Properties of the Turbulence and Topology in a Turbulent Magnetic Reconnection

Magnetic reconnection is a fundamental plasma process (Vasyliunas 1975) responsible for many explosive phenomena in the universe, such as solar flares (Masuda et al. 1994), magnetospheric substorms (Priest & Forbes 2000; Angelopoulos et al. 2008; Fu et al. 2013b; Burch et al. 2016), and disruptions in fusion experiments (Yamada et al. 1994; Ji et al. 1998). During magnetic reconnection, magnetic energy is converted into particle energy (Guo et al. 2016; Fu et al. 2017; Voros et al. 2017; Zhang et al. 2021) as magnetic topology changes (Cao et al. 2006; Gosling 2012; Fu et al. 2015), and simultaneously electrons can be accelerated to several times their thermal energy (Fu et al. 2011, 2013b, 2019a, 2019b; Khotyaintsev & Cully 2011; Li et al. 2017; Xu et al. 2018). Turbulence is intimately associated with magnetic reconnection (Matthaeus & Lamkin 1986; Tu & Marsch 1995; Eastwood et al. 2009; Chen et al. 2017; Fu et al. 2017; Zhao et al. 2019), facilitating energy cascade and transfer (Leonard 1975; Huang et al. 2012; Karimabadi et al. 2013; Zhao et al. 2020; Wang et al. 2022).

In turbulent magnetic reconnection, a wide set of studies has indicated that energy can cascade from large to small scales and dissipated at the dissipation scales, following a universal −5/3 power law in the inertial region (Lazarian et al. 2012; Franci et al. 2017; Zhao et al. 2020), associated with different plasma wave modes (Cao et al. 2006; He et al. 2011; Salem et al. 2012; Grigorenko et al. 2023) and magnetic nulls (Fu et al. 2016; Fu 2019c; Wang et al. 2020; Wang 2023a). However, few of the previous studies disclosed the energy cascade, intrinsic turbulence wave, and magnetic field topologies in different spatial areas (outflow/flow reversal area) (TenBarge & Howes 2013), although these properties can be very distinguishable and dynamic in these regions (Daughton et al. 2011; Fu et al. 2013a; Cao et al. 2013; Higashimori et al. 2013; Fu et al. 2014). Being aware of this point, we provide new perspectives to study turbulent magnetic reconnection in this study. Specifically, here we aim to accomplish three objectives: (1) to explore the properties of energy dissipation in different spatial areas; (2) to resolve which waves are responsible for the energy deposition in different spatial areas (dispersion relations of the wave mode) (Cobelli et al. 2009; Sahraoui et al. 2010; Narita et al. 2016); and (3) to demonstrate what a dynamic picture of turbulent reconnection looks like in different spatial areas (e.g., spatial-temporal evolution of the energy distribution in 3D K -spectrum and the topology of the magnetic field of the X-line/O-line) (Fu et al. 2014; Wang et al. 2020).

To answer these questions, identifying dispersion relations and magnetic nulls in turbulent magnetic reconnection should be the first step. Here, combining the data from the Magnetospheric Multiscale (MMS) mission (Burch et al. 2016), the first-order Taylor expansion (FOTE) technique (Fu et al. 2015), and the dispersion relation from timing (DRAFT) technique (Zhang et al. 2022) enables us to solve these problems. Respectively, MMS is designed to study magnetic reconnection, turbulence, and particle acceleration (Burch et al. 2016); the FOTE method has been widely used to detect the magnetic nulls, identify the null types, and further reconstruct 3D magnetic topology of nulls (Fu et al. 2015); the DRAFT method is applied for analyzing plasma waves and turbulence, which can experimentally determine the dispersion relation of plasma waves and obtain its 3D K -spectrum (Zhang et al. 2022). In this study, these two advanced methods and MMS data have been used to comprehensively analyze one magnetic reconnection event associated with strong magnetic field fluctuations in the Earth's plasma sheet.

