Direct Observations of Electron Firehose Fluctuations in the Magnetic Reconnection Outflow

Electron temperature anisotropy‐driven instabilities such as the electron firehose instability (EFI) are especially significant in space collisionless plasmas, where collisions are so scarce that wave–particle interactions are the leading mechanisms in the isotropization of the distribution function and energy transfer. Observational statistical studies provided convincing evidence in favor of the EFI constraining the electron distribution function and limiting the electron temperature anisotropy. Magnetic reconnection is characterized by regions of enhanced temperature anisotropy that could drive instabilities—including the electron firehose instability—affecting the particle dynamics and the energy conversion. However, in situ observations of the fluctuations generated by the EFI are still lacking and the interplay between magnetic reconnection and EFI is still largely unknown. In this study, we use high‐resolution in situ measurements by the Magnetospheric Multiscale spacecraft to identify and investigate EFI fluctuations in the magnetic reconnection exhaust in the Earth's magnetotail. We find that the wave properties of the observed fluctuations largely agree with theoretical predictions of the non‐propagating EF mode. These findings are further supported by comparison with the linear kinetic dispersion relation. Our results demonstrate that the magnetic reconnection outflow can be the seedbed of EFI and provide the first direct in situ observations of EFI‐generated fluctuations.

In the past decades, the electron firehose instability has been investigated in particular in the context of solar wind plasmas (Verscharen et al., 2022, and references therein) since the EFI is invoked as one of the most significant possible isotropization mechanisms to explain the quasi-isotropic state of the solar wind electrons. Indeed, the electron distribution functions observed in the solar wind are much closer to isotropic distributions than expected by considering the Chew-Goldberger-Low (CGL) model (Chew et al., 1956) of a spherically expanding solar wind (Štverák et al., 2008). Hence, the development of temperature-anisotropy-driven instabilities could explain the discrepancy between the model and the observed quasi-isotropic electron distributions. Statistical observational studies have confirmed the scenario of the EFI being crucial for isotropization by showing that the temperature anisotropy is well constrained by the thresholds of temperature-anisotropy-driven instabilities, notably the whistler instability and the EFI (Cattell et al., 2022;Štverák et al., 2008). Recently, several studies were devoted to investigating the EFI by modeling the solar wind electron distribution with more accuracy (both focusing on the propagating EF mode only Shaaban et al., 2021) or including also the non-propagating mode (Shaaban et al., 2019)). This includes going beyond the bi-Maxwellian approximation and taking into account the complex structure of the solar wind electron distribution function-consisting of a thermal core, a suprathermal halo, and a field-aligned beam (Feldman et al., 1975;W. G. Pilipp et al., 1987). Other efforts have been devoted to the investigation of the EFI onset (Innocenti et al., 2019) and evolution (Camporeale & Burgess, 2008;Hellinger et al., 2014;Innocenti et al., 2019). These studies focus on the non-propagating EF mode, as it arises self-consistently in the simulations of expanding solar wind (Innocenti et al., 2019) and has the larger growth rate in all simulations, consistently with the predictions of the linear theory.
Despite the majority of the work having been devoted to the study of the EFI in the solar wind context, the EFI can arise in any space environment where the plasma is unstable to the instability. Statistical studies collected and analyzed electron distribution functions in different near-Earth plasmas. Gary et al. (2005) used Cluster data to investigate electron distributions in the magnetosheath, while Zhang et al. (2018) used THEMIS observations to study electron distributions at dipolarization fronts in the magnetotail. These studies show that the electron distribution functions are constrained by the EFI threshold, suggesting that the EFI plays an important role in shaping the distribution functions.
Magnetic reconnection is a fundamental plasma process that plays a key role in energy conversion, plasma heating, and particle energization in a variety of plasma environments (Biskamp, 2000). The magnetic reconnection process is characterized by regions of enhanced temperature anisotropy (Egedal et al., 2013) that can be the seedbed for temperature anisotropy-driven instabilities. Indeed, a 3D PIC simulation study recently reported the presence of EFI-generated fluctuations in the reconnection outflow region (Le et al., 2019). The particle scattering and wave-particle interaction processes induced by the development of the EFI could potentially affect the energy conversion and acceleration produced by the reconnection process. However, little is known about the interplay between magnetic reconnection and the EFI. More importantly, direct observations of the EFI-generated fluctuations are currently lacking.
In previous studies focusing on near-Earth plasmas the presence of the EFI has been detected somewhat indirectly by looking at the limited anisotropy of the electron distribution functions (Gary et al., 2005;Zhang et al., 2018). The effect of the EFI is commonly inferred from the fact that the electron distribution is bounded by the instability COZZANI ET AL.

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3 of 21 threshold. This approach is suitable for statistical studies but it does not allow for direct observations of the EF wave modes. In this study, we use high-resolution measurements of the Magnetospheric Multiscale mission (MMS)  to shed light on the EFI-generated waves in the Earth's magnetotail. We report MMS observations of the non-propagating EF mode in the magnetic reconnection outflow region observed by MMS during a current sheet flapping event in the magnetotail. We show that the observed electron temperature anisotropy is constrained by the EFI threshold and we present direct in situ observations of the EFI-generated fluctuations.
This paper is organized as follows: In Section 2, we review the properties of the EF modes based on linear dispersion theory, focusing in particular on the non-propagating EF mode. In Section 3, we introduce the MMS data products used in this study. In Section 4, we present an overview of the current sheet flapping event in the Earth's magnetotail that we used for the analysis and we discuss the selection criteria for the EF events. Then, we present the detailed analysis of the EF fluctuations observed during two of the selected EF events in Section 5. In Section 6 we compare the results of the in situ spacecraft observations with a numerical solver. Sections 8 and 9 present the discussion and the conclusions respectively.

