Electromagnetic viscosity supported anomalous electric field in the electron diffusion region of collisionless magnetic reconnection

Domain: physics
Published: 2025-11-26
Editorial outcome: minor_revision

Abstract
Kinetic-scale electromagnetic fluctuations are frequently observed in the reconnection electron diffusion region. However, their potential to accelerate magnetic reconnection through anomalous effects remains a topic of debate, with a lack of direct in-situ observational evidence. Using the unprecedented high-resolution data from Magnetospheric Multiscale mission, we directly and systematically calculate the secular field-particle energy exchange rate and anomalous electric fields associated with electromagnetic fluctuations within 13 electron diffusion regions observed in Earth’s magnetotail. Our findings reveal that the electromagnetic anomalous viscosity is the primary contributor to the anomalous electric field induced by electromagnetic fluctuations within the electron diffusion region. The maximum contribution of the anomalous viscosity can account for up to ~20% of the fast reconnection electric field, though it is typically less than 5% in most cases. We further find that locally growing electromagnetic fluctuations tend to accelerate reconnection, while locally damping electromagnetic fluctuations inhibit it. These results offer insights into the coupling between kinetic-scale electromagnetic fluctuations and magnetic reconnection in collisionless plasmas.

Introduction
Magnetic reconnection is a fundamental process responsible for fast energy conversion in astrophysical, space, and laboratory plasma systems1–3. It is implicated in explosive phenomena such as fast radio bursts and X-ray bursts4–6, solar flares7,8, and planetary substorms9,10. A long-standing question in magnetic reconnection is how the plasma decouples from magnetic fields in the electron diffusion region (EDR). In collisional plasmas, Coulomb collisions provide the resistivity necessary for the decoupling of magnetic fields and plasmas, and for energy dissipation within the diffusion region11–13. However, in collisionless plasmas, Coulomb collisions between particles are negligible due to the large mean free path compared to the characteristic system size14–17. Therefore, alternative dissipation mechanisms must be considered in collisionless magnetic reconnection.

The electron pressure gradient and electron inertial effect in the electron momentum equation are two candidate mechanisms for breaking magnetic field lines in collisionless EDR18,19. Recent particle-in-cell (PIC) simulation have shown that the electron pressure gradient term can support the reconnection electric field in fast collisionless reconnection20,21. This finding has been corroborated by high-resolution spacecraft observations22–24. In addition, wave-particle interactions have been proposed as another possible dissipation mechanism. Field-particle interactions associated with electromagnetic (EM) fluctuations can participate in reconnection via effective particle scattering and diffusion, providing anomalous resistivity resembling Coulomb collisions25. These interactions, termed anomalous effects, include anomalous drag, viscosity, and transport, and are proposed to affect the temporal behavior of fast reconnection in collisionless plasmas26–30. The relevant waves or instabilities that contribute to anomalous effects involve lower-hybrid drift instabilities31,32, whistler waves28,29,33,34, drift kink instabilities35, Buneman instabilities36, electron Kelvin-Helmholtz instabilities37–39, and plasmoid-induced turbulence40,41.

Fluctuations near the lower-hybrid frequency (LHF) have drawn considerable attention as a candidate responsible for anomalous effects42. Electrostatic (ES) LHF fluctuations are typically observed at boundaries with a strong density gradient, both in symmetric and asymmetric reconnection. These ES fluctuations primarily emerge in the low plasma \beta region away from the EDR. Although ES LHF fluctuations can also survive at the center of reconnecting current sheets when a large guide field reduces the \beta value therein32,43, EM LHF fluctuations are more frequently detected near and within the EDR28,29,39,44–48. It is commonly recognized that anomalous electric field contributed by LHF fluctuations are negligible compared to the electron pressure gradient term within the EDR21,49. However, there are still some studies that suggest that an anomalous electric field arising from LHF fluctuations is significant28–30,32,50,51. One of the reasons for the emergence of these different viewpoints is that the PIC simulations are constrained by artificial parameters, making it difficult to achieve realistic ion-to-electron mass ratios and large electron plasma to gyrofrequency ratios (\omega_{pe}/\omega_{ce})42,52. Although the anomalous effects have been quantified in asymmetric reconnection by in-situ observation53, comparable evidence for their importance within the EDR of symmetric reconnection is still lacking. Therefore, it is essential to directly quantify and systematically assess the anomalous effects within the EDR of symmetric reconnection using high-resolution in-situ measurements, in order to better understand the role of EM fluctuations in reconnection.

Here, we use data from the Magnetospheric Multiscale (MMS) mission54 (see Methods section) to directly characterize the secular field-particle energy exchange rate and anomalous effects of EM fluctuations within 13 EDRs of symmetric reconnection observed in the Earth’s magnetotail. These EM fluctuations appear near the local LHF range. The result shows that the EM fluctuations can provide anomalous effects within EDRs and are primarily through the EM anomalous viscosity term T_{EM} = -<\delta(n_e V_e) \times \delta B> / <n_e>. The maximum contribution of T_{EM} is up to approximately 20% of the reconnection electric field, while it is less than 5% inside most EDRs. The positive/negative effects of these EM fluctuations on the reconnection electric field are closely related to the growing/damping state of these EM fluctuations.

Results
Observations of electromagnetic fluctuations inside EDR
Table 1 presents 13 EDR events associated with EM fluctuations in the magnetotail (see Methods section)55. An EDR of magnetotail reconnection was encountered by the MMS spacecraft at approximately 20:24:07 UT on June 17, 2017, when the spacecraft were positioned around [-19.3, -11.1, 3.6] R_E in Geocentric Solar Magnetospheric (GSM) coordinates56–58. Figure 1b–i illustrate MMS 2 observations during this reconnecting current sheet crossing in the local current sheet (LMN) coordinate system. The reversal of B_L from negative to positive (Fig. 1b) and the positive electron jet V_{eL} (Fig. 1c) suggest that MMS crossed the reconnecting current sheet from the +L side of the reconnection X-line, as depicted in Fig. 1a. The reconnection electric field E_R is directed along the +M direction in this LMN coordinate system, and a large out-of-plane electron jet was observed along the -M direction with V_{eM} ~ -1800 km/s. The magenta rectangle marks the EDR of this reconnection event (see Methods section), where the electron frozen-in condition is broken, i.e., E_0 = E + V_e \times B \neq 0, and concomitant observed non-zero energy dissipation J \cdot E_0, crescent-shaped electron velocity distribution56, the enhancement of electron agyrotropy and thermal Mach number59.

EM fluctuations with frequencies between ~1 and 16 Hz, near the local LHF, were enhanced in the vicinity of the EDR (Fig. 1e, g). The electric field fluctuations \delta E are dominated by \delta E_N with a maximum amplitude of ~5 mV/m (Fig. 1d), while the magnetic field fluctuations \delta B show similar amplitudes in three directions with a maximum amplitude of ~1 nT (Fig. 1f). These EM fluctuations are mainly inclined to right-handed circular polarization (Fig. 1h), indicating they may be whistler-branch waves. The EM fluctuations primarily exist in the center of the reconnecting current sheet, where \beta_e is greater than 1 (Fig. 1i). These features are consistent with the EM turbulence in the EDR observed in three-dimensional PIC simulations29 and laboratory plasmas28.