The event that we consider was detected by MMS on 2017 July 26, at about 07:30 UT, when it was located at [−23.0, 7.7, 5.0] R_E in GSE coordinates, a region associated with strong turbulence in the plasma sheet. In this event, the four MMS spacecraft form a regular tetrahedron 16 km in size. Figure 1 shows observations of the magnetic field (Figure 1(a)) from a Fluxgate Magnetometer (FGM) experiment (Russell et al. 2016) together with particle observations (Figures 1(b)–(e)) from a Fast Plasma Investigation (FPI) experiment (Pollock et al. 2016). Concretely, Figure 1(a) displays the magnetic field; Figure 1(b) represents the ion velocity; Figure 1(c) represents the electron density; Figures 1(d) and (e) illustrate the electron and ion differential energy fluxes from ∼10 ev to ∼25 kev. From 07:21:13–07:38:42 UT, strong, and turbulent fluctuations in the magnetic field appeared (Figure 1(a)), and the high irregularity in the behavior of the magnetic field could suggest chaotic magnetic field lines. Concurrently, electron and ion fluxes showed characteristic energies with significant variations (Figures 1(d) and (e)). The density was depleted (Figure 1(c)). The ion flow velocity (see Figure 1(b)) reverses from tailward to earthward, suggesting an ongoing magnetic reconnection process.

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In fact, this event has been reported recently as strong turbulence driven by magnetic reconnection, carrying charged particle heating and substantial nonthermal acceleration (Ergun et al. 2018; Wang 2023b). The ingredient for the observed heating appears to be the strong, localized currents (J) and large-amplitude electric fields E, which result in a significant net positive mean of JE, signifies particle energization, and makes ion and electron heating rates experience a fourfold increase from their initial temperature (Ergun et al. 2020). The strong, intermittent turbulence also generates magnetic holes or depletions in ∣ B ∣ that can trap particles. Trapping considerably increases the dwell time of a subset of particles in the turbulent region, which results in significant nonthermal particle acceleration (not shown here).

To better understand the properties of energy dissipation in different spatial areas, here we divide the whole turbulence region (07:21:13–07:38:42 UT) into three areas according to ion flow velocity (Figure 1(b)): tailward flow region (07:21:13–07:26:57 UT); reversal flow region (07:26:57–07:30:00 UT); and earthward flow region (07:30:00–07:38:42 UT); see the green, red, and blue time stamps in Figure 1.

Figure 2 then shows omnidirectional power spectra of the magnetic field averaged over these four intervals and three frequency ranges. We used Welch's averaged periodogram method to estimate the frequency and power spectral density (PSD) of the turbulent magnetic field (Welch 1967). The low-frequency B spectrum is computed from an FGM with a sampling rate of 128 Hz. In Figures 2(a) and (b), the black line denotes the measured spectrum, whereas the navy, green, red, and blue lines denote power-law fits to specific frequency ranges. Respectively, Figure 2(a) represents the spectra in the whole time interval, and Figure 2(b) represents the spectra in the other three regions mentioned above (the measured spectra in Figure 2(b) are multiplied by the appropriate coefficient for better comparison). We can clearly see that the power spectrum broke at two frequencies, according to which we divided the spectra into three frequency ranges. The power spectra at specific frequency ranges are used to fit the corresponding slopes. Over the whole event, in the lowest frequency range ∼0.016 to ∼0.15 Hz (below the ion cyclotron frequency f_ci), the magnetic spectrum shows a power-law index that is consistent with an inertial cascade (−5/3); the frequency range between ion cyclotron frequency f_ci and the average lower hybrid frequency (f_lh ∼ 8 Hz) has a steeper spectral index (–2.43); and at the highest frequencies (> 8Hz), the magnetic spectral index steepens to −3.63. The observed spectral break in the power spectra around f_ci and f_lh indicate the local cyclotron resonance, which pertains to wave–particle interactions and is an important energy conversion channel for turbulence dissipation (Woodham 2018; Chen et al. 2019). These observations are consistent with previously reported turbulence spectra in the magnetosphere.

