Fast Magnetic Field Annihilation in Magnetotail Electron-scale Current Sheet

We present observations in Earth’s magnetotail by the Magnetospheric Multiscale spacecraft that are consistent with magnetic ﬁeld annihilation, rather than magnetic topology change, causing fast magnetic-to-electron energy conversion in an elongated electron-scale current sheet. This energy conversion process is in contrast to that in the standard electron diﬀusion region (EDR) of a reconnecting current sheet where the energy of antiparallel magnetic ﬁelds is mostly converted to electron bulk-ﬂow energy. Fully kinetic simulation also demonstrates that an elongated EDR is subject to the formation of electron-scale magnetic islands in which fast energy conversion is dominated by the annihilation. Consistent with the observations and simulation, theoretical analysis shows that fast magnetic diﬀusion can occur in an elongated EDR in the presence of nongyrotropic electron eﬀects. The discovery of the annihilation-dominated EDR reveals a new form of energy conversion in the collisionless reconnection process.


articles/599365-fast-magnetic-field-annihilation-in-magnetotail-electron-scale-currentsheet
Hosted file essoar.10507891.1.docx available at https://authorea.com/users/523446/articles/599365-fastmagnetic-field-annihilation-in-magnetotail-electron-scale-current-sheet • Theoretical analysis suggests fast collisionless magnetic diffusion in the elongated portion of EDR with nongyrotropic electrons Abstract We present observations in Earth's magnetotail by the Magnetospheric Multiscale spacecraft that are consistent with magnetic field annihilation, rather than magnetic topology change, causing fast magnetic-to-electron energy conversion in an elongated electron-scale current sheet. This energy conversion process is in contrast to that in the standard electron diffusion region (EDR) of a reconnecting current sheet where the energy of antiparallel magnetic fields is mostly converted to electron bulk-flow energy. Fully kinetic simulation also demonstrates that an elongated EDR is subject to the formation of electronscale magnetic islands in which fast energy conversion is dominated by the annihilation. Consistent with the observations and simulation, theoretical analysis shows that fast magnetic diffusion can occur in an elongated EDR in the presence of nongyrotropic electron effects. The discovery of the annihilationdominated EDR reveals a new form of energy conversion in the collisionless reconnection process.

Plain Language Summary
Magnetic reconnection in electric current sheets is the key to fast release of magnetic energy in many space and astrophysical plasma systems, such as during magnetospheric substorms and solar flares. Establishing the mechanism by which magnetic energy is converted to particle energy in the reconnection process is the key to understanding the large-scale impacts of reconnection, including energy partition and particle acceleration. It is generally believed that an electron-scale diffusion region (EDR), where a magnetic-to-electron energy conversion occurs, has an X-type magnetic field geometry around which the energy of antiparallel magnetic fields injected into the EDR is mostly converted to the bulk-flow energy of electrons by magnetic tension of reconnected field lines. Contrary to this standard X-type magnetic field geometry of reconnection, we report observations in Earth's magnetotail by NASA's Magnetospheric Multiscale spacecraft showing that the EDR can be highly elongated. The important and surprising consequence of the observed elongated shape of the EDR is that the fast energy conversion in the EDR is caused mostly by magnetic field annihilation, rather than magnetic topology change. The fast collisionless annihilation that we discovered is fundamentally different from the classical magnetic field annihilation due to collisional and wave-induced resistivity.

