# MMS Analysis of a Dayside Compressed Magnetospheric Separatrix in the Presence of Cold Ions and a Moderate Guide Field

M. Baraka, O. Le Contel, P. Canu, S W Alqeeq, J. Dargent, A. Beck, G. Cozzani, A. Retinò, T. Chust, L. Mirioni, et al.

► To cite this version:

M. Baraka, O. Le Contel, P. Canu, S W Alqeeq, J. Dargent, et al., MMS Analysis of a Dayside Compressed Magnetospheric Separatrix in the Presence of Cold Ions and a Moderate Guide Field. Journal of Geophysical Research Space Physics, 2025, 130 (4), 10.1029/2024JA033234. [https://doi.org/10.1029/2024JA033235]

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Distributed under a Creative Commons Attribution 4.0 International License
# JGR Space Physics

## RESEARCH ARTICLE

10.1029/2024JA033234

**Key Points:**

* Far from diffusion regions, the drifting cold ions support the normal electric field in the magnetospheric inflow region

* In the fluid frame, energy conversion reverses due to the parallel electric field related to the divergence of the electron pressure tensor

* Partition of thermal and flow energy conversion is mostly driven by the compressible term of the electron pressure strain

**Correspondence to:**

M. Baraka,
mohammed.baraka@lpp.polytechnique.fr

**Citation:**

Baraka, M., Le Contel, O., Canu, P., Alqeeq, S. W., Dargent, J., Beck, A., et al. (2025). MMS analysis of a dayside compressed magnetospheric separatrix in the presence of cold ions and a moderate guide field. *Journal of Geophysical Research: Space Physics*, 130, e2024JA033234. https://doi.org/10.1029/2024JA033234

Received 28 AUG 2024

Accepted 8 APR 2025

**Author Contributions:**

**Conceptualization:** M. Baraka, O. Le Contel

**Data curation:** M. Baraka, N. Ahmadi, H. Y. Wei, D. Fischer, D. J. Gershman

**Formal analysis:** M. Baraka

**Investigation:** M. Baraka

**Methodology:** M. Baraka

**Supervision:** O. Le Contel, P. Canu

**Validation:** O. Le Contel

**Writing – original draft:** M. Baraka

**Writing – review & editing:** O. Le Contel, S. W. Alqeeq, J. Dargent,

G. Cozzani, A. Retinò, T. Chust,

L. Mirioni, S. Toledo-Redondo,

M. Akhavan-Tafti, R. Bandyopadhyay, A. Chasapis, C. Norgren, Y. Khotyaintsev, N. Ahmadi, H. Y. Wei, D. Fischer, D. J. Gershman, J. L. Burch, R. B. Torbert,

## MMS Analysis of a Dayside Compressed Magnetospheric Separatrix in the Presence of Cold Ions and a Moderate Guide Field

<figure><img src="image_2.png" /></figure>

M. Baraka¹ ID, O. Le Contel¹ ID, P. Canu¹ ID, S. W. Alqeeq¹, J. Dargent¹ ID, A. Beck² ID, G. Cozzani³ ID, A. Retinò¹ ID, T. Chust¹, L. Mirioni¹ ID, S. Toledo-Redondo⁴ ID, M. Akhavan-Tafti⁵ ID, R. Bandyopadhyay⁶ ID, A. Chasapis⁷, C. Norgren⁸, Y. Khotyaintsev⁸ ID, N. Ahmadi⁷ ID, H. Y. Wei⁹ ID, D. Fischer¹⁰ ID, D. J. Gershman¹¹ ID, J. L. Burch¹² ID, R. B. Torbert¹³, B. L. Giles¹¹ ID, S. A. Fuselier¹² ID, R. E. Ergun⁹ ID, P.-A. Lindqvist¹⁴ ID, C. T. Russell⁹ ID, R. J. Strangeway⁹ ID, and K. R. Bromund¹¹ ID

¹Laboratoire de Physique des Plasmas, Paris, France, ²Laboratoire Leprince-Ringuet, Palaiseau, France, ³Laboratoire de Physique et Chimie de l'Environnement et de l'Espace (LCP2E), Orléans, France, ⁴Departamento Electromagnetismo y Electronica, Universidad de Murcia, Murcia, Spain, ⁵Climate and Space Sciences and Engineering, Ann Arbor, MI, USA, ⁶Department of Astrophysical Sciences, Princeton University, Princeton, NJ, USA, ⁷Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO, USA, ⁸Swedish Institute of Space Physics, Uppsala, Sweden, ⁹Institute of Geophysics and Planetary Physics, Los Angeles, CA, USA, ¹⁰Space Research Institute, Austrian, Academy of Sciences, Graz, Austria, ¹¹NASA Goddard Space Flight Center, Greenbelt, MD, USA, ¹²Southwest Research Institute, San Antonio, TX, USA, ¹³Space Science Center and Department of Physics, Durham, NH, USA, ¹⁴Space and Plasma Group, Royal Institute of Technology, Stockholm, Sweden

**Abstract** This study reports on a dayside magnetic reconnection event detected by the Magnetospheric Multiscale mission in the presence of a moderate guide field (≈ 0.5 times the reconnecting magnetic field on the magnetosphere side) and assumed to be present in the whole reconnection region. The spacecraft traversed the compressed magnetospheric separatrix region, observing cold ions with densities up to 10 cm⁻³ and a large magnetosheath density up to 150 cm⁻³. We provide a detailed analysis of current densities, generalized Ohm's law, and energy conversion processes in both the spacecraft (J·E) and the fluid (J·E') frames during the separatrix crossing. The normal electric field is directed away from the separatrix due to the cold ion drift on the magnetosphere side and to the magnetosheath ion drift in the presence of a guide field in the exhaust region. In the spacecraft frame, energy transfers from the plasma to the fields (J·E ≈ J⊥·E⊥ < 0) due to the convective field associated with the earthward motion of the magnetopause and the ion diamagnetic current associated with the large density gradient. In the fluid frame, energy conversion reverses due to the magnetic field-aligned current density and electric field produced by the divergence of the electron pressure tensor (J·E' ≈ J‖·E'‖). Additionally, we give insights into the local changes in electromagnetic, bulk flow, and thermal energies. We show that flow and thermal energy variations of the plasma are mostly driven by the compressible term of the electron pressure strain at the separatrix.

## 1. Introduction

Magnetic reconnection is a fundamental process in plasma physics and plays a crucial role in various astrophysical and space plasma phenomena. It involves the rearrangement of magnetic field lines, converting magnetic energy into kinetic and thermal energy, and accelerating charged particles (e.g., Fujimoto et al., 2011; Hesse & Cassak, 2020; Priest & Forbes, 2000). The magnetopause, the outer boundary of the Earth's magnetosphere that interacts with the solar wind, often exhibits a significant asymmetry in magnetic field strength, plasma density, and temperature on either side. For such an asymmetric reconnection process, the quadrupolar Hall magnetic field pattern evolves to a more dipolar pattern with larger Hall magnetic field signatures on the magnetosheath side than on the magnetosphere side as the Hall currents on the magnetosheath side are stronger due to the larger plasma density (e.g., P. A. Cassak & Shay, 2007; Mozer et al., 2008). In the presence of an out-of-plane magnetic field component (a guide field, hereafter B<sub>gf</sub>), a new North-South asymmetry of the reconnection region is produced (e.g., Swisdak et al., 2003; Pritchett, 2008; Pritchett & Mozer, 2009; J. P. Eastwood et al., 2013). More recently, the Magnetospheric Multiscale (MMS) mission has significantly advanced our understanding of magnetic reconnection by providing direct and detailed measurements of the key plasma parameters and electromagnetic

© 2025. The Author(s).

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
B. L. Giles, S. A. Fuselier, R. E. Ergun, P.-A. Lindqvist, C. T. Russell, R. J. Strangeway, K. R. Bromund

fields, especially in the vicinity of dayside reconnection sites at electron scales (e.g., Burch et al., 2016; S. A. Fuselier et al., 2016; Webster et al., 2018).

Much attention has been paid to the diffusion region due to its role in reconnection and energy dissipation while separatrices have received comparatively less attention. The separatrix region is a kinetic-scale boundary layer (e.g., André et al., 2016; Khotyaintsev et al., 2006; Retinò et al., 2006; Vaivads et al., 2004), originating from the diffusion regions and extending far away from the X-line. It separates the inflow and outflow regions in magnetic reconnection. For dayside asymmetric reconnection, we distinguish magnetosheath and magnetosphere separatrices, which separate magnetosheath and magnetosphere inflow regions respectively from the outflow regions containing a mixing of the two plasmas. Detailed investigations of the separatrix region permit the definition of substructures of the separatrix region (Jiang et al., 2022; Khotyaintsev et al., 2006; Retinò et al., 2006; Steinvall et al., 2021). One can distinguish the separatrix boundary layer, defined as the thin layer separating the electron inflow region where electrons are moving toward the reconnection site and the electron outflow region where electrons are moving away. This thin layer is embedded in the broader separatrix region where the field-aligned current is significant and changes sign (Huang et al., 2021). Using MMS data gathered on the 21st of October 2015, we investigate in detail a case study of a magnetosphere separatrix crossing occurring far from the diffusion regions and during specific conditions related to a large solar wind density and the presence of magnetospheric cold ions. Such conditions are often related to the impact of large-scale perturbations transported by the solar wind. The geomagnetic activity associated with such perturbations usually leads to the leak of cold plasma from the inner magnetosphere to the magnetopause through plasmaspheric plumes (Chappell, 1974).

