Characteristics of Turbulence Driven by Transient Magnetic Reconnection in the Terrestrial Magnetotail

This paper investigates the evolution of turbulence within the magnetotail reconnection exhaust observed by the Magnetospheric Multiscale spacecraft. The reconnection was in an unsteady state that caused significant temporal variations in the outflow speed. By dividing the exhaust into nine fast flows, we analyzed and compared the characteristics of turbulence in these nine flows. We find that the strength of the intermittency has a good relationship with the peak speed of the fast flows. The higher-order analysis of magnetic field fluctuations reveals that the turbulence is multifractal in the inertial range for these flows except one with the highest peak speed. Moreover, the turbulence is monofractal on kinetic scales in all of these fast flows. The magnetic energy was intermittently dissipated in these turbulent flows, predominantly occurred in the coherent structures. Since the coherent structures with the largest energy dissipation in these flows are different, we suggest that the mechanism of energy dissipation may be different among these flows.


Introduction
Turbulence and reconnection are two fundamental processes in astrophysics plasma (Parker 1979;Taylor 1986). They are not independent processes but intimately associated with each other. Magnetic reconnection is the reconfiguration of the topology of two magnetic field tubes, which results in the conversion of magnetic energy into various forms of plasma energy. Turbulence is an inherently nonlinear phenomenon characterized by the presence of broadband perturbations. It is universally observed in space plasmas such as the solar wind, the Earth's magnetosheath, the magnetotail, the polar cusp region, and the high-latitude ionosphere (Lagoutte et al. 1992;Matthaeus et al. 1995;Borovsky et al. 1997;Zimbardo et al. 2010;Wang et al. 2014;Cranmer et al. 2015;Ergun et al. 2015Ergun et al. , 2018Burch et al. 2016;Torbert et al. 2018). A crucial element of turbulence is the energy cascade, which transports energy from large to small scales until it reaches the dissipation scale, in which the energy goes to heat and accelerate particles and is finally dissipated.
In recent decades, there has been interest in studying turbulence and reconnection together. The formation of kineticscale current sheets in turbulence provides favorable conditions for the onset of reconnection (Servidio et al. 2009(Servidio et al. , 2011. Recent satellite observations have identified reconnection in the turbulent high-speed solar wind, magnetosheath, and the transition layer of the bow shock (Gosling 2007;Retinò et al. 2007;Yordanova et al. 2016;Vörös et al. 2017;Phan et al. 2018;Lu et al. 2020). On the other hand, three-dimensional kinetic simulations demonstrate that magnetic reconnection evolves from a laminar current sheet into a turbulent state, accompanied by the generation and coalescence of magnetic flux ropes (MFRs; Daughton et al. 2011;Wang et al. 2016) or complex current filaments (Che et al. 2011). Dissipation in turbulent reconnection proceeds in an intermittent cascade manner and produces multifractal structures, which is similar to the phenomenology of fluid-like turbulence (Leonardis et al. 2013).
Magnetic reconnection in the terrestrial magnetotail is deemed a principal element of triggering auroral substorms and driving fast plasma flows (Baumjohann et al. 1989;Angelopoulos et al. 1992Angelopoulos et al. , 2008Nakamura et al. 2001;Ergun et al. 2018;Zhou et al. 2019). The consequence of magnetic reconnection in the magnetotail is to leave a depleted plasma sheet, a region with a strong magnetic field and plasma fluctuations (Borovsky et al. 1997). The characteristics and evolution of turbulence driven by magnetotail reconnection have become a hot topic of interest recently. Eastwood et al. (2009) found that large-amplitude fluctuations in a reconnection ion diffusion region are the fast whistler turbulence. While Huang et al. (2012) pointed out that the turbulence is dominated by the Alfvén-whistler mode once there is a moderate guide field in the diffusion region. They estimated that the turbulence supported a sufficient anomalous resistivity for the reconnection (Huang et al. 2012). Large-scale exhausts in the magnetotail are the hotbed for turbulence, within which magnetic energy is converted to kinetic energy in an intermittent manner (Osman et al. 2015). Fu et al. (2017) found that the energy dissipation in turbulent reconnection occurs at the o-lines instead of at the x-lines. Ergun et al. (2018) studied a turbulent reconnection in the magnetotail by using the high-resolution Magnetospheric Multiscale (MMS) data. They found that large-amplitude parallel electric fields with frequencies above the ion cyclotron frequency provide significant dissipation and can result in energetic electron acceleration (Ergun et al. 2018). Turbulence cascade also plays an important role as a pathway of energy dissipation during reconnection in the magnetotail (Bandyopadhyay et al. 2021). Zhou et al. (2021) observed multiple electron-only reconnecting current filaments in the turbulent outflows driven by a primary reconnection in the magnetotail. They suggest that these secondary reconnections (SRs) in the outflow contribute significantly to the overall energy release during the large-scale primary reconnection.