Introduction
Magnetic reconnection in Earth's magnetotail is fast, with an inflowing plasma speed of~0.1 times the Alfvén speed in the inflow region, and has large-scale impacts, for example, explosive release of magnetic energy during magnetospheric substorms (Angelopoulos et al., 2008;Nagai et al., 2011). Magnetotail reconnection occurs under such simple boundary conditions that on the two sides of the current sheet the magnetic field is approximately oppositely-directed with a comparable intensity, and plasmas have similar densities and temperatures. Such nearly antiparallel and symmetric reconnection is ideal for studying intrinsic properties of magnetic reconnection in the collisionless regime.
In the standard model of reconnection, the electron diffusion region (EDR), where both ions and electrons violate the 'frozen-in condition', is localized with an X-type field geometry where a magnetic-to-electron energy conversion occurs by changing the magnetic field topology (Torbert et al., 2018) (Figure 1a). It is generally accepted that in such EDRs most of the energy of antiparallel magnetic fields injected by the inflow is converted to bulk-flow energy of electrons ejected into the regions downstream of the EDR (Shay et al., 2007), called the exhausts, by magnetic tension (Lorentz force) of reconnected field lines. Numerical simulations suggest that the EDR may be elongated to a planar geometry (Figure 1b), resulting in a reduction of the reconnection rate (Daughton et al., 2006;Fujimoto, 2006). An elongated EDR is subject to the formation of electronscale magnetic islands, in and around which significant energy conversion may persist through magnetic field annihilation (Text S6; Nakamura et al., 2021) ( Figure 1e,f). However, there has been a lack of observations to understand if and how fast energy conversion occurs in such spatially extended EDRs with negligibly small reconnected field components across the current sheet. This is not only because unambiguous identification of EDR structures in space requires high-spatiotemporal-resolution plasma measurements, but also because there has been no data analysis method to distinguish whether the magnetic field injected into the EDR is reconnected or annihilated. Standard EDR with an X-type magnetic field geometry where the magnetic-toparticle energy conversion is mostly due to magnetic topology change (Torbert et al., 2018). (b) EDR after elongation along L, as seen in kinetic simulations (Daughton et al., 2006), where the energy conversion may be mostly due to magnetic field annihilation (Text S7 in the Supporting Information; see also Nakamura et al., 2021). (c) Initial stage of the magnetic island formation in the EDR, as observed by MMS. (d) Possible later stage of the island growth in exhausts of the EDRs. (e) Time evolution of an EDR from a fully kinetic simulation of turbulent reconnection , showing EDR elongation and island growth. In-plane magnetic field lines are shown by black curves and energy conversion rate j • (E + v × B)  in color, where E and B are the electric and magnetic fields, respectively, j current density, and v electron flow velocity. (f) Simulated vector potential (flux function) along = 0 for selected times. It shows that at the O-point (local maximum) continuously decreases from = 55.6Ω −1 to 56.0Ω −1 at a rate comparable to that at the primary X-point ( minimum), where Ω is ion gyrofrequency, demonstrating that magnetic field annihilation occurs in and around the island (Text S6).
In this letter, we present in-depth analysis of the fortuitous multi-spacecraft observations by the Magnetospheric Multiscale (MMS) mission (Burch et al., 2016) of a magnetotail EDR on 10 August 2017 (Zhou et al., 2019), in which the magnetic-to-particle energy conversion rate was consistent with fast reconnection. Contrary to the standard model of reconnection with an X-type field geometry at the EDR, however, our analysis suggests that the EDR was elongated in the outflow direction and the fast energy conversion observed in the EDR was supported by magnetic field annihilation, rather than magnetic topology change. Consistent with this interpretation, our theoretical analysis shows that fast magnetic diffusion can occur in an elongated EDR in the presence of nongyrotropic electron effects.  Figure 2c) and a relatively slow duskward ion flow with no enhancement at the current sheet show that its electric current was supported by electrons. These features indicate that MMS crossed an electron-scale current sheet (ECS) embedded inside a region of large-scale reconnection when traversing from its anti-sunward exhaust to sunward exhaust. Earlier studies of this current sheet (Zhou et al., 2019;Li et al., 2019) reported established signatures of EDRs (Burch et al., 2016;Torbert et al., 2018), including oppositely-directed electron jets with a speed exceeding iA (Figure 2c), crescent-shaped electron velocity distributions, and magnetic-to-particle energy conversion.