Figure 1j–o presents the estimated dispersion relation (f-k) spectra of the EM fluctuations using spectral phase differences analysis based on multiple spacecraft measurements34,47,60–62. Figure 1j–l displays the estimated f-k spectra and phase velocity V_{ph} of EM fluctuations on the -N (B_L < 0) side of the current sheet along the L, M, and N directions, respectively. Figure 1m–o has the same format as Fig. 1j–l except for the +N (B_L > 0) side of the current sheet. These results show that EM fluctuations primarily propagate in the -N (+N) direction on the B_L < 0 (B_L > 0) side with V_{phN} ~ -1500 km/s (V_{phN} ~ 800 km/s) as shown in Fig. 1a. Additionally, there is another weaker wave mode at a higher frequency range (f ~ 6–15 Hz, marked by a red dashed line), suggesting that EM fluctuations are composed of several wave modes or the phase speeds may be quite uncertain. The wavenumber k \rho_e \rho_i \approx 1 (Fig. 1j–o) of these EM fluctuations indicates they are long-wavelength LHF wave modes47,63. Furthermore, the EM fluctuations propagate along the +L and -M directions, matching the direction of the outflow and out-of-plane electron jets, respectively. The phase velocity of EM fluctuations along the -M direction, V_{phM} ~ -2500 km/s, is close to the ion-electron relative drift velocity in the -M direction, V_{d} \approx V_{eM} ~ -2000 km/s (Fig. 2b), which implies a possible interaction between the EM fluctuations and the out-of-plane electron jet28.

Secular field-particle energy exchange
To elucidate the field-particle interaction, we calculated the secular field-particle energy exchange rate between the EM fluctuations and plasma, denoted as <\delta J \cdot \delta E>(t,f) (see Methods section). This metric helps determine whether these EM fluctuations are growing or damping in the EDR64–66. Figure 2d, e present the spectra of <\delta J \cdot \delta E>(t,f), estimated from the current density derived via the curlometer technique, J_c = \nabla \times B = \mu_0, and from plasma moments, J_p = en(V_i - V_e), respectively. Both spectra show significantly negative <\delta J \cdot \delta E>(t,f) values associated with EM fluctuations in the 1–16 Hz range. Figure 2c displays the secular field-particle energy exchange rates, <\delta J_c \cdot \delta E>(t) (red curve) and <\delta J_p \cdot \delta E>(t) (black curve), integrated from 1 to 16 Hz, derived from the <\delta J_c \cdot \delta E>(t,f) and <\delta J_p \cdot \delta E>(t,f) spectra, respectively. Both the black and red curves exhibit negative values with a peak around -5 pW/m³, which indicates that the plasma moments data and results are reliable, and the EM fluctuations gain energy from the plasma. Wang et al.55 proposed that these EM fluctuations, initially generated along the direction of the electric current (the Y direction in the GSM coordinates), propagate toward a tilted current sheet with the normal direction of the current sheet points primarily along Y_{GSM}, hence the waves appear to propagate along this local normal direction. In contrast, our results show that the LHF EM fluctuations grow locally within the EDR, suggesting that they originate from the center of the reconnecting current sheet and then propagate outward (Fig. 1a). This local growth may be related to the pinching of the current sheet as proposed by Yoon et al.67, which could act as a source of fluctuating energy.

The integrated secular field-electron energy exchange rates, <\delta J_e \cdot \delta E>(t) (green curve in Fig. 2c, f), estimated from the electron current density J_e, are primarily with the black curve, as the current density in the EDR is primarily carried by the electrons. This suggests that the EM fluctuations gain energy directly from electrons rather than ions in the EDR. To identify electrons in which phase space are involved in these interactions, Fig. 2g–i shows the distributions of the secular energy exchange rates in one-dimensional electron velocity phase space along the L, M, and N directions (see Methods section). We see that intense secular field-electron energy exchange occurs within a broad range of [-3.5, 3.5] \times 10^4 km/s in all three directions in velocity phase space. This contrasts with the classical resonance interaction scenario, where energy exchange occurs only around resonance velocities; hence, we suggest that non-resonant interactions are important in the EDR68. The intense field-electron energy exchange in the EDR may result in efficient electron scattering and associated anomalous effects in reconnection28,29,39.

Anomalous effects of electromagnetic fluctuations
The anomalous effects induced by wave-particle interactions can be quantified by examining ensemble averaged second-order terms in the electron momentum equation, which can be written as34,53,69:
<E> + <V_e> \times <B> = - \nabla \cdot <P_e> / e<n_e> - m_e \nabla \cdot <n_e V_e V_e> / e<n_e> + D + T + I (1)
where D, T = T_{ES} + T_{EM}, and I are the anomalous drag (resistivity), anomalous viscosity (momentum transport), and anomalous Reynolds’ stress, respectively. These quantities are defined as:
D = - <\delta n_e \delta E> / <n_e> (2)
T_{ES} = - <\delta n_e \delta V_e> \times <B> / <n_e> (3)
T_{EM} = - <\delta(n_e V_e) \times \delta B> / <n_e> (4)
I = - m_e [\nabla \cdot <n_e V_e V_e> - \nabla \cdot (<n_e> <V_e> <V_e>)] / e<n_e> (5)
The contributions of I are neglected in our analyses since it is generally much smaller than D and T due to it being related to the specific charge of the electron42. The anomalous viscosity T is split into two components T_{ES} and T_{EM}, here. T_{ES} (the ES anomalous viscosity term) is the anomalous Lorentz force due to the background magnetic field <B> and this term exactly cancels with D for frozen-in electrons, such as in the ES lower-hybrid drift waves53,70. T_{EM} (the EM anomalous viscosity term) corresponds to the anomalous Lorentz force due to fluctuating magnetic field \delta B, i.e., the classical EM anomalous viscosity term41. The total anomalous electric field is defined as R = D + T_{ES} + T_{EM}.

Figure 3 illustrates the anomalous effects of EM fluctuations along the out-of-plane direction (M direction), estimated using the spectral average method (see Methods section). The EM anomalous viscosity term <n_e> T_{EM,M} (Fig. 3e) exhibits significant positive values corresponding to EM fluctuations in the EDR, whereas <n_e> D_M (Fig. 3c) and <n_e> T_{ES,M} (Fig. 3d) do not show clear enhancement. Figure 3b presents the anomalous electric fields in the M direction, which are integrated from the spectra within the 1–16 Hz frequency range (see Table 1). The total anomalous electric field R_M reaches a positive peak of approximately 0.025 mV/m inside the EDR. It is in the same direction as the out-of-plane reconnection field; however, its magnitude is only 3% of the reconnection electric field (see Table 1). The increase in R_M is primarily contributed by T_{EM,M} of EM fluctuations at frequencies near the lower hybrid frequency (f_{LH}). The anomalous drag D_M and the ES anomalous viscosity term T_{ES,M} are negligible in comparison to T_{EM,M}. This is different from previous observations of ES fluctuations53 in asymmetric reconnection at the magnetopause. Notably, D_M and T_{ES,M} show an anti-correlated change within the EDR, even though electrons are not entirely frozen-in, similar to the ideal electron-magnetohydrodynamic nature seen in PIC simulations70 and recent observations53.

