Non-Maxwellianity of electron distributions near Earth's magnetopause
D. B. Graham, Yu. V. Khotyaintsev, M. André, A. Vaivads, A. Chasapis, W. H. Matthaeus, A. Retino, F. Valentini, D. J. Gershman
CConfidential manuscript submitted to
JGR-Space Physics
Non-Maxwellianity of electron distributions near Earth’smagnetopause
D. B. Graham , Yu. V. Khotyaintsev , M. André , A. Vaivads , A. Chasapis , W. H.Matthaeus , A. Retino , F. Valentini , D. J. Gershman , Swedish Institute of Space Physics, Uppsala, Sweden. Space and Plasma Physics, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology,Stockholm, Sweden. Laboratory of Atmospheric and Space Physics, University of Colorado, Boulder, CO, USA. Department of Physics and Astronomy, University of Delaware, Newark, DE, USA. Laboratoire de Physique des Plasmas, CNRS/Ecole Polytechnique/Sorbonne Université/ Université Paris-Sud/Observatoirede Paris, Paris, France. Dipartimento di Fisica, Università della Calabria, Arcavacata di Rende, Italy. NASA Goddard Space Flight Center, Greenbelt, MD, USA. Department of Astronomy, University of Maryland, College Park, MD, USA.
Key Points: •
1: Electron non-Maxwellianity is computed for 6 months of data ( ∼
85 million elec-tron distributions). •
2: Electron non-Maxwellianity is typically large in the magnetosphere due to hot andcold electron populations. •
3: Enhanced non-Maxwellianity is found in reconnection regions, the bowshock, andmagnetosheath turbulence.
Corresponding author: D. B. Graham, [email protected] –1– a r X i v : . [ phy s i c s . p l a s m - ph ] F e b onfidential manuscript submitted to JGR-Space Physics
Abstract
Plasmas in Earth’s outer magnetosphere, magnetosheath, and solar wind are essentially col-lisionless. This means particle distributions are not typically in thermodynamic equilibriumand deviate significantly from Maxwellian distributions. The deviations of these distribu-tions can be further enhanced by plasma processes, such as shocks, turbulence, and magneticreconnection. Such distributions can be unstable to a wide variety of kinetic plasma insta-bilities, which in turn modify the electron distributions. In this paper the deviations of theobserved electron distributions from a bi-Maxwellian distribution function is calculated andquantified using data from the Magnetospheric Multiscale (MMS) spacecraft. A statisticalstudy from tens of millions of electron distributions shows that the primary source of theobserved non-Maxwellianity are electron distributions consisting of distinct hot and coldcomponents in Earth’s low-density magnetosphere. This results in large non-Maxwellianitiesin at low densities. However, after performing a stastical study we find regions where largenon-Maxwellianities are observed for a given density. Highly non-Maxwellian distributionsare routinely found are Earth’s bowshock, in Earth’s outer magnetosphere, and in the electrondiffusion regions of magnetic reconnection. Enhanced non-Maxwellianities are observed inthe turbulent magnetosheath, but are intermittent and are not correlated with local processes.The causes of enhanced non-Maxwellianities are investigated.
Many space and astrophysical plasmas are essentially collisionless so Coulomb col-lisions are unlikely to be efficient in keeping particle distributions close to thermal equalib-rium, i.e., a Maxwellian distribution. As a result non-Maxwellian distributions can readilydevelop and are indeed frequently observed in space plasmas. In collisionless plasmas non-Maxwellian distributions can remain kinetically stable, which need not relax to Maxwelliandistributions. However, non-Maxwellian distributions can be important source of instabili-ties and can potentially generate a variety of electrostatic and electromagnetic waves. Plasmaprocesses such as shocks, magnetic reconnection, and turbulence can further increase thedeformations in the particle distributions from a Maxwellian distribution. Quantifying thedeviation in particle distributions is crucial to understanding the effects of both large-scaleprocesses, such as shocks and magnetic reconnection, and kinetic-scale processes, such aswave-particle interactions.At present various papers have considered the non-Maxwellianity of particle distribu-tions in both simulations and observations [
Greco et al. , 2012;
Valentini et al. , 2016;
Chas-apis et al. , 2018;
Perri et al. , 2020;
Liang et al. , 2020]. These studies have focused on plasmaturbulence in Earth’s magnetosheath and magnetic reconnection. The simulation resultsshowed that ion non-Maxwellianity was spatially non-uniform and was associated with strongcurrents and temperature anisotropy [
Greco et al. , 2012;
Valentini et al. , 2016]. Similarly,electron non-Maxwellianity was found to increase the electron diffusion region (EDR) andseparatrices of magnetic reconnection [
Liang et al. , 2020]. In kinetic simulations the back-ground distributions, such as in modeling of magnetosheath turbulence and magnetic re-connection, are assumed to be Maxwellian. Such distributions are not necessarily valid inEarth’s magnetosheath and magnetopause, where the background distributions can differsignificantly from a Maxwellian distribution while remaining kinetically stable. Recentobservations from the Magnetospheric Multiscale (MMS) spacecraft suggest that ion non-Maxwellianity is weakly correlated with the local current sheets in the turbulent magne-tosheath [
Perri et al. , 2020]. In contrast,
Chasapis et al. [2018] found that electron non-Maxwellianity tended to increase in regions of strong currents. Estimates of the non-Maxwellianityof particle distributions based on recent MMS observations have only focused on very shorttime intervals. Thus, it is unclear if such deviations from Maxwellianity are statistically sig-nificant when compared with a large volume of data. Therefore, a statistical study of the non-Maxwellianity is required. –2–onfidential manuscript submitted to
JGR-Space Physics
The purpose of this paper is to investigate and quantify the non-Maxwellianity of elec-tron distribution functions in the near Earth plasma environment, specifically near Earth’smagnetopause, in the magnetosheath, and near the bowshock. In this paper we propose ameasure of the deviation of the observed electron distribution function from a bi-Maxwelliandistribution function, where temperature anisotropy is included. We show that statisticallythe non-Maxwellianity of electron distributions increases as plasma density decreases. Largedeviations of observed electron distributions from a bi-Maxwellian distribution function areobserved in the outer magnetosphere, in magnetic reconnection electron diffusion regions,at the bowshock, and intermittently in magnetosheath turbulence. The outline of this paperis as follows: Section 2 the data used is stated, section 3 states the theory and methods usedto calculate electron non-Maxwellianity, section 4 presents the statistical results, section 5presents case studies of the reconnection ion and electron diffusion regions, the bowshock,and magnetosheath turbulence, and section 6 states the conclusions.