A recent 3D particle simulation illustrates that the property of turbulence is related to the distance from the reconnection x-line (Pucci et al. 2017), which is believed as the source of turbulence in reconnection. Moreover, Yordanova et al. (2008) showed that the properties of turbulence varied as a function of distance to the source in the magnetosheath. Motivated by these studies, we observationally investigate the variation of the turbulence in a transient magnetic reconnection to understand the characteristics and evolution of turbulence driven by reconnection.

Event Overview
This study used the data from the MMS mission. The 3D magnetic field vectors are measured by the fluxgate magnetometer (FGM) with a cadence of 128 samples s −1 in the burst mode . The 3D electric field vectors are measured by the electric double probe with a time resolution of 8192 samples s −1 in the burst mode (Ergun et al. 2016;Lindqvist et al. 2016;Torbert et al. 2016). Particle moments are measured by the Fast Plasma Investigation (FPI) with a time resolution of 0.03 s for electrons and 0.15 s for ions in the burst mode (Pollock et al. 2016). The background subtractions and spin tone corrections have been implemented in the released level-2 moments (Gershman et al. 2019). An FGM-SCM (search-coil magnetometer) merged (FSM) data with a time resolution of 8192 samples s −1 is used to calculate the structure function (Argall et al. 2018). Figure 1 displays the plasma and magnetic field data from MMS 1 in a one and a half an hour interval encompassing a reconnection x-line on 2017 May 28, when MMS was in the Earth's magnetotail at around [−19.5, −12, and −3.4] R E (Earth radii) in geocentric solar magnetospheric (GSM) coordinates. The satellite was far away from the magnetopause boundary layer. The average spacecraft separation was about 60 km ∼ 0.12 d i ∼ 0.04 ρ ci ∼ 3ρ ce ∼ 0.07λ De , where d i is the ion inertial length, ρ ci is the ion gyro-radius, ρ ce is the electron gyro-radius, and λ De is the Debye length. These are the averaged values in the time interval plotted in Figure 1. MMS observed an active x-line at around 04:00 UT, which is manifested by the bidirectional ion flows away from the x-line and the accompanied reversal of magnetic field B z component. All the vectors in this paper are presented in GSM coordinates unless otherwise noted. Tailward flow lasted about 10 minutes from 03:50-04:00 UT, while the earthward flow persisted for more than 1 hr. Magnetic fields exhibit strong fluctuations near the x-line and in the outflow (shown in Figures 1(b), (c)). The magnetic field power spectrum has a power-law relationship in both the inertial and dissipation range, with the slope close to −5/3 in the inertial range (Zhou et al. 2021), which suggests that this reconnection excited turbulence. More details about this reconnection event can be found in Zhou et al. (2021). Figure 1(e) displays that plasma β is almost larger than 1 in the plotted interval except for a few instants that β drops to 0.1. The average β is larger than 10, indicating that MMS was deep inside the plasma sheet, which is characterized by intense fluxes of the thermal population shown in Figures 1(f) and (g). Figure 1(a) clearly shows that the bulk flow speed V ix varies significantly during the whole interval, especially earthward of the x-line. Taken 100 km s −1 as the threshold for the fast flow, we have identified nine fast flows, which are marked by the red bars at the top of Figure 1. The emergence of these different fast flows was probably caused by the change of the reconnection rate, i.e., this reconnection was in a transient stage (Zhou et al. 2017a). In the following, we focus on the characteristics of turbulence and energy dissipation in these nine fast flows.