Properties of Electron Firehose Modes
Linear kinetic dispersion theory predicts that a magnetized plasma can be unstable to the development of the EFI under the condition of presenting a sufficiently large electron temperature anisotropy and being sufficiently warm, that is, with β e,‖ > 2 (β e,‖ = 2μ 0 n e T e,‖ /B 2 , where μ 0 is the vacuum magnetic permeability, n e is the electron number density and B is the ambient magnetic field). As mentioned in the Introduction, the linear theory predicts the presence of two distinct branches of the EFI. One is propagating (real frequency ω ≠ 0) and it is characterized by parallel propagation at small θ kB (where θ kB is the angle between the wave vector k and the background magnetic field); the other mode is non-propagating and predicted to develop for oblique wave-normal angles, θ kB . For θ kB > 30° the mode was defined as oblique by several studies (Gary & Nishimura, 2003;Li & Habbal, 2000), while more recently Camporeale and Burgess (2008) considered a higher threshold of θ kB ∼ 50° to discriminate between the parallel and oblique mode.
It is established by both analytical and numerical studies that the non-propagating (oblique, resonant) mode is characterized by a lower threshold and higher growth rate than the propagating (parallel, non-resonant) mode. Indeed, the growth rate γ of the non-propagating mode is expected to be Ω ci < γ < Ω ce , while γ < Ω ci for the propagating mode (Gary & Nishimura, 2003) (here Ω α = eB/m α is the cyclotron frequency, e the elementary charge and m α the mass, α = e, i indicates the electron and ion species). For this reason, in the following, we will focus on the non-propagating EF mode only.
The EF instability threshold is predicted by the linear dispersion theory. The threshold depends upon the electron temperature anisotropy T e,‖ /T e,⊥ and the parallel electron beta β e,‖ and in the following we will use the formulation reported by Gary and Nishimura (2003), which reads The two primed quantities are dimensionless fitting parameters with 1 ≲ S ′ ≲ 2 and ′ ≲ 1 which are defined for 2 ≤ β e,‖ ≤ 50. For an instability growth rate γ/Ω ce = 0.001, S ′ = 1.29 and ′ = 0.97 .
The non-propagating EF mode is resonant with both ions and electrons. To establish if a mode is resonant or non-resonant with a plasma species, one can evaluate the Landau resonance factor ζα = ∕ √ 2| ‖| th, and the cyclotron resonance factor ± = | ± Ωc |∕ √ 2| ‖| th, . Here, |k ‖ | is the magnitude of the wave vector component parallel to the background magnetic field and v th,α is the thermal speed. In particular, for resonant species, which strongly interact with the waves, the resonant velocity is expected to lay within a thermal speed of the distribution function peak, satisfying the condition , ± ≲ 1 . Instead, for non-resonant species , ± ≫ 1 (Gary et al., 1984). For a non-propagating mode, the Landau resonant factor is Re(ζ α ) = 0. Figure 1 shows the properties of the non-propagating EF mode for β e,‖ = 9 and T e,‖ /T e,⊥ = 2. The value of β e,‖ = 9 is representative of the magnetotail plasma sheet conditions. Figure 1 is obtained with the numerical solver Plasma Dispersion Relation Kinetics (PDRK, (Xie & Xiao, 2016)) which solves the kinetic linear dispersion COZZANI ET AL.
10.1029/2022JA031128 4 of 21 relation for multi-species plasmas in the magnetized electromagnetic case. The model implemented in the solver assumes homogeneous plasma conditions. The properties are shown in the parameter space composed of the normalized wave vector kρ e and the wave-normal angle θ kB (ρ e is the electron Larmor radius). Figure 1a shows that this choice of input parameters leads to positive growth with a maximum rate γ max /Ω ce ∼ 0.13. A positive growth rate is found for kρ e ≲ 1 and the wave vector at maximum γ = γ max is kρ e = 0.66. As discussed above, the non-propagating EF mode is associated with oblique wave-normal angle θ kB and, for the chosen set of parameters, the wave-normal angle at the maximum growth rate is θ kB = 69° (see Figure 1a). Figure 1b confirms that the mode is non-propagating, as ω = 0 in all points of the parameter space. To quantify the waves' electrostatic and electromagnetic components we use the parameter long = | ⋅̂| 2 ∕| | 2 which is equal to 1 for a purely longitudinal electrostatic wave and equal to 0 for a transverse electromagnetic wave. Figure 1c shows that long < 0.5 in the region of significant positive growth rate, meaning that the non-propagating EF mode is electromagnetic. In Figure 1d we show the ratio δB ‖ /δB where δB ‖ is the fluctuating magnetic field parallel to the background magnetic field and δB is the total fluctuating magnetic field. The magnetic field fluctuations are predominantly transverse that is, |δB ⊥ | 2 ≫|δB ‖ | 2 . The δE ‖ /δE ratio (Figure 1e) indicates that the electric field fluctuations are dominated by the component aligned with the background magnetic field. Then, Figure 1f shows the polarization of the magnetic field fluctuations. For non-propagating waves, the polarization can be defined as = B x B y , where Bx and By are two components of the magnetic field fluctuations. In the solver, the background magnetic field is along the z direction while the wave vector k = (k x , 0, k z ). As the polarization is 0 for all the values of kρ e and θ kB in Figure 1f, the waves are expected to have a linear polarization.
In Section 5 we will consider several of the characteristics discussed above to identify fluctuations consistent with the non-propagating EF mode in MMS in situ observations. In particular, EFI-generated waves are expected to have zero real frequency and a wave vector kρ e ≲ 1 directed obliquely with respect to the background magnetic field. The fluctuations are also expected to have a significant electromagnetic component (quantified via long ) and to be resonant with electrons. Figure 1. Properties of the non-propagating EF mode computed with the PDRK numerical solver. The input parameters used in the numerical solver are T e,‖ = 1,000 eV, T e,⊥ = 500 eV, the background magnetic field B = 3 nT and density n e = n i = n = 0.2 cm −3 while the isotropic ion temperature is T i = T i,‖ = T i,⊥ = 4,000 eV. The panels show the parameters space kρ e -θ kB versus (a) imaginary frequency γ/Ω ce (b) real frequency ω/Ω ce (c) . The quantities in panels (b)-(f) are shown for values of the growth rate exceeding the marginal stability condition, which is usually set at 10 −3 . (Camporeale & Burgess, 2008) 10.1029/2022JA031128 5 of 21

Magnetospheric MultiScale (MMS) Data
We use data from the Magnetospheric MultiScale (MMS) spacecraft . In particular, we use the magnetic field B data from the fluxgate magnetometer (FGM) (Russell et al., 2016), electric field data E from the spin-plane double probes (SDP) (Lindqvist et al., 2016) and the axial double probe (ADP) (Ergun et al., 2016), and particle data from the fast plasma investigation (FPI) (Pollock et al., 2016). All data presented in this paper are high-resolution burst mode data. During the time interval selected for this study (15:24:00.0-15:58:00.0 UTC on 2017-07-06), the spacecraft were in a tetrahedral configuration with inter-spacecraft separation of ∼16 km. In the interval of interest, the average electron inertial length is 14 km, so the inter-spacecraft separation is comparable with the electron scales. Data from the MMS1 spacecraft are shown throughout the paper, as the observations are similar for the four spacecraft.