In the subsolar magnetosheath (hereafter $M_{sh}$), the most probable density ($n_{sh}$) is about $20 \text{ cm}^{-3}$ (e.g., P. Cassak & Fuselier, 2016). In our observations, the density peak after the separatrix crossing reached $\approx 100 \text{ cm}^{-3}$. The large density and the $B_y$ component (in Geocentric Solar Ecliptic coordinate, hereafter GSE) detected by MMS in the magnetosheath are consequences of the arrival of a compressed solar wind. This compressed solar wind region is ahead of a shock-less Stream Interaction Region (SIR) event (e.g., Jian et al., 2006) observed by OMNI at L1 (not shown). A corresponding large pressure gradient occurs at the magnetopause. Such pressure gradients give rise to diamagnetic currents, which flow perpendicular to the magnetic field (e.g., Ganushkina et al., 2018; Hasegawa, 2012).

On the magnetosphere side (hereafter $M_{sp}$), cold ions, characterized by their low energy of a few electron volts (eV), are abundant in the Earth's magnetosphere (André & Cully, 2012). During the MMS mission, comprehensive investigations into cold ions have been conducted (S. Fuselier et al., 2017; André et al., 2021). They can be distinguished by their energy, composition, and, pitch angle distribution. Consequently, observable fluctuations occur in their fluxes and energy levels, introducing additional complexity to selecting an appropriate energy range for analysis for different events. The variability in ion behavior near the magnetopause necessitates a thorough understanding of the associated dynamics. As cold ions approach the magnetopause, dynamic interactions may lead to their acceleration or heating (e.g., Song, Zhou, Yi, Deng, & Zhong, 2023; Song, Zhou, Yi, Zhong, & Deng, 2023; Toledo-Redondo, André, Vaivads, et al., 2016; Toledo-Redondo et al., 2017). They can modify the Hall physics when crossing the separatrix region (André et al., 2016; Toledo-Redondo et al., 2015) and they introduce a new characteristic length scale determined by cold ion gyroradius, creating a cold-ion diffusion region (Toledo-Redondo, André, Khotyaintsev, et al., 2016; Toledo-Redondo et al., 2021). In both observations and simulations, a perpendicular current reduction caused by cold ions is demonstrated by (Toledo-Redondo et al., 2018).

The investigation of the process leading to the generation of the electric field normal to the magnetopause, independent of the Hall electric field, has been explored in both recent and earlier studies. Initially, theory explained the presence of this electric field in the non-reconnecting magnetopause by considering the finite Larmor radius effect of the $M_{sh}$ ions entering the $M_{sp}$ side and making their U-turn leading to out-of-plane velocity and creating the associated electric field (Cowley, 1995). During magnetopause reconnection, this behavior of the normal component of the electric field is only expected in the vicinity of the X-line on the $M_{sp}$ side defined as an in-plane electric field independent of the Hall electric field. This behavior, as introduced by Malakit et al. (2013), was initially linked to the Larmor electric field generated by the early escape of a small fraction of magnetospheric ions into the dissipation region, leading to charge separation. However, it turned out this explanation was not accurate. Further studies (Dargent et al., 2017; Shay et al., 2016) demonstrated that this finite Larmor radius effect is actually due to the entry of the magnetosheath ions inside the magnetosphere, consistent with in-situ
observations (Graham et al., 2017). Far from the diffusion region along the magnetospheric separatrix, the boundary layer is thickened due to the formation of the exhaust region, which results in the absence of this electric field. Therefore, the Larmor electric field is only measured in the magnetospheric inflow region of the X-line just upstream of the Hall electric field region, directed toward the Earth and away from the X-line (Pritchett & Mozer, 2009). However, in the presence of magnetospheric cold ions, kinetic simulations have shown that such a normal electric field signature can be maintained far away from the X-line. Indeed, their low energy enables them to $E \times B$ drift (where $E$ and $B$ are the electric and magnetic field vectors, respectively) in the electric field structure and to sustain this normal electric field all along the vertical direction of the magnetopause boundary despite the increase of the separatrix thickness above the Larmor radius scale of the magnetosheath ions (Dargent et al., 2017).

We investigate the normal electric field in the presence of cold ions and a $B_{gf}$ component, identified as a constant out-of-plane background magnetic field in both $M_{sp}$ and $M_{sh}$ regions where no Hall effects are present. This guide field component is assumed to be present in the whole reconnection region. In our observations, we found that the normal electric field signature is maintained far away from the X-line in the presence of the cold ions that are $E \times B$ drifting in the electric field structure in agreement with Dargent et al. (2017) findings. Additionally, a normal electric field appears in the outflow region due to the jet's convective motion and the guide field's presence.

Furthermore, we discuss the energy conversion associated with the crossing of the compressed magnetospheric separatrix. Poynting's theorem demonstrates that the energy conversion between electromagnetic fields and plasma particles happens through $J \cdot E$ (where $J$ is the current density). We found that in the spacecraft frame, the energy is transferred from the plasma to the fields ($J \cdot E < 0$) mostly due to the convective electric field associated with the earthward motion of the magnetopause pushed by the high dynamic pressure of the shocked solar wind. In the fluid frame, we found a bipolar signature of the energy conversion ($J \cdot E'$, where $E' = E + v_e \times B$ is the electric field in the electron fluid frame) mostly related to the magnetic field-aligned current and electric field ($J_\| \cdot E'_\|$), $E'_\|$ being produced by the parallel component of the divergence of the electron pressure tensor. In this frame, the energy is transferred from the fields to the plasma on the magnetosphere side and from the plasma to the fields on the outflow region side.

We also provide insights into the partition of energy conversion at the magnetosphere separatrix. Our investigation was inspired by the work of Bandyopadhyay et al. (2021), who conducted an analysis of the pressure strain calculations at the electron diffusion region (EDR) of global (magnetopause) and local (magnetosheath current sheets including electron-only reconnection) reconnection events in both observations and simulations. Here we extend this analysis to the magnetospheric separatrix region, showing that the pressure strain values at the separatrix are not significantly different from those reported by Bandyopadhyay et al. (2021).

The compressible term of the electron pressure strain mostly drives flow and thermal energy conversion. From the analysis of the flow energy equation using a different formulation which only groups together time variation and transport terms of flow and thermal energies, we suggest that ion flow can be accelerated by the work of the pressure force on the magnetosphere side and decelerated by the work of the electric field just after the separatrix crossing.

This paper is organized with an overview of the event in Section 2. Current densities are estimated in Section 3. Generalized Ohm's law and partial moments are analyzed in Section 4. Energy conversion and partitioning are discussed in Section 5. Finally, the summary and conclusion are given in Section 6.

## 2. MMS Event Overview

We use measurements of the magnetic field from the fluxgate magnetometer (FGM) (C. Russell et al., 2016), plasma parameters, and the corresponding integrated particle moments from the fast plasma instrument (FPI) (Pollock et al., 2016), electric field components from the electric double probe (EDP) (Torbert et al., 2016), the Spin-Plane Double probe instrument (SDP) (Lindqvist et al., 2016) and, the Axial Double Probe (ADP) (Ergun et al., 2016). The four satellites were in a tetrahedral formation, with a separation distance of ~11 km located slightly on the dusk side of the dayside magnetopause at [8.8, 2.8, -0.2] $R_E$ in GSE. Such a satellite separation larger than the electron inertial length (average values of ≈0.5 km on the $M_{sh}$ side and ≈2 km on the $M_{sp}$) and much smaller than the ion inertial length (≈24 km on the $M_{sh}$ side and ≈76 km on the $M_{sp}$) allows to obtain
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Magnetospheric Multiscale observation of (21 October 2015) event in the local boundary normal (LMN) coordinate system. (a) Magnetic field components, (b) ion density (black) and electron density (blue), (c) and (d) are ion and electron velocities, (e) and (f) are the ion and electron omnidirectional spectrograms, (g) the electric double probe Electric field, and (h) Current density from particle moments. The black and red lines in (e) and (f) are perpendicular and parallel temperatures plotted respectively. The white line in (e) is energy corresponding to **E** × **B** drift of protons. The blue-dotted line represents the ion edge. The red-dotted line represents the separatrix.</figcaption></figure>

accurate gradient estimates from 4-spacecraft methods for ions but can lead to an underestimate of electron gradients (e.g., Dahani et al., 2024).

Figure 1 shows data from the field and plasma instruments from 04:39:00 to 04:40:00 UT on 21st of October 2015. All data with a higher time resolution than FPI electron measurement (0.03 s) including current calculations from the curlometer are time-averaged at 0.03 s and interpolated to the FPI electron resolution. Vector data is displayed using the local boundary coordinates system (LMN) obtained from minimum variance analysis on the magnetic field (MVAB) (B. U. Sonnerup & Scheible, 1998) averaged over the four spacecraft measurements and during the separatrix crossing between 04:39:00 to 04:40:00 UT. The LMN coordinates relative to GSE co-ordinates with eigenvalue ratios of $\lambda_L/\lambda_M = 18.9$ and $\lambda_M/\lambda_N = 9.7$ are $L = [0.11,0.20,0.97]$ pointing northward, $M = [0.24,-0.95,0.16]$ pointing dawnward, and $N = [0.96,0.22,-0.16]$ normal to the separatrix boundary pointing sunward. Thus, this local LMN frame is defined at the magnetospheric separatrix crossing. Although distinct from the LMN frame of the diffusion regions, we assume and verify a posteriori that it remains relevant for analyzing the dynamics at the magnetospheric separatrix. We applied the four spacecraft timing analysis (C. Russell et al., 1983; Schwartz, 1998) to the crossing of this boundary based on the $B_L$ rotation between 04:39:23
and 04:39:27 UT and found that the normal direction $N = [0.97,0.15,-0.16]$ in GSE is consistent with MVAB normal. We found an earthward motion of the magnetopause with $V_N = -47$ km/s.