Figure 4 details the other two EDRs (events 7 and 10 in Table 1), which are associated with strong EM fluctuations near the LHF observed in the magnetotail on 2018-08-27 (Fig. 4a–h) and 2017-07-11 (Fig. 4i–p), respectively. The secular field-particle energy exchange <\delta J \cdot \delta E>(t,f) in EDR 7 is predominantly negative at the central region and slightly positive at the edges (Fig. 4c, d), indicating that the EM fluctuations are growing within this EDR. The total anomalous electric field in the M direction R_M during EDR 7 shows a positive peak of ~0.371 mV/m at the center, with smaller negative peaks at the edges of the EDR (Fig. 4e). This feature resembles the anomalous electric fields observed in recent three-dimensional simulations41,48. The positive peak magnitude of R_M is up to 18% of the reconnection electric field in this EDR, much larger than that of EDR 2. Figures 4f–h present the time-frequency spectra of the anomalous effects, with the positive peak of T_{EM,M} attributed to EM fluctuations around the LHF.

In contrast, the secular field-particle energy exchange <\delta J \cdot \delta E>(t,f) for EDR 10 is positive (Fig. 4k, l), hence the EM fluctuations are damping within this EDR. The anomalous electric field in the M direction R_M for EDR 10 shows a positive peak of ~0.036 mV/m, only ~1% of the reconnection electric field, at the center and negative values at the EDR edges as well (Fig. 4m). One can see that fluctuations in the anomalous electric fields are observed both inside and outside the EDR. The intense secular field-particle energy exchange and associated larger anomalous electric field occur outside the EDR in this event, differing from the above two EDRs. Interestingly, in these events, R_M is predominantly contributed by T_{EM,M}, while D_M is small and counterbalanced by T_{ES,M} in most regions.

Statistical results
Columns 5 and 6 of Table 1 display the maximum total anomalous electric field R_{M,max} and the maximum average non-ideal electric field <E'_0>_{M,max} in the M direction for all EDR events, where <E'_0> = <E> + <V_e> \times <B>. In all events, the EM fluctuations contribute a positive anomalous electric field in the +M direction, which can support the reconnection electric field. To assess the relative significance of the anomalous electric field to the reconnection electric field, we calculate the ratio of R_{M,max} to <E'_0>_{M,max}, denoted as P = R_{M,max} / <E'_0>_{M,max}, as shown in Column 7 of Table 1. The maximum contribution of the anomalous electric field associated with EM fluctuations can reach up to ~20% of the reconnection electric field, but it is less than 5% in approximately 70% of the events.

For each event, we normalize R_M, T_{EM,M}, T_{ES,M} and D_M by the maximum absolute values of the total anomalous electric field |R_M|_{max}. Figure 5a shows the statistical results of the normalized anomalous electric field R_M = |R_M| / |R_M|_{max} and the normalized anomalous viscosity term T_{EM,M} = |T_{EM,M}| / |R_M|_{max} for all EDRs. All data points are clustered around the line of R_M = |R_M| / |R_M|_{max} = T_{EM,M} = |T_{EM,M}| / |R_M|_{max} (black line), with a positive correlation coefficient of approximately r=0.98, indicating that the anomalous electric fields within all EDRs are mainly contributed by the EM viscosity term T_{EM}. Figure 5b shows the normalized anomalous drag D_M = |D_M| / |R_M|_{max} and the normalized ES viscosity term T_{ES,M} = |T_{ES,M}| / |R_M|_{max}. These two terms are smaller than T_{EM,M} and nearly cancel each other out within the EDRs, with a correlation coefficient of approximately r=-0.64 between them.

Does the evolution stage (growing or damping) of these EM fluctuations correlate with their roles in the reconnection process? Fig. 5c, d examines the statistical relationship between <\delta J \cdot \delta E> and R_M across all EDR events. For each event, we normalize <\delta J \cdot \delta E> and R_M by their maximum absolute values, <\delta J \cdot \delta E> = |<\delta J \cdot \delta E>| / |<\delta J \cdot \delta E>|_{max} and R_M = |R_M| / |R_M|_{max}. Figure 5c shows a scatter plot of <\delta J \cdot \delta E> = |<\delta J \cdot \delta E>| / |<\delta J \cdot \delta E>|_{max} and R_M = |R_M| / |R_M|_{max} from events 1–9. A clear inverse correlation is observed, with a correlation coefficient of approximately -0.68. In these events, negative (positive) secular field-particle energy exchange correlates with a positive (negative) anomalous electric field in the M direction, and the intensity of <\delta J \cdot \delta E> = |<\delta J \cdot \delta E>| / |<\delta J \cdot \delta E>|_{max} can influence the magnitude of R_M = |R_M| / |R_M|_{max}. This suggests that the growing or damping state of EM fluctuations is a key factor in determining whether they contribute to or inhibit the reconnection electric field. In contrast, Fig. 5d shows the scatter plot of <\delta J \cdot \delta E> = |<\delta J \cdot \delta E>| / |<\delta J \cdot \delta E>|_{max} and R_M = |R_M| / |R_M|_{max} from events 10–13, where no clear correlation is evident. These findings indicate that while the growing or damping state of EM fluctuations significantly influences their anomalous effects on the reconnection electric field, it is not the sole determinant.

Methods
MMS data
The data used in this study are from the following instruments onboard MMS: the Fluxgate Magnetometer (FGM)72; the Fast Plasma Investigation (FPI)73; the Electric Double Probes (EDP)74,75. Since the frequency ranges of these EM fluctuations in the EDR events are mainly less than 16 Hz, the burst mode magnetic fields from FGM with a sampling rate of 128 Hz, the fast mode electric fields from EDP with a sampling rate of 32 Hz, and the burst mode electron data from FPI with the resolution of 0.03 s–33 Hz are used in this paper.

The magnetic fields, electric fields, and plasma moments used to calculate the spectra of the time-averaged quantities in this study have been averaged over multiple spacecraft. Note that not all four MMS spacecraft have valid data for each event. For example, the FPI on MMS4 was out of order after June 2018, resulting in a lack of electron moments data from MMS4 after June 2018. Therefore, for events 1, 2, 5, 10, 11, and 12, the fields and plasma moments were averaged over all four MMS spacecraft, while for the other events, the average values are calculated using the data from only three MMS spacecraft.

In the magnetotail, the electron counts are low, and internal photoelectrons may significantly affect the reliability of the plasma moments. The L2 electron moments data from MMS have been corrected to eliminate photoelectron contamination. The L2 data are deemed reliable in events 1–5 and 8–12, as the electron moments data are almost identical to the partial moments data, which excludes low-energy electrons that are likely influenced by photoelectrons. For these events, both the electron moments and the corresponding uncertainty data are used. However, the L2 data for events 6, 7, and 13 are considered unreliable. Therefore, partial spacecraft moments data with energy greater than 50 eV are used for these events to enhance the quality of the electron moments. One should be noted that there is no uncertainty data available in the partial moments data product.