The average flow speed of these nine fast bulk flows V f is 400-800 km s −1 , while the Alfvén speed V A is 300-400 km s −1 ; thus, it is not obvious that Taylor's hypothesis (Taylor 1938) is applicable here as the case in the magnetosheath and solar wind. However, if the fluctuations represent well-developed turbulence, this condition (V f ? V A ) is somewhat relaxed (Matthaeus & Goldstein 1982). The fluctuations are frozen in if the characteristic time, t λ ≈ 1/(b k k), for dynamical evolution of structure of size λ is much greater than the convection time of the structure with speed V f : t ≈ λ/V f , where b k is the Fourier-series amplitude of magnetic field measured in Alfvén units at wavenumber k (Bandyopadhyay et al. 2021). Once V f ? b k , t λ will be much larger than t and the fluctuations can be viewed as frozen-into the flow. Here we find that the rms of the Alfvén speed δV A in these fast flows is about 0.05-0.2 V f . Because the magnitude of b k is generally smaller than δV A , the frozen-in approximation is valid for these fluctuations. Moreover, we employed the second-order structure function to test the Taylor hypothesis (Chen & Boldyrev 2017) in these fast bulk flows. The second-order structure function estimated by single-spacecraft under the Taylor hypothesis generally agrees with that obtained from pairs of spacecraft (not shown), further proving that the Taylor hypothesis is valid in this study.

Results
First, we investigate the probability density distribution (PDF) of the current density, nonideal electric field, and energy dissipation in the nine fast flows. Here we illustrate the results of three fast flows as examples. Figure 2 shows the PDFs of the nonideal electric field E E E d ¢ = ¢ -á ¢ñ , current density δj x,y,z = j x,y,z − 〈j〉 x,y,z , and energy dissipation δD e = D e − 〈D e 〉, D e = j · (E + V e × B) in the fast flow from 04:09:10-04:13:00 UT, where E E V B e ¢ = +´is the nonideal electric field or electric field in the electron frame and 〈 〉 means average in the given period. The electromagnetic fields and electron velocity have been averaged among the four spacecraft, while the current density is calculated at the barycenter of the spacecraft tetrahedron by the curlometer technique (Dunlop et al. 2002). The elongation of the MMS tetrahedron was about 0.12 and the planarity was nearly 0.1. In addition, we find that the average | ∇ ·B|/| ∇ ×B| is less than 0.2; thus, we believe the current density calculated by the curlometer technique is reliable. Note that the current density can also be calculated from the plasma moments. We also compare the current density estimated by the curlometer technique and that calculated from the plasma moments. We find a systematic difference between the two (not shown), which is primarily due to the ion measurements. The ion data suffers from penetration radiation from the Sun, which causes contamination in the ion energy spectrum. In addition, hot ions in the magnetotail occasionally have energy exceeding the upper energy limit of the FPI instrument, thus ion moments measured by FPI are less reliable than electron moments, which leads to less reliable current density calculated from plasma data. Therefore, we decide to use the current density estimated by the curlometer technique in this study.
The black curves in each panel represent the normalized Gaussian distribution. The peak speed of this fast flow is the largest one among the nine flows. It is also the one closest to the x-line. Here we assume that the x-line continuously moved tailward; hence, a given point with a larger time difference to the observational time of the x-line (∼03:59 UT) is regarded as located further from the x-line. We find the PDFs of three components of E¢ and j are closest to the Gaussian distribution in the flows that is nearest to the X-line. The corresponding kurtosis of E¢ and j are around 4 and 5. The PDFs of energy dissipation D e deviates from Gaussian, implying the intermittent nature of the energy dissipation in this flow. Generally, when we multiply two Gaussian variables A and B into C, the distribution of C is no longer a normal distribution but a chi-square distribution. It is simple to verify that the kurtosis of the Gaussian distribution is 3, while the kurtosis of C is 9 (Matthaeus 2021). Figure 3 displays the PDFs of the current density, electron frame electric field, and the nonideal energy conversion in the fast flow with the smallest peak speed. Figure 4 presents the results in the fast flow, which is further away from the x-line than flows 1 and 2, while the peak flow speed is slightly faster than that of flow 2. From Figures 2-4, we can see that the PDFs of E¢ deviate from the Gaussian distribution,; however, the PDFs of the variation in the current density is hardly noticeable. Moreover, the PDFs of D e are always non-Gaussian in these fast flows, which indicates that the energy exchange between the magnetic field and plasma is in an intermittent manner (Leonardis et al. 2013;Osman et al. 2015;Pucci et al. 2017). In addition to these three periods, we have analyzed the other six fast flows using the same method mentioned above. The results show that, compared with the first flow, which has the largest speed and is the closest one to the x-line, the flows further away from the x-line or with smaller bulk speed tend to have heavier tails in the PDF of D e , while the PDF of the current density in all of these flows are almost Gaussian. The integrated values of the energy dissipation in each flow is shown in Table 1.