Figure 1a shows the magnetic field components and magnitude while Figure 1b illustrates the ion and electron densities. Ion and electron velocities are shown in Figures 1c and 1d, respectively. Figure 1e displays the ion spectrogram from MMS1 with the perpendicular and parallel ion temperature marked by black and red lines respectively, along with $E \times B$ drift energy represented by the white line. Figure 1f shows the electron spectrogram also from MMS1, with the perpendicular and parallel electron temperature also marked by black and red lines, respectively. Finally, electric field and current density components are displayed in Figures 1g and 1h, respectively.

We divide the interval into three distinct light-shaded regions: the magnetosphere region where cold ions are dominating (blue-shaded region) from 04:39:06 to 04:39:20 UT, the $M_{sp}-M_{sh}$ mixed ion region (yellow-shaded region) from 04:39:20 to 04:39:25 UT, and the outflow region (red-shaded region) from 04:39:25 to 04:39:51 UT. We separate each analyzed FPI ion distribution function into the cold magnetospheric population (10–200 eV) considering the lower-energy channel in the FPI detector, medium (200–4,000 eV) including the $M_{sh}$ ions, and hot magnetospheric ions within (4–30 keV). The different energy ranges for each population are chosen as follows: The lower energy limit of the cold ion population was set to 10 eV, the lowest FPI energy channel. Then we identified the cold ion region (blue-shaded) where cold ions are observed moving perpendicularly to the background magnetic field. The upper limit of energy was increased until it included most of the ions moving in a perpendicular direction in this region, resulting in a stable range between 10 and 200 eV. Hot ions, being from magnetospheric origin, were isolated by setting the lower limit to 4 keV and extending to the highest limit of energy range of FPI at 30 keV. The criteria for selecting the lower limit of energy was that the hot ion density strongly decreases on the magnetosheath side as the average energy of magnetosheath ions is much smaller (≈ 250 eV). The energy range of medium ions from (0.2–4 keV) was selected to correspond to the ion populations on the magnetosheath side. Using this energy range, the medium ion population density dominates in the magnetosheath region and strongly decreases on the magnetosphere side. By definition, in the mixing region, all ion populations are detected and hardly distinguished. To determine which population is decoupled from the magnetic field, we will check, in Section 4, the generalized Ohm's law (André et al., 2016).

In the blue-shaded region, on the magnetospheric side of the boundary, the initial peak of the density (~10 $cm^{-3}$) is primarily dominated by cold ions as shown in both density and ion omnidirectional spectrum in (Figures 1b and 1e). On the opposite side of the boundary, the plasma density peak is ~10 times larger due to the arrival of the compressed solar wind region. Meanwhile MMS observed the reversal of $B_L$ (Figure 1a) and clear evidence of southward reconnection ion outflow region observed by FPI ($V_L \sim -200$ km/s) (Figure 1c). The electron velocity has similar values as the ion velocity in the outflow region (Figure 1d). As reminded in the introduction, for an asymmetric reconnection process, the quadrupolar Hall magnetic field pattern evolves to a more dipolar pattern with a larger Hall magnetic field signature on the magnetosheath side than on the magnetosphere side as the Hall currents on the magnetosheath side are stronger due to the larger plasma density (P. A. Cassak & Shay, 2007; Mozer et al., 2008). Without a guide field southward of the reconnection region, the Hall magnetic field signature is expected to be smaller and directed dawnward ($\delta B_M > 0$) on the magnetospheric side and larger and duskward ($\delta B_M < 0$) on the magnetosheath side. In the presence of a moderate dawnward guide field, the expected small Hall magnetic signature on the magnetospheric side should be reinforced whereas on the magnetosheath it should be reduced. In some events, the guide field could even reverse the expected out-of-plane Hall magnetic field signature (Mozer et al., 2008). From MVAB, the guide field ($B_{gf}$) is identified as the constant background out-of-plane magnetic field across the boundary (e.g., Eastwood et al., 2018; Eriksson et al., 2016). In our case study, this $B_{gf}$ is measured everywhere in the $M_{sh}$ and $M_{sp}$ ($B_{gf} \approx 38$ nT), where there is no expected Hall magnetic field signature (i.e., $B_M = B_{gf}$). In the Msp, $B_M/B_L \sim 0.5$ (measured in the blue-shaded region away from Hall effects), although in the $M_{sh}$, the ratio is much larger since the reconnecting component $B_L$ is very small. This event is therefore considered an asymmetric reconnection event with a moderate guide field, taking for reference the ratio determined on the magnetosphere side. The guide field introduces a new North/South asymmetry and modifies the Hall magnetic field pattern (reinforcing or reducing the Hall magnetic signature depending on its orientation) (Pritchett & Mozer, 2009). Starting from the magnetosphere in Figure 1a, we first see a decrease of $B_M$ ($\delta B_M < 0$) associated with a positive parallel current. This signature does not correspond to the expected Hall signature. It seems to be related to localized accelerated electrons coming from the reconnection
region and producing this positive parallel current. Such a signature has already been reported by (Fu et al., 2024) (see their Figures 2 and 3). It could be related to the earthward shifting of the stagnation point, which is no longer co-located with the X-point and occurs for asymmetric reconnections (P. A. Cassak & Shay, 2007). Then $B_M$ reaches again a value closer to $B_{gf}$ just at the separatrix crossing which is consistent with the expected positive Hall magnetic field $\delta B_M$ signature. In the outflow region, we first see that $B_M$ fluctuates between 04:39:27 and 04:39:30 UT which could be related to the crossing of non-linear structures (flux ropes) formed at the magnetospheric separatrix (e.g., Daughton et al., 2011) and then decreases ($\delta B_M < 0$), which is consistent with the reduction produced by the expected negative Hall magnetic field on the outflow side. The $B_N$ component is sometimes slightly negative as in (Fu et al., 2024; Graham et al., 2017) while a positive value is expected. However, due to the very small amplitude, local fluctuations can easily modify its signature.

To support the identification of the measured large $V_L$ as a jet produced by a northward reconnection region, we use the scaling law for inflow parameters (P. A. Cassak & Shay, 2007). Plasma conditions are extremely asymmetric since averaged magnetic field and plasma density in the $M_{sp}$ region are $B_{sp} = 79.5$ nT and $n_{sp} = 10.5$ cm$^{-3}$ whereas in the $M_{sh}$ region (after the outflow region and averaged between 04:39:51 and 04:40:15 UT) $B_{sh} = 38$ nT and $n_{sh} = 59.4$ cm$^{-3}$. Using these inflow conditions, the predicted outflow speed for an asymmetric reconnection process can be obtained from the following equation (P. A. Cassak & Shay, 2007):

$$V_{\text{out}} = \left( \frac{B_{sp}B_{sh}}{\mu_0} \frac{B_{sp} + B_{sh}}{\rho_{sp}B_{sh} + \rho_{sh}B_{sp}} \right)^{1/2} \quad (1)$$

where $B_{sp}$, $\rho_{sp}$ and $B_{sh}$, $\rho_{sh}$ are magnetic field magnitudes and mass densities in the magnetosphere and the magnetosheath, respectively. The obtained $V_{out} \sim 188$ km/s, is very close to the measured $V_L$ value therefore consistent with an ongoing reconnection process in such asymmetric conditions. Interestingly, we can note that the predicted outflow density is $\approx 44.4$ cm$^{-3}$ whereas the measured density is $\approx 100$ cm$^{-3}$ (see Figure 1b). However, the Cassak and Shay predictions are obtained assuming a steady state and homogeneous inflow regions. As we mentioned, this dayside reconnection event is related to the arrival of a large-scale solar wind perturbation developing an increase in the solar wind density which compresses the magnetopause toward the Earth. The steady-state and homogeneity of inflow regions are likely not fulfilled. In particular, the compression of the reconnection region and the density gradient in the magnetosheath ahead of the magnetopause could explain a larger outflow density than the predicted value while the predicted velocity agrees quite well with the measurements.

To confirm in another way that the observed fast flow is a reconnection jet, we calculated the predicted flow for a rotational discontinuity using the relation for the tangential velocity jump, neglecting the effect of pressure anisotropy (Paschmann et al., 1986): $v_{2t} - v_{1t} = (B_{2t}/\rho_2 - B_{1t}/\rho_1)$. Here, “1” indicates the time of reference, and “2” is the prediction for subsequent times. Our measurements indicate that the ratio between the measured and predicted velocity is 0.3 (not shown) when the plasma beta $\beta > 2$ ($\beta$ being the ratio between the thermal and magnetic pressures) in agreement with results found by (Paschmann et al., 1986). The consistency between predicted and measured flow velocities allows us to confirm a posteriori that the local LMN frame, although determined at the separatrix crossing, is relevant for analyzing this reconnection event.

The cold ions show up as a narrow trace in the low energy part of the omnidirectional spectrogram (Figure 1e) and their energy varies similarly to the white line on the spectrogram, which shows the changes of the kinetic energy of ion convective motion ($E \times B/B^2$).

Figure 1g shows a reversal in the direction of the normal component of the electric field ($E_N$) (directed away from the magnetopause). It reverses from $E_N < 0$ on the $M_{sp}$ side to $E_N > 0$ on the $M_{sh}$ side. The interpretation of this behavior and its correlation with the existence of the cold ions and the guide field are discussed in Section 4. Understanding this behavior is one of the main objectives of this study.