Definition and time window of EDRs
Most EDR events in our list have been reported in previous studies, as shown by Wang et al.55, and the references therein. In this paper, we employ six parameters to identify the EDR: (1) electron agyrotropy as defined by Swisdak76; (2) agyrotropy of the measured electron pressure tensor based on Scudder et al.59; (3) electron thermal Mach number following Scudder et al.59; (4) energy dissipation as proposed by Zenitani et al.77; (5) energy gain per cyclotron period based on Scudder et al.59; and (6) the relative strength of electric and magnetic forces in the electron fluid rest frame, also based on Scudder et al.59. Although not every case exhibits increase in all of the six parameters, at least three parameters show significant enhancements. These observations indicate that all the analyzed events correspond to EDR crossings.

Moreover, we find that the electron thermal Mach number increases significantly in all EDR events. Therefore, we primarily use the enhancement of this parameter to determine the time windows for each EDR. Considering that the EM fluctuations near the EDRs typically have a minimum frequency close to 1 Hz, we determine the EDR windows (i.e., begin and end time) with a temporal resolution of 1 s.

Calculate the spectrum of the time-averaged quantities
The spectral average method. The spectrum of secular filed-particle energy exchange rate as a function of time and frequency is defined as64–66
<\delta J \cdot \delta E>(t,f) = 1/4 [\delta J(t,f) \cdot \delta E(t,f)^* + \delta J(t,f)^* \cdot \delta E(t,f)] (6)
where \delta J(t,f), \delta J(t,f)^*, \delta E(t,f), \delta E(t,f)^* represent the wavelet spectra and corresponding conjugate counterparts of the current density J(t) and electric field E(t), respectively.

The secular energy exchange rates in one-dimensional electron velocity phase space. The time series of the reduced one-dimensional (1D) electron velocity distribution functions are expressed as f_e(t, V_k) = \int f_e(t, V) dV^2, where k=L, M, N denotes the directions in the LMN coordinate system. Based on the definition of electron current density, J_{e,k} = -enV_{e,k}, the time series of 1D electron current density distribution in velocity phase space can be expressed as
J_{e,k}(t, V_k) = -eV_k \int f_e(t, V_k) dV_k (7)
Taking k=M as an example, for each velocity V_M = V_0, we have an associated electron current density J_{e,M}(t, V_0) and its wavelet spectrum \delta J_{e,M}(t, V_0, f). Using Eq. (6) in the main text, we derive the secular energy conversion spectrum corresponding to this current density, <\delta J_{e,M} \cdot \delta E>(t, V_0, f). Integrating this spectrum over the frequency range of the EM fluctuations yields the secular energy conversion rate <\delta J_{e,M} \cdot \delta E>(t, V_0), representing the contribution from electrons with velocity V_M = V_0.

Repeating the above process across all velocities in the 1D velocity phase space yields the 1D distribution of field–electron secular energy exchange rates in the 1D velocity phase space, <\delta J_{e,M} \cdot \delta E>(t, V_M), as shown in Fig. 2h. The same computational procedure is applied to the k=L, N to obtain <\delta J_{e,L} \cdot \delta E>(t, V_L) and <\delta J_{e,N} \cdot \delta E>(t, V_N) as shown in Fig. 2g, i, respectively.

Spectra of anomalous terms. The quantities of anomalous effects in Eq. (1) are computed from an ensemble average ideally. In this work, we use the four-spacecraft average and the spectra time-average method provided in Eq. (6) to estimate the spectra of anomalous terms. First, we calculate the four-spacecraft average fields and plasma moments, i.e., E, B, n_e, V_e, and (n_e V_e). Second, we calculate their wavelet spectra, i.e., \delta E, \delta B, \delta n_e, \delta V_e, and \delta(n_e V_e) and corresponding conjugate counterparts, i.e., \delta E^*, \delta B^*, \delta n_e^*, \delta V_e^*, and \delta(n_e V_e)^*. Then, calculate the spectra of anomalous terms via
<n_e> D(t,f) = - 1/4 [\delta n_e \delta E^* + \delta n_e^* \delta E] (8)
<n_e> T_{ES}(t,f) = - 1/4 [\delta n_e \delta V_e^* + \delta n_e^* \delta V_e] \times <B> (9)
<n_e> T_{EM}(t,f) = - 1/4 [\delta(n_e V_e) \times \delta B^* + \delta(n_e V_e)^* \times \delta B] (10)
where <n_e> and <B> are the background electron density and magnetic field calculated via the low-pass filter below the frequency of EM fluctuations53, for example, it is 1 Hz for event 2.