To describe quantitatively the intermittency in these nine fast flows, we employed higher-order moments to characterize the  scaling characteristics of turbulence (Pagel et al. 2001). A structure function of order m is defined as follows: , 1 where τ indicates the time lag between adjacent points and 〈 〉 indicates the time average. Here the magnetic field B is obtained from the FSM data, which has higher time resolution than the FGM data. In a system exhibiting scaling, the structure function satisfies a power-law scaling: If the scaling exponent satisfies a linear relationship with the order m: g(m) = q × m, then it implies monofractal with a single index q, i.e., a global scale invariance. In multifractal turbulence, the scaling exponent g(m) is a nonlinear function of the order m (Frisch & Kolmogorov 1995;Pagel & Balogh 2001;Bruno & Carbone 2013). Figure 5(a) shows the structure function of the magnetic field B z averaged over the four spacecraft and the corresponding scaling exponents in the fast flow observed from 04:08:50-04:13:26 UT. Generally, the reconnection outflow in the magnetotail is mainly in the X-direction so B x is the main component along the lag direction. However, B x involves the fluctuations caused by current sheet flapping which is not a direct effect of reconnection. Here we choose B z because it is the reconnected magnetic field component in the magnetotail, the fluctuation of which is directly related to the reconnection process (e.g., Huang et al. 2012).
A nominal rule for the maximum order of moment m that can be determined reliably from a sample of N points is m max = log 10 N − 1 (De Wit 2004;De Wit et al. 2013). Since the number of data points in each flow is generally larger than 1 million but less than 10 million, the maximum order of moment m can be determined reliably is 5; thus we focus on the scaling exponents in the range mä [1:5] in this paper. The inertial and dissipation range is separated by a time lag of 3 s, which is consistent with the breakpoint in the power spectral density shown in Figure 5(b). Linear fits for the inertial and dissipation ranges are represented by the black lines in Figure 5(a). We estimate the scaling exponents g(m) by the gradients fitted in Figure 5(a). The relations between g(m) and m in the dissipation and inertial range are displayed in Figures 5(b) and (d), respectively. The error bars of g(m) is the sum of the regression error from Figure 5(a) (Kiyani et al. 2006). One can see that g(m) has a linear relation with m both in the inertial and dissipation range, implying that the turbulence is globally scale invariant, i.e., corresponds to monofractal turbulence. This contradicts with recent studies showing that inertial range fluctuation is spatial multifractal, and distributes highly nonuniform in the reconnection outflow (Osman et al. 2015) and fast solar wind (Kiyani et al. 2009).
However, other flows exhibit different characteristics. Figure 6 presents the structure function in the fast flow during 04:48:36-04:52:06 UT. Similar to the flow shown in Figure 6, the scaling exponents in the dissipation range have a linear relation with the order m, whereas g(m) is a nonlinear function of m in the inertial range; hence, the turbulence is multifractal in the inertial range in this fast flow.
In order to get a better understanding of the intermittent strength, we employ the intermittency models to describe the scaling laws of the intermittent turbulence (Carbone et al. 1996). Here we use the p model to quantify the strength of the intermittency, which is measured by the parameter p. It is suggested that the p model is the best to describe the fully developed solar wind turbulence (Carbone 1994;Horbury & Balogh 1997;Pagel & Balogh 2001). The above equations become a linear function when p = 1/ 2, just like the characteristics of the dissipative range, while p = 1 corresponds to a fully developed turbulence.