Finally, we are interested in the whole separatrix region defined as the region where MMS measures significant $J_M$ and $J_L$ current densities (mostly perpendicular and parallel currents, respectively). As seen in (Figure 1h) at 04:39:24 UT, there are two negative peaks of the current density associated with the separatrix crossing in both parallel and perpendicular directions. The main component is in the out-of-plane direction ($J_M < 0$) with a peak
<figure><img src="image_4.png" /><figcaption>**Figure 2.** A 2D schematic illustration of magnetic reconnection. Solid lines represent the magnetic field, black lines for the magnetosheath, and red lines for the magnetosphere. Dotted lines represent the separatrices. The green arrow shows the Magnetospheric Multiscale trajectory traversing the southern reconnection outflow.</figcaption></figure>

value of ~1200 nA/m² and a peak value of ~900 nA/m² in the L direction ($J_L < 0$). The structure of the southward outflow region and magnetospheric inflow region is illustrated in Figure 2. As shown by the green-line spacecraft trajectory in Figure 2, MMS traverses the current layer, isolating the $M_{sp}$ plasma and the outflow region, referred to as $M_{sp}$ separatrix region. The separatrix boundary, defined as the thin layer where the electron velocity changes sign (separating the electron inflow region where electrons are moving toward the reconnection site from the electron outflow region where electrons are moving away) is marked by the red-dotted line in Figure 2 and at approximately 04:39:24.9 UT across all figures. This current region, with its carriers, and the identification of the separatrix boundary are discussed in detail in Section 3.

### 3. Analysis of the Current Densities

In this section, we investigate the current densities at the magnetosphere separatrix in the presence of a large-density gradient. We calculate the current density at the time resolution of the electron moment measurements (0.03 s, interpolating ion velocity) and average them over all MMS spacecraft ($J_p = en_e(v_i - v_e)$, where $e$ is the elemental charge, $n_e$ is the electron density and $v_i, v_e$ the ion and electron velocities). We verify the consistency with the values ($J_{curl} = (\nabla \times B)/\mu_0$, where $\mu_0$ is the magnetic permeability of the vacuum) obtained from the curlometer technique which provides a linear estimate of the current flowing through the tetrahedron from curl $B$ (Chanteur & Harvey, 1998; Dunlop et al., 2002). We can also consider the currents from plasma measurements on each spacecraft separately, and this approach typically shows differences in signatures at each spacecraft therefore it offers insights into the spatial structure of the current within the tetrahedron, limited to the cadence of the plasma moments. However, only the four-spacecraft average of the current density will be used in the next sections.

We investigate the current sources along the crossing of the magnetospheric separatrix region in both parallel ($J_\|$, Equation 2) and perpendicular ($J_\perp$, Equation 3) to the local magnetic field direction:
$$J_{\parallel} = \frac{(\mathbf{J} \cdot \mathbf{B})\mathbf{B}}{|\mathbf{B}|^2} \qquad (2)$$

$$J_{\perp} = J - J_{\parallel} \qquad (3)$$

In the non-reconnecting magnetopause current layer, the primary source of the total $J_{\perp}$ is the diamagnetic current ($J_{dia}$ computed from particle pressure gradients in Equation 4) and mostly carried by the ion diamagnetic current ($J_{dia,i}$) (e.g., Beedle et al., 2022; Dong et al., 2018).

$$J_{\perp} \approx J_{dia} = en(\mathbf{v}_{i,dia} - \mathbf{v}_{e,dia}) = \frac{-[\nabla \cdot (\mathbf{P}_i + \mathbf{P}_e)] \times \mathbf{B}}{B^2} \qquad (4)$$

where $n = n_e = n_i$ is the plasma density, $\mathbf{v}_{i,dia}$ and $\mathbf{v}_{e,dia}$ are the ion and electron diamagnetic velocities, and $\mathbf{P}_i$ and $\mathbf{P}_e$ are the ion and electron pressure tensors, respectively.

To determine if the current layer can be described by the two-fluid theory, we verified whether the ion Larmor radius is smaller than the length scale of the current layer. Using the magnetopause velocity obtained by timing analysis ($V_N = -47$ km/s) and assuming a time crossing of the current layer of 2.75 s (between 04:39:23.50 to 04:39:26.25 UT), the width of the current layer is $\approx 129$ km. Since the ion Larmor radius of dominating magnetosheath ions is $\sim 20$ km (for ion energy $\approx 250$ eV and $B \approx 60$ nT), smaller than the width of the boundary, the current layer of the separatrix can be described by the two-fluid theory as in the analyzed case study of a non-reconnecting magnetopause layer by (Dong et al., 2018).

Figure 3 illustrates the electric current densities as well as the individual contributions for M and L components from both electrons and ions. Figure 3a shows the magnetic field components where the constant $B_{gf}$ has been subtracted. Figures 3b and 3c show that the total current densities derived from the curlometer technique align closely with the corresponding current density computed using particle measurements. Figure 3d shows that the dominant M component of the diamagnetic current $J_{dia}$ is consistent with $J_{\perp}$. The comparison between the ion diamagnetic current ($J_{dia,i}$) and the electron diamagnetic current ($J_{dia,e}$) shows that the ion diamagnetic current contributes the most to the total $J_{dia}$.

As illustrated in the schematic in Figure 2 and shown in Figures 3e and 3f, just before the magnetospheric separatrix boundary crossing (indicated by the red-dotted line in both figures), there is a strong magnetic field-aligned current. This current is initially negative (anti-parallel) and carried by both ions and electrons on the magnetospheric side (between 04:39:24 and 04:39:25 UT), a bit later after the separatrix boundary crossing the current reverses and is dominated by electrons (at 04:39:26 UT). This behavior is consistent with the electron motion flowing toward the diffusion regions on the magnetospheric side in the inflow region and flowing away from the diffusion regions in the outflow region. The current carried by ions is also consistent with the motion of ions dominated by $M_{sh}$ ions flowing away from the diffusion regions on the outflow region side as well as on the magnetospheric side (between 04:39:24 and 04:39:25 UT). As explained in the event overview, the corresponding Hall magnetic field signature $\delta B_M > 0$ at the separatrix is observed (Figure 3a). It is worth mentioning that an additional out-of-plane magnetic field signature is observed between the ion edge at 04:39:21 UT and overlaps with the Hall magnetic field signature $\delta B_M > 0$ at the separatrix crossing (see Figure 2) and prevents from exceeding the constant $B_{gf}$ value. The magenta bar on top of Figure 3 marks the whole separatrix region, which we identify by the increase in the currents $J_M$ and $J_L$ starting at 04:39:23.50 and continuing until the end of the reversed parallel current $J_{\parallel}$ at 04:39:26.25 UT. The separatrix region includes the separatrix boundary defined, as we have already mentioned above, by the thin layer where the electron velocity (i.e., electron current) changes sign (separating electron inflow region where electrons are moving toward the reconnection site and electron outflow region where electrons are moving away). Note that due to the contribution of the penetrating magnetosheath ions to the parallel current, the region where the parallel current changes sign is shifted from the region where electron velocity is reversed. Therefore, the determination of the separatrix boundary is not based on the parallel current reversal as usual but on the electron velocity/current reversal.

These results are consistent with those shown by Graham et al. (2017) who investigated a magnetopause reconnection event during an ion diffusion crossing by MMS northward of the reconnection region, with a smaller
<figure><img src="image_5.png" /><figcaption>**Figure 3.** (a) Magnetic field components, (b) and (c) *J*<sub>M</sub> and *J*<sub>L</sub> current components from plasma moments (black) and curlometer (red), (d) out of plane perpendicular current component (black), diamagnetic current (red), ion diamagnetic current (green), and electron diamagnetic current (blue), (e) parallel current (black), ion parallel current (red), med-energy parallel ion current (blue), and parallel electron current (green), and (f) parallel current, the magenta horizontal line on top of the panel marks the separatrix region.</figcaption></figure>

density of cold ions and without a guide field. As for our event detected far from the diffusion region, the positive peak of the parallel current (*J*<sub>L</sub>) and the negative peak of the perpendicular diamagnetic current (*J*<sub>M</sub>) are measured at the boundary (black dashed line in their Figure 3) and the negative peak of the parallel current is measured on the magnetosheath side just after the boundary crossing.
Finally, we found that, near the magnetospheric separatrix boundary, as in the non-reconnecting magnetopause current layer, the ion diamagnetic current is the dominant source of the perpendicular current (Beedle et al., 2022; Dong et al., 2018). The values are close to the results obtained by Dong et al. (2018), where the ion diamagnetic current provides 85% of the total diamagnetic current, while in our case study, it provides 75%. However, regarding the parallel current, Dong et al. (2018) found that electrons are the dominant current carriers in the parallel direction on both sides of the magnetopause layer while ions only contribute as a small background current.

## 4. Generalized Ohm's Law and Partial Moments

We analyze the contribution of each population to the terms of generalized Ohm's law, considering different ion populations and electron terms and neglecting both time variations and the contribution of the electron inertia as demonstrated by Toledo-Redondo et al. (2015) and André et al. (2016):

$$ \mathbf{E} = \frac{1}{en} \mathbf{J} \times \mathbf{B} - \frac{1}{en} \nabla \cdot \mathbf{P}_e - \frac{n_h}{n} \mathbf{v}_h \times \mathbf{B} - \frac{n_m}{n} \mathbf{v}_m \times \mathbf{B} - \frac{n_c}{n} \mathbf{v}_c \times \mathbf{B} \quad (5) $$

Here, $n_h$, $n_m$, and $n_c$ are hot, medium, and cold ion densities where $n = n_h + n_m + n_c$, $\mathbf{v}_h$, $\mathbf{v}_m$, and, $\mathbf{v}_c$ are the bulk velocities of hot, medium, and cold ions.

