Abstract
Detection of a separator line that connects magnetic nulls and the determination of the dynamics and plasma environment of such a structure can improve our understanding of the three-dimensional (3D) magnetic reconnection process1,2,3,4,5,6,7,8,9. However, this type of field and particle configuration has not been directly observed in space plasmas. Here we report the identification of a pair of nulls, the null–null line that connects them, and associated fans and spines in the magnetotail of the Earth using data from the four Cluster spacecraft. With di and de designating the ion and electron inertial lengths, respectively, the separation between the nulls is found to be ∼0.7±0.3di and an associated oscillation is identified as a lower-hybrid wave with wavelength ∼de. This in situ evidence of the full 3D reconnection geometry and associated dynamics provides an important step towards establishing an observational framework of 3D reconnection.
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In general, 3D magnetic reconnection occurs on a separator line that is analogous to a two-dimensional (2D) X-line. The legs of this line, called separatrices, correspond to fans (Σ-surfaces) bounded by spines (γ-lines) that emerge from the nulls1,2,3,4. 3D reconnection models are fundamentally based on chains of A–B null pairs1,2,3,4,8,9. (The classification of a null depends on whether the field along the separator line converges (A) or diverges (B) from the null point, as shown in Fig. 1d.) Identification of A–B null pairs and identification of the separator and associated structures such as spines and fans are fundamental to increasing our understanding of 3D reconnection.
Identification of null–null lines and their neighbouring 3D structure and associated dynamics has been attempted for solar coronal plasmas10,11. Recently, Xiao et al. successfully identified an isolated magnetic null point related to a reconnection region in the magnetotail of the earth9. These authors made use of measurements provided by the Cluster mission12, which provides data from four similarly instrumented spacecraft that form a tetrahedron in space, thereby providing unique opportunities to detect small-scale 3D plasma structures. In their analysis, these authors applied the Poincaré-index method13,14 to infer the presence of a true magnetic null point.
Here we analyse an event on 1 October 2001 between 09:36 UT and 09:55 UT, during which time the Cluster spacecraft meandered several times around a reconnection region in the magnetotail, near ∼(−16.3,7.9,0.9) RE GSM (in geocentric solar magnetospheric coordinates), where RE is the Earth’s radius. As shown in Fig. 1a, the four spacecraft were separated by less than 2,250 km. Four-second-average data from the Fluxgate Magnetometer (FGM)15 for the magnetic field (B) and the Cluster Ion Spectrometer (CIS)16 for the plasma density (N) and velocity (V) during the interval 09:45–09:51 UT are plotted for spacecraft C4 in Fig. 2a–c. The event has previously been extensively studied assuming a 2D structure for reconnection17,18,19,20. A tailward passage of the X-line with respect to the Cluster tetrahedron from 09:47 to 09:51 UT is shown. The measured magnetic field and flow velocity patterns match the 2D Hall reconnection picture21,22.
Vector field theory23 implies that X-lines are structurally unstable and that any small perturbation would break them apart into pairs of nulls2. We, therefore, considered that this X-line passage event could be a potential candidate for detecting a pair of nulls. In the interval 09:48:20–09:48:50 UT, the Cluster tetrahedron encountered the ‘X-line’ as indicated by the observed flow reversal and the quadrupolar Hall magnetic field perturbations. To identify a null pair in the reconnection region during the period under consideration, we apply the Poincaré index method13,14, which has been successfully used in previous work for identifying magnetic nulls9. For this purpose we use high resolution (0.04 s) magnetic-field data from all four spacecraft and find that the Poincaré index jumps up first to +1, then jumps down to −1, and returns afterwards to +1 again (see Fig. 2d–g). We may, therefore, conclude that some magnetic nulls were present within the Cluster tetrahedron. We can also verify the result by linear interpolation as shown in Fig. 1a.
The total relative error in calculating δ B can be estimated from the ratio (refs 9,24,25). In this event is a few hundredths at the times when the Poincaré index is ±1. The types of these nulls can be identified from three eigenvalues of the corresponding δ B matrix around the nulls. If the real parts of two eigenvalues for a null are negative, we call it a negative or A type. Otherwise, we call it a positive or B type1,2,3,4. We find that the Cluster tetrahedron first encounters an A null in the interval 09:48:24.166–09:48:25.682 UT, then a B null in the interval 09:48:26.975–09:48:29.830 UT. Characteristics of the null pair observed are shown in Table 1.
The detailed analysis of the null-pair characteristics is consistent with a 3D reconnection configuration such as that plotted in Fig. 1d, where the A–B null line is analogous to the X-line in the 2D model (as illustrated in Fig. 1c), with the legs replaced by the fan planes ΣA and ΣB that are bounded by the spines γB and γA respectively1,2,3,4. The surfaces ΣA and ΣB are determined by the eigenvectors a2 and a3, and b2 and b3, in Table 1 respectively; whereas the spines γA and γB are determined by the eigenvectors a1 and b1. The field lines above the upper halves of the Σ surfaces and those below the lower halves merge and cross-link at the null–null line.