In this paper we use high-resolution burst mode data from the four MMS spacecraft[
Burch et al. , 2016]. We use particle distributions and moments from Fast Plasma Investiga-tion (FPI) [
Pollock et al. , 2016]. Three-dimensional electron distributions and moments aresampled every 30 ms. The electron distributions are sampled over 32 energy channels rang-ing from 10 eV to 30 keV, which covers the thermal electron energy range in the outer mag-netosphere and magnetosheath. Ion distributions and moments are sampled every 150 ms.We use electric field E and magnetic field B data from the Electric field Double Probes (EDP)[ Lindqvist et al. , 2016;
Ergun et al. , 2016] and Fluxgate Magnetometer (FGM) [
Russellet al. , 2016], respectively. The spacecraft potential 𝑉 𝑠𝑐 is computed from the Spin-planeDouble Probes (SDP). In this section we define the non-Maxwellianity parameter 𝜖 . The non-Maxwellianity 𝜖 is defined as the velocity space integrated absolute difference between the observed distri-bution function and a model bi-Maxwellian distribution function given by the particle mo-ments: 𝜖 = 𝑛 𝑒 ∫ 𝑣,𝜃,𝜙 | 𝑓 𝑒 ( 𝑣, 𝜃, 𝜙 ) − 𝑓 model ( 𝑣, 𝜃, 𝜙 )| 𝑣 sin 𝜃𝑑𝑣𝑑𝜃𝑑𝜙, (1)where 𝑣 is the electron speed, 𝜃 is the polar angle, 𝜙 is the azimuthal angle, 𝑛 𝑒 is the elec-tron number density, 𝑓 𝑒 ( 𝑣, 𝜃, 𝜙 ) is the observed velocity-space density, and 𝑓 model ( 𝑣, 𝜃, 𝜙 ) isthe velocity space density of the model distribution. The factor 1 /( 𝑛 𝑒 ) normalizes 𝜖 to a di-mensionless quantity with domain 𝜖 ∈ [ , ] , where 𝜖 = 𝑓 𝑒 ( 𝑣, 𝜃, 𝜙 ) from 𝑓 model ( 𝑣, 𝜃, 𝜙 ) and 𝜖 = 𝑓 𝑒 ( 𝑣, 𝜃, 𝜙 ) and 𝑓 model ( 𝑣, 𝜃, 𝜙 ) in velocity space. For 𝑓 model we use a drifting bi-Maxwellian distribution, given by: 𝑓 model ( v ) = 𝑛 𝑒 𝜋 / 𝑣 𝑒, (cid:107) 𝑇 𝑒, (cid:107) 𝑇 𝑒, ⊥ exp (cid:32) − ( 𝑣 (cid:107) − 𝑉 (cid:107) ) 𝑣 𝑒, (cid:107) − ( 𝑣 ⊥ , − 𝑉 ⊥ ) + 𝑣 ⊥ , 𝑣 𝑒, (cid:107) ( 𝑇 𝑒, ⊥ / 𝑇 𝑒, (cid:107) ) (cid:33) , (2)where 𝑇 𝑒, (cid:107) and 𝑇 𝑒, ⊥ are the parallel and perpendicular electron temperatures, 𝑣 𝑒, (cid:107) = √︁ 𝑘 𝐵 𝑇 𝑒, (cid:107) / 𝑚 𝑒 is the parallel electron thermal speed, 𝑉 (cid:107) is the bulk velocity parallel to B , 𝑉 ⊥ is the magni-tude of the bulk velocity perpendicular to B , and 𝑘 𝐵 is the Boltzmann constant. This calcula-tion of 𝜖 corresponds to the lowest order moment, i.e., number density, so finite 𝜖 will resultfrom deviations from a bi-Maxwellian at thermal electron energies. The velocity coordinates ( 𝑣 (cid:107) , 𝑣 ⊥ , , 𝑣 ⊥ , ) used in equation (2) are defined such that 𝑣 (cid:107) is the speed along the magneticfield direction, 𝑣 ⊥ , is the speed along the perpendicular bulk velocity direction, and 𝑣 ⊥ , isorthogonal to 𝑣 (cid:107) and 𝑣 ⊥ , . The parameters 𝑛 𝑒 , V 𝑒 , and T 𝑒 , used to calculate 𝑓 model are theFPI-DES electron moments [ Pollock et al. , 2016]. To compute 𝜖 we transform 𝑓 model into the –3–onfidential manuscript submitted to JGR-Space Physics same coordinate system and discretize to the same energy and angle channels as the observed 𝑓 𝑒 for direct comparison. We note that this definition of 𝜖 differs from the definition used inprevious studies [e.g., Greco et al. , 2012]. The definition of
Greco et al. [2012] is closelyrelated to the enstrophy of the particle distribution [
Servidio et al. , 2017]. We have used thedefinition in equation (1) because: (1) The definition used in
Greco et al. [2012] is not di-mensionless, which is problematic when there are large changes in density, such as acrossthe magnetopause. (2) We have used a drifting bi-Maxwellian distribution as 𝑓 model , ratherthan an isotropic Maxwellian distribution, so that increases in 𝜖 do not simply correspondto large bulk electron flows in the spacecraft frame, or large temperature anisotropies, whichare straightforward to obtain from the particle moments. In equation (2) we have assumed asingle perpendicular electron temperature 𝑇 𝑒, ⊥ meaning 𝑓 model is gyrotropic, so agyrotropicfeatures of the observed electron distribution will not be captured by the model distribution.Thus, agyrotropic distributions, such as those found in the electron diffusion regions of mag-netic reconnection, should result in an increased 𝜖 .At low electron energies, there are several effects that can artificially increase 𝜖 . Theseinclude:(1) Spacecraft photoelectrons are detected when the Active Spacecraft Potential Con-trol (ASPOC) [ Torkar et al. , 2016] is off and the spacecraft potential is larger than ≈
10 eV.The energy channels affected by spacecraft photoelectrons are removed and the remainingenergy channels are corrected for when calculating 𝜖 , so the effect of spacecraft photoelec-trons should be small.(2) Photoelectrons generated inside the electron detectors produce enhancements in thephase space density [ Gershman et al. , 2017]. This affects the Sunward pointing detectors,and can occur at energies exceeding 𝑒𝑉 𝑠𝑐 .(3) Secondary photoelectrons can occur within the detector, resulting in artifically largephase-space densities at low energies. These electrons can occur at energies exceeding 𝑒𝑉 𝑠𝑐 .(4) Low-energy electrons can be focused along the spin-plane and axial booms, whichare positively charged [e.g., Toledo-Redondo et al. , 2019]. In addition, when ASPOC is onthe ion plumes are emitted from the spacraft, modifying the motion of low-energy electrons[
Barrie et al. , 2019]. This can distort the measured electron distribution at low energies.While there is a model for internal photoelectrons [
Gershman et al. , 2017], whichcan approximately remove these effects, the other effects are not straightforward to remove.Therefore, we simply perform the calculation of 𝜖 for electron energies 𝐸 >
28 eV. Thiscorresponds to neglecting the lower 4 energy channels in the FPI-DES distribution functionsfor phase 1a of MMS operations when evaluating equation (1). In addition, energy chan-nels with eV/e < 𝑉 𝑠𝑐 are removed, and the energy channels are corrected by − 𝑒𝑉 𝑠𝑐 whencalculating 𝑓 model and 𝜖 . The spacecraft potential 𝑉 𝑠𝑐 is computed from the average probe-to-spacecraft potentials of the four spin-plane probes. This average probe-to-spacecraft poten-tial was compared to the cutoff energies of photoelectrons seen in FPI-DES data to calibrate 𝑉 𝑠𝑐 [ Graham et al. , 2018].As examples of the calculation of 𝜖 , Figure 1 shows three observed electron distribu-tions, the associated 𝑓 model , and values of 𝜖 . The distributions are from a reconnection eventobserved at the magnetopause on 30 October 2015 [ Graham et al. , 2016a]. The first distribu-tion is in the magnetosphere close to the magnetopause (Figures 1a–1c), the second distribu-tion is close to the reconnection ion diffusion region where 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ peaks (Figures 1d–1f),and the third distribution is in the magnetosheath where 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ < 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ = . 𝜖 = .