Figures 4c and 4d, shows the agreement between the electric field from EDP and the electron convective term $-\mathbf{v}_e \times \mathbf{B}$ in both $E_N$ and $E_M$ components, which indicates that electrons remain frozen-in during the crossing of the separatrix. For ions, decoupling from the magnetic field occurs within the Hall electric field region just before the separatrix boundary crossing (red-dotted line). Indeed in this region, the ideal term $-\mathbf{v}_i \times \mathbf{B}$ is larger in magnitude than the observed electric field and compensated by the Hall term $\mathbf{J} \times \mathbf{B}$ leading to the decoupling between ion and electron bulk motions (e.g., B. U. Å. Sonnerup, 1979; Terasawa, 1983; Toledo-Redondo et al., 2015). Figures 4e and 4f, shows the decomposition of the ion convective term in different regions. In the cold ion region (blue shaded), the drifting cold ions (out-of-plane velocity $V_M < 0$ on the magnetosphere side) causing the normal convective term ($V_M B_L$) associated with the negative normal $E_N$ electric field. In the outflow region, the reconnection jet's motion and the guide field's presence result in the second term ($-V_L B_M$) dominated by $M_{sh}$ medium energy ions and associated with the positive normal $E_N$ electric field.

Again these results are consistent with those obtained by (Graham et al., 2017). By comparing their Figure 4, we can see that the peak of the Hall electric field is measured at the same location along $N$, on the magnetospheric side just before the boundary crossing (although in their case it is not canceled by the convective motion of ions). The plasma density is illustrated in (Figure 5a) for various energy ranges, the blue-shaded region indicates an abundance of the cold ions on the magnetospheric side of the separatrix $n_c/n \approx 0.82$. This population is observed to move in a perpendicular direction denoted as $\mathbf{V}_{\perp,i} \approx \mathbf{V}_{\perp,ic} = \mathbf{E} \times \mathbf{B}/B^2$, which explains the out-of-plane velocity in this region (Figures 5b and 5c) where $\mathbf{V}_{\perp,ic}$ is the perpendicular velocity of the cold ions population. Due to their high density (~10 cm⁻³), low energy cold ions contribute significantly to the generalized Ohm's law $\mathbf{E} \approx -n_c/n(\mathbf{v}_c \times \mathbf{B})$ in this region (Figures 5e and 5f).

At the ion edge, which is the boundary where the first $M_{sh}$ ions are observed in the $M_{sp}$ region, marked by a blue-dotted line separating the blue (magnetosphere) and yellow (mixing region) shaded regions, we observe anti-parallel $M_{sh}$ ions at the boundary as seen in the med-ion pitch angle spectrogram at 04:39:20 UT (Figure 5d). The cold ions between the ion edge and the separatrix are indistinguishable from the $M_{sh}$ ions as we see the density ratio of cold ions starts to decrease in the mixing region $n_c/n \approx 0.47$, accompanied by an increase of the med-ions density $n_m/n \approx 0.51$.

As mentioned in the introduction, the existence of a normal electric field directed earthward on the magnetosphere side far from the diffusion region and associated with cold ions was shown using a 2D particle-in-cell (PIC) simulation by Dargent et al. (2017). To facilitate the comparison, we first reproduce the same simulation (using the same setup) described in (Dargent et al., 2017). The simulations are performed using the fully kinetic, collaborative, open source, object-oriented ($C^{++}$), 2D PIC code SMILEI (Derouillat et al., 2018). The magnetic field and density are normalized to $\mathbf{B}_0$ and $n_0$. Masses and charges are normalized to the proton mass ($m_p$) and charge ($e$). Time and length are normalized to the inverse of the proton gyrofrequency ($\omega_i^{-1}$) and proton inertial
<figure><img src="image_6.png" /><figcaption>**Figure 4.** (a) Magnetic field components, (b) ion velocity, (c) the N component of electric field (E_N) from the electric double probe instrument and of the generalized Ohm's law terms: the convective terms for ion (electron) blue (red), the Hall term (yellow), and, the electron pressure gradient computed using four spacecraft technique (cyan), (d) the M component of the electric field (E_M) and the convective terms, (e) and (f) the decomposed ion convective terms for E_N and E_M components respectively.</figcaption></figure>

length ($\delta_i$). The domain size is (320, 128)$\delta_i$ with 6,400 cells in the x direction and 5,120 cells in the y direction, initializing 50 particles per population. The boundary conditions are periodic in both x and y directions. Furthermore, we used an asymmetric antiparallel magnetic field $B_{sh}/B_{sp} = 0.5$, no guide field, a density ratio $n_r = n_{sh}/n_{sp} = 3$, and a mass ratio of $m_i/me = 25$ (see Dargent et al., 2017, for more details). In this simulation, which includes cold ions, ions are split into a population of density $n_{sh}$ on the magnetosheath side, and two populations on the magnetosphere side $n_h$ (hot) and $n_c$ (cold), satisfying $n_{sh} = 3n_{sp} = 3(n_c + n_h)$. The magnetosheath ion density profile is fixed, while on the magnetospheric side, the hot and cold ion densities are determined by setting their asymptotic density and temperature ratios as $n_h/n_c = 0.5$ and $T_h/T_c = 500$. The temperature and density profiles are calculated to ensure pressure balance.

Figure 6a shows the two-dimensional color-coded map of the normal electric field obtained from the PIC simulation. Blue areas (respectively red areas) correspond to earthward (respectively sunward) normal electric fields. The black lines represent the magnetic field lines. Note that the range, in x ~ L = 120$\delta_i$ (between 100 and 220 $\delta_i$, where $\delta_i$ is the proton inertial length), is larger compared to typical PIC simulations, which usually cover around ~40$\delta_i$ (e.g., Shay et al., 2016). A cut is performed at x = 130$\delta_i$ to investigate the physics far from the
<figure><img src="image_7.png" /><figcaption>**Figure 5.** (a) Total (black), hot (red), medium (green), and, cold (blue) ion densities observed by fast plasma instrument, (b) Out of plane perpendicular (red), ExB drift (black), cold (cyan), medium (yellow), hot (green) ion velocity, (c) and (d) Pitch angle distributions for cold and med-ions respectively, (e) and (f) N and M components of the generalized Ohm's law including the partial moments: the electric double probe electric field (black), the convective term (red), cold ions convective term (cyan), med-ions convective term (yellow), hot ions convective term (green), the hall term in (e) (blue).</figcaption></figure>

diffusion region and to avoid mixing the cold ions effects with the finite Larmor radius effects near the x-line. The white dashed line in Figure 6a represents the cut at $x = 130\delta_i$, corresponding to the MMS trajectory crossing the separatrix, with the *N* direction from observation aligning with the *y* direction in the simulation. This simulation reaches a quasi-steady state, reconnecting a similar amount of magnetic flux with a comparable reconnection rate by $t = 120$ (t being normalized to the ion gyrofrequency). Figure 6b shows the different terms of the generalized Ohm's law. We show consistency between observations and simulation, particularly in the cold ion region (the blue-shaded area) where cold ions are observed drifting in the structure of the electric field $-(n_c/n) \mathbf{v}_e \times \mathbf{B}$ (cyan line in Figure 6b between 22.5 and 25 $\delta_i$) as in the in-situ measurements (cyan line in Figure 6c). In the blue-shaded area, observations show that the convective term for cold ions is approximately equal to the measured normal electric field, primarily due to the high density of cold ions and the significantly lower density of hot ions compared to cold ions. However, in the simulation, the density ratio of cold ions is two-thirds that of the total magnetospheric plasma density compared with 0.82 in the observations, thus making the contribution from hot ions notable where $-(n_h/n) \mathbf{v}_h \times \mathbf{B}$ (green line) in Figure 6b is contributing significantly to the total convective term (red line). The results obtained from observations and simulation differ when approaching the Hall electric field region. Indeed, at 04:39:24 UT, MMS observed the Hall peak $(\mathbf{J} \times \mathbf{B})/(en) \simeq 9$ mV/m, and the convective
<figure><img src="image_8.png" /><figcaption>**Figure 6.** (a) The white vertical dashed line at $x = 130\delta_i$ indicates the separatrix crossing where we compare the simulation to Magnetospheric Multiscale observations, (b) the generalized ohm's law components are compared to Figure 5e, and (c) same as Figure 5e. A Gaussian filter has been applied to the simulation data.</figcaption></figure>

term peak $-v_i \times B \approx -9$ mV/m opposing the Hall term. This compensation leads to $E_N$ inside the separatrix being nearly zero, primarily because the *M* component of the ion perpendicular velocity is dominated by the diamagnetic part, and the total *M* component of the perpendicular current density is dominated by the ion diamagnetic current (see Figure 3d). In the simulation, the contribution from the magnetosheath ions is also dominant and opposes the Hall term at the separatrix. However, the larger peak of the Hall term is due to the current at the separatrix being mostly carried by electrons (not shown) and not dominated by the ion diamagnetic current as in observations. Finally, the large discrepancy between observation and simulation in the outflow region is explained by the fact that the simulation was performed without a guide field, which is responsible for the sunward electric field in observations. Despite these limitations, to our knowledge, this case study is the first study confirming the role of magnetospheric cold ions in maintaining an earthward normal electric field far from
the diffusion regions. In the next section, energy conversion and partitioning at the vicinity of the magnetosphere separatrix are investigated.

# 5. Energy Conversion and Partitioning

## 5.1. Theoretical Background

We investigate energy conversion processes in the form of $J \cdot E$ associated with the separatrix crossing. The release and transport of electromagnetic energy is governed by Poynting's theorem (neglecting electric field energy as we consider transport and wave speeds much slower than the speed of light):

$$ \frac{\partial}{\partial t} \left[ \frac{\mathbf{B}^2}{2\mu_0} \right] + \nabla \cdot \left[ \frac{\mathbf{E} \times \mathbf{B}}{\mu_0} \right] = -\mathbf{J} \cdot \mathbf{E} \qquad (6) $$

where $\mathbf{B}^2/(2\mu_0)$ is the energy stored in the magnetic field, and $(\mathbf{E} \times \mathbf{B})/\mu_0$ is the Poynting vector. In Equation 6 when $\mathbf{J} \cdot \mathbf{E} > 0$, the energy is converted from the electromagnetic field to plasma (dissipation or loading measure) when $\mathbf{J} \cdot \mathbf{E} < 0$ the energy is transferred from the plasma to the electromagnetic field (dynamo or generator measure).