Reconstruction of the Electron-scale Current Sheet
We investigate the ECS structure in detail by use of two sophisticated data analysis techniques that can reconstruct multi-dimensional magnetic field structures in regions around the spacecraft from in situ measurements of the magnetic field and plasma bulk parameters. One is a single-spacecraft method based on electron-magnetohydrodynamics (EMHD) equations that can recover quasi-steady, two-dimensional (2-D) magnetic, electrostatic, and electron velocity fields around the path of the observing spacecraft, hereafter called EMHD reconstruction Text S4). The other is a multi-spacecraft method based on polynomial (second-order Taylor) expansion of the magnetic field that can recover three-dimensional (3-D) magnetic field using instantaneous measurements by the four spacecraft of the magnetic field and particle current density, called polynomial reconstruction (Denton et al., 2020;Text S5).
To analyze the dimensionality of the ECS, the Maximum Directional Derivative (MDD) method (Shi et al., 2019) was applied to four-spacecraft measurements of the magnetic field for an interval 12:18:29-12:18:37 UT surrounding the reconstruction interval (Figure 3). Figure 3e,f shows that the maximum eigenvalue is much larger than the other two eigenvalues and the first dimension number index 1 (Rezeau et al., 2018) is much larger than the other two indices, suggesting that the magnetic structure of the ECS was locally nearly 1-D during the reconstruction interval. Figure 3g shows that k max , which can be taken as the ECS normal direction, was stably northward throughout the reconstruction interval. These results indicate that the ECS was approximately planar with no significant undulation on the scale of the spacecraft separation~18 km which was comparable to the electron inertial length ( ∼ 14 km). It is also seen that the 2-D assumption made in the EMHD reconstruction is well satisfied in this nearly 1-D ECS.
Figures 2e-2k show the magnetic field and plasma data from MMS, used as input for the two reconstruction techniques, in a current-sheet ('LMN') coordinate system (Text S2): the current-sheet normal points along N (roughly northward in the magnetotail), the reconnecting antiparallel magnetic field component is along L (roughly sunward), and M = N × L is along the 'X-line' direction (roughly duskward). A weak guide field (~2 nT) (Zhou et al., 2019), the compo-nent ( ) along the X-line of the magnetic field external to the entire largerscale current sheet or, in this case, at the center of the ECS (Figure 2f), confirms that reconnection occurred under nearly antiparallel magnetic field conditions. Negative to positive variations of ( Figure 2g) is consistent with MMS moving from the anti-sunward to sunward side of the reconnection site; the X-line was moving anti-sunward (Text S3). The assumptions of constant density and constant and isotropic electron temperature made in the EMHD reconstruction are approximately satisfied for an intense current density interval at 12:18:32.1-12:18:34.0 UT (Figure 2i,k), to which the method is applied. , that can be used as measures of the dimensionality of the structure encountered by the spacecraft. (g-i) Eigenvectors corresponding to the three eigenvalues (g; k max , h; k mid , i; k min ). (j) |∇•B| |∇×B| as an error proxy. Both the EMHD and polynomial reconstruction results (Figures 4a-4d and 4f-4i) clearly show that an electron-scale magnetic island was forming and growing in the ECS with a thickness of about one . This is consistent with the MDD result suggesting that the ECS had a planar and elongated configuration (Figure 3), ideal for island generation (Daughton et al., 2006). Figures 4a-4d show that over a~0.4 s interval of 12:18:32.6-12:18:33.0 UT, during which the current sheet was crossed in the order of MMS 1, MMS 3, MMS 4, and MMS 2 ( Figure  2e), both the length (along L) and width (along N) of the island grew. An entirely consistent feature is seen in Figures 4f-4i. We also note that the time scale (~0.4) of the ECS crossing is comparable to the proton cyclotron period, so that the island growth was in fact slow on the electron time scale; the quasi-steady assumption of the EMHD reconstruction is approximately satisfied. Furthermore, the energy-conversion rate j • E ′ = j • (E+v ×B)   (Figure 4e), where the particle current density is j = (v −v ) with the elementary charge of and E the electric field measured in the spacecraft frame, is strongly positive around an X-point closer to the center of the reconstruction domain. Its magnitude (~0.5 nW −3 ) is comparable to the value (~0.2 nW −3 ) expected for fast reconnection, based on the current density~150 nA m -2 ( Figure 2h) and reconnection electric field~1.3 mV −1 for ≈ 15 nT and inflow ion speed of ∼ 0.1 iA ≈ 85 km −1 (Figure 2b). We thus conclude that energy conversion at the X-point was ongoing. Note that the active X-point was captured inside the MMS tetrahedron during the ECS crossing (Figures 4a-4d, 4g, and 4h), reinforcing the conclusions based on the reconstruction results. In order to reveal how fast the magnetic island was growing, we calculated the amount of magnetic flux per unit length along M embedded between the reconstructed X-and O-points closer to the center of the reconstruction domain. Consistent with the island growth, the flux increases with time ( Figure 5a) and the measured component of the electric field along M ( ) appears to vary in space and time (Figure 5c). However, the estimated rates of flux accumulation inside the island (Figure 5b) are about one order of magnitude smaller than the measured (flux injection rate) that is comparable to the expected reconnection electric field of~1.3 mV −1 . Tests of our reconstruction methods using simulated data show that the estimated fluxes may differ by a factor of 4 at most ( Figure S3 and Texts S4 and S5). Thus, the result suggests that the in-plane magnetic field injected into the ECS was mostly annihilated, rather than ejected out of the X-point into the island, at the time and portion of the ECS observed by MMS. of the electric field from MMS 1, MMS 2, and MMS 3 that made reliable electric field measurements, transformed into the frame comoving with the structure (see Text S3 for details), along with Rec . The upper and lower levels of the error bars are based on results from the EMHD reconstructions with an offset of = +0.1 nT or −0.1 nT in GSE added to the measurements, considering that the error in the magnetic field measurements is~0.1 nT . The flux values from the polynomial reconstruction are shown only when both the X-and O-points are within twice the spacecraft separation of the centroid of the MMS spacecraft .