Event Overview and Data Selection
We consider a 34-min-long interval on 2017-07-06 when MMS was located at [−24.1, 1.5, 4.4] R E (in Geocentric Solar Magnetospheric GSM coordinate system) in the Earth's magnetotail. During this interval, MMS observes multiple crossings of the magnetotail current sheet, identified by the frequent B x reversals (see Figure 2a). The plasma density (see Figure 2c) shows variations that are associated with the magnetic field. Higher values of the magnetic field (e.g., |B| ∼ 20 nT at 15:40:02.7) correspond to lower densities (n ∼ 0.1 cm −3 ), indicating that MMS is sampling the lobe region, while lower values of magnetic field (e.g., |B| ∼ 1.5 nT at 15:40:50.0) are associated with higher densities in the plasma sheet (n ∼ 0.26 cm −3 ). These observations indicate that the current sheet is flapping (e.g., Gao et al., 2018;Richard et al., 2021). During this interval, MMS often observes fast plasma flows. As shown in Figure 2b, the x component of the ion velocity reaches values of |V i,x | ∼ 1,000 km/s. The highest values are observed close to the neutral line B x ∼ 0 while the value of V i,x decreases toward zero when B x increases, which corresponds to MMS entering the lobe region. In the first part of the interval V i,x < 0, so the flow is directed tailward. At ∼15:46:41 MMS observes a flow reversal followed by strong Earthward flow with V i,x ∼ 1,000 km/s. We also observe that the V i,x reversal is associated with the reversal of B z . In particular, B z is predominantly negative in the interval of tailward flow, while B z is predominantly positive in the interval of Earthward flow. The observed characteristics suggest that the fast flows are associated with magnetic reconnection. Specifically, MMS is sampling the magnetic reconnection outflow regions, tailward outflow first and then Earthward outflow, corresponding to a tailward-propagating reconnection site. Similar conclusions were drawn in a study by Leonenko et al. (2021) focusing on the properties of super thin current sheets (sub-ion scale thickness) observed during the flapping event. We conclude that MMS observed a tailward retreating X-line in the magnetotail.
As the main goal of this study is the investigation of the EFI and the associated waves, we compute the instability threshold to identify the intervals in which the instability could develop. Figures 2d and 2e shows that there are several data points where T e,‖ /T e,⊥ > 1 and β e,‖ > 2 at the same time, which is a necessary condition for the development of the EFI. Then, Figure 2f shows the quantity EFI = ) −1 which is obtained recasting Equation 1. If EFI > 0 the threshold for the firehose instability is exceeded, and the generation of waves is expected. We find 24 intervals with EFI > 0 . Two time points t 1 and t 2 for which EFI > 0 are considered to be part of the same interval if t 2 − t 1 < τ where τ = 0.3 s. This value of τ corresponds to sub-ion time scales. In particular, it corresponds to one-third of an ion time scale computed with dimensional analysis. For the dimensional analysis, we consider the average ion bulk velocity (500 km/s) as the characteristic speed and the average ion inertial length d i (n = 0.2 cm −3 ) = 500 km as the characteristic spatial scale (the average is computed over the interval of Figure 2). We note that the number of intervals does not change for τ = 0.5 s. Figure 3 shows the distribution of the data points of the interval shown in Figure 2 in the parameter space β e‖ -T e‖ /T e⊥ , together with the EFI thresholds corresponding to growth rates γ/Ω ce = 0.001 (dark red curve), 0.01 (orange curve), and 0.1 (yellow curve) (see Gary and Nishimura (2003) for the values of the parameters used in the curves for different γ values). As mentioned in the Introduction, the whistler instability (WI) develops when T e‖ /T e⊥ < 1. So, for completeness, we plot the WI thresholds corresponding to growth rates γ WI /Ω ce = 0.01 (lilac curve) and 0.1 (dark blue curve), computed following Gary and Wang (1996). Figure 3 also shows the theoretical threshold of the ordinary-mode instability (OMI), which, similarly to the EFI, can develop when T e‖ /T e⊥ > 1 (Ibscher et al., 2012;Lazar et al., 2014) and which shares characteristics with the Weibel instability in field-free plasmas (Weibel, 1959). The OMI and the characteristics of the OMI-generated waves are further discussed in Section 8. Note that, to avoid confusion with the other instabilities, the EFI growth rates are named γ EFI in Figure 3 while, in the following, we will continue to use to notation γ to indicate the EFI growth rate for brevity. Only a few data points exceed the EFI thresholds corresponding to γ/Ω ce = 0.001 and 0.01, while no points are found above the γ/Ω ce = 0.1 threshold, suggesting that the EFI plays a key role in shaping the electron distribution function. We also note that the OMI threshold is well above all the data points composing the distribution, indicating that the observed plasma is stable with respect to OMI. Analogously as for the EFI, the shape of the distribution appears to be constrained by the theoretical WI thresholds.
From all the intervals where the EFI threshold is exceeded, we select the ones composed of at least two data points and for which β e,‖ < 30. We exclude intervals with large β e,‖ because, as it can be inferred from Figure 3, even small fluctuations of T e,‖ /T e,⊥ due to instrumental noise can yield to EFI > 0 when β e,‖ is large, even though T e,‖ /T e,⊥ ∼ 1 so that the available free energy would not be enough for the instability to develop. In addition, Figure 3. (a) Electron distribution in the parameter space β e‖ -T e‖ /T e⊥ . The counts are scaled with bin size. The bin number is N bin = 100 for both β e‖ and T e⊥ /T e‖ . The gray bins are bins with low counts (in the range of 1-12), which are less significant statistically. The red, orange, and yellow curves correspond to the EFI threshold (see Equation 1) for growth rates γ/Ω ce = 0.001, 0.01, and 0.1 respectively. The lilac and dark blue curves correspond to the WI threshold for growth rates γ WI /Ω ce = 0.01 and 0.1, following Gary and Wang (1996). The green line corresponds to the Weibel ordinary-mode instability (OMI) threshold according to Equation 58 in Ibscher et al. (2012). Unlike the EFI and WI threshold, the OMI threshold does not depend on fitting parameters (Ibscher et al., 2012) and the green line corresponds to the OMI marginal condition of stability with γ OMI /Ω ce = 0. The colored stars mark the average value of β e,‖ and T e,‖ /T e,⊥ during the intervals of the selected events identified with the correspondingly color-coded vertical lines in Figure 2.

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we select the intervals where magnetic field fluctuations could be identified by visual inspection, allowing us to thoroughly perform the wave analysis. Using these selection criteria, we retain seven intervals Δt EFI >0 , with EFI > 0 . They are marked with the vertical lines in Figure 2. The colored stars in Figure 3 mark β e,‖ and T e,‖ /T e,⊥ averaged during the intervals identified with the correspondingly color-coded vertical lines in Figure 2. We use a slightly different approach for Event #4. The interval Δt EFI >0 of Event #4 fulfills the selection criteria but it also contains several points with EFI < 0 . The points with EFI > 0 are present in two subintervals (15:30:59.690-15:31:00.145 and 15:31:01.224-15:31:01.340) spaced by few data points with EFI < 0 . Since the two subintervals with all the data points EFI>0 are both observed during the interval of wave activity (15:30:58.5-15:31:01.5), they are still part of the same event. To have meaningful averaged values of β e,‖ and T e,‖ /T e,⊥ associated with the unstable plasma, we average only the data points that are actually above the EFI threshold in Δt EFI >0 . The time intervals of the seven selected events are summarized in Table 1, together with the corresponding averaged plasma parameters.
In summary, we identify several intervals in which the EFI threshold is exceeded while MMS is sampling the outflow reconnection region in the Earth's magnetotail during a current sheet flapping event. After applying the selection criteria discussed above, we select seven events exhibiting wave activity at the time when the EFI threshold is exceeded. In the following, we will investigate the wave properties and establish whether the observed fluctuations are compatible with EFI-originated waves.