Theory predicts that A and B nulls that connect to form a 3D reconnection geometry obey the following features2. (1) a3 corresponding to the smaller negative eigenvalue of the A null and b3 corresponding to the smaller positive eigenvalue of the B null form an A–B null line. (2) The null–null line is the intersection of the two fan planes ΣA and ΣB. Therefore the null–null line (b3) and γB(b1) as the boundary of ΣA should be on the same surface ΣA. (3) Similarly, both γA(a1) and the null–null line a3 should be on the surface ΣB. In other words, if the Σ surfaces are approximately planar, the angles between γA(a1) and ΣB, and between a3 and ΣB, should be small.
Indeed, the observation shows that in this event: (1) a3 and b3 approximately coincide, creating the A–B line with an angle of ∼10∘ between them. (2) As shown in Table 1, γB(b1) and b3 are approximately located on ΣA with inclinations less than 4∘ and 2∘, respectively. (3) γA(a1) and a3 are approximately on ΣB with inclinations less than 2∘ and 0∘, respectively. We thus conclude that the data meet the criteria of 3D reconnection, and the Cluster spacecraft encountered an A–B null pair.
The null–null line is found almost to align with the Y direction as shown in Fig. 1d, and the angles between the separatrices (fans, or legs of the ‘X-point’) calculated from the data are ∼42∘–46∘, as shown in Table 1. This is also in reasonable agreement with the result estimated from 2D electric-potential patterns18. Therefore the X–Z-plane projection of the 3D reconnection configuration observed is very similar to the previously reported 2D ‘X-line’ reconnection geometry of this event17,18.
The length of the null–null line cannot be measured precisely. However, the time interval from the Cluster tetrahedron’s first encounter with the A null at 09:48:24.166 UT to its first encounter to the B null at 09:48:26.974 UT is 2.808 s. Because the time resolution of the velocity data is only 4 s, we must apply other means to calculate the drift speed of the magnetic configuration relative to the Cluster tetrahedron. An appropriate approach is to calculate the velocity on the basis of the null positions at adjacent moments. This can be done by locating successive null positions in the Cluster frame using linear interpolation13 and then calculating the average drift speed and its mean square error to obtain 310±120 km s−1. Multiplying the drift velocity by the above separation time, we find for the approximate null–null separation 860±340 km≈0.7±0.3di, where di=1,320 km is the ion inertial length for the ion density of 0.03 cm−3 shown in Fig. 2a.
In examining the high-resolution data to characterize the null trajectory we also detected oscillations in the lower-hybrid frequency range of 13 Hz (see Fig. 3b). Lower-hybrid oscillations have been reported in conjunction with reconnection in other observations26,27, as well as in experiments and simulations5,6,7,28. In this event, the oscillation is found on the separator line. The polarization of the wave is found in the L–M plane, where L and M are the directions of maximum and intermediate variances of magnetic field25, with the preferable direction almost along the line diagonal to the L and M axes. The wavelength can be calculated from the trajectory and drift velocity of the null as 24±10 km∼de=30 km. From cross-wavelet analysis of the magnetic field around the null–null line29,30, one finds that the wavevector is almost perpendicular to the magnetic field with . This suggests that the dynamics in the neighbourhood of the null–null line is characterized by lower-hybrid waves in the magnetized electron–unmagnetized ion regime. From the fact that 1/k|| is at least an order of magnitude larger than de, lying in the range of the null–null separation, it is likely that the null pair is related to the lower-hybrid perturbation. The observed lower-hybrid wave in the vicinity of the null–null line is consistent with the density dip at the X-point reported in observations and simulations21,22.
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Acknowledgements
This work is supported by the NSFC Programs (grant Nos 40390150, 40504021, 10233050, 10575018, 40536030, 40425004 and 40228006) and the National Basic Research Programme of China (grant No 2006CB806300), as well as the CAS Project KJCX2-YW-T04 and the China Double Star-Cluster Science Team. C.J.X. also thanks A. Grinsted at the University of Lapland for supporting the MatLab wavelet coherence package.
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C.J.X., X.G.W., and Z.Y.P. are first authors with equal contributions to theoretical and data analysis. Z.W.M. has also made important contributions to the paper, and H.Z., G.P.Z. and J.X.W. have developed analysis tools and participated in the analysis. Other coauthors have provided the Cluster observation data and been involved in discussions. M.G.K. has also made significant contributions to the final revision of the paper.
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Xiao, C., Wang, X., Pu, Z. et al. Satellite observations of separator-line geometry of three-dimensional magnetic reconnection. Nature Phys 3, 609–613 (2007). https://doi.org/10.1038/nphys650
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DOI: https://doi.org/10.1038/nphys650
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