20. The modeled distribution (Figure 1b) is an approximately isotropic distri-bution. One of the differences between the observed and modeled distribution is the fluctu-ations in 𝑓 𝑒 due to the counting statistics of the particle instrument at relatively low 𝑛 𝑒 (cid:46) –4–onfidential manuscript submitted to JGR-Space Physics − , which results in an increase in 𝜖 . This can be seen by comparing Figures 1a and 1b;the observed 𝑓 𝑒 shows fluctuations as functions of speed and angle, while the modeled 𝑓 𝑒 smoothly varies. Figure 1c shows that quantitively there is some deviation of the observeddistribution from 𝑓 model . In particular, at pitch angle 𝜃 = ◦ , the shape of 𝑓 𝑒 differs from 𝑓 model in the thermal energy range.For the second distribution both 𝑓 𝑒 and 𝑓 model are qualitatively similar. However, Fig-ure 1f shows that there is significant deviation in 𝑓 𝑒 from 𝑓 model . For 𝜃 = ◦ , 𝑓 𝑒 is close toMaxwellian at thermal energies, while for 𝜃 = ◦ and 180 ◦ a flat-top distribution is observed,which deviates from the shape of a Maxwellian. Thus, the flat-top distribution correspondsto an enhanced non-Maxwellianity of 𝜖 = . 𝑓 𝑒 because the particle counts are high. Here, 𝑓 𝑒 and 𝑓 model are very similar, although thereare some deviations in 𝑓 𝑒 from 𝑓 model , as seen in Figure 1i. For this distribution 𝜖 = . In this section we investigate the statistical properties of 𝜖 . We calculate 𝜖 for all burstmode electron data from September 2015 to March 2016 (the first magnetopause phase ofthe MMS mission), corresponding to ∼
85 million distributions from the four spacecraft.We first consider the dependence of 𝜖 on 𝑛 𝑒 . Low-densities correspond to the outer mag-netosphere, while high densities typically correspond to the magnetosheath. In Figure 2 weplot two-dimensional histograms of log ( 𝜖 ) versus log ( 𝑛 𝑒 ) for data when ASPOC is off,ASPOC is on, and all data in panels (a)–(c), respectively. In all three panels we see that for 𝑛 𝑒 (cid:46)
10 cm − there is statistically an increase in 𝜖 as 𝑛 𝑒 decreases, which approximatelyscales as 𝜖 ∝ /√ 𝑛 𝑒 . For 𝑛 𝑒 (cid:38)
10 cm − , typically corresponding to the magnetosheath, wefind that 𝜖 does not depend strongly on 𝑛 𝑒 . We find that 𝜖 is not strongly affected by whetheror not ASPOC is on, except at low 𝑛 𝑒 , where 𝜖 is slightly larger when ASPOC is on. Thismight be due to the internal photoelectron emission in the FPI detectors, which is more sig-nificant at low 𝑛 𝑒 and the spacecraft potential is low, or due to distortions in the observedelectron distribution by the plume of ions around the spacecraft when ASPOC is on [ Bar-rie et al. , 2019]. We do not find any statistical differences between the four spacecraft (notshown).There are two main reasons for the increase in 𝜖 at low 𝑛 𝑒 in the magnetosphere:(1) In the magnetosphere distinct cold and hot electron populations are frequentlypresent at the same time [ Walsh et al. , 2020]. In these cases the total effective electron tem-perature is 𝑇 𝑒 ≈ 𝑛 𝑐 𝑇 𝑐 + 𝑛 ℎ 𝑇 ℎ 𝑛 𝑐 + 𝑛 ℎ , (3)where the subscripts 𝑐 and ℎ refer to the cold and hot electron components. When 𝑛 𝑐 and 𝑛 ℎ are comparable 𝑇 𝑒 differs significantly from 𝑇 𝑐 and 𝑇 ℎ , and large 𝜖 develop. In the mag-netosheath the electron distributions are characterized by a single temperature, although theshape can differ from bi-Maxwellian distribution. This results in a smaller 𝜖 in the magne-tosheath compared to the magnetosphere.(2) As 𝑛 𝑒 decreases the particle counts measured in each energy and angle bin of FPI-DES decreases. This results in increased statistical uncertainty, corresponding to a moregrainy looking distribution at lower density, which differs from the smooth 𝑓 model distribu-tion, resulting in an increase in 𝜖 .The first effect iss physical and due to differences in typical magnetospheric and mag-netosheath distributions, while the second effect is an instrumental effect. Both effects re-sult in the statistical increase in 𝜖 as 𝑛 𝑒 decreases, seen in Figure 2. To illustrate these two –5–onfidential manuscript submitted to JGR-Space Physics
Figure 1.
Three examples of electron distributions and the predicted bi-Maxwellian distribution based onthe electron moments from MMS1 on 30 October 2015. Panels (a), (d), and (g) show a two-dimensional sliceof the observed three-dimensional electron distribution in the B and E × B plane. Panels (b), (e), and (h) showthe modeled bi-Maxwellian distributions in the same plane. Panels (c), (f), and (i) show the phase-space den-sities at pitch angles 0 ◦ (black), 90 ◦ (red), and 180 ◦ (blue) for the observed distributions (circles) and modeledbi-Maxwellian (solid lines). The distribution in panels (a)–(c) is in the magnetosphere, (d)–(f) is near the iondiffusion region where 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ peaks, (g)–(i) is in the magnetosheath where 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ is minimal. Theelectron distribution properties and 𝜖 of the three distributions are given in panels (c), (f), and (i).–6–onfidential manuscript submitted to JGR-Space Physics
Figure 2.