Wave Analysis
In this Section, we present the detailed wave analysis of two of the seven selected events (event #6 and #7), which we use to illustrate the typical wave properties. The other events are discussed later in Section 7. Event #6 exhibits very clear wave activity and a significant electron temperature anisotropy peaking at T e,‖ /T e,⊥ ∼ 1.48. However, the analyzed waves are not co-located with the interval where the EFI threshold is exceeded. So, we show also the detailed analysis of another event, event #7, during which we identify two intervals of wave activity. One is co-located with the interval with EFI > 0 and the other, similarly to event #6, is observed immediately after the interval where the EFI threshold is exceeded. Also, event #6 is characterized by V i,x < 0, meaning that MMS is observing the tailward reconnection outflow, while event #7 is observed in the Earthward outflow region. Hence, choosing these two events allows us to show the observed waves' properties in Earthward and tailward outflow regions. We aim to compare the observed wave characteristics to the theoretical expectations for EFI-generated fluctuations. As previously discussed, we focus on the non-propagating (oblique, resonant) EF mode as it is predicted to have a lower instability threshold and a larger growth rate with respect to the propagating (parallel, non-resonant) mode.

Event #6
An overview of event #6 is shown in Figure 4. Figure 4e shows that during the interval Δt EFI >0 = 15:38:07.400 -15:38:07.890, highlighted with the red-shaded area, the temperature anisotropy T e,‖ /T e,⊥ exceeds the EFI threshold (red line, see Equation 1) and it reaches a maximum value of ∼1.5 . In addition, β e,‖ has moderate values (β e,‖ ∼ 7 is the average β e,‖ in the interval where the instability threshold is reached, see Figure 4d). The magnetic field is shown in Figure 4a and the relatively low magnitude of |B| ∼ 6 nT suggests that MMS is sampling the plasma sheet. MMS also observed a strong electron (and ion, not shown) flow mainly directed along the x GSM direction with |v e,x | ∼ 1,500 km/s suggesting that MMS is sampling the reconnection outflow region (see Figure 4c).

Figures 4f and 4g
show the wavelet spectrograms of the electric and magnetic field power. Both electric and magnetic field power increase in the yellow-shaded interval. The fluctuations are rather broadband but they exhibit √ fcifce (f ci and f ce are respectively the ion and electron cyclotron frequency). As a first step, we isolate the high-frequency fluctuations from the lower-frequency variations of the magnetic field. We define the filtering frequency f filt by requiring that the magnetic field signal filtered in the frequency range f < f filt exhibits all the variations of the background magnetic field. In this case we choose f filt = 2.6 Hz (see Figure 4a). The magnetic field exhibits low frequency variations (f < f filt , Figure 4a) and, interestingly, higher frequency fluctuations (f > f filt showing wave activity Figure 4b). The interval with enhanced wave activity Δt = 15:38:08.0-15:38:11.0 is highlighted by the yellow-shaded area. The magnetic field fluctuations δB have similar amplitude in all three components, both in the GSM coordinate system (see Figure 5b) and in field-aligned coordinates (see Figure 4b).  (i) and (j) contains all the points with power P(k x , k z ) (and P(k y , k z )) larger than 10% of the maximum power P max , that is, P(k x , k z ) > 0.1P max and P(k y , k z ) > 0.1P max . To better characterize the observed waves, we compute the dispersion relation from the phase differences of δB z between spacecraft pairs, applying the multi-spacecraft interferometry method (Graham et al., 2016(Graham et al., , 2019 to the time interval Δt. Figure 4h shows that the normalized power P(f,k)/P max increases in the frequency range 2.6 Hz < f < 3.8 Hz (black dashed lines) with a peak at f = f obs = 3.2 Hz (black star). The wave number at the P(f,k)/P max peak is kρ e ∼ 0.4 (ρ e ∼ 26 km is the electron gyroradius averaged over Δt) which corresponds to the wave phase speed in the spacecraft reference frame of v ph ∼ 900 km/s. Figures 4i and 4j show that the wave vector k is directed mainly along the x direction, that is, aligned with the direction of the plasma flow. The average wave vector direction is ̂= [−0.82, 0.43, −0.38] GSM.
In addition, we estimate the uncertainty of the wave vector Δkρ e . Even though the P(k x , k z )/P max and P(k y , k z )/P max distributions exhibit a clear peak (Figures 4i-4j), they are characterized by a certain spread in the (k x , k z ) and (k y , k z ) parameter space respectively. To compute the observed wave vector uncertainty Δkρ e , we consider all the points for which the power P(k i , k j ) is above 10% of the maximum power P max in Figures 4i-4j, where i, j = x,y,z. The area selected with this criterion is shown in brighter colors in Figures 4i-4j. For each wave vector component k j , the minimum k j for which the power P(k i , k j ) is larger than 10% of the maximum power P max is k j, min (P = 0.1P max ). Analogously, k j, max (P = 0.1P max ) is the maximum value of k j for which the power P(k i , k j ) is equal or larger than 10% of the maximum power P max . In general, k j, min (P = 0.1P max ) and k j, max (P = 0.1P max ) are asymmetric with respect to k j corresponding to the maximum power. A simple way to symmetrize the uncertainty with respect to k j is to use the average between the two uncertainties k j, min (P= 0.1P max ) and k j, max (P = 0.1P max ) so that the uncertainty Δk j ρ e of the wave vector jth component is . We then compute the uncertainty of the wave vector magnitude Δkρ e . We obtain Δkρ e ∼ 0.17 ∼ 0.41kρ e , which is quite significant but expected, taking into account the considerable variability of the observed quantities. Figure 5 shows additional characteristics of the observed fluctuations that are crucial to establishing whether the observed waves are indeed associated with the EFI. As discussed in Section 2, the non-propagating EF mode is characterized by zero real frequency f = ω/2π = 0, kρ e ≲ 1, the wave vector is directed obliquely with respect to the background magnetic field, and it is an electromagnetic mode. In addition, theoretical expectations about the non-propagating EF mode include ± ≲ 1 , that is, the mode is resonant with electrons. Figure 5 shows that the characteristics of the observed fluctuations are compatible with the theoretical predictions listed above. First, we establish that the observed mode is non-propagating in the plasma reference frame, that is, the Doppler-shifted frequency is zero (f obs − f DS = f obs − (v e · k)/2π = 0, where f DS = (v e · k)/2π is the Doppler shift frequency) or, equivalently, f obs = f DS . To do that, we compare the observed frequency of the fluctuations (f obs , red solid thick line in Figure 5c) to the Doppler shift frequency f DS (black solid thick line in Figure 5c) in the time interval Δt where the waves are observed (yellow shaded interval in Figure 4). The Doppler shift frequency f DS is significant as the wave vector k is quite aligned with the electron velocity v e . In particular, Figure 5d shows that kv e < 60 • during the considered time interval and the average ⟨ kv e ⟩Δ ∼ 38 • , where kv e is the angle between k and v e . The time series of the Doppler shift frequency f DS displays significant variations, which are due to the variations of the electron velocity v e . To account for the variability of f DS , we compute f DS which includes the wave vector uncertainty Δkρ e and the standard deviation of v e computed across the interval Δt. The quantity