We estimate the p parameter by fitting the exponents g(m) of the structure function through the K41 and KI65 models. We vary the value of p in the range of [0, 1] by a step of 0.01 to find the p corresponding to the smallest goodness (Pagel et al. 2001): where o(m) is the observed exponents and g(m) is the one obtained from the model. The fit is limited in the interval m = (1, 5). In this test, both the Kolmogorov theory and Kraichnan theory are applied and the best fit is chosen as the final result. As is shown in Figure 5(d), the blue solid line represents the best fit by the K41 model with p = 0.58 and goodness = 2 × 10 −3 . This is consistent with the monofractal feature of the turbulence in the inertial range. We employ the above method to calculate p and the goodness for the nine fast flows. Figure 7(a) shows the relation between the parameter p and the peak bulk speed for the seven fast flows. The values of p and the time gap dt and distance dL between the observation time of the flow and the X-line are shown in Table 1. We do not list the results in two of the fast flows because no obvious breakpoint is found between the inertial and dissipation range in the structure function within these two fast flows, which prevents us from finding the scaling exponents accurately. As shown in Figure 7(a), the parameter p generally decreases as the increment of peak flow speed. This means that the intermittency is reduced with the increase of the flow speed. However, two abnormal points disobey this trend. We find that the two corresponding flows (04:48:36-04:52:06 and 05:02:26-05:11:10 UT) are both further away from the x-line than the adjacent flow with smaller bulk speed (04:16:40-04:25:48 UT). It is shown in a 3D kinetic simulation that the properties of the turbulence produced in the outflows vary in space (Pucci et al. 2017). In particular, they showed that the intermittency becomes stronger in the flow further away from the reconnection x-line (Pucci et al. 2017). Therefore, we also examine the relationship between the parameter p and the distance away from the x-line, as shown in Figure 7(b). To convert between space and time, we refer to the typical X-line retreating speed of ∼100 km s −1 in the magnetotail (Russell & McPherron 1973;Imada et al. 2007;Oka et al. 2011;Zhou et al. 2017b). This velocity is then used to calculate the distance between the MMS spacecraft and the X-line, as shown in the top of Figure 7(b). We find that there is a clear trend when dt < 40 minutes or dL < 38 R E (443 d i ), that is, further away from the reconnection x-line, the greater the parameter p is, i.e., the stronger the intermittency. This result is similar to the result discussed by Pucci et al. (2017). However, the intermittency remains nearly constant after dt ∼ 40 minutes or dL ∼ 38 R E (443 d i ) even though the distance from the x-line increases, which suggests the possibility that the intermittency only increases to a certain distance. The above results indicate that the intermittency strength has a good relationship with the peak flow speed, while its relation with the distance from the x-line is not obvious.

Discussion and Summary
We have shown that the energy is intermittently dissipated in kinetic scale in these fast flows. It has been shown that fluctuations in turbulence are manifested as small-scale coherent structures, which essentially contribute to energy dissipation (Leonardis et al. 2013). To understand where does the energy dissipation mainly occurs, we examine the dissipation of different fast flows. We find that most flows involve coherent structures via intermittent cascade and these coherent structures are largely different in different flows. Here we present three flows as examples. Figure 8 compares the magnetic field, ion and electron flows, current density, and energy dissipation in three fast flows in detail. The first flow is characterized by multiple MFRs. This flow is the one presented in Figures 2 and 5. There were nine MFRs (indicated by the black arrows) embedded within this flow, each one is manifested by the bipolar variation in B z and enhancement in B y and |B|. Some of the MFRs were coalescing as there is an intense negative current J y between the two MFRs (Wang et al. 2016;Zhou et al. 2017a), such as MFR7 and MFR8. In addition, two SRs were observed in this flow (Zhou et al. 2021). Figure 9a6 illustrates that large De (either positive or negative) mainly occurs in/around these MFRs and at the secondary reconnection sites, implying that energy conversion is primarily contributed by these MFRs and the SR, which is partially consistent with the results of Fu et al. (2017).