Theoretical analysis
Is fast annihilation of the magnetic field as detected by MMS physically possible in an ECS? For quasi-steady 2-D reconnection in collisionless plasmas, electron demagnetization (violation of the electron frozen-in condition) at the X-line occurs when off-diagonal terms of the electron pressure tensor are significant . Consistently, nongyrotropic electron velocity distributions as a manifestation of electron demagnetization have been observed in the present (Li et al., 2019) as well as other EDRs (Burch et al., 2016;Torbert et al., 2018). Earlier studies also demonstrated that the nongyrotropic electron pressure term can quantitatively account for the electric field ( ) of fast reconnection as observed (Egedal et al., 2019). Our theoretical analysis shows that when the generalized Ohm's law has a term equivalent to the nongyrotropic electron pressure tensor and the ECS is elongated in the exhaust direction, a diffusion equation for (Eq. (S7)), ≈ ge ∞ 2 2 , (1) is applicable, where ge is the gyroradius of thermal electrons and ∞ is the electron inflow speed (| |) immediately outside of the EDR (Text S7). This equation is supported by a solution of the EMHD equations ( Figure S7) and is consistent with the observed fast annihilation (Text S7). The fast collisionless annihilation is in stark contrast with the classic (resistive magnetohydrodynamics) model (Parker, 1957;Sweet, 1958) of reconnection in an elongated current sheet in which the annihilation is negligibly small under magnetotail conditions. However, we do not exclude a likely possibility that at around the sunward and anti-sunward ends of the present ECS, where the magnetic field may have a Y-type geometry (Figure 1b,c), the magnetic field was efficiently reconnected and ejected downstream, as observed for other magnetotail reconnection events (Nakamura et al., 2018).

Conclusions
The main results of our study can be summarized as follows. (1) The MMS observations reported in the present letter are consistent with fast energy conversion in an elongated EDR dominated by magnetic field annihilation; (2) the fully kinetic simulation shows annihilation-dominated energy conversion in and around electron-scale islands formed in the EDR elongated in the exhaust direction (Figure 1e,f); and (3) theoretical analysis suggests fast collisionless magnetic diffusion in a planar EDR with nongyrotropic electrons (Text S7). Thus, three different approaches, namely, our MMS event analysis, simulation, and theoretical analysis all support magnetic field annihilation in an elongated EDR. The discovery of the annihilation in a reconnecting ECS could have far-reaching implications for how magnetic energy is dissipated in plasma turbulence in the collisionless regime, because reconnection has been suggested to play a role in the dissipation process (Matthaeus and Lamkin, 1986;Retinò et al., 2007;Servidio et al., 2009).
An important question remains about whether the magnetic energy dissipated by the annihilation is partitioned to thermal or nonthermal electrons, if not to the electron bulk flow. Recently, Nakamura et al. (2021) have shown, based on a fully kinetic simulation, that electrons are strongly heated in electronscale magnetic islands where the annihilation is ongoing. In the present event, however, no clear signature of electron heating or energization was found likely because electron beta outside the EDR was so high (~5) that available magnetic energy was probably too small for energization effects to be identified; even if all the magnetic energy was converted to electron thermal energy, the temperature increase would be only about 20% of the ambient value. It is noted that for typical reconnection, only~2% of the available magnetic energy is partitioned to thermal electrons in reconnection exhausts (Phan et al., 2013). Thus, the process of energy partition through the annihilation will need to be quantitatively assessed in the future.