Event #7
As shown in Figure 4, during event #6 the interval where the EFI threshold is exceeded (Δt EFI >0 = 15:38:07.400-15:38:07.890) and the interval exhibiting the strong wave activity (Δt = 15:38:08.000-15:38:11.000) are not co-located, albeit the waves are observed immediately after the region with EFI > 0 . In this Section, we present the detailed analysis of event #7 which exhibits wave activity both co-located with and, like event #6, immediately after the interval with EFI > 0 . The observed fluctuations during event #7 are very similar to the ones reported in event #6 and are also consistent with EFI-generated waves. Figure 6 is analogous to Figure 4 for event #6 and it shows that during event #7 the EFI threshold is exceeded in interval Δt EFI >0 =15:53:47.700-15:53:48.430 between the vertical red lines (see in particular Figure 6e), where β e,‖ increases to a maximum value of 28 (Figure 6d) as MMS is located close to the neutral line. The magnetic field magnitude is |B| ∼ 2 nT ( Figure 6a) and MMS observes a strong electron flow, mainly along the outflow in the GSM x direction reaching |v e,x | ∼ 1,200 km/s (Figure 6c). Figure 6b shows the magnetic field fluctuations δB (f filt = 2.5 Hz) which have similar amplitude in all three components in both intervals of wave activity. Both magnetic and electric field power increase in the intervals with wave activity (Figures 6f and 6g). As mentioned above, we identify two intervals characterized by wave activity: interval 7A (Δt A = 15:53:47.0-15:53:50.0), which encloses the interval with EFI > 0 and interval 7B (Δt B = 15:53:50.5-15:53:53.0). The fluctuations have larger amplitude in interval 7B, which is not co-located with the interval where the instability threshold is exceeded. In the following, we will focus in particular on the analysis of the fluctuations observed in interval Δt A .
We use the multi-spacecraft interferometry method (Graham et al., 2016(Graham et al., , 2019 to establish the characteristics of the fluctuations in Δt A . The normalized power of the magnetic field fluctuations P(f,k)/P max increases in the frequency range Δf = [2.5, 4.0] Hz (black dashed lines in Figure 6h) and peaks at f = f obs = 3.2 Hz (black star). The wave number at the peak of δB z normalized power P(f,k)/P max is kρ e ∼ 0.66 (ρ e ∼ 22 km is the electron gyroradius averaged over interval 7A) which corresponds to phase speed in the spacecraft reference frame of v ph ∼ 710 km/s. Figures 6i and 6j shows that the wave vector k is directed mainly along x GSM and aligned with the direction of the outflow (̂= [0.78, 0.61, 0.03] GSM). Analogously to event #6, we estimate the uncertainty of the wave vector magnitude Δkρ e and we obtain Δkρ e ∼ 0.22 ∼ 0.33 kρ e .
Similarly as Figure 5 for event #6, Figure 7 shows the property of the fluctuations in interval 7A to establish whether the observations are consistent with theoretical expectations for the EF fluctuations. Figure 7c indicates that the waves observed in Δt A can be considered as non propagating, as f obs lies between ⟨fDS⟩Δt − f DS and ⟨fDS⟩Δ + f DS and for the majority of the time points f obs lies in the variability range (gray area of Figure 7a) of f DS . Also in this case, the contribution of f DS to the Doppler shifted frequency is significant as ⟨ kv e ⟩Δ ∼ 36 • in interval 7A (see Figure 7d). Other characteristics of the fluctuations in interval 7A include ⟨long⟩Δ Δ ∼ 0.23 , indicating that they are electromagnetic (in this case Δf = [2.5, 4.0] Hz). The spectrogram of long is shown in Figure 7e and despite exhibiting some variability, it never reaches values close to 1 in the considered Δf during interval 7A. Also, electrons are resonant since ⟨ ± ⟩Δ ∼ 1.2 (not shown). The angle between the wave vector and the background magnetic field θ kB changes significantly in interval 7A, going from a minimum value of θ kB ∼ 30° to values close to 90° (Figure 7d), while the time-averaged value of the wave normal angle is ⟨ kB⟩Δ ∼ 69 • . The strong variation of θ kB across Δt A is due to the changing background magnetic field direction. In particular B y goes from negative B y ∼ −1 nT to positive B y ∼ 5 nT in the considered interval. However, for the majority of the interval θ kB > 30°, so that the wave vector can be considered to be oblique with respect to the background magnetic field.
In summary, we observe non-propagating fluctuations with wave vector kρ e ∼ 0.66 directed obliquely with respect to the background magnetic field. The fluctuations have a significant electromagnetic component and are resonant with electrons. We conclude that the observed fluctuations are generated by the EFI instability as they exhibit the characteristics associated with the non-propagating EF mode. As mentioned above, event #7 presents two intervals with wave activity. We have shown the detailed wave analysis of the fluctuations in interval 7A, which are co-located with the region where the EFI threshold is exceeded. The fluctuations with larger amplitude observed in interval 7B have similar characteristics (not shown) and we conclude that they are also EFI-generated waves. It is reasonable to expect that the development of the waves and the increase in the wave amplitude results in a decrease in the temperature anisotropy, which is reduced to a value close to isotropic.