Using the electric field in the electron rest frame $\mathbf{E}' = \mathbf{E} + \mathbf{v}_e \times \mathbf{B}$, Equation 6 becomes:

$$ \frac{\partial}{\partial t} \left[ \frac{\mathbf{B}^2}{2\mu_0} \right] + \nabla \cdot \left[ \frac{\mathbf{E} \times \mathbf{B}}{\mu_0} \right] = -\mathbf{v}_i \cdot (\mathbf{J} \times \mathbf{B}) - \mathbf{J} \cdot \mathbf{E}' \qquad (7) $$

The first term on the right-hand side of Equation 7 stands for the work done by the Lorentz force to the ion or electron fluids. The second term is the dissipation measure utilized to identify the EDR in the non-relativistic case, as introduced by (Zenitani et al., 2011) with $\mathbf{E}' = \mathbf{E} + \mathbf{v}_e \times \mathbf{B}$ given by the generalized Ohm's law. It's worth mentioning that Birn and Hesse (2010) discussed the energy transfer, conversion, and dissipation for a symmetric magnetic reconnection with a guide field based on the one-fluid equations showing that $\mathbf{J} \cdot \mathbf{E}'$, where $\mathbf{E}' = \mathbf{E} + \mathbf{v} \times \mathbf{B}$ (plasma rest frame) directly converts electromagnetic energy into thermal energy, while in multi-fluid approach the link appears in a more complicated way. Indeed, in a multi-fluid framework, the electromagnetic work ($\mathbf{J} \cdot \mathbf{E}$) is not present in the equation of evolution of internal energy. Hence, there's no direct link between $\mathbf{J} \cdot \mathbf{E}$ and changes in internal energy or temperature enhancement.

Now, using the multi-fluid approach, we discuss the energy conversion terms and how energy is converted between fields and plasma and partitioned between electrons and ions. Following recent works (e.g., Chasapis et al., 2018; Matthaeus et al., 2020; Yang et al., 2017) used notably to quantify the energy channels in the EDR during magnetopause reconnection (Bandyopadhyay et al., 2021; Zhou et al., 2021), we analyze the flow and thermal energy equations, once all different transport terms are grouped on the same side of equations:

$$ \frac{\partial K_{\alpha}}{\partial t} + \nabla \cdot (K_{\alpha} \mathbf{v}_{\alpha}) + \nabla \cdot (\mathbf{P}_{\alpha} \cdot \mathbf{v}_{\alpha}) = (\mathbf{P}_{\alpha} \cdot \nabla) \cdot \mathbf{v}_{\alpha} + n_{\alpha} q_{\alpha} \mathbf{v}_{\alpha} \cdot \mathbf{E} \qquad (8) $$

$$ \frac{\partial U_{\alpha}}{\partial t} + \nabla \cdot (U_{\alpha} \mathbf{v}_{\alpha}) + \nabla \cdot \mathbf{h}_{\alpha} = -(\mathbf{P}_{\alpha} \cdot \nabla) \cdot \mathbf{v}_{\alpha} \qquad (9) $$

where $K_{\alpha} = \frac{1}{2} n_{\alpha} m_{\alpha} |\mathbf{v}_{\alpha}|^2$ is the fluid kinetic energy of species $\alpha$, $U_{\alpha} = 3p^{\alpha}/2 = P_{ii}^{\alpha}/2$ is the internal energy of species $\alpha$, with $P_{ii}^{\alpha}$ as the trace of the pressure tensor, $\mathbf{h}_{\alpha}$ is the heat flux, $\mathbf{P}_{\alpha}$ is the pressure tensor, and $\mathbf{v}_{\alpha}$ is the fluid velocity of the $\alpha$ species. In a closed and periodic system, such as often used in simulations, the divergence terms are often neglected by integrating the energy budget over the entire simulation domain. This simplification makes the divergence of the fluxes vanish (Genestreti et al., 2018; J. Eastwood et al., 2020).

In in-situ observations, since measurements are taken point by point in space and time, the divergence of fluxes must be considered in principle, although for four-satellite missions evolving in a tetrahedral configuration such as Cluster or MMS, the same kind of assumptions can be made at the scale of the tetrahedron (as a closed surface). Furthermore, following Bandyopadhyay et al. (2021) who focused on energy conversion processes more than
absolute energy balance, we only discuss the right-hand side terms of Equations 8 and 9 and do not consider divergence terms which transport energy from one location to another. Energy conversion related to the flow energy equation is produced by the right-hand side terms of Equation 8, that is, the pressure strain and the work of the electric field. Hence, the **J**·**E** term in Equation 6 appears directly in the change of the total plasma flow energy after summing over the ion and electron species.

In the internal energy Equation 9, energy conversion takes place only through the pressure strain $-(\mathbf{P}_\alpha \cdot \nabla) \cdot \mathbf{v}_\alpha$. Then, the pressure strain can be decomposed into two components: compressive term $(-p_\alpha \nabla \cdot \mathbf{v}_\alpha)$ and incompressive term $(-\Pi_\alpha \cdot D_\alpha)$, where $p_\alpha = \mathbf{P}_{\alpha,ii}/3$, $\Pi_{\alpha,ij} = \mathbf{P}_{\alpha,ij} - p_\alpha \delta_{ij}$, and $\mathbf{D}_{\alpha,ij} = (\partial_i v_{\alpha,j} + \partial_j v_{\alpha,i})/2$.

For analyzing the energy conversion and partition, it can be also interesting to adopt a different formulation, which only groups together time variation and transport terms of flow and thermal energies and write:

$$ \frac{\partial K_{\alpha}}{\partial t} + \nabla \cdot (K_{\alpha} \mathbf{v}_{\alpha}) = -\mathbf{v}_{\alpha} \cdot (\nabla \cdot \mathbf{P}_{\alpha}) + n_{\alpha} q_{\alpha} \mathbf{v}_{\alpha} \cdot \mathbf{E} \quad (10) $$

$$ \frac{\partial U_{\alpha}}{\partial t} + \nabla \cdot (U_{\alpha} \mathbf{v}_{\alpha}) = -\nabla \cdot \mathbf{h}_{\alpha} - (\mathbf{P}_{\alpha} \cdot \nabla) \cdot \mathbf{v}_{\alpha} \quad (11) $$

In Equation 10, the local kinetic/flow energy variation of species $\alpha$ takes place through two terms on the right-hand side defined as $D(K_\alpha) = -\mathbf{v}_\alpha \cdot (\nabla \cdot \mathbf{P}_\alpha) + n_\alpha q_\alpha \mathbf{v}_\alpha \cdot \mathbf{E}$. The first term is related to the work of the pressure term $(-\mathbf{v}_\alpha \cdot (\nabla \cdot \mathbf{P}_\alpha))$. The second term $\mathbf{J}_\alpha \cdot \mathbf{E}$ corresponds to the work of the electric force applied on each fluid. When $D(K_\alpha)$ is positive (negative), the species is accelerated (decelerated).

In the same way, heating (cooling) could be examined from Equation 11, but due to large errors in the divergence of the heating flux for both ions and electrons, it is difficult to conclude the respective role of the heat flux and the pressure strain. That is why the first formulation has been preferred when analyzing the MMS measurements.

In the next section, we analyze energy conversion both in the spacecraft frame and fluid-frame and we use this theoretical framework in these two different formulations to analyze the energy conversion and partitioning occurring far from the diffusion regions and across the magnetosphere separatrix.

## 5.2. Observations

### 5.2.1. Energy Conversion in the Spacecraft Frame

Figures 7a–7d show for reference, magnetic field components and magnitude, ion and electron densities, electric field, and current density in the LMN frame. The primary contribution to the total energy conversion, represented by $\mathbf{J} \cdot \mathbf{E} = J_N E_N + J_M E_M + J_L E_L$ shown in Figure 7e, comes from the out-of-plane $J_M E_M$ component, which is negative. The L direction contributes to the total $\mathbf{J} \cdot \mathbf{E}$ while the N component contribution is the smallest despite the large normal electric field, as the normal current is weak in this region (Figure 7d). Note that during the crossing, the out-of-plane current that is dominated by the ion diamagnetic current was directed duskward $J_M < 0$ (Figure 7d), and the out-of-plane convective electric field $E_M$ component is positively oriented (dawnward) within the $M_{sp}$ (Figure 7c). The positive $E_M$ that appears in the $M_{sp}$ is due to the earthward motion of the magnetopause ($V_N = -47$ km/s from timing analysis) and plasma as shown from the contribution to the ideal term component $E_M \sim -V_{N,i} B_L$ (Figure 4f) $V_{N,i}$ being slightly negative in the $M_{sp}$. In the mixing region, the main contribution to $E_M$ comes from the $M_{sh}$ ions $(-n_m/n) \mathbf{v}_m \times \mathbf{B}$ that also move earthward. Figure 7f shows consistency in $\mathbf{J} \cdot \mathbf{E} \approx \mathbf{J} \cdot (-\mathbf{v}_i \times \mathbf{B}) \approx -\mathbf{J}_{i,m} \cdot (n_m/n) \mathbf{v}_m \times \mathbf{B}$, where $\mathbf{J}_{i,m}$ is the diamagnetic current density carried by medium energy (magnetosheath) ions. In summary, in the spacecraft frame, the energy is transferred to the electromagnetic field from the plasma mainly due to the work of the Lorentz force during the earthward motion of the magnetosheath ions. An opposite behavior ($\mathbf{J} \cdot \mathbf{E} > 0$) is expected in the case of an expanding magnetopause moving sunward as the convective electric field is reversed while the perpendicular current is unchanged.