The uncertainties of anomalous terms are estimated using the spectral average method as well, which incorporates uncertainties from the electron moments and assumes a 10% uncertainty in the gain of the electric field53. The results indicate that the uncertainty of T_{EM} is generally smaller than that of T_{ES} and D terms, as shown in Figs. 3b and 4e, m. The values of T_{EM} within the EDR are greater than their associated uncertainties, suggesting that the results are reliable.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript explores the contribution of plasma waves within the electron diffusion region (EDR) to the reconnection electric field using MMS observations. While the study presents interesting findings, the manuscript requires several revisions before it can be considered for publication. The authors should address the following points:\n1. Motivation and Context: The motivation and the cited references need refinement. Early debates on the role of anomalous terms like anomalous resistivity in driving fast reconnection have been largely resolved. Particle-in-cell (PIC) simulations, supported by observations, demonstrate that fast reconnection in collisionless plasmas occurs without requiring anomalous effects. Even in collisional plasmas, regimes that allow plasmoid formation can also exhibit fast reconnection. The findings of this study align with this established understanding. While outlier publications may suggest otherwise, they are often not well-regarded and insufficient to call the prevailing consensus into question. The authors should revise the introduction and discussion to reflect this. The statement that “The mechanism for breaking magnetic field lines inside EDR in collisionless plasmas remains an open issue” is inaccurate. A more appropriate motivation would acknowledge that full PIC simulations are constrained by artificial parameters, making it challenging to simulate realistic mass ratios and large wpe/wce.\n2. Turbulence and Anomalous Effects: The manuscript cites references attributing anomalous effects to self-generated turbulence. However, prior research shows that while turbulence affects the temporal behavior of reconnection, it does not significantly influence its onset or rate. The authors should clarify this distinction and ensure their discussion aligns with these findings.\n3. Definition of EDR: The authors define the EDR by the breaking of the simple frozen-in condition. However, Scudder et al. have emphasized that this condition is an oversimplification and highlighted the importance of considering electron agyrotropy (e.g., doi: 10.1103/PhysRevLett.108.225005). The authors should evaluate how agyrotropy effects might impact their analysis and include this consideration in their discussion.\n4. Guide Field Strength and Bn: The manuscript does not mention the guide field strength or Bn in the analyzed EDR cases. If these values are negligible, this should be explicitly stated. Including this information will provide greater clarity and context to the results.\n5. Clarification of Anomalous Effects: The statement, “Moreover, although the anomalous electric fields provided by EM fluctuations are much smaller than the reconnection electric field in most events, they could provide the seed for triggering the onset of reconnection, i.e., the anomalous effects could break the electron frozen-in condition,” should be removed. Collisionless reconnection, particularly in the absence of a significant Bn, does not require anomalous effects. Furthermore, the authors’ findings do not establish whether such fluctuations precede the onset of reconnection. The onset of reconnection itself may induce fluctuations within the EDR, complicating the question of causation.\n6. Minor Typographical Error: Replace “collision plasmas” with “collisional plasmas” in the text.", "reviewer_2": "The manuscript “Electromagnetic viscosity supported anomalous electric field in the electron diffusion region of collisionless magnetic reconnection,” by Zhong and others, presents a statistical analysis of the role of anomalous viscosity during magnetic reconnection, showing details of several EDRs as examples. The subject matter is indeed interesting and important and is especially receiving recent attention. I believe the manuscript can indeed be an importance advance in this regard, but I have some doubts about parts of the manuscript. My following questions must be addressed adequately before I can recommend publication of this manuscript.\n1. The EDR crossing that the authors are showing in detail is a case where the anomalous effects contribute only 3% of the reconnection electric field. However, in event #7 in Table 1, the contribution is 18%, which the authors emphasize in the abstract and conclusion. Why are the authors not analyzing this event in detail and instead choosing EDR #2? Even EDR #1 has 10% contribution from anomalous effects. The authors should provide a reason for choosing to show an event with lackluster contribution. I would also like the authors to provide a similar analysis (like those in Figures 1 and 2) for event #7 or maybe #1, either in the main text or the Supplementary Materials, as they have already done so in EDR 9 and 11.\n2. The authors emphasize in various places the difference between their work and, for example, reference 42 or reference 31 in that in the latter works, the electrostatic contribution was significant whereas here it is negligible. However, in the latter cases there were rather strong density gradients and so electrostatic drift waves are expected to be significant. It is well-known that asymmetric and symmetric reconnections (and the existence of guide fields) are quite different in their characteristics, and so the authors should discuss the dependency of the anomalous diffusion characteristics on these factors and perform analyses if necessary.\n3. I am not sure I am convinced on the calculation of secular field-electron energy exchange in velocity space (Figures 2g-i). An immediate question arises just from looking at the plots; the energy exchange apparently occurs for velocities of up to several 10,000 km/s, but the average electron velocity from Figure 2b is only a few 1,000 km/s. It is hard to comprehend how this could be so. Several 10,000km/s thermal velocity would correspond to a temperature of around a few keV, but EDR 2 was not this hot. Also, I am not sure I quite understand Method 2.2, namely how to use a 1D distribution and derive a 3D vector from it. For instance, taking the x-component (assuming the authors have defined 𝑓𝑒(𝑡,𝑉) to satisfy ∫𝑓𝑒(𝑡,𝑉)𝑑𝑉 = 𝑛𝑒 and not 4𝜋∫𝑓𝑒(𝑡,𝑉)𝑉2𝑑𝑉; please clarify this in the paper as well),\n𝑑𝐽𝑥(𝑡,𝑉) = −𝑒𝑉𝑥𝑓𝑒(𝑡,𝑉)𝑑𝑉,\nbut at a particular 𝑉 = 𝑉0, 𝑉𝑥 is degenerate, i.e., it can have multiple values for the same 𝑉0, so I do not understand how the subsequent calculation can be carried out. This secular energy transfer is one of the important points in the manuscript so this point must be addressed satisfactorily. Also, why are there three components as in Figure 2g-i? This whole calculation is very confusing and does not make sense to me. The authors should first double-check their calculations and if they are indeed correct, they should explain the procedure in much more detail and also be able to justify the large interaction space in velocity space.\n4. How were the windows for EDR selected in Figure 5? Presumably each EDR has different configurations such as density asymmetry and guide fields, as well as different numbers of data points. This makes putting all the data from all EDR crossing in one plot questionable. For example, if EDR #1 contained 100 data points and EDR #2 contained 10 points, the plot is highly skewed towards EDR #1. This also affects the credibility of plots like Figure 5b; Most of the points look concentrated near the center while some points follow the red line (inverse correlation). Those that follow the red line may be only from a couple of EDR crossings. Also, why is there no R value for Figures 5a and 5b? The authors should consider ways to distinguish different EDR crossings on the plot, e.g., using different colors or shapes for each crossing event.\nMinor points:\n1. The plot labels should clarify the deltas, e.g., < 𝛿𝐽⋅𝛿𝐸 > instead of < 𝐽⋅𝐸 > in Figure 2d-f\n2. What are the origin of these waves? The authors suggest whistlers which are readily observed in collisionless reconnection, but Figure 1a shows quasi-perpendicular propagation. This current sheet is bifurcated, which means that pinching has occurred [c.f., Yoon et al., Nat. Commun. 