Comparison Between In Situ Observations and Model
To corroborate our conclusion that the observed fluctuations are EFI-generated, we compare the MMS observations with the results of the numerical solver PDRK (Xie & Xiao, 2016), which has been used to obtain Figure 1. We consider a quasi-neutral plasma composed of electrons and protons. In the following, we will refer to the protons as ions, for consistency with MMS notation. We use a non-drifting bi-Maxwellian distribution function with T e,‖ /T e,⊥ > 1 for electrons and a non-drifting Maxwellian distribution function for ions as input. Since we focus on the EFI, we choose to perform the analysis in the plasma frame, in order to have the electron temperature anisotropy T e,‖ /T e,⊥ > 1 as the only source of free energy in the solver. The ion temperature is assumed to be isotropic T i = T i,‖ = T i,⊥ and this approximation is motivated by the fact that the non-propagating EF mode is not affected by the ion temperature anisotropy (López et al., 2022;Maneva et al., 2016). The PDRK solver input parameters are obtained by averaging the relevant observed quantities in the interval Δt EFI >0 , where the EFI threshold is exceeded. The input parameters for the seven observed events are collected in Table 1. To avoid confusion, in this section the quantities that resulted from the analysis of in situ spacecraft observations are labeled with the subscript [obs].
The model implemented in the solver assumes that the plasma is homogeneous, as well as the background magnetic field. To check whether it is reasonable to compare this model to the MMS observations, we estimated the mean value of the density ⟨n⟩Δt  EFI >0 and its standard deviation σ n for each of the events in the intervals Δt EFI >0 and we computed the ratio n∕⟨n⟩Δt  EFI >0 . We find that the ratio ranges between n∕⟨n⟩Δt  EFI >0 = 0.01 and n∕⟨n⟩Δt  EFI >0 = 0.06 , indicating that the density is quite constant during the intervals where the EFI threshold is exceeded. Computing the same quantities for the magnetic field magnitude we obtain |B|∕⟨|B|⟩Δt  EFI >0 with values ranging from 0.01 to 0.22. To quantify the variation of the magnetic field direction b = B/|B|, we computed the maximum angular deviation of the magnetic field direction in Δt EFI >0 from the average direction, ⟨ ⟩Δt  EFI >0 . The maximum angular deviation during the intervals Δt EFI>0 ranges from 5° to 29°. Since the observed variations of the density and background magnetic field are quite limited, we conclude that the results of the PDRK solver can be meaningfully compared with the MMS observations. Figure 8 shows the results of the PDRK solver with input parameters mimicking the in situ observations of event #6. A positive growth rate γ is obtained for several points in the parameter space kρ e -θ kB with the maximum growth rate γ max /Ω ce ∼ 0.025 at [kρ e , θ kB ] = [0.54, 58°] (see Figure 8a). The unstable wave mode is characterized by zero real frequency (see Figure 8b). The values of θ kB associated with highest wave growth range between 52° and 64° and indicate that the mode is oblique (see Figure 8a). The values of long , which are below 0.8 for the majority of the points in the area of the parameter space with positive growth rate, indicate that the mode is electromagnetic (see Figure 8d). We conclude that the unstable mode is the non-propagating EF mode, as expected considering the imposed input electron distribution function with T e,‖ /T e,⊥ > 1. Figure 8 shows that the results of the numerical solver are consistent with in situ observations, providing further evidence that the observed fluctuations are associated with the EFI. The observed [ kB]obs ∼ 61 • and [k e]obs ∼ 0.41 , corresponding to maximum magnetic field fluctuations normalized power P(k,f)/P max in Figure 5h, are marked with red stars in Figure 8. The red-shaded area corresponds to the points in the parameters space which lay within [Δk e] Figure 8 shows a good agreement between the numerical results and the in situ observations, as the observational points composing the red-shaded area significantly overlap with the EFI unstable region predicted by the numerical solver. The  Table 1). These values correspond to the average over the interval where the EFI threshold is exceeded (Δt  EFI >0 =15:38:07.400-15:38:07.890). kρ e and θ kB versus (a) imaginary frequency γ/ Ω ce (b) real frequency ω/Ω ce (c) δB ‖ /δB (d) long . The quantities in panels (b)-(d) are shown for values of the growth rate exceeding the marginal stability condition, which is usually set at 10 −3 (Camporeale & Burgess, 2008). The values listed above panel (a) and (b) correspond to the values observed in situ. In each subplot, the red star corresponds to the observed kρ e and θ kB at the peak of normalized power of the fluctuations (see Figures 4h-4j). The red-shaded area represents the uncertainty of these measurements, Δkρ e and Δθ kB . COZZANI ET AL. Analogously to Figure 8 for event #6, Figure 9 shows a good agreement between the in situ observations and the numerical solver results for interval 7A of event #7. Figure 9a shows that a positive growth rate γ is obtained for several points in the parameter space kρ e -θ kB . The growth rate peaks (γ max /Ω ce ∼ 0.01) at [kρ e , θ kB ] = [0.56, 56°] so the growing mode is rather oblique with respect to the background magnetic field. Figure 9b shows that all the points associated with γ > 0 have zero real frequency, so the mode is non-propagating. Also, long ≲ 0.5 for the majority of the points in the area of the parameter space with γ > 0, suggesting that the mode is electromagnetic (Figure 9d). Similar to what we concluded for event #6, these characteristics suggest that the unstable mode presented in Figure 9 is the non-propagating EF mode.
The wave analysis results of the observed fluctuations in interval 7A of event #7 are shown in Figure 9. In this case, the wave analysis of in situ observations gives [k e]obs ∼ 0.66 and [ kB]obs ∼ 64 • and the associated uncertainties [Δk e]obs ∼ 0.22 ∼ 0.33 [k e]obs and [Δ kB]obs ∼ 8 • ∼ 0.13 [ kB]obs . During event #7 (interval 7A), as well as for event #6, we observe a good agreement between the in situ observations and the results of the numerical solver, reinforcing the conclusion that the observed fluctuations are indeed consistent with the non-propagating EF mode.

Other Events
As discussed in Section 4, during the interval shown in Figure 2 we have identified seven intervals fulfilling EFI > 0 together with the selection criteria involving the number of data points with EFI > 0 , the value of β e,‖ and the presence of wave activity. For each of the events, we perform the detailed wave analysis presented in Section 5 and we compare the in situ observations with the numerical solver results, using the input parameters reported in Table 1. Each event is defined by the interval where the EFI threshold is exceeded (Δt  EFI >0 , see Table 1) and  Table 1). COZZANI ET AL.

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16 of 21 by the interval where the wave activity is observed (Δt, see Table 2). As already discussed in Section 5, event #7 presents two intervals (7A and 7B) with enhanced wave activity.
For all the selected events, the observed fluctuations have characteristics consistent with the non-propagating EF mode. The results of the analysis of the seven events are summarized in Figure 10 and Table 2. In Figure 10, the abscissa shows the event number # and the quantities are averaged in the intervals of wave activity. Figure 10a shows the observed frequency f obs (black star) and the Doppler shift frequency ⟨fDS⟩Δt (gray star) with the error bars corresponding to the variability f DS for each of the selected events. For all the events, f obs lies in the variability range of f DS so that the Doppler shifted frequency is close to zero and the fluctuations can be considered as non-propagating. An exception is event #4 since f obs lies outside (but still very close to) the variability range of f DS . We still include event #4 in the list of EF events as the other characteristics of the observed waves are