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Based on the analysis presented in Figure 2, it appears that there may be differences in spectral behavior between different regions in turbulent reconnection. Specifically, Figure 2(b) shows that in the tailward flow region (green time stamp), magnetic spectrum exhibits indexes that change from −1.66 to −2.56, and then to −3.29. In the earthward flow region (blue time stamp), the magnetic spectrum shows indexes changing from −1.46 to −2.32, and then to −3.37. However, in the reversal flow region (red time stamp), the magnetic spectrum represents indexes changing from −1 to −2.53, and then to −4.05. It is noteworthy that at breakpoints, the spectral slopes change more prominently in the flow reversal region than in the outflow regions (tailward and earthward flow regions). This suggests that the ongoing energy cascade and conversion physical process are more pronounced in the flow reversal region, and the corresponding cyclotron resonance mechanism is more effective in this region compared to the outflow region. The steeper spectra breaking observed in the reversal flow region provides evidence for the effectiveness of the cyclotron resonance mechanism in this region.

To investigate the nature of the turbulence during this turbulent reconnection, we perform a DRAFT analysis (Zhang et al. 2022) to determine in which waves the energy is deposited crossing different spacetime scales. Such analysis, calculating the phase velocity of a single-frequency wave by using the timing analysis, dividing the wave frequency by the phase velocity to obtain the wavevector, solving the dispersion relation (ω–k relation) by considering all frequency channels, provides the wave mode of the turbulence and the comprehensive analysis of turbulence (including the spectrum of wave normal angle, wavevector, wave phase velocity, and 3D K -spectrum). We have quantitatively defined three error parameters, the match of amplitude (MOA), ratio of half-wavelength to spacecraft separation (λ/2Rsc), and correlation coefficient (CC) to judge the reliability of the DRAFT technique. These three criteria could be applied to the current study because they could guarantee that the four spacecraft detect the same fluctuations and that the time shift is accurate enough. Generally speaking, the larger these parameters, the more accurate the results. Empirically, we set the threshold of the three parameters as MOA > 0.5, λ/2Rsc > 1, CC > 0.8 to guarantee that the results derived from the DRAFT method are accurate (see the validation in Zhang et al. 2022).

The results of the analysis are shown in Figure 3. The left panel illustrates the properties of the turbulence and the right panel the dispersion relation of the waves. As can be seen, there are strong wave emissions roughly between frequencies 0.01 and 8 Hz (Figure 3(a)) during the whole turbulence region (07:21:13–07:38:42 UT). Such wave turbulence has a phase-velocity magnitude of around 200–1500 km s⁻¹ (Figure 3(b)), a wave normal angle greater than 80° (Figure 3(c)), and a wavenumber near 0.005 km ⁻¹ (Figure 3(d)). As can be seen in Figures 3(e)–(g), these properties should be derived accurately because the three parameters we defined are considerably large in the frequency range of 0.01–8 Hz (λ/2Rsc ≫ 2, CC > 0.8, MOA > 0.7), meeting the criteria we set before. We can reorganize the spectrum data of the wave power, wave normal angle, and wavevector, i.e., showing the wave power P(f, k ) as functions of frequency (f) and wavevector ( k ), to obtain the dispersion relation. Specifically speaking, for each data point of the wave power P(f, t) shown in Figure 3(a), we can obtain its frequency f, wave normal angle θ (f, t) shown in Figure 3(c), and wavevector k (f, t) shown in Figure 3(d). Then we can project this P(f, t) point onto a grid composed of frequency f and wavevectors k , which constitutes one sample of the dispersion relation with the given wave normal angle shown in Figures 3(h)–(o). Considering all frequencies and intervals of interest, we can obtain the dispersion relation of the turbulence wave mode. Since multiple data points of wave power P(f, t) can appear at the same (f, k ) grid, here we take the median of them in as shown in Figures 3(h)–(o). During the reconnection, such reorganization of data is presented in the right panel of Figure 3, with the color bar denoting the samples. All these dispersion relations are in GSE coordinates for waves with propagation angles 60 < θ < 90° because those counts with 0° < θ < 60° are in small quantity and neglectable (see Figure 3(c)). To guarantee that the results are accurate, during reorganization, we removed those sampling counts with small MOA (MOA < 0.5), small λ/2Rsc (λ/2Rsc < 1), and small CC (CC < 0.8), even though such type of counts is very low (see Figures 3(e)–(g)). During the whole turbulence region (07:21:13–07:38:42 UT), we calculated the arithmetic mean value of the total magnetic field and electron density (9.92 nT and 0.053 cm⁻³), and then rounded them to the nearest, as averaged plasma condition (< ∣ B ∣ > ≈ 10 nT and < n > ≈0.05 cm⁻³), by which we estimated the local Alfvén velocity V_A ≈1000 km s⁻¹. We overplot the local Alfvén velocity in Figures 3(h)–(o) to better understand the wave dispersion relations in this event (see the white solid lines). It should also be mentioned that during the turbulent region, a substantial part of the ion energy flux exceeds the maximum energy (25 keV) of the FPI instrument (see Figure 1(e)), making N_i difficult to determine accurately. Therefore, here we cautiously consider electron density N_e, but not ion density N_i to calculate the Alfvén velocity, since the above calculations would dramatically change the resulting Alfvén velocity.