Two-dimensional histograms of log ( 𝑛 𝑒 ) versus log ( 𝜖 ) . (a) Histogram for data when ASPOCis off, (b) histogram for data when ASPOC is on, and (c) histogram for all data. The color shading indicatesthe counts. The black line indicates the median (50th percentile) of 𝜖 as a function 𝑛 𝑒 and the lower and upperred curves indicate the 10th and 90th percentiles as a function of 𝑛 𝑒 , respectively. effects, we plot two electron distributions in Figure 3. The first distribution (top row) is inthe magnetosphere close to the magnetopause. For this distribution we calculate 𝜖 = . 𝜖 = . Khotyaintsev et al. ,2016]. Figures 3a and 3b show 2D slices of the observed 𝑓 𝑒 in the B and E × B plane and 𝑓 model calculated from the particle moments. The distribution in Figure 3a is characterizedby a very cold component, with 𝑛 𝑐 = . − and 𝑇 𝑐 =
27 eV, and a hot component, with 𝑛 ℎ = .
06 cm − and 𝑇 ℎ = 𝑛 𝑒 = .
36 cm − and 𝑇 𝑒 = 𝑓 model shown in Figure 3b differs from both the cold and hot components of 𝑓 𝑒 , result-ing in a large 𝜖 . In Figure 3c we plot ( 𝑓 𝑒 − 𝑓 model ) 𝑣 , which indicates the regions of velocityspace that contribute most significantly to 𝜖 . The largest contribution is from low energies,where 𝑓 𝑒 (cid:29) 𝑓 model and most of the particles are located. At intermediate energies ∼ 𝑓 𝑒 (cid:28) 𝑓 model because 𝑓 model has 𝑇 𝑒 = 𝑓 𝑒 is negligible due to the temperaturesof the two components. For 𝐸 ∼ 𝑓 𝑒 (cid:29) 𝑓 model due to 𝑇 ℎ > 𝑇 𝑒 . As a result | 𝑓 𝑒 − 𝑓 model | is large over almost all velocity space making 𝜖 large.At low densities the counts per energy and angle bin are small and in many cases nocounts are measured, as indicated by the white regions in Figure 3a. This results in a moregrainy looking 𝑓 𝑒 , in contrast to the smooth 𝑓 model (Figure 3b). This results in 𝜖 tending toincrease as 𝑛 𝑒 decreases. At high densities the counts are very high in the thermal energyrange (for example, the distribution in Figure 3d), so the effects of the finite counting statis-tics are small. In Appendix A: we show that the most significant contribution to 𝜖 in themagnetosphere is the distinct cold and hot electron populations. The effect of the countingstatistics on 𝜖 is smaller when hot and cold electrons are present.For the distribution in Figure 3d only a single electron population is observed, charac-terized by 𝑛 𝑒 =
21 cm − and 𝑇 𝑒 =
74 eV. As a result 𝑓 model (Figure 3e) is very similar to 𝑓 𝑒 ,and 𝜖 = . 𝑓 𝑒 and 𝑓 model looking similar, non-Maxwellian features are observedin the thermal energy range, as shown in Figure 3f.In Figure 2c we overplot the median 𝜖 ( 𝜖 ) in black, and the 10th and 90th percentiles 𝜖 and 𝜖 (lower and upper red curves, respectively) as functions of 𝑛 𝑒 . We find that 𝜖 ≈ . − (cid:46) 𝑛 𝑒 (cid:46)
100 cm − . Because of the strong depen-dence of 𝜖 on 𝑛 𝑒 , we need to consider the statistical median and percentiles when consideringspecific events to determine if the electron distributions are unusually non-Maxwellian com- –7–onfidential manuscript submitted to JGR-Space Physics
Figure 3.
Two examples of electron distributions and the modeled bi-Maxwellian distribution based onthe electron moments from MMS1 on 06 December 2015 (cf.,
Khotyaintsev et al. [2016]). Panels (a) and(d) show a two-dimensional slice of the observed three-dimensional electron distribution in the B and E × B plane. Panels (b) and (e) show the modeled bi-Maxwellian distributions in the same plane. Panels (c) and (f)show ( 𝑓 𝑒 − 𝑓 model ) 𝑣 in the same plane, which indicates the regions of velocity space that contribute most to 𝜖 . The distribution in panels (a)–(c) is in the magnetosphere close to the magnetopause, and (d)–(f) is in themagnetosheath close to the magnetopause. –8–onfidential manuscript submitted to JGR-Space Physics pared with the median values of 𝜖 . We will use these percentiles as a function of density todetermine whether electron distributions in specific regions significantly deviate from a bi-Maxwellian distributions or not.In Figure 4a we plot the histogram of log 𝜖 versus log 𝑇 𝑒 . We find that statistically 𝜖 increases as 𝑇 𝑒 increases. This is primarily due to the statistical increase in 𝑇 𝑒 as 𝑛 𝑒 de-creases. Low-density higher-temperature regions correspond to the outer magnetosphere,while high-density lower-temperature regions typically correspond to the magnetosheath. InFigure 4b we plot 𝜖 versus log 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ . Overall, we find that the statistical dependence of 𝜖 on 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ is relatively weak. However, the smallest 𝜖 are found for 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ ∼
1. In thisstudy there are more distributions with 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ > 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ <
1, with a median andmean 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ of 1.1 and 1.2, respectively.We now compare 𝜖 with the agyrotropy of the electron distribution. We use the agy-rotropy measure √ 𝑄 as defined in Swisdak [2016]. This measure is based on the off-diagonalcomponents of the electron pressure tensor P 𝑒 and is given by √︁ 𝑄 = (cid:32) 𝑃 + 𝑃 + 𝑃 𝑃 ⊥ + 𝑃 ⊥ 𝑃 (cid:107) (cid:33) / , (4)where we have rotated the measured P 𝑒 into the field-aligned coordinates of the form: P 𝑒 = (cid:169)(cid:173)(cid:171) 𝑃 (cid:107) 𝑃 𝑃 𝑃 𝑃 ⊥ 𝑃 𝑃 𝑃 𝑃 ⊥ (cid:170)(cid:174)(cid:172) . (5)The agyrotropic measure √ 𝑄 has values between 0 and 1, with 0 corresponding to a gy-rotropic distribution and 1 to maximum agyrotropy. In Figure 4c we plot the histogram of 𝜖 versus √ 𝑄 for 𝑛 𝑒 > − (at lower 𝑛 𝑒 the agyrotropy measures tend to be unreliable).We find that the vast majority of distributions are approximately gyrotropic, with medianand mean values of √ 𝑄 of 0 .
008 and 0 . √ 𝑄 observedare ∼ .