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17 of 21 consistent with the EF mode. Also, it is worth clarifying that the so-called f DS variability range, f DS , does not have to be interpreted as a rigorously defined error of f DS , but rather a qualitative estimation of the uncertainty. The same quantities shown in Figure 10a, this time normalized by the lower-hybrid frequency f LH , are shown in Figure 10b. In all the events, the observed frequency is comparable with the local f LH . Figures 10c-10f show other characteristics that we take into account for the wave analysis in Section 5. In all the events, the wave characteristics are quite similar. Notably, kρ e ranges between 0.30 and 0.74 ( Figure 10c); θ kB ranges between 38° and 82° indicating that the observed mode is oblique ( Figure 10d); ⟨long⟩Δ Δ ranges between 0.23 and 0.57 meaning that the observed waves have a significant electromagnetic component (Figure 10e). The parameter ⟨ ± ⟩Δ has a minimum value of 0.9 for event #4 and a maximum of 2.4 for event #7 (Figure 10f). Another common feature of the fluctuations observed in all the events is that all three δB components have similar amplitude (see Figure 5b and Figure 7b for event #6 and #7). Also, during all the events the electron and ion velocity, notably in the GSM x direction, are large (|v e,x | ≳ 800 km/s and |V i,x | ≳ 500 km/s, see Figure 2), indicating that all the intervals with EF waves and where the EFI threshold is exceeded are located in the magnetic reconnection outflow region.
We then compare the in situ observations of each event with the results of the numerical solver PDRK, analogously to Sections 5 and 6 for event #6 and #7 (interval 7A). The PDRK solver is run with initial parameters such as background magnetic field, density, and temperatures, tailored to each event (see parameters in Table 1). For event #4, the temperature anisotropy has been artificially increased in the solver by 6% (from the value T e,‖ / T e,⊥ = 1.18 observed in situ to T e,‖ /T e,⊥ = 1.25) in order to obtain an unstable EF mode. The fact that it is needed to consider a higher T e,‖ /T e,⊥ value to obtain wave growth is not surprising as it is expected for the anisotropy to decrease as the instability develops and the waves grow. Since waves are directly observed in situ, the electron temperature anisotropy at the time of the observations is likely lower than the T e,‖ /T e,⊥ at the time of the instability onset. For each event, we find a good agreement between in situ observations and the model (not shown) suggesting that the waves observed in the selected events are fluctuations generated by the EFI developing in the reconnection outflow.

Discussion
In this study, we investigate a current sheet flapping event in the Earth's magnetotail associated with strong flows in the x GSM direction indicative of ongoing magnetic reconnection. The flow is directed tailward during the first part of the interval and Earthward at the end of the interval, indicating that MMS observed a magnetic reconnection X-line retreating tailward. Magnetic reconnection regions such as the outflow can be characterized by strong temperature anisotropy so that temperature anisotropy-driven instabilities, such as the EFI, can develop at those locations.
Even though the EFI has been invoked to explain the constrained electron temperature anisotropy in a variety of plasma environments, direct observations of the EFI-generated waves were lacking. In this study, we report in situ MMS observations of EF waves in the reconnection outflow region. There are two distinct EF modes but, as specified above, we focus exclusively on the non-propagating EF mode since it has a larger growth rate and a lower instability threshold with respect to the propagating EF mode. While being located in the reconnection outflow, MMS observes several time intervals during which the EFI threshold is exceeded (EFI > 0) . Taking into account the selection criteria discussed in Section 4, we finally select seven events that are characterized by both EFI > 0 and wave activity. We presented a detailed wave analysis of two of those events, showing that the observed wave characteristics are in agreement with the properties of the non-propagating EF mode.
Even though the non-propagating EF mode has distinct characteristics, it shares a few properties with the electromagnetic part of the lower hybrid mode. Lower hybrid drift waves (LHDW) are commonly observed in plasma regions characterized by strong spatial gradients in various quantities such as the density or the magnetic field. For example, the characteristics of the LHDW have been thoroughly investigated at the Earth's magnetopause (e.g., Graham et al., 2019). In the context of a current sheet, LHDW can be triggered by the lower hybrid drift wave instability (LHDI) and while an electrostatic, short wavelength (kρ e ∼ 1) mode will be localized at the edges of the current sheet, an electromagnetic, longer wavelength ( k √ e i ∼ 1 ) mode can be present at the center (Daughton, 2003;Yoon et al., 2002). The electrostatic mode is characterized by a larger growth rate but it stays confined at the edges of the current sheet, while the electromagnetic mode develops at later times and is present at the current sheet center (Daughton, 2003). The electromagnetic LHD mode is characterized by oblique propagation with respect to the background magnetic field and by frequency of the order of the lower hybrid frequency 18 of 21 f LH . So, both the electromagnetic LHD mode and the non-propagating EF mode are electromagnetic and characterized by large wave-normal angles. Despite these similarities, the two modes are of course distinct. First, the EF mode is non-propagating so it has zero real frequency, while LHDW have a frequency of the order of f LH . Also, EFI-generated waves are expected to have a quite low δB ‖ /δB, while for obliquely propagating LHDW δB ‖ is the largest component of the fluctuating magnetic field.
To further corroborate our results, we make sure that the fluctuations that we have identified as the EF waves are not the electromagnetic lower hybrid mode, which has been reported in several studies investigating magnetic reconnection in the Earth's magnetotail and at the magnetopause (Chen et al., 2020;Cozzani et al., 2021;Wang et al., 2022;Yoo et al., 2020). This further check is motivated by the fact that the observations are complex and characterized by significant uncertainties. The direct comparison with the LHD mode-which shares characteristics with the EF waves-will demonstrate that we are not mislabeling the observed waves and provide further robustness to our results. Thus, we consider two LHDW events corresponding to reconnection electron diffusion region (EDR) crossings in the magnetotail reported by Cozzani et al. (2021) (on 2017-08-10 at 12:18:33.0) and Chen et al. (2020) (on 2017-07-03 at 05:27:07.5). As for the seven events discussed in previous sections, we computed the EFI threshold and we performed the wave analysis. The results are summarized in Figure 10  . We note that while the EFI threshold is reached during event LHDW2, it is never reached for LHDW1, neither during the interval of wave activity nor considering an interval of several seconds centered around the interval of wave activity. For this reason, we could not define Δ EFI >0 for event LHDW1. Both events present characteristics that are similar to the EF events (kρ e ≲ 1, oblique θ kB and long ≲ 0.5 ). However, for LHDW2 we observe a non-zero frequency (see Figures 10a and 10b), so the observed waves could not be identified as non-propagating EF waves. Concerning event LHDW1, while the observed frequency (black diamond in Figures 10a and 10b) lies inside the variability range f DS , we note that f DS is at least four times larger than any f DS computed for the EF events, indicating that the measurement is not reliable in this case. Also, the behavior of f DS is drastically different in LHDW1 and the EF events. During the EF events, we observe the Doppler shift frequency f DS fluctuating around the value of the observed frequency f obs so that for several points in the time interval with wave activity f DS = f obs (see e.g., Figure 7c). In contrast, during the wave activity interval of event LHDW1, f DS does not fluctuate around f obs (not shown); it varies approximately linearly during the considered interval and it takes the value f obs only twice. More importantly, the EFI instability threshold is never exceeded during event LHDW1. Hence, it is unlikely that EFI-generated waves would be observed during event LHDW1. We conclude that, while the observed EF and LHDW waves share some similarities, it is possible to distinguish between the two modes. This comparison further confirms that the reported events are reliably identified as EF fluctuations.
Another instability that, analogously to the EFI, can develop when T e,‖ /T e,⊥ > 1 is the ordinary-mode instability (OMI) (Ibscher et al., 2012;Lazar et al., 2014). As the OMI generates non-propagating fluctuations, it is often associated with the Weibel instability in magnetic field-free plasmas. The OMI instability characteristics have been examined in detail by Ibscher et al. (2012). The study by Ibscher et al. (2012) also provides the theoretical threshold for the OMI instability, which has been shown in Figure 3 (green line). The OMI shares a few properties with the non-propagating EFI. Notably, the OMI-generated fluctuations are non-propagating, and the characteristic wave vector kρ e ∼ 1. However, in the case of OMI, the wave-normal angle is expected to be θ kB = 90° while it is expected in the range 30° < θ kB < 90° for the oblique, non-propagating EFI. The observed wave-normal angles belong to the expected θ kB range of EFI for all the considered events (the maximum θ kB = 82° is observed during event #4), suggesting that the observed fluctuations are indeed consistent with EFI-generated waves. In addition, Lazar et al. (2014) investigated the interplay between the EFI and the OMI and their results show a dominance of the non-propagating EFI, which is characterized by a lower threshold with respect to the OMI. Indeed, Figure 3 shows that the electron distribution in the β e,‖ -T e,‖ /T e,⊥ parameter space is bounded by the non-propagating EFI threshold while none of data points approach the OMI threshold. Hence, despite the similarities between the non-propagating EF mode and the OMI, we confirm that the observed fluctuations are identified as generated by the EFI.
As mentioned in previous sections, during several of the EF events, the waves that we have identified as EFI-generated are not observed in correspondence of the EF unstable intervals where EFI > 0 , but rather immediately before or after. This may be unexpected as we might expect to observe the EF waves in the source region, 19 of 21 as they are non-propagating fluctuations. At the same time, we expect the electron temperature anisotropy to decrease as the waves grow and the instability proceeds to the non-linear stage leading to electron isotropization. This means that MMS could observe a region with unstable plasma without (prior to) wave development and observe clear wave activity in a region where the instability has already saturated and reduced the anisotropy of the plasma, so it is stable to EFI at the time of the observations. The validity of this interpretation depends on the time scales associated with the development and saturation of EFI compared to the duration of the observed intervals with EFI > 0 and of the intervals with wave activity. The time scales of interest are related to the wave growth rate γ, T γ = 2π/γ and to the time required to reach the maximum fluctuations amplitude T peak . These two quantities cannot be easily computed with in situ measurements. However, we can obtain an estimation of T γ from the results of the linear solver. The time scale T peak has been evaluated in simulation studies. The value of T peak is quite similar in simulation studies by Gary and Nishimura (2003); Camporeale and Burgess (2008); Hellinger et al. (2014) and corresponds to Tpeak ≈ 5 − 10 T max , where T max = 2 ∕ max is computed for the maximum growth rate. In the case of event #6, the interval where the EFI threshold is exceeded, Δt EFI >0 , has a duration of 0.49 s. The maximum growth rate is γ max = 0.025 Ω ce (see Figure 8a) so that T max = 2 ∕ max = 0.43 s (here Ω ce = 580 rad/s for a background magnetic field of 3 nT). Considering the estimate value of T peak based on simulations results, Tpeak ≈ 5 − 10 max ≈ 2.15 − 4.3 . Hence, Tpeak = 4.4 − 8.7 Δt EFI >0 , meaning that the time spent by MMS in the unstable region is not enough to observe the wave development. At the same time, it is not surprising that the waves remain in the region where the temperature anisotropy is already being reduced, as the waves are non-propagating. This estimation yields to similar results also for the other events that have the wave activity not co-located with Δt EFI >0 . This simple qualitative estimation, despite its inherent limitations, can help us understand the lack of wave observations in the intervals with EFI > 0 . The observed EF fluctuations are located in the reconnection outflow, which is characterized by strong flow. It is worth underlining that the presence of this strong electron flow is crucial for observing the non-propagating EF mode as it allows for a significant Doppler shift frequency that, in the case of non-propagating modes, will coincide with the observed frequency ( fobs = fDS ± f DS ) . We note, however, that a non-negligible Doppler shift frequency depends not only upon the magnitude of v e but also on the angle between v e and k. In all considered events, v e has a significant component along the wave vector yielding significant Doppler shift frequency.
We have observed EF fluctuations in both the Earthward and tailward outflow. Notably, the wave analysis of Section 5 is focused on an event located in the tailward outflow region (event #6) and an event located in the Earthward outflow region (event #7). Despite the difference in the location with respect to the reconnection site, the characteristics of the two events are similar. However, the limited number of events would prevent us to draw any conclusion about the possible differences (or similarities) due to the different location relative to the X-line.
Interestingly, for all the EF events the observed waves are more complex than predicted by linear dispersion theory. The observed EF waves exhibit magnetic field fluctuations of similar amplitude for all three components in both GSM and field-aligned (FAC) coordinate systems (see Figure 4b and Figure 5b for event #6; Figure 6b and Figure 7b for event #7). This is in contrast with the linear theory predicting low δB ‖ /δB, meaning that the components perpendicular to the background magnetic field are dominating the fluctuations (see Figures 1d,  8c, and 9c). Also, while all the observed waves have a clear electromagnetic component, for several events ⟨long⟩Δ Δ ∼ 0.5 further indicating that the observed waves are quite complex as they are not fully electromagnetic or electrostatic.