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More concretely, Figures 3(h)–(k) and Figures 3(l)–(o) respectively show dispersion relations of the turbulence wave mode in the tailward flow, reversal flow, earthward flow, and whole turbulence region when f < f_ci and f_ci < f < f_lh. As can be seen in Figures 3(h)–(k), when f < f_ci, dispersion relations of the waves in different spatial regions show similar properties consistent with KAWs (Hasegawa & Mima 1978; Salem et al. 2012): wave phase velocities are slightly smaller than Alfvén speed, and wave normal angles are roughly close to 90° (Figure 3(c)). However, in Figures 3(l)–(o), when f_ci < f < f_lh, the dispersion relations of the waves in different spatial regions show different properties. Specifically, in the tailward region (07:21:13–07:26:57 UT), Figure 3(l) shows a dispersion relation with wave phase velocity clearly lower than V_A, which is consistent with slow mode waves (Howes et al. 2012; Cao et al. 2013; Wang et al. 2016); in reversal and earthward flow regions (07:26:57–07:30:00 UT and 07:30:00–07:38:42 UT), Figures 3(m) and (n) show dispersion relation with wave phase velocity is clearly higher than V_A, which is consistent with fast mode waves (Chandran 2005). Interestingly, although KAWs are often detected in turbulence, we first experimentally verify the KAWs' mode in turbulent magnetic reconnection. Here we reveal the dispersion relations associated with the analysis of the turbulence from the cross scale in turbulent magnetic reconnection, which will help us better understand exactly in which waves the energy is deposited during turbulence.

Certainly, we can reorganize the data in Figures 3(a)–(d) in a different format, i.e., showing the wave power P(f, k ) as functions of k_x , k_y , k_z to obtain the three-dimensional K -spectrum of these waves. We consider all frequency waves during the whole interval 07:21:13–07:38:42 UT, but again, we remove those sampling counts with small MOA (MOA < 0.5), small λ/2Rsc (λ/2Rsc < 1), and small CC (CC < 0.8). Such type of data reorganization is shown in Figure 4, as an estimation of magnetic field energy distribution in the wavevector space. Using the same technique, we continuously reconstruct the K spectrum from 07:21:13–07:38:42 UT and show them in the Figure 4 animation, which can be considered as the spatial-temporal evolution of the magnetic field energy distribution. The 3D view of Figure 4 is obtained by displaying the magnetic wave field energy in (k_y , k_z ) plane for 21 different values of k_x ranging from −0.1 to 0.1 km⁻¹. As can be seen in Figure 4 and its animation, the large-scale regions (k_x ∼ 0 km⁻¹) correspond to most significant energies; the small-scale regions (k_x ∼ 0.02 km⁻¹) correspond to significant but slightly weaker energies, while the smallest-scale regions (k_x ∼ 0.1 km⁻¹) correspond to weakest energies. This kind of energy distribution, following the nature of the trend of PSD, could be consistent with turbulent theory (Leonard 1975).