1, which correspond to values observed in the EDRs of magnetopause reconnection[
Graham et al. , 2017;
Norgren et al. , 2016;
Webster et al. , 2018]. For these large values of √ 𝑄 there is a tendency of 𝜖 to increase with √ 𝑄 . However, only a tiny fraction of the distri-butions have large √ 𝑄 , and for most distributions there is little dependence of 𝜖 on √ 𝑄 . Thisis not surprising because: (1) gyrotropic distributions can deviate significantly from a bi-Maxwellian distribution function, and (2) √ 𝑄 is based on the pressure tensor, so large largeoff-diagonal pressure terms may not correspond to large changes in phase-space density.In Figure 4d we plot the histogram of 𝜖 versus the magnitude of the current density | J | calculated from the particle moments. We find no correlation between 𝜖 and | J | , whichsuggests that regions of large | J | , such as narrow current sheets, do not significantly enhance 𝜖 above background values. Similarly, we do not find any correlation between 𝜖 and E · J (notshown). Overall, aside from 𝜖 scaling with 𝑇 𝑒 due to the correlation between 𝑇 𝑒 and 𝑛 𝑒 , thereis little statistical dependency of 𝜖 on the local plasma conditions.To further investigate the relationship between 𝜖 and 𝑛 𝑒 and 𝑇 𝑒 we plot the 2D his-togram of 𝑛 𝑒 and 𝑇 𝑒 in Figure 5a. We see that there is a statistical increase in 𝑇 𝑒 as 𝑛 𝑒 de-creases, as expected from electron distributions in the outer magnetosphere, magnetopause,and in the magnetosheath. However, for a given 𝑛 𝑒 there is a large range of 𝑇 𝑒 , in particularfor 𝑛 𝑒 (cid:46)
1, corresponding to the outer magnetosphere. This larger range of 𝑇 𝑒 in the magne-tosphere can result from the relative densities of cold and hot components 𝑛 𝑐 and 𝑛 ℎ varyingsignificantly. In Figures 5b and 5c we plot the mean and standard deviation of 𝜖 versus 𝑛 𝑒 and 𝑇 𝑒 . These values are computed on the same grid as in Figure 5a, so Figure 5a indicatesthe number of values used to calculate each mean and standard deviation. Figure 5b showsthe strong dependence of 𝜖 on 𝑛 𝑒 , as well as a weaker dependence on 𝑇 𝑒 . In particular, themean 𝜖 increases as 𝑇 𝑒 increases for a given 𝑛 𝑒 . This is likely in part due to the lowest en-ergy part of the electron distribution being excluded to avoid contamination from internal –9–onfidential manuscript submitted to JGR-Space Physics
Figure 4.
Two-dimensional histograms of 𝜖 versus different plasma conditions. (a) log 𝜖 versus log 𝑇 𝑒 .(b) 𝜖 versus log 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ for 𝑛 𝑒 > − . (c) 𝜖 versus agyrotropy measure √ 𝑄 for 𝑛 𝑒 > − . (d) 𝜖 versus | J | for 𝑛 𝑒 > − . –10–onfidential manuscript submitted to JGR-Space Physics photoelectrons and spacecraft charging effects. Figure 5c shows that the standard deviationof 𝜖 tends to increase with decreasing 𝑛 𝑒 and increasing 𝑇 𝑒 . This might correspond to morevariable electron distributions in the outer magnetosphere. Figure 5.
Two-dimensional plots of counts and 𝜖 versus 𝑇 𝑒 and 𝑛 𝑒 , and 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ and 𝜖 . (a) Two-dimensional histograms of 𝑇 𝑒 versus 𝑛 𝑒 . (b) Mean 𝜖 versus 𝑇 𝑒 and 𝑛 𝑒 . (c) Standard deviation of 𝜖 versus 𝑇 𝑒 and 𝑛 𝑒 . (d) Two-dimensional histogram of 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ versus 𝛽 𝑒, (cid:107) . (e) Mean 𝜖 versus 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ versus 𝛽 𝑒, (cid:107) . (f) Standard deviation of 𝜖 versus 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ versus 𝛽 𝑒, (cid:107) . We now investigate the relationship between the parallel electron plasma 𝛽 , 𝛽 𝑒, (cid:107) , and 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ and the instability of these electron distributions. In Figure 5d we plot the 2D his-togram of 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ and 𝛽 𝑒, (cid:107) . We find that for 𝛽 𝑒, (cid:107) (cid:46) 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ ,although most data are found near 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ =
1. For 𝛽 𝑒, (cid:107) (cid:38) 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ de-creases as 𝛽 𝑒, (cid:107) increases.We can compare this histogram with the thresholds for the oblique resonant electronfirehose instability [ Li and Habbal , 2000] and the whistler temperature anisotropy instability[
Kennel and Petschek , 1966]. The oblique electron firehose instability can occur in a high 𝛽 𝑒, (cid:107) plasma with 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ >
1. Theoretical work has shown that the oblique resonant elec-tron firehose instability has a lower threshold than the field-aligned non-resonant electronfireshose instability [
Li and Habbal , 2000], so we only consider the former case here. Thewhistler temperature anisotropy instability can occur for 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ <
1. These instabilitiesare proposed to constrain the values of 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ that can develop by pitch-angle scatteringelectrons so the electron distribution becomes more isotropic. Numerical studies have shownthat the threshold 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ for these instabilities scales with 𝛽 𝑒, (cid:107) and has the form [ Garyand Wang , 1996] 𝑇 𝑒, (cid:107) 𝑇 𝑒, ⊥ = (cid:16) + 𝑆 𝑒 𝛽 − 𝛼 𝑒 𝑒, (cid:107) (cid:17) − , (6)where 𝑆 𝑒 and 𝛼 𝑒 are constants. We consider the thresholds corresponding to growth rates 𝛾 / Ω 𝑐𝑒 = .
01 and 0 .
1. For the electron firehose instability we use 𝑆 𝑒 = − .
23 and 𝛼 𝑒 = . 𝑆 𝑒 = − .
32 and 𝛼 𝑒 = .
61 for 𝛾 / Ω 𝑐𝑒 = .
01 and 0 .
1, respectively [
Gary and Nishimura ,2003]. For the whistler instability we use 𝑆 𝑒 = .
36 and 𝛼 𝑒 = .
55, and 𝑆 𝑒 = . 𝛼 𝑒 = .
49 for 𝛾 / Ω 𝑐𝑒 = .
01 and 0 .
1, respectively [
Gary and Wang , 1996]. These thresholds –11–onfidential manuscript submitted to
JGR-Space Physics are plotted in Figures 5d–5f, where the black and red curves correspond to the firehose andwhistler thresholds, respectively. The thick lines correspond to 𝛾 / Ω 𝑐𝑒 = .
01 and the thinlines correspond to 𝛾 / Ω 𝑐𝑒 = . 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ > 𝛾 / Ω 𝑐𝑒 = .