Conclusions
We used high-resolution in situ measurements by MMS to investigate EFI-generated fluctuations in the outflow region of magnetic reconnection. We considered a current sheet flapping event in the Earth's magnetotail when MMS was almost continuously measuring the reconnection exhaust (both tailward and Earthward flow). We identified seven events characterized by wave activity during which the EFI threshold is exceeded.
Our results show that the observed waves have properties consistent with the non-propagating EF mode as predicted by the linear kinetic dispersion theory. In particular, we observe non-propagating fluctuations (i.e., zero real frequency) characterized by a wave vector kρ e ≲ 1 directed obliquely with respect to the background magnetic field, with significant electromagnetic component and resonant with electrons. However, there are also COZZANI ET AL.
10.1029/2022JA031128 20 of 21 some differences between the observed fluctuations and the prediction of the linear theory. Notably, all three fluctuating magnetic field components have similar amplitude; the waves are not fully electromagnetic or electrostatic, that is, ⟨long⟩Δ Δ ∼ 0.5 . This study, reporting for the first time direct observations of the EFI-generated fluctuations in the reconnection outflow region, represents the first step toward a more complete understanding of the EFI and its possible interplay with reconnection. Further investigation of the EF modes in the reconnection outflow region will be crucial to improve our knowledge of the global energy conversion associated with reconnection. Indeed, a significant fraction of the total energy conversion associated with reconnection occurs outside of the reconnection site proper, notably in the outflow and separatrix regions (e.g., Lapenta et al., 2016). In the outflow, the EFI-generated fluctuations could lead to particle scattering and enhanced wave-particle interaction which in turn can affect particle energization and energy conversion during reconnection, ultimately altering the global energy budget of the magnetic reconnection process. The results of this work are also beneficial to the study of the EFI in other plasma environments and regimes. In particular, the EFI is thought to play a key role in electron distribution isotropization in the solar wind but direct observation of the EF mode is currently prevented by the limited time resolution of particle measurements and lack of multi-spacecraft observations.