows the trajectory of null point inside the tetrahedron and their projection on three surfaces.


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Figure 6 Position and trajectory of null point. Left part shows the x, y, z coordinate of null point during the interval. Right part shows the trajectory of null point inside the tetrahedron as well as their projection on three surfaces.

and form a null pair. We should note the calculation of direction of spine and fan is based on beeline and 2D plane assumption, however, the fan could be a curved surface and spine could also be a curved line, thus there a might be some deviation from real situation. Moreover, errors in calculation could lead to degree larger than zero. As the observation of two nulls is not observed at the same time and the structure of two nulls might change during this period, which might also lead to some deviation from real situation.

3 Conclusion and discussion

Studying the characteristics and dynamics of null points in diffusion region is quite important for understanding 3-D magnetic reconnection. In this paper, we investi-gated the structures of null points in diffusion region, and estimated the velocity and trajectory. One null pair might exist inside diffusion region. We found that at the time when the Poincaré index equals zero, the diver-gence and gradient of magnetic field is similar at the time when the Poincaré index equals –1 or 1. When the Poincaré index equals 1 or −1, null points are inside tet-rahedron. During the period when the Poincaré index equals zero, it is highly possible that the null points still exist but outside the tetrahedron and approach one of the surfaces. The Poincaré index changes to zero because the null runs outside tetrahedron. The trajectory of null point in Figure 6 supports our assumption.

Priest et al.[3,20] proposed three null point reconnec-

tion models, including spine reconnection, fan reconnec-tion and separator reconnection. In spine reconnection the electric current is along the spine while in fan re-connection it will be in the fan plane. In separator re-connection, the electric current flows along the separator line connecting a pair of null points. The separator line is the intersect line of two fan surfaces of the two null points. When fan is a curved surface, the separator can-not remain as a beeline but a curve. We try to determine the reconnection type by using electric current deduced from curl of magnetic field of the four satellites, and it is found that the angle between current and spine of As null is 41° and the angle between current and fan is 35°. For Bs null, the angles between current and spine and fan is 14° and 35°, respectively. It seems that reconnection observed here does not belong to any above-mentioned reconnection model alone, but the combining of differ-ent types of reconnections[21]. However, the electric cur-rent is not absolutely accurate as it is calculated based on linear assumption, and when plasma flow is consid-ered, there will be a singular point, and magnetic recon-nection will become more complex. In nonsteady case, the position of X points, magnetic islands and plasma bubbles will be changed due to the interaction between magnetic field and plasma flow, and in 3D nonsteady case, reconnection could lead to formation of flux ropes. It is quite important and necessary to combine satellites observations with computer simulations to investigate the structures and evolution of null points and the related dynamic process of 3-D reconnection.

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EDITOR

SUN Shu Institute of Geology and Geophysics Chinese Academy of Sciences Beijing 100029, China

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Structures of magnetic null points in reconnection diffusion region: Cluster observations

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