01 provides an approximateboundary for the largest 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ for 𝛽 𝑒, (cid:107) (cid:38)
2. Only 0 .
016 % of the data exceed the 𝛾 / Ω 𝑐𝑒 = .
01 threshold, so regions unstable to the firehose instability are very rare. For 𝛽 𝑒, (cid:107) (cid:46) 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ . Thismeans that for 𝛽 𝑒, (cid:107) (cid:46)
2, such as on the magnetospheric side of the magnetopause and in themagnetosphere, the firehose instability is unlikely to occur, and thus cannot limit the valuesof 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ found there. Regions where the firehose instability is more likely to occur arein the magnetosheath, and potentially at the magnetopause boundary and in reconnectionregions, where the magnetic field B becomes very small.For 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ < 𝛽 𝑒, (cid:107) (cid:38) . 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ . For the 𝛾 / Ω 𝑐𝑒 = .
01 thresholdapproximately 0 .
12 % of the data exceed the threshold, nearly an order of magnitude higherthan for the firehose instability. We find that for 𝛽 𝑒, (cid:107) (cid:38) 𝛾 / Ω 𝑐𝑒 = . 𝛾 / Ω 𝑐𝑒 = . 𝜖 as functions of 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ and 𝛽 𝑒, (cid:107) .Figure 5e shows that when 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ approaches or exceeds the whistler and firehose thresh-olds the values of 𝜖 remain relatively small. This suggests that the thresholds are reasonableindicators of instability. Figure 5f shows that the standard deviation is relatively small whenthe thresholds are exceeded, although there are regions with enhanced standard deviationsclose to both the firehose and whistler instabilities. This might indicate that some distribu-tions in these regions may be unstable.For magnetosheath plasma, electron distributions often exhibit flat-top distributionfunctions so they will often differ from a bi-Maxwellian, producing a finite 𝜖 . It is unclear towhat to what extent such distributions affect the firehose and whistler thresholds, althoughthe good agreement between the threshold conditions and the cutoff in the observed data sug-gests that the numerical thresholds for the instabilities are reasonable. For magnetosphericplasmas previous observations have shown that non-Maxwellian distributions develop. Oneexample is the magnetospheric plasmas consist of both hot and cold electron distributions,which will modify the threshold for whistler waves [e.g. Gary et al. , 2012]. Similarly, in themagnetospheric separatrices of reconnection complex electron distributions develop, whichcan excite whistler waves [
Graham et al. , 2016b;
Khotyaintsev et al. , 2019]. Thus, we ex-pect that whistlers can be generated if they do not exceed the threshold conditions due to thedeviations from the bi-Maxwellian distribution function. It is unclear if deviations from abi-Maxwellian distributions can enhance the instability of the oblique electron firehose insta-bility.
Using the results in Figure 2, we can use the statistical percentiles of 𝜖 as a functionof 𝑛 𝑒 to find regions of localized enhanced or reduced 𝜖 and investigate their source regions.We specifically focus on the magnetic reconnection ion and electron diffusion regions, thebowshock, and the turbulent magnetosheath. –12–onfidential manuscript submitted to JGR-Space Physics
We investigate the ion diffusion region (IDR) observed 2015 December 30 by
Gra-ham et al. [2016a] at Earth’s dayside magnetopause. The IDR was identified by a strong Hallelectric field and intense parallel electron heating. Figure 6 provides an overview of the elec-tron behavior for this event from MMS1. The spacecraft crossed the magnetopause from themagnetosphere to the magnetosheath, indicated by the reversal in the magnetic field B andsubstantial increase in 𝑛 𝑒 (Figures 6a and 6b). At the magnetopause, we see large fluctua-tions in the electron bulk velocity V 𝑒 (Figure 6c), intense parallel electron heating (Figure6f), and an increase in √ 𝑄 (Figure 6g). We find that √ 𝑄 peaks at 0 .
06 near the center of thecurrent sheet, which may indicate close proximity to the EDR. The peak in electron heatingwas observed on the magnetospheric side of the magnetopause in the magnetospheric in-flow region next to the Hall region, where ions decouple from the electron motion [
Grahamet al. , 2016a]. Here 𝛽 𝑒, (cid:107) is low so the peak 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ = . 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ found in the magnetopause current sheet. On the magnetosheath side ofthe current sheet we find that some short intervals have 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ < 𝜖 is plotted in Figure 6h. Overall, we find that 𝜖 de-creases from about 0 . . 𝑛 𝑒 , so it is difficult to identity regions of enhanced non-Maxwellianity by simply plotting 𝜖 . Using the measured 𝑛 𝑒 we calculate the 10th, 50th (me-dian), and 90th percentiles of 𝜖 , 𝜖 , 𝜖 , and 𝜖 , respectively, from the statistical results inFigure 2c. These quantities are plotted in Figure 6h. In the magnetosphere (low density)we find that 𝜖 remains close to 𝜖 , indicating that statistically the distribution is close toMaxwellian for a magnetospheric distribution. In this case there is only a single colder elec-tron population with 𝑇 𝑒 ≈
40 eV, and negligible hot electrons. This results in the relativelylow 𝜖 observed in the magnetosphere. In the magnetosheath (high density) 𝜖 has values be-tween 𝜖 and 𝜖 , suggesting that while the electron distributions are non-Maxwellian, butthe non-Maxwellianity is not particularly large. However, in the IDR where large 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ occurs there is an increase in √ 𝑄 , and 𝜖 approaches and exceeds 𝜖 , meaning that statis-tically the non-Maxwellianity is high in this region. This is most clearly seen in Figure 6i,where we plot the percentile that the observed 𝜖 belongs to as a function of 𝑛 𝑒 based on thestatistical results in Figure 2c. We find that the percentile increases where 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ and √ 𝑄 peak, while in the magnetosphere the percentile is low and in the magnetosheath the per-centile remains close to 50. The magenta dashed lines in Figure 6 indicate the times of thethree electron distributions in Figure 1. The source of the increased 𝜖 in the ion diffusion re-gion is the flat-top shape of the distribution parallel and antiparallel to B (Figure 1d). Figure1 shows that the shapes of the magnetospheric and magnetosheath distributions in the ther-mal energy range deviate from a bi-Maxwellian distribution.These results indicate that calculating 𝜖 alone is not sufficient to identify regions of un-usually large non-Maxwellianity when considering magnetopause crossings. However, usingthe statistical results derived from Figure 2c, we can identify regions where the calculated 𝜖 correspond to unusually large deviations from a bi-Maxwellian distribution function. For thisexample enhanced deviations in the electron distributions from a bi-Maxwellian are foundin the ion diffusion region of asymmetric reconnection, although there is little change in thevalue of 𝜖 across the magnetopause. As an exampe of an EDR crossing, Figure 7 shows the magnetopause crossings ob-served on 2015 October 22 by MMS1 [
Phan et al. , 2016;
Toledo-Redondo et al. , 2016]. TheEDR is observed at 06:05:22 UT, indicated by the yellow-shaded region in Figure 7. Over –13–onfidential manuscript submitted to
JGR-Space Physics
Figure 6.