The magnetic field is less turbulent in the fast flow during 04:38:44-04:41:54 UT than the former one. The coherent structures in this fast flow include the secondary reconnecting current sheet and dipolarization front (DF), which is a kinetic-scale vertical current sheet (e.g., Zhou et al. 2009Zhou et al. , 2019. The largest |D e | occurs near or in these coherent structures. D e at the secondary reconnection site is mostly positive while it is mostly negative near the DF. The rightmost column of Figure 8 shows that three SRs, a DF, and a MFR are detected within the fast flow during 05:22:07-05:27:16 UT. Similar to the former two flows, the largest |D e | occurs in the coherent structures, with the SRs providing a stronger energy exchange than the other structures. All of these reveal that coherent structures involved in strong turbulent fluctuations can produce large dissipation, contributing greatly to the total energy dissipation.
Additionally, the kurtosis of D e in the first fast flow is about 9.8, which is consistent with the previous conclusion that the kurtosis of the product of two Gaussian variables is 9. The kurtosis of D e in the other two flows is 57 and 40, respectively. This indicates that more coherent structures do not result in greater kurtosis since the first flow involves more coherent structures than the other two flows. In other words, the intermittent intensity is not proportional to the number of coherent structures.
In summary, we have used MMS observation to study the characteristics of turbulence in outflows of a transient reconnection. Statistical analysis confirms that the energy conversion and dissipation proceed in an intermittent manner,   which is consistent with the simulation results (Leonardis et al. 2013). However, the PDF of current density is close to the Gaussian distribution and exhibits no obvious intermittency in these fast flows, while the PDF of the nonideal electric field deviates slightly from the Gaussian distribution in the flows away from the x-line. The magnetic field fluctuations within different fast flows show different fractal characters. For instance, a multifractal scaling was shown in a fully developed turbulence and a monofractal scaling was shown in weak intermittent turbulence within the inertial range.
The evolution of intermittent turbulence and its source has attracted much attention in recent years (Yordanova et al. 2008). It has been shown that the intermittency increases with the distance from the source of turbulence in the heliosheath (Burlaga et al. 2006). Pagel et al. (2001) observed a high degree of intermittence in the fast solar wind and the low intermittency region of the slow wind is adjacent to the high intermittency region, suggesting that the slow wind has no unified origin. Hence, the intermittency intensity of turbulence may provide powerful evidence for the source of turbulence. Magnetic reconnection is commonly recognized as an important source of turbulence, whereas there are few studies on the evolution of intermittent turbulent reconnection within the magnetotail. Here we employ the intermittency models to calculate the intensity of the intermittency in different fast flows produced by magnetotail reconnection. We find that intermittent strength is a good function of fast flow velocity; however, its relationship with the distance to the reconnection x-line is not clear. A direct comparison with the results of Pucci et al. (2017) must be carefully considered since in Pucci et al. (2017) the comparison is performed at the same time in simulation, whereas the different flows were observed at different times in this study. Temporal evolution of the flow and reconnection may be one factor causing the difference between our observation and simulation in Pucci et al. (2017).
We show that intense energy dissipation in turbulent flows primarily occurs in/near coherent structures. This is at odds with recent results demonstrating that dissipation occurs near current sheets and not in them (Parashar & Matthaeus 2016;Bandyopadhyay et al. 2020), which is partially caused by the different dissipation proxies used. Here we used j · E' while Bandyopadhyay et al. (2020) used Pi-D (the pressure strain interaction) to quantify the dissipation. Since j · E' is large in regions with large current density j, the intense dissipation identified in this paper is correlated well with the strong current sheets or filaments, which are primarily within or nearby the coherent structures. Previous studies showed that intermittent turbulence generates small-scale coherent structures that are responsible for nonuniform dissipation (Frisch et al. 1995;Biskamp 2003). The coherent structures of these intense energy dissipation events are entirely distinct in different fast flows within the same reconnection outflow region, in which the dissipation mechanisms may be also different among the flows. Our results shed new light on the nature of turbulence in transient magnetic reconnection.
We are grateful to the entire MMS team for providing highquality, high-precision data to complete this work. This work is supported by the National Natural Science Foundation of China (NSFC) under grant Nos. 41774154, 42074197, and 42130211. All data used in this study were from the MMS Scientific Data Center, which can be found at https://lasp.colorado.edu/mms/ sdc/public/.