12, 3774 (2021)), which is a possible source of perpendicular waves. I suggest the authors discuss the possible sources of such perpendicular waves."}, "author_response": "We thank the reviewer for carefully reading our manuscript and for the constructive comments and suggestions. We have made significant improvements to the manuscript based on the reviewer’s comments and suggestions. All revisions have been highlighted in the revised version of the manuscript. Below, we provide point-by-point responses to each of the reviewer’s comments.\nThis manuscript explores the contribution of plasma waves within the electron diffusion region (EDR) to the reconnection electric field using MMS observations. While the study presents interesting findings, the manuscript requires several revisions before it can be considered for publication. The authors should address the following points:\nMotivation and Context: The motivation and the cited references need refinement. Early debates on the role of anomalous terms like anomalous resistivity in driving fast reconnection have been largely resolved. Particle-in-cell (PIC) simulations, supported by observations, demonstrate that fast reconnection in collisionless plasmas occurs without requiring anomalous effects. Even in collisional plasmas, regimes that allow plasmoid formation can also exhibit fast reconnection. The findings of this study align with this established understanding. While outlier publications may suggest otherwise, they are often not well-regarded and insufficient to call the prevailing consensus into question. The authors should revise the introduction and discussion to reflect this. The statement that “The mechanism for breaking magnetic field lines inside EDR in collisionless plasmas remains an open issue” is inaccurate. A more appropriate motivation would acknowledge that full PIC simulations are constrained by artificial parameters, making it challenging to simulate realistic mass ratios and large wpe/wce.\nResponse: Thank you for your suggestion! We have revised the Introduction and Discussion sections to reflect the prevailing consensus that anomalous effects do not significantly influence the rate of fast collisionless magnetic reconnection. We also clarified that the motivation for this study acknowledges the limitations of full PIC simulations, which are constrained by artificial parameters and often cannot realistically reproduce key physical conditions, such as realistic mass ratios and large values of ω /ω . Therefore, it is essential to directly estimate and systematically assess the anomalous effects within the EDR using in-situ spacecraft measurements in order to better understand the role of lower-hybrid frequency electromagnetic fluctuations. Please see the revised content in the manuscript (lines 40–76, lines 304-315).\nTurbulence and Anomalous Effects: The manuscript cites references attributing anomalous effects to self-generated turbulence. However, prior research shows that while turbulence affects the temporal behavior of reconnection, it does not significantly influence its onset or rate. The authors should clarify this distinction and ensure their discussion aligns with these findings.\nResponse: Thank you for your comment! We have revised the Introduction and Discussion sections to reflect the prevailing consensus that anomalous effects do not significantly influence the rate of fast collisionless magnetic reconnection. In addition, we clarify that the self-generated turbulence is proposed to affect the temporal behavior of reconnection. Please see the revised content in the manuscript (lines 51–53, lines 64-66, lines 304-310).\nDefinition of EDR: The authors define the EDR by the breaking of the simple frozen-in condition. However, Scudder et al. have emphasized that this condition is an oversimplification and highlighted the importance of considering electron agyrotropy (e.g., doi: 10.1103/PhysRevLett.108.225005). The authors should evaluate how agyrotropy effects might impact their analysis and include this consideration in their discussion.\nResponse: Thank you for your comments! The electron diffusion region (EDR) #2–#8 and #10-#12 in our list have been reported in previous studies, as shown by Wang et al. (2022) and the references therein. Specifically, Wang et al. (2022) noted that these EDRs are identified by features that V (in GSM coordinates) reverse sign or changes ix from above a few hundred km/s to near-zero values, while V (especially along Y ) e GSM has a much larger amplitude than V. In each event, the ion bulk motion is small and far i from the E×B drift; the large-amplitude electron flow along the M direction supports the current sheet, and the L component of the velocity is significant with deviations from the E×B drift close to the mid-plane. They have also confirmed that non- gyrotropic electron distributions exist during the current sheet crossing (Wang et al., 2022). Furthermore, the EDR #1, #9, and #13 in our list also exhibit features consistent with the above criteria.\nIn addition, we define EDR signatures using six additional parameters: (1) electron agyrotropy as defined by Swisdak (2016); (2) agyrotropy of the measured electron pressure tensor based on Scudder et al. (2012); (3) electron thermal Mach number following Scudder et al. (2012); (4) energy dissipation as proposed by Zenitani et al. (2011); (5) energy gain per cyclotron period based on Scudder et al. (2012); and (6) the relative strength of electric and magnetic forces in the electron fluid rest frame, also based on Scudder et al. (2012).\nTaking EDR #7 as an example, we plot these six parameters in Figure R1. It can be seen that all six parameters significantly increase within this EDR. For other events, though not all case exhibits increase in all of the six parameters, at least three parameters show evident enhancements. In particular, electron agyrotropy, as defined by Swisdak (2016) or Scudder et al. (2012), shows clear enhancement in events #1–#4, #6, #7, and #10– #13. Although events #5, #8, and #9 do not exhibit enhanced agyrotropy by these definitions, non-gyrotropic electron distributions are still present in the velocity phase space during the current sheet crossings (Wang et al., 2022). These observations indicate that all the analyzed events correspond to EDR crossings.\nMoreover, we find that the electron thermal Mach number increases obviously in all the EDRs. Therefore, we primarily use the enhancement of this parameter to determine the time windows for each EDR. Considering that the electromagnetic fluctuations near the EDRs typically have a minimum frequency close to 1 Hz, we determine the EDR windows with a temporal resolution of 1 second, as indicated by the yellow shading in Figure R1.\nWe have revised the relevant content in the manuscript (lines 432–448).\nFigure R1. Definition of electron diffusion region (EDR): take EDR #7 as an example. (a) magnetic field in LMN coordinates, (b) electron bulk velocity in LMN coordinates, (c) electron agyrotropy as defined by Swisdak (2016), (d) agyrotropy of the measured electron pressure tensor based on Scudder et al. (2012), (e) electron thermal Mach number following Scudder et al. (2012), (f) energy dissipation as proposed by Zenitani et al. (2011), (g) energy gain per cyclotron period based on Scudder et al. (2012), (h) the relative strength of electric and magnetic force in the electron fluid rest frame base on Scudder et al. (2012).\nReference\nWang, S. et al. Lower‐Hybrid Wave Structures and Interactions With Electrons Observed in Magnetotail Reconnection Diffusion Regions. JGR Space Physics 127, (2022).\nScudder, J. D. et al. First Resolved Observations of the Demagnetized Electron- Diffusion Region of an Astrophysical Magnetic-Reconnection Site. Phys. Rev. Lett. 108, 225005 (2012).\nSwisdak, M. Quantifying gyrotropy in magnetic reconnection. Geophys. Res. Lett. 43, 43–49 (2016).\nGuide Field Strength and Bn: The manuscript does not mention the guide field strength or Bn in the analyzed EDR cases. If these values are negligible, this should be explicitly stated. Including this information will provide greater clarity and context to the results.