After identifying the properties of the turbulence of dispersion relation and energy dissipation mechanisms, in this section, we focus on the properties of the topology of this turbulent magnetic reconnection. Magnetic nulls, where magnetic field strength vanishes and plasmas become unmagnetized, are commonly found in 3D magnetic reconnection (Priest & Titov 1996) and are considered structures where the reconnection process happens (Priest & Forbes 2000). Investigating the properties of such structures can improve our understanding of the 3D picture of turbulent magnetic reconnection (Chen et al. 2019; Wang et al. 2020). Interestingly, we noticed that during spacecraft crossing the strong turbulent magnetic reconnection region, MMS sometimes measured magnetic field strength B_t approaching zero (Figure 1(a)), indicating the potential existence of magnetic nulls in this event. Particularly we search for magnetic nulls and analyze null properties resolved by the FOTE technique during two short intervals 07:23:55.30–07:23:55.45 UT and 07:31:47.30–07:31:47.60 UT (see the black dashed lines in Figures 1(a)–(e)). Such properties, derived from the Jacobian matrix δ B measured by the four spacecraft, include the distance from magnetic null to each spacecraft (see Figure 5(a), (b) and (e), (f)), null types (see Figure 5(b), (f)), and dimensionality of the structure (see Figure 5(d), (h)). Specifically, the Jacobian matrix δ B has three eigenvectors, v ₁, v ₂, v ₃, and three eigenvalues, λ₁, λ₂, λ₃. The null types shown in Figure 5(b) and (f) are identified using these eigenvalues: when all of them are real, the null is radial type, including A-type null (one positive eigenvalue) and B-type null (one negative eigenvalue); when two of them are complex, the null is spiral type, including As-type null (positive real eigenvalue) and Bs-type null (negative real eigenvalue); and when their relationship is unclear (due to nonlinearity of the magnetic field), the null type cannot be identified and therefore is unknown. The A- and B-type nulls can be simplified to X-null (2D) if one of the eigenvalues is very small (Fu et al. 2015). Similarly, the As- and Bs-type nulls can be simplified to O-null (2D), if the real part of the eigenvalues is very small. The dimensionality of the structure can thus be defined as F_3D ≡ ∣λ∣_min/∣λ∣_max, with F_3D < 0.2 indicating a 2D structure and F_3D > 0.5 indicating a 3D structure.

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To guarantee that the null positions are accurately resolved, we require the null-spacecraft distance (r_i ) to be less than 1000 km (1 d_i ; see the quantitative validation in Fu et al. 2015). Additionally, for examining the reliability of null properties, we define two parameters (η ≡ ∣∇ · B ∣/∣∇ × B ∣ and ξ ≡ ∣(λ₁+λ₂+λ₃)/∣λ∣_max). Here the two parameters (η and ξ) actually reflect the divergence of the magnetic field due to linear assumption. The larger η and ξ are, the more unreliable results we will get. Following previous studies (e.g., Chen et al. 2017; Wang et al. 2020), we require these two parameters to be smaller than 40% to guarantee the null properties are accurately identified.

As can be seen from the results of the analysis shown in Figure 5, during these two short periods, both spiral nulls (A and X) and radial nulls (Bs) are detected (Figure 5(b), (f)); both 3D and 2D structures exist (Figure 5(d), (h)). Because the values of η and ξ are small (η < 40%, ξ < 40%; see Figure 5(c), (g)) and the null-spacecraft distance is short (∣r∣ < 1000 km; see Figure 5(a), (e)), these results should be accurate. We trace and inverse-trace magnetic fields to obtain the null topologies at 07:23:55.32 and 07:31:46.13 UT as shown in Figures 6(a) and (b). Such topologies are shown in the eigenvector coordinates. The Bs-type null (see Figure 6(b)) at 07:23:55.32 UT is an example of a 3D structure with F_3D > 0.5 (Figure 5(d)) and the X-null (see Figure 6(a)) at 07:31:46.13 UT is an example of a 2D structure with F_3D < 0.2 (Figure 5(h)). We find that the magnetic topologies of Bs-type null and X-null, respectively, have radial and spiral features, consistent with the theoretical model of magnetic nulls (Lau & Finn 1991; Pontin 2011).

