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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.

Figure 1: The 3D reconnection configuration.
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a, The locations of the Cluster tetrahedron at two times separated by 2.808 s, in a frame moving with the average drift speed of nulls and origin at the 09:48:25.593 UT position of spacecraft C3, (−16.2, 7.9, 0.5) RE in GSM coordinates. The yellow dots indicate the locations of two nulls calculated from linear interpolation at moments shown in Table 1. The red arrows from the dots represent the directions of movement of the nulls and the black arrows are a3 for the A null and b3 for the B null, showing the trend of the null–null line. b, The reconnection region in the magnetotail (red rectangle). c, The X–Z-plane 2D projection of the 3D reconnection geometry. d, Topological structure of 3D reconnection with observed A and B nulls, fan surfaces ΣA and ΣB and spine lines γA and γB in GSM. The fans ΣA (bounded by γB) and ΣB (bounded by γA) intersect along the null–null line (shown in purple). The characteristic directions of all vectors and surfaces were calculated from Cluster data (Table 1).

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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.

Figure 2: The Cluster measurements and Poincaré index calculation in the reconnection event on 1 October 2001.
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a–c, Overview of the 4-s-resolution data of the plasma density N (a), magnetic field B (b) and velocity V (c) for C4 from 09:45 to 09:51 UT, where the Vx>0 intervals are masked red and the Vx<0 interval is masked yellow. The blue-masked area is the flow-reversal interval where the nulls are detected. d–f, The intercalibrated high-resolution (0.04 s) magnetic-field data for C1–C4 for a period in which nulls were identified; g, the corresponding Poincaré index calculated during the period 09:48:20–09:48:50 UT.

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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.

Table 1 Characteristics of the null pair observed on 1 October 2001.
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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.

Figure 3: The null’s drift velocity and the lower-hybrid oscillation signatures.
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a, The velocity of the null drifting inside the Cluster tetrahedron in L M N coordinates with L=(−0.22,0.97,−0.05), M=(0.97,0.23,0.11), N=(0.12,−0.03,−0.99) in GSM. The L-axis is almost aligned with the null–null line shown in Table 1. b, The fast Fourier transform (FFT) spectrum of the null-point oscillation shows maximum power at 10–15 Hz, consistent with the lower-hybrid frequency (, with magnetic field 20 nT). c, Hodogram of the velocity of the null drifting in the L–M plane starting at the blue spot. d, The wavevector in L M N coordinates calculated by cross-wavelet analysis shows that kN>kM≫kL in the frequency range of 10–13 Hz (a.u. is the abbreviation of arbitrary units).

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