\nResponse: We have examined the guide field strength B and the normal component B g N for these EDRs. Figure R2a shows the relationship between the ratio P = R /<E’> *100% (values in column 7 of Table 1) and B , which is normalized M,max M,max g to the asymptotic inflow magnetic field B . Here P denotes the percentage of the 0 anomalous electric field in the reconnection electric field. We see that B for these g events ranges from 0 to 0.5, and the percentage of anomalous effects appears unrelated to the value of B (correlation coefficient 𝑟 ~ -0.02).\ng\nFigure R2b shows the relationship between the percentage of anomalous effects and B N , which is also normalized to the asymptotic inflow magnetic field B . The normalized 0 B for all EDR events is less than 0.13, with most events (77%) having values below N 0.05, hence B can be considered negligible. Furthermore, the correlation coefficient N between the percentage of anomalous effects and B is about 0.3, suggesting that there N is no clear correlation between them.\nWe have added the values of B to table 1 and this content to the manuscript (lines 283– g 288).\nFigure R2. Normalized B and B of the 13 EDR events and their relationship with the g N proportion of anomalous electric field. (a) Correlation between normalized B and the g proportion of anomalous electric field. (b) Correlation between normalized B and the N proportion of anomalous electric field. The value of 𝑟 indicates the correlation coefficient.\nClarification of Anomalous Effects: The statement, “Moreover, although the anomalous electric fields provided by EM fluctuations are much smaller than the reconnection electric field in most events, they could provide the seed for triggering the onset of reconnection, i.e., the anomalous effects could break the electron frozen-in condition,” should be removed. Collisionless reconnection, particularly in the absence of a significant Bn, does not require anomalous effects. Furthermore, the authors’ findings do not establish whether such fluctuations precede the onset of reconnection. The onset of reconnection itself may induce fluctuations within the EDR, complicating the question of causation.\nResponse: We have removed this sentence according to your suggestion.\nMinor Typographical Error: Replace “collision plasmas” with “collisional plasmas” in the text.\nResponse: Revised.\nReviewer #2:\nWe thank the reviewer for carefully reading our manuscript and for the constructive comments and suggestions. We have made significant improvements to the manuscript based on the reviewer’s comments and suggestions. All revisions have been highlighted in the revised version of the manuscript. Below, we provide point-by-point responses to each of the reviewer’s comments.\nThe manuscript “Electromagnetic viscosity supported anomalous electric field in the electron diffusion region of collisionless magnetic reconnection,” by Zhong and others, presents a statistical analysis of the role of anomalous viscosity during magnetic reconnection, showing details of several EDRs as examples. The subject matter is indeed interesting and important and is especially receiving recent attention. I believe the manuscript can indeed be an importance advance in this regard, but I have some doubts about parts of the manuscript. My following questions must be addressed adequately before I can recommend publication of this manuscript.\nThe EDR crossing that the authors are showing in detail is a case where the anomalous effects contribute only 3% of the reconnection electric field. However, in event #7 in Table 1, the contribution is 18%, which the authors emphasize in the abstract and conclusion. Why are the authors not analyzing this event in detail and instead choosing EDR #2? Even EDR #1 has 10% contribution from anomalous effects. The authors should provide a reason for choosing to show an event with lackluster contribution. I would also like the authors to provide a similar analysis (like those in Figures 1 and 2) for event #7 or maybe #1, either in the main text or the Supplementary Materials, as they have already done so in EDR 9 and 11.\nResponse: Thank you for your comment! The reason we selected Event #2 for detailed presentation is that it is a particularly unique case. It is considered being observed at the early stage (or onset phase) of the reconnection (Lu et al., 2020), so we aimed to examine whether anomalous effects play significant role for reconnection onset. On the other hand, the basic features of electromagnetic fluctuations during this event have been analyzed in previous studies (e.g., Wang et al., 2022). However, those analyses were unable to determine whether the fluctuations were generated locally within the EDR or not. In this paper, using spectral methods, we demonstrate directly that the electromagnetic fluctuations grow locally within this EDR.\nFurthermore, though not fully presented in the main text, we have conducted the same analysis for all EDR events. To better highlight the contribution of anomalous effects, we have replaced Event #11 in the main text with Event #7. Event #9 (The new number of the event is #10 in revised Table 1) is retained because, unlike the other two events, it exhibits electromagnetic disturbances both inside and outside the EDR, and < 𝛿𝐽⋅ 𝛿𝐸 > associated with the electromagnetic fluctuations is mostly positive, which implies that the fluctuations were damping. Therefore, the sources of fluctuations near the EDRs are complex.\nWe have revised the relevant content in the manuscript (lines 209–221).\nReference\nLu, S. et al. Magnetotail reconnection onset caused by electron kinetics with a strong external driver. Nat Commun 11, 5049 (2020).\nWang, S. et al. Lower‐Hybrid Wave Structures and Interactions With Electrons Observed in Magnetotail Reconnection Diffusion Regions. JGR Space Physics 127, (2022).\nThe authors emphasize in various places the difference between their work and, for example, reference 42 or reference 31 in that in the latter works, the electrostatic contribution was significant whereas here it is negligible. However, in the latter cases there were rather strong density gradients and so electrostatic drift waves are expected to be significant. It is well-known that asymmetric and symmetric reconnections (and the existence of guide fields) are quite different in their characteristics, and so the authors should discuss the dependency of the anomalous diffusion characteristics on these factors and perform analyses if necessary.\nResponse: Thank you for your comment! We have added a discussion of the dependency of the anomalous effects on the asymmetric and guide field of reconnecting current sheet. Please see the revised content in the manuscript (lines 283–298).\nI am not sure I am convinced on the calculation of secular field-electron energy exchange in velocity space (Figures 2g-i). An immediate question arises just from looking at the plots; the energy exchange apparently occurs for velocities of up to several 10,000 km/s, but the average electron velocity from Figure 2b is only a few 1,000 km/s. It is hard to comprehend how this could be so. Several 10,000km/s thermal velocity would correspond to a temperature of around a few keV, but EDR 2 was not this hot. Also, I am not sure I quite understand Method 2.2, namely how to use a 1D distribution and derive a 3D vector from it. For instance, taking the x-component (assuming the authors have defined 𝑓𝑒(𝑡,𝑉) to satisfy ∫𝑓𝑒(𝑡,𝑉)𝑑𝑉=𝑛𝑒 and not 4𝜋∫𝑓𝑒(𝑡,𝑉)𝑉2𝑑𝑉; please clarify this in the paper as well),\n𝑑𝐽𝑥(𝑡,𝑉)=−𝑒𝑉𝑥𝑓𝑒(𝑡,𝑉)𝑑𝑉,\nbut at a particular 𝑉=𝑉0, 𝑉𝑥 is degenerate, i.e., it can have multiple values for the same 𝑉0, so I do not understand how the subsequent calculation can be carried out. This secular energy transfer is one of the important points in the manuscript so this point must be addressed satisfactorily. Also, why are there three components as in Figure 2g-i? This whole calculation is very confusing and does not make sense to me. The authors should first double-check their calculations and if they are indeed correct, they should explain the procedure in much more detail and also be able to justify the large interaction space in velocity space.\nResponse: Thank you for your comment! We have double-checked the results and confirmed that they are indeed correct. We have revised the relevant content to make the description more detailed and clearer (lines 457-475).