### 5.2.2. Energy Conversion in the Fluid Frame

We also investigate $\mathbf{J} \cdot \mathbf{E}'$ associated with the separatrix crossing. Figure 7g displays the magnetic field-aligned current density obtained from the particle measurements (black line) and from the curlometer (red line). Figure 7h
<figure><img src="image_9.png" /><figcaption>**Figure 7.** (a) Magnetic field components, (b) ion density (red) and electron density (blue), (c) electric double probe Electric field, (d) Current density from particle moments, (e) **J**·**E** components, (f) Total **J**·**E**, (g) Total parallel current density, (h) Parallel component of the electric field of **E** (red) and its corresponding error bars, parallel component of the divergence of the electron pressure tensor (green) and its corresponding error bars, and (i) **J**·**E**' (Black), parallel component of **J**·**E** (red), parallel component of **J**·[**E**̇] (Green). The magenta horizontal lines on top of the figure and the sub-figure mark the separatrix region.</figcaption></figure>
shows the parallel electric field (E_||) provided by the EDP team as separate L2 data (red line) and its corresponding error bars. Error bars are provided in the cdf files for each spacecraft then the four spacecraft average error is calculated as follows: E<sub>err,avg</sub> = 1/4√E<sub>err,1</sub><sup>2</sup> + E<sub>err,2</sub><sup>2</sup> + E<sub>err,3</sub><sup>2</sup> + E<sub>err,4</sub><sup>2</sup>. The parallel electric field related to the parallel component of the divergence of the electron pressure tensor -(∇·P_e)_||/(ne) and its corresponding error bars are also plotted (green line) in Figure 7h. These error bars are calculated as follows (Paschmann & Daly, 1998):

$$ \delta \left[ \frac{\nabla \cdot \mathbf{P}_e}{en_e} \right] = \left| \frac{\nabla \cdot \mathbf{P}_e}{en_e} \right| \sqrt{\left( \frac{\delta[\nabla \cdot \mathbf{P}_e]}{\nabla \cdot \mathbf{P}_e} \right)^2 + \left( \frac{\delta[n_e]}{n_e} \right)^2} \quad (12) $$

where to simplify, δ[∇·P_e] ≈ ΔP/Δz√((2δP/ΔP)<sup>2</sup> + (δz/Δz)<sup>2</sup>), with P is the four spacecraft average of the isotropic pressure, and Δz is the distance between two spacecraft separated along z as it is mostly the direction of the background magnetic field. Error bars δP corresponds to the statistical error (counting statistics) for the isotropic pressure obtained from error bars provided with L2 data by the FPI team (Gershman et al., 2015; Pollock et al., 2016). Error bars δz of the spacecraft positions is assumed to be ~100 m (Roberts et al., 2023). The quality factor of the tetrahedron being high (0.85), we neglect this source of error. As a result, one can note that error bars for the electric field obtained from particle measurements (green line) are much smaller than those obtained for the electric antenna measurements. The two terms agree quite well at the separatrix boundary (vertical red-dotted line) although discrepancies can be noted farther from the separatrix. Energy conversion terms J·E' (black line), J_||·E_|| (red line) and J_⊥·(∇·P_e/(en))_|| (green line) are displayed Figure 7i. At the separatrix, all terms reverse. A better agreement is found between the three quantities for the positive peak than for the smaller negative peak. It is important to note that this energy conversion reversal is due to the reversal of the parallel electric field related to the parallel component of the divergence of the electron pressure tensor. On the magnetospheric side, the parallel electric field (respectively the divergence of the pressure tensor) is directed away from (respectively toward) the diffusion regions, accelerating electrons toward the diffusion regions. On the outflow region side, the parallel electric field is directed toward the diffusion region, accelerating electrons in the opposite direction. Thus, even if the four spacecraft average of the parallel electric field amplitude from EDP is of the order of the error bar (±0.8 mV/m), the fact that measurements from the different instruments (electric field antennas and electron detectors) provide consistent signatures (error bars of the electric field related to the divergence of the pressure tensor are small, yet they are included within the larger error bars of the electric field antenna) suggests that the observed bipolar signature of the parallel electric field and the corresponding energy conversion reversal (J_||·E_||) at the magnetospheric separatrix are real. However, such a consistency is not verified deeper on the magnetosphere side therefore this result has to be confirmed in a further study by carrying out a statistical survey and comparing it with results of kinetic simulations carried out with relevant initial conditions.

### 5.2.3. Energy Partitioning

In this section, we begin by analyzing the flow energy Equation 8. While Figure 8a provides magnetic field signatures for reference, Figure 8b highlights the energy conversion rate related to the flow energy equation, which is caused by the total pressure strain and the work done by the electric field summed over species. Figures 8c and 8d show this rate separately for each species. Figure 8b indicates that the energy conversion is dominated by the pressure strain term oscillating on the magnetospheric side with a large positive peak at the separatrix crossing suggesting an increase in the flow energy. In the outflow region, it fluctuates more rapidly. Analyzing ions and electrons separately, Figure 8c shows that ion flow energy is reduced on the magnetospheric side by pressure strain and electric work. However, at the separatrix pressure strain effects dominate whereas just after the crossing electric work takes over. Depending on the effects of the transport terms, these results suggest that ion flow energy could reverse at the separatrix crossing, being first increased by the pressure strain and then reduced mostly due to the electric work. For electrons, Figure 8d shows that the pressure strain term always dominates the energy conversion and that electron contribution drives the total energy conversion rate.

Then we analyze the effects of the pressure strain interactions in the thermal energy Equation 9. While Figure 9a provides magnetic field signatures for reference again, Figure 9b highlights the energy conversion rate to random
<figure><img src="image_10.png" /><figcaption>**Figure 8.** (a) Magnetic field components, (b) The total plasma pressure strain interaction (red) and the work done by the electric field (green), summed (black), (c) The ion pressure strain interaction (red) and the work done by the electric field on ions (green), summed (black), and (d) The electron pressure strain interaction (red) and the work done by the electric field on electrons (green), summed (black). The magenta horizontal line on top of the figure marks the separatrix region.</figcaption></figure>

energy measured by the total pressure strain. In the magnetosphere region, total pressure strain shows enhancement and fluctuations oscillating between positive and negative values, with an overall negative trend, particularly at the separatrix (≈ −8 nW/m³). Figure 9c illustrates the total pressure-strain rate for both ions and electrons. As we have mentioned, electrons are the main contributors, with a rate of −7.5 nW/m³ whereas ions also contribute significantly, with a rate of −2 nW/m³ at the vicinity of the separatrix. As shown in Figure 9d, the compressive term is the dominant component for both electrons and ions, whereas the incompressible term is small at the separatrix. These results suggest that the thermal energy is reduced at the separatrix crossing mostly due to the compressible part of the electron pressure strain. This partition between electrons and ions and the predominance of the compressive effects, as well as their magnitudes, are consistent with results found by (Bandyopadhyay et al., 2021; Zhou et al., 2021) regarding thermal energy variations in the EDR of a large-scale reconnection event at the magnetopause (Burch et al., 2020), small-scale reconnection event with a moderate guide field at a magnetosheath current sheet (Wilder et al., 2018).

Finally, the dominant role of the compressible component of the electron pressure strain in flow and thermal energy conversions suggests that thermal energy could be directly transferred into bulk flow energy at the magnetospheric separatrix far from the diffusion regions.
<figure><img src="image_11.png" /><figcaption>**Figure 9.** (a) Magnetic field components, (b) The total pressure strain, (c) The total pressure strain for electrons (black) and ions (red), (d) Compressive pressure strain interaction, and (e) Incompressive pressure strain interaction. The magenta horizontal line on top of the figure marks the separatrix region.</figcaption></figure>

Additionally, we investigate the energy conversion into or out of flow energy in a different formulation given by Equation 10. Figure 10 shows the plasma flow energy variation measure on the right-hand side of Equation 10. In Figure 10a, the ion flow energy variation $D(K_i)$ exhibits a bipolar signature at the separatrix. This is characterized by a positive peak due to the work of the pressure term ($-v_i \cdot \nabla \cdot \mathbf{P}_i$) and a negative peak due to the work of the electric force $\mathbf{J}_i \cdot \mathbf{E}$ at the vicinity of the separatrix on the exhaust region side. Just after, ions are accelerated again by the pressure effects. In Figure 10b, the decomposition of the deceleration term due to the electric field ($\mathbf{J}_i \cdot \mathbf{E} = v_i \cdot (\mathbf{J} \times \mathbf{B}) + \mathbf{J}_i \cdot \mathbf{E}'$) at the separatrix is shown, $\mathbf{J}_i \cdot \mathbf{E}'$ is positive which indicates that the ions are accelerated, while $v_i \cdot (\mathbf{J} \times \mathbf{B})$ is negative, indicating that the ions are decelerated. This decomposition shows that the ion deceleration occurring after the separatrix seen in Figure 10a is not produced by the work of the electric field ($\mathbf{J}_i \cdot \mathbf{E}'$) but by the work of the Lorentz force $v_i \cdot (\mathbf{J} \times \mathbf{B})$. Figure 10c shows the electron flow energy variation $D(K_e)$ with slightly positive values on the magnetospheric side and larger negative values on the outflow side of the magnetospheric separatrix which suggest that electrons are decelerated in the outflow region far from the diffusion regions whereas ions are still accelerated. Pressure effects are insufficient to overcome the negative work of the electric force applied to electrons. Finally, Figure 10d displays the total plasma acceleration measure $\sum_{\alpha=i,e} D(K_\alpha)$, which consists of a bipolar signature (ranging +2 nW.m⁻³ to -4 nW.m⁻³) at the separatrix
<figure><img src="image_12.png" /><figcaption>**Figure 10.** (a) Ions flow energy acceleration measures, (b) Ion acceleration measures due to the electric field, (c) Electrons flow energy acceleration measures, and (d) Total flow energy acceleration measures. The magenta horizontal line on top of the figure marks the separatrix region.</figcaption></figure>