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To better understand the properties of these magnetic nulls, a statistical investigation is necessary. We consider the magnetic field data during the whole turbulent magnetic reconnection and divide them into four periods T1, T2, T3, and T4 as mentioned before (see Figures 6(c), (d)). We apply the FOTE method to these periods to find magnetic nulls. Again, to make sure the results of the analysis are reliable, we only consider the magnetic nulls with null-SC distance ∣ r ∣ ≤ 1000 km, η < 40%, and ξ < 40% as mentioned before. Respectively, Figures 6(c) and (d) show the types and dimensionality of these nulls, with the color bar denoting the number of magnetic nulls. In total, we find 1816 nulls in the turbulent sheet from 07:21:13–07:38:42 UT. Generally, (1) among these nulls, 1423 nulls are of spiral type (As, Bs, and O), while 393 nulls are of radial type (A, B, and X). Obviously, the number of spiral nulls is about 3.6 times as large as the radial nulls; in other words, spiral nulls contribute 78% to the total magnetic nulls, while radial nulls contribute 22%. (2) Among these nulls, 1032 nulls are three-dimensional structures (F_3D > 0.5), while 127 nulls are two-dimensional ones (F_3D < 0.2). Also, no matter in which period, the number of 3D nulls is always about 8 times larger than that of 2D nulls.

We can see that our statistical results of null numbers are consistent with the results of percentages reported by Chen et al. (2017) and Olshevsky et al. (2016) in turbulent magnetosheath plasma, where spiral nulls contribute around 80% to the total magnetic nulls in turbulent magnetosheath. The ratio is also close to what Eriksson et al. (2015) find in a fully random magnetic field, which suggests the physical processes responsible for the null formation may not favor the formation of particular null types. Besides, owing to the complex condition in the upstream region and the strong turbulence in the current sheet, magnetic reconnection in the magnetosphere is essentially three-dimensional and super-dynamic. The real two-dimensional structures in turbulent magnetic reconnection are few, contributing about 7% to the total magnetic nulls in this study, proving the complex 3D aspects of the dynamic magnetic reconnection that essentially exist in space plasmas. In theory, the properties of magnetic nulls are determined by the currents surrounding them (Parnell et al. 1996, 1997). Therefore, investigating the intrinsic nature of currents and 3D configurations of magnetic nulls, which are closely related to the energy dissipation process, will help better understand the 3D turbulent reconnection. We will focus on this topic in our future work.

In summary, using the DRAFT method, FOTE method, and MMS data, we present observations of the turbulent fluctuations within a magnetic reconnection in the Earth's magnetotail. We find that the cyclotron resonance mechanism is more effective in the flow reversal region than in the outflow region of turbulent magnetic reconnection; energy injected by the reconnection propagates through the turbulent cascade, respectively, along the fast/slow mode branches at low-hybrid scale and the kinetic Alfvén mode branch at ion cyclotron scale; energy distribution indicates weakening wave power with increasing scale in 3D K -spectrum; spiral nulls (O-lines) were about 3.6 times more than radial ones (X-lines), and 3D structures were about 8 times more than 2D ones during reconnection. Our observations provide new sights to study turbulent magnetic reconnection processes.

We thank the MMS Data Center (https://lasp.colorado.edu/mms/sdc/public/) for providing the data. This work was supported by NSFC grants Nos. 42125403 and 41821003, and the Fundamental Research Funds for the Central Universities.