\nThe time series of the reduced one-dimensional (1D) electron velocity distribution functions are expressed as 𝑓(𝑡,𝑉 ) = ∬𝑓 (𝑡,𝑽)𝑑𝑉2, where 𝑘 = 𝐿,𝑀,𝑁 denotes the 𝑒 𝑘 𝑒 directions in the LMN coordinate system. Figure R4b shows that the electron bulk velocity in the M direction is approximately −2,000 km/s within the EDR. However, as shown in Figure R4c, electrons are distributed over a broad velocity range (mainly between −30,000 and 30,000 km/s) along the M direction of velocity phase space, which is much larger than the electron bulk velocity V (blue curve in Figure R4c). eM\nThe interaction signatures between electrons and waves cannot be directly observed in 1D electron velocity distribution functions 𝑓 (𝑡,𝑉 ) (Figure R4c). To better visualize 𝑒 𝑘 the particles that respond to fluctuations in velocity phase space, we first calculate the electron current density distribution in velocity phase space. Based on the definition of electron current density, 𝐽 = −𝑒𝑛𝑉 , the time series of the 1D electron current 𝑒,𝑘 𝑒,𝑘 density distribution in velocity phase space can be expressed as:\n𝐽 (𝑡,𝑉 ) = −𝑒𝑉 𝑓(𝑡,𝑉 )𝑑𝑉 (1)\n𝑒,𝑘 𝑘 𝑘 𝑒 𝑘 𝑘\nwhere 𝑒 is the elementary charge.\nTaking the M direction as an example, the current density 𝐽 (𝑡,𝑉 ) was band-pass 𝑒,𝑀 𝑀 filtered in the 1–16 Hz range to isolate the relevant fluctuation components, as shown in Figure R4e. Quasi-periodic current density perturbations are clearly visible in the velocity phase space. These fluctuations span a broad velocity range (mainly between −30,000 and 30,000 km/s). Figure R4d shows that the wave-induced current densities (band-pass filtered during 1–16 Hz) directly estimated from the electron moment data 𝐽 = −𝑒𝑛𝑉 (black curve) and that integrated from the velocity phase space 𝐽 = 𝑒,𝑀 𝑒,𝑀 𝑒,𝑀 ∫−𝑒𝑉 𝑓(𝑡,𝑉 )𝑑𝑉 (blue curve) agree well, indicating the reliability of the results.\n𝑀 𝑒 𝑀 𝑀\nNext, we obtained the distribution of secular energy conversion in velocity phase space. For each velocity V =V , we have an associated electron current density 𝐽 (𝑡,𝑉 ) M 0 𝑒,𝑀 0 and its wavelet spectrum 𝛿𝐽 (𝑡,𝑉 ,𝑓) . ssing Equation (6) in the manuscript, we 𝑒,𝑀 0 derive the secular energy conversion spectrum corresponding to this current density, < 𝛿𝐽 𝛿𝐸 > (𝑡, 𝑉 ,𝑓). Integrating this spectrum over the frequency range of 1–16 Hz 𝑒,𝑀 𝑀 0 gives the total secular energy conversion rate < 𝛿𝐽 𝛿𝐸 > (𝑡,𝑉 ), representing the 𝑒,𝑀 𝑀 0 contribution from electrons with velocity V =V .\nM 0\nRepeating this process across all velocities in the range of −55,000 to 55,000 km/s for this event yields the 1D distribution of field–electron secular energy exchange rates in the velocity phase space, < 𝛿𝐽 𝛿𝐸 > (𝑡,𝑉 ), as shown in Figure R4f. The same 𝑒,𝑀 𝑀 𝑀 computational procedure was applied to the L and N directions to obtain < 𝛿𝐽 𝛿𝐸 > 𝑒,𝐿 𝐿 (𝑡,𝑉 ) and < 𝛿𝐽 𝛿𝐸 > (𝑡,𝑉 ), respectively. We can see that, even though the bulk 𝐿 𝑒,𝑁 𝑁 𝑁 electron velocity is only a few thousand km/s, interactions between electrons and waves can occur over a much broader velocity range in velocity phase space.\nFigure R4. Response of electrons to fluctuations in the velocity phase space observed by MMS1. (a) Magnetic field in LMN coordinates. (b) Electron bulk velocity. (c) Reduced 1D electron velocity distribution in the M direction and the M component of the electron bulk velocity (V ). (d) Wave-induced current density (1-16 Hz) estimated eM from the electron moment data (black curve) and that integrated from the velocity phase space (blue curve). (e) Bandpass-filtered (1–16 Hz) electron current fluctuations in velocity phase space in the M direction. (f) Secular energy conversion < 𝛿𝐽 𝛿𝐸 > 𝑒,𝑀 𝑀 (𝑡,𝑉 ).\nM\nHow were the windows for EDR selected in Figure 5? Presumably each EDR has different configurations such as density asymmetry and guide fields, as well as different numbers of data points. This makes putting all the data from all EDR crossing in one plot questionable. For example, if EDR #1 contained 100 data points and EDR #2 contained 10 points, the plot is highly skewed towards EDR #1. This also affects the credibility of plots like Figure 5b; Most of the points look concentrated near the center while some points follow the red line (inverse correlation). Those that follow the red line may be only from a couple of EDR crossings. Also, why is there no R value for Figures 5a and 5b? The authors should consider ways to distinguish different EDR crossings on the plot, e.g., using different colors or shapes for each crossing event.\nResponse: Thank you for your comment! As shown in Figure R1, we computed six parameters to identify the EDR. We find that the electron thermal Mach number 𝑀 𝑒,⊥ increases significantly in all EDRs. Therefore, we primarily use the enhancement of 𝑀 to determine the time windows for each EDR. Considering that the 𝑒,⊥ electromagnetic fluctuations near the EDRs typically have a minimum frequency close to 1 Hz, we determine the EDR windows with a temporal resolution of 1 second, as indicated by the yellow shading in Figure R1. As a result, the EDR windows span from 3 to 8 seconds, as listed in Table 1.\nFurthermore, we use different colors and markers to represent each event in Figure R5 (corresponding to revised Figure 5 in the manuscript) and have added the correlation coefficient R values in Figures R5a and R5b (corresponding to Figures 5a and 5b in the manuscript). These observed patterns are not driven by any single event, but rather reflect collective characteristics shared by the majority of events.\nPlease see the revised content in the manuscript (lines 243–253).\nFigure R5. Correlation analysis. (a) relation between the total anomalous electric field 𝑅 /|𝑅 | and EM viscous 𝑇 /|𝑅 | , (b) the relation between anomalous 𝑀 𝑀 𝑚𝑎𝑥 𝐸𝑀,𝑀 𝑀 𝑚𝑎𝑥 drag 𝐷 /|𝑅 | and ES viscosity 𝑇 /|𝑅 | . Data points in panels (a) and (b) 𝑀 𝑀 𝑚𝑎𝑥 𝐸𝑆,𝑀 𝑀 𝑚𝑎𝑥 are from all 13 EDRs. The black dashed lines in (a) and (b) denote 𝑅 /|𝑅 | = 𝑀 𝑀 𝑚𝑎𝑥 𝑇 /|𝑅 | and 𝑇 /|𝑅 | = −𝐷 /|𝑅 | , respectively. (c) and (d) show 𝐸𝑀 𝑀 𝑚𝑎𝑥 𝐸𝑆,𝑀 𝑀 𝑚𝑎𝑥 𝑀 𝑀 𝑚𝑎𝑥 the correlation between the normalized secular field-particle energy exchange and anomalous electric field, i.e., < 𝛿𝐽⋅𝛿𝐸 >/| < 𝛿𝐽⋅𝛿𝐸 > | and 𝑅 /|𝑅 | . 𝑚𝑎𝑥 𝑀 𝑀 𝑚𝑎𝑥 Data points in panel (c) are from EDR events #1-#8 and #13, while data points in panel (d) are from EDR events #9-#12. Different colors and markers are used to represent different EDR event data.\nMinor points:\nThe plot labels should clarify the deltas, e.g., <𝛿𝐽⋅𝛿𝐸> instead of <𝐽⋅𝐸> in Figure 2d-f\nResponse: revised.\nWhat are the origin of these waves? The authors suggest whistlers which are readily observed in collisionless reconnection, but Figure 1a shows quasi-perpendicular propagation. This current sheet is bifurcated, which means that pinching has occurred [c.f., Yoon et al., Nat. Commun. 12, 3774 (2021)), which is a possible source of perpendicular waves. I suggest the authors discuss the possible sources of such perpendicular waves.\nResponse: The basic features of electromagnetic waves in the vicinity of most EDR events in our list have been analyzed by Wang et al. (2022). Our analysis of the wave propagation direction in EDR #2 is consistent with the findings reported by Wang et al. (2022). They proposed a possible explanation in which electromagnetic waves, initially generated along the direction of the electric current, propagate toward a tilted current sheet with the normal direction of the current sheet points primarily along Y , hence GSM the waves appear to propagate along this local normal direction.\nHowever, our secular energy conversion analysis reveals that these electromagnetic waves can grow locally within EDR #2 (Figures 2d and 2e in manuscript). Specifically, electrons within EDR #2 transfer energy to the waves. This local growth may be associated with the pinching of the current sheet, as proposed by Yoon et al. (2021), which could serve as a source of fluctuating energy.\nWe have added relevant discussion to the manuscript (lines 147–155).\nReference\nYoon, Y. D., Yun, G. S., Wendel, D. E. & Burch, J. L. Collisionless relaxation of a disequilibrated current sheet and implications for bifurcated structures. Nat Commun 12, 3774 (2021)."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have addressed the issues and made appropriate revisions to the manuscript. I recommend publication of the revised manuscript.", "reviewer_2": "I thank the authors for their meticulous revision of the manuscript. All my questions have been addressed appropriately and I now deem the manuscript suitable for publication."}, "author_response": ""}]

Decision letter