crossing. The positive peak is due to the work of the pressure force $-\sum_{\alpha=i,e} (v_{\alpha} \cdot \nabla \cdot \mathbf{P}_{\alpha})$ and suggests a plasma acceleration on the magnetospheric side. Then it is followed by a negative peak due to $\mathbf{J} \cdot \mathbf{E}$ on the outflow region, side indicating a plasma deceleration due to the electric field. Later, the plasma (ions and electrons) flow energy no longer seems to be strongly changed, with ion acceleration effects compensated by electron deceleration effects. Thus, the same bipolar trend is observed for the bulk flow energy variation from Equations 8 and 10 but not with the same magnitude, suggesting that the transport term $(\nabla \cdot (\mathbf{P}_{\alpha} \cdot \mathbf{v}_{\alpha}))$ neglected in Equation 8 can play a significant role. As for the energy conversion results, these results of energy partitioning require further investigation based on statistical studies using more observations and comparison with kinetic simulations. However, estimates of small error calculations of the pressure strain at the magnetopause, where high plasma densities ensure high counting statistics, were obtained by Roberts et al. (2023), which makes these case study results reliable.

**6. Summary and Conclusions**

In the present case study, we have investigated a magnetic reconnection event detected by MMS far from the diffusion regions at the dayside magnetopause on the 21st of October 2015. This event occurred with highly asymmetric conditions due to the arrival of a compressed solar wind. Furthermore, a moderate guide field ($B_{gt}$)
and magnetospheric cold ions were present. The guide field $B_{gf}$ was identified as a constant background out-of-plane magnetic field measured both in $M_{sp}$ and in $M_{sh}$ regions far from the Hall effect regions, and estimated 0.47 times the reconnecting magnetic field component on the magnetosphere side.

Our findings demonstrate that in such conditions, the movement of magnetospheric cold ions and the existence of the guide field exert notable influences on the electric field normal to the magnetopause. Based on a detailed generalized Ohms's law analysis, we showed that negative values (directed earthward) of the normal component of the electric field ($E_N$) on the magnetospheric side are due to the drift motion of the cold ions in the out-of-plane direction. This behavior is consistent with Dargent et al. (2017) findings, which showed that the initial Larmor electric field present along the magnetopause before the starting of reconnection can be maintained during the ongoing reconnection process even far from the diffusion regions (where the boundary layer is thicker than the $M_{sh}$ ion Larmor radius) in the presence of cold ions. To our knowledge, these in-situ measurements are the first that confirm this behavior highlighted by the kinetic simulations. The positive values of $E_N$ (directed sunward) in the outflow region on the $M_{sh}$ side are due to the existence of a moderate guide field ($-V_L B_M$).

Furthermore, we investigated the current density signatures and found that the perpendicular current's source ($-J_M$) at the magnetospheric separatrix is primarily the diamagnetic current, with the ion diamagnetic current playing a dominant role in agreement with previous studies on the magnetopause current layer (e.g., Dong et al., 2018; Beedle et al., 2022). The Hall electric field ($\propto J \times B$) term was mainly supported by this large diamagnetic current generated from the density asymmetry in the separatrix region. During the magnetospheric separatrix crossing, a strong anti-parallel current is measured ($\sim -1000$ nA/m²), carried by both ions and electrons on the magnetosphere side and mostly by electrons on the magnetosheath side. Thus, the separatrix boundary, defined as the thin layer where electron velocity changes sign, is shifted from the region where the parallel current changes sign due to this magnetosheath ion contribution to the parallel current. The expected reversal of the parallel current at the separatrix is observed, with the corresponding Hall magnetic field signature ($\delta B_M > 0$). Interestingly, an additional out-of-plane magnetic signature is measured between the ion edge and the classical Hall signature at the separatrix producing a local magnetic shear of the out-of-plane magnetic field. This signature could be generated by the accelerated magnetosheath electrons coming from the reconnection region observed at the ion edge. We interpret this arrival of magnetosheath electrons due to the shifting of the stagnation point inside the magnetosphere occurring for asymmetric reconnections. Such a signature could be also related to a secondary reconnection region forming in the out-of-plane magnetic field shear as suggested by (Fu et al., 2024; Zhao et al., 2021). This additional current system needs more investigation.

Then the energy conversion processes occurring in the vicinity of the magnetospheric separatrix were scrutinized. In the spacecraft frame, it was found that the energy conversion corresponds to a transfer of energy from the plasma to the electromagnetic fields ($J \cdot E < 0$) driven by the convective electric field that is, mainly by the product of perpendicular current and electric field. Indeed, the energy conversion term is dominated by $J_M E_M$ at the magnetospheric separatrix due to the earthward motion of the compressed magnetopause ($E_M \approx V_N B_L$) and the strong out-of-plane diamagnetic current density mostly carried by $M_{sh}$ ions. One can note that the contribution of cold ions to this process diminishes as they begin to deplete within the ion edge.

We further investigated the energy conversion in the fluid frame ($J \cdot E'$). It is mainly due to the parallel electric field associated with the divergence of the electron pressure tensor and the large anti-parallel current flowing at the separatrix. Moreover, we showed that the energy conversion reverses at the separatrix, being positive on the magnetosphere side (energy transferred from the fields to the plasma) and negative on the outflow region side. It is worth pointing out that this reversal is due to the reversal of the parallel electric field related to the divergence of the electron pressure tensor. On the magnetospheric side, the parallel electric field (respectively the divergence of the pressure tensor) is directed away from (respectively toward) the diffusion regions, electrons being accelerated toward the diffusion regions. The opposite is found on the other side, electrons being accelerated away from the diffusion regions. Since the values of the parallel electric field are of the order of the error bar ($\pm 0.8$ mV/m), these results have to be confirmed by further statistical investigation and comparisons with kinetic simulations. However, the good agreement at the separatrix between the parallel electric field measured by the electric field antennas and that estimated from the divergence of the electron pressure tensor measured by the electron detectors strongly suggests that these signatures are real.
Finally, we analyzed the change of flow and thermal energies through the pressure strain calculations. We found a dominant role of the electron pressure strain term in both flow and thermal energy variations. Flow energy would increase whereas thermal energy would decrease on the magnetospheric side just before the separatrix crossing suggesting a direct energy transfer from thermal into the flow. The compressive term of the pressure strain is the dominant component for both electrons and ions, whereas the incompressible term is small at the separatrix. These results found for the thermal energy variations are comparable with events investigated near the EDR (Bandyopadhyay et al., 2021; Hasegawa et al., 2023) suggesting that the same amount of energy conversion occurs near the separatrix.

Additionally, we also checked the change in the flow energy using a different formulation, which considers the work of the pressure gradient instead of the pressure strain. We obtained that the energy flow oscillates showing an increase on the magnetospheric side due to the work of the pressure gradient and a sharper decrease in the outflow region due to the work of the electric force. The same bipolar trend but different magnitudes of the flow energy variation obtained from the two different formulations indicate that the transport term, $\nabla \cdot (P_\alpha \cdot v_\alpha)$, neglected in the first formulation, could play a significant role.

These results obtained from a case study will be verified based on a statistical study of reconnection events detected by MMS in similar conditions (far from diffusion regions, compressed vs. expanding magnetopause, presence of cold ions, and guide field). To further our understanding of the partitioning of the energy, a kinetic simulation using initial conditions as close as possible to these in-situ space measurements (asymmetry conditions in magnetic field and density, moderate guide field, higher cold ion density, etc) will be carried out to extend the present results and those obtained by Dargent et al. (2017).

## Data Availability Statement

We thank the entire MMS team and instrument PIs for data access and the PYSPEDAS software team. Magnetospheric Multiscale data are available with free access at (https://cdaweb.gsfc.nasa.gov/). We use version 3.3 of the FPI level 2 moments, version 4.18 of the FGM data, version 2.2 of the EDP data, and version 2.0 of MEC data. The FPI electron data, for MMS 1,2,3, and 4 are available through (Gershman et al., 2022a, 2022c, 2022e, 2022g). The FPI ion data can be accessed via (Gershman et al., 2022b, 2022d, 2022f, 2022h). The FGM data can be accessed via (C. T. Russell et al., 2022a; C. T. Russell et al., 2022b, 2022c, 2022d). The EDP data can be accessed via (Ergun et al., 2022a, 2022b, 2022c, 2022d). The MEC data can be accessed via (Henderson et al., 2022a, 2022b, 2022c, 2022d). The PYSPEDAS software is available at (https://github.com/spedas/pyspedas). The simulation was provided with computing HPC and storage resources by GENCI at CINES thanks to the Grant 2023-A0150510062 on the supercomputer Adastra's MI250x partition.

## Acknowledgments

We are grateful for the entire MMS team and instrument PIs for data access. STR acknowledges the support of MCIN/AEI/ PRTR 10.13039/501100011033 (Grant TED2021-129357A-100). The simulation was provided thanks to the Grant 2023-A0150510062 on the supercomputer Adastra's MI250x partition. M. Baraka acknowledges the support provided by the Campus France grant. This work has benefited from a grant managed by the Agence Nationale de la Recherche (ANR, as part of the program "Investissements d'Avenir" under the reference (ANR-18-EURE-0014).

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