Overview of the electron behavior from MMS1 associated with the ion diffusion region observedon 2015 October 30. (a) B . (b) 𝑛 𝑒 . (c) V 𝑒 . ( d) Electron omnidirectional differential energy flux (blue lineindicates 𝑇 𝑒 ). (e) Spectrogram of the electron pitch-angle distribution for energies 20 eV < 𝐸 <
500 eV.(f) 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ (black), and firehose (red) and whistler (green) thresholds (for 𝛾 / Ω 𝑐𝑒 = . √ 𝑄 . (h) 𝜖 (black), and the 10th and 90th percentiles of 𝜖 as a function of 𝑛 𝑒 (red) and median 𝜖 as a function 𝑛 𝑒 . (i)Percentile of the observed 𝜖 as a function of 𝑛 𝑒 . The magenta dashed lines indicate the times the electrondistributions in Figure 1 are taken. –14–onfidential manuscript submitted to JGR-Space Physics the entire interval we observe three partial magnetopause crossings, where 𝐵 𝑧 decreasesand 𝑛 𝑒 increases (Figures 7a and 7b). Between the first and second magnetopause cross-ings there is a southward ion outflow and after the third magnetopause crossing, where theEDR is observed, there is a northward outflow. The spacecraft enters the magnetosheath atapproximately 06:05:35 UT. The outflows are here indicated by the large-scale changes inthe electron flow 𝑉 𝑒,𝑧 in Figure 7c. In the magnetosphere 𝑇 𝑒 is very low due to cold electronsdominating the electron distributions (Figure 7d). The electron pitch-angle distribution inFigure 7e shows that there are complex changes in the electron distribution functions acrossthe magnetopause boundaries. Similarily, 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ varies significantly across the interval(Figure 7f). At the third magnetopause crossing we observe the largest V 𝑒 (Figure 7c) and √ 𝑄 (Figure 7g), which peaks just above 0 .
1, signifying the EDR.In Figure 7h we plot the timeseries of 𝜖 , as well as 𝜖 , 𝜖 , and 𝜖 . We note that afterthe first magnetopause crossing 𝜖 , 𝜖 , and 𝜖 remain relatively constant. In the EDR wefind a large enhancement in 𝜖 (well above 𝜖 ), which is colocated with the peak in √ 𝑄 . Forthis event we also observe very large enhancements in 𝜖 in both the northward and southwardion outflows (both have extended regions where 𝜖 > 𝜖 ). Indeed the largest 𝜖 is observed inthe northward reconnection outflow, rather than at the EDR. Here, √ 𝑄 is negligible so thesedeviations from bi-Maxwellianity are gyrotropic in nature. This event illustrates why there isstatistically a lack of correlation between 𝜖 and √ 𝑄 .In Figure 7f we compare 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ with the local electron firehose and whistler thresh-olds. Throughout the interval 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ remains below the threshold of the electron fire-hose instability. For the whistler instability we find that in some regions 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ exceedsthe threshold. Figure 7i, which shows the frequency-time spectrogram of B , showing thatwhistler waves are found near these regions. However, whistlers are found where 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ does not satisfy the threshold for instability, such as in the magnetosphere and throughoutthe northward outflow. In the magnetosphere we observe distinct hot and cold electron pop-ulations (Figure 7d), where the cold population is characterized by 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ >
1, result-ing in the overall distribution having 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ >
1, while the hot population with energies
𝐸 > 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ <
1, which is the likely source of the whistler waves. In this casewe estimate 𝑛 ℎ / 𝑛 𝑐 ≈ × − , meaning that the hot component has only a small effect on 𝜖 .In the outflow we observe complex electron distributions (an example is shown below in Fig-ure 8). Therefore, in these regions there tends to be a significant 𝜖 , so the whistler instabilitythresholds predicted from a single bi-Maxweliian distribution are no longer valid.In Figure 8 we investigate three electron distributions observed in the EDR where √ 𝑄 peaks (Figures 8a–8c), in the reconnection outflow where 𝜖 peaks (Figures 8d–8f), anda point in the ouflow where 𝜖 is small (Figures 8g–8i). These points are indicated by thethree dashed magenta lines in Figure 7. Figure 8a shows a distribution consisting of a near-stationary core and a crescent propagating in the E × B direction, which is responsible forthe peak in √ 𝑄 and the large V 𝑒 . Such distributions are typical of EDRs at the magnetopause[ Burch et al. , 2016;
Graham et al. , 2017;
Norgren et al. , 2016]. As expected the model dis-tribution does not capture the observed features, and is simply a bi-Maxwellian drifting inthe E × B direction, which results in the enhanced 𝜖 . Note that the pitch-angle distribution(Figure 8c) does not capture the agyrotropy, so in this plot the source of 𝜖 becomes unclear.In Figure 8d the observed distribution prominantly feaatures counter-streaming elec-tron beams parallel and antiparallel to B , and higher-energy electrons perpendicular to B .For electron energies 𝐸 (cid:46)
200 eV the counter-streaming result in 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ >
1, while for 𝐸 (cid:38)
200 eV we find 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ <
1. The total 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ is close to 1, so the model distri-bution is approximately Maxwellian (Figure 8e). Figure 8f shows that 𝑓 𝑒 at pitch angles 0 ◦ ,90 ◦ , and 180 ◦ all differ significantly from 𝑓 model and as a result 𝜖 is relatively large. Thesedistributions can account for the observed whistler waves, where 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ does not satisfythe instability threshold. –15–onfidential manuscript submitted to JGR-Space Physics
Figure 7.
Overview of the electron behavior from MMS1 associated with the electron diffusion region ob-served on 2015 October 22. (a) B . (b) 𝑛 𝑒 . (c) V 𝑒 . (d) Electron omnidirectional differential energy flux (blueline indicates 𝑇 𝑒 ). (e) Spectrogram of the electron pitch-angle distribution for energies 20 eV < 𝐸 <
500 eV.(f) 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ (black), and firehose (red) and whistler (green) thresholds (for 𝛾 / Ω 𝑐𝑒 = . √ 𝑄 . (h) 𝜖 (black), and the 10th and 90th percentiles of 𝜖 as a function of 𝑛 𝑒 (red) and median 𝜖 as a function 𝑛 𝑒 . (h)Percentile of the observed 𝜖 as a function of 𝑛 𝑒 . –16–onfidential manuscript submitted to JGR-Space Physics
Figure 8.
Three examples of electron distributions and the predicted bi-Maxwellian distribution based onthe electron moments from the EDR crossing observed by MMS1 on 22 October 2015. Panels (a), (d), and (g)show a two-dimensional slice of the observed three-dimensional electron distribution in the B and E × B plane.Panels (b), (e), and (h) show the modeled bi-Maxwellian distributions in the same plane. Panels (c), (f), and(i) show the phase-space densities at pitch angles 0 ◦ (black), 90 ◦ (red), and 180 ◦ (blue) for the observed dis-tributions (circles) and modeled bi-Maxwellian (solid lines). The distribution in panels (a)–(c) is in the EDRwhere √ 𝑄 peaks, (d)–(f) is in the ion outflow where 𝜖 peaks, (g)–(i) is close to the magnetosheath where 𝜖 issmall. The electron distribution properties and 𝜖 of the three distributions are given in panels (c), (f), and (i).–17–onfidential manuscript submitted to JGR-Space Physics
The distribution in Figure 8g is close to Maxwellian and very similar to 𝑓 model (Figure8h). Figure 8i shows that the profiles of 𝑓 𝑒 at 𝜃 = ◦ , 90 ◦ , and 180 ◦ match well 𝑓 model , sothe distribution is approximately Maxwellian. Figures 7 and 8 show that the shape electrondistributions can vary significantly throughout the reconnection outflow, with both complexdistributions and approximately Maxwellian distributions developing.More generally we have investigated the non-Maxwellianity of the electron distribu-tions associated with the 11 EDRs observed in the first phase of the MMS mission [ Fuselieret al. , 2017;
Webster et al. , 2018]. We find that in each case there is an enhancement in 𝜖 inor near the EDR, which typically exceeds 𝜖 . In some outflow regions large 𝜖 are observed,although the values of 𝜖 , and whether these values exceed 𝜖 , varies between events. Thissuggests that large values of 𝜖 are a necessary, but not sufficient criterion for identification ofEDRs. In this subsection we investigate the non-Maxwellianity of electron distributions at thebowshock. In particular, we investigate a quasi-perpendicular bowshock in detail. We findthat 𝜖 is strongly enhanced at the bowshock but typically returns to nominal values within themagnetosheath. Figure 9 shows an example of a quasi-perpendicular bowshock observed on04 November 2015 around 05:58:00 UT. This bowshock was previously studied by Oka et al. [2017]. At this time the spacecraft were located at [10.4, 2.1, -0.5] 𝑅 𝐸 (GSE). For this shockwe estimate the shock-normal angle to be 𝜃 𝐵𝑛 = ◦ , corresponding to a quasi-perpendicularshock, and the Alfven Mach number is 𝑀 𝐴 ≈
11. Based on the width of the shock foot weestimate that the shock moves ∼
30 km s − Sunward in the spacecraft frame.Figure 9 shows that the spacecraft was initially in a solar wind-like plasma, with low B , fast Earthward flow 𝑉 𝑒,𝑥 = −
700 km s − , and density 𝑛 𝑒 = . − . We see that theelectron fluxes vary near 100 eV and observe large-amplitude emission of Langmuir wavesat the local plasma frequency (not shown), indicating that the spacecraft is in the electronforeshock region. The shock foot begins at approximately 05.58:00 UT, and is seen as theincrease in 𝑛 𝑒 and | 𝐵 𝑦 | , and the decrease in | 𝑉 𝑒,𝑥 | . The shock ramp begins at approximately05:58:14 UT, and a series of shock ripples are observed in B and 𝑛 𝑒 . The overshoot is ob-served at 05:58:19 UT.In Figure 9g we plot 𝜖 along with 𝜖 , 𝜖 , and 𝜖 . At the beginning of the interval 𝜖 is very low, but increases beginning at 05:57:46 UT. At this time 𝜖 likely varies becausethe spacecraft is in the electron foreshock, where electrons reflected and accelerated at thebowshock can cause 𝜖 to increase above the values in the unperturbed solar wind. In theshock foot 𝜖 begins to exceed 𝜖 , which extends for about 13 seconds until just after theshock ramp (yellow-shaded interval in Figure 9). Based on the shock speed this region ofenhanced 𝜖 has a spatial scale of ∼
400 km, which corresponds to 2 𝜌 𝑖 or 5 𝑑 𝑖 based onthe magnetosheath parameters, where 𝜌 𝑖 and 𝑑 𝑖 are the ion Larmor radius and ion inertiallength, respectively. Thus, the region of enhanced 𝜖 extends over ion spatial scales. Down-stream of the shock 𝜖 typically remains between 𝜖 and 𝜖 , with some local enhancementsin 𝜖 . Throughout this region we observe rapid fluctuations in 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ , although 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ does not exceed the threshold for the firehose instability and rarely satisfies the threshold forwhistler waves.In Figures 9h and 9i we plot the fluctuating electric and magnetic fields, 𝛿 E and 𝛿 B ,in field-aligned coordinates above 5 Hz. The region of enhanced 𝜖 coincides with the mostintense 𝛿 E and 𝛿 B . Fluctuations both parallel and perpendicular to the background mag-netic field are observed. For 𝛿 E the parallel fluctuations are typically observed near the ionplasma frequency 𝑓 𝑝𝑖 , while the perpendicular fluctuations are observed at lower frequen-cies. The magnetic field fluctuations are primarily seen at low frequencies below the lowerhybrid frequency. In the magnetosheath 𝛿 B are significantly reduced, while 𝛿 E parallel tothe background magnetic field continue to be observed intermittently. The large-amplitude –18–onfidential manuscript submitted to JGR-Space Physics
Figure 9.
Overview of the electron behavior from MMS1 associated a bowshock crossing observed on2015 November 4 observed by MMS1. (a) B . (b) 𝑛 𝑒 . (c) V 𝑒 . (d) Electron omnidirectional differentialenergy flux (blue line indicates 𝑇 𝑒 ). (e) Spectrogram of the electron pitch-angle distribution for ener-gies 20 eV < 𝐸 <
500 eV. (f) 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ (black), and firehose (red) and whistler (green) thresholds (for 𝛾 / Ω 𝑐𝑒 = . 𝜖 (black), and the 10th and 90th percentiles of 𝜖 as a function of 𝑛 𝑒 (red) and median 𝜖 as a function 𝑛 𝑒 . (h) Fluctuating electric field 𝛿 E above 𝑓 = 𝛿 B above 𝑓 = JGR-Space Physics electrostatic and electromagnetic fluctuations in the bowshock may result in the electron dis-tribution with large 𝜖 becoming more Maxwellian in the magnetosheath due to wave-particleinteractions.In Figure 10 we plot three electron distributions at the times indicated by the magentadashed lines in Figure 9 and compare these distributions with the modeled bi-Maxwelliandistribution function. The distributions are observed in the shock foot [panels (a)–(c)], nearthe shock ramp [panels (d)–(f)], and in the magnetosheath [panels (g)–(i)]. In each observeddistribution there is a super-thermal tail in the electron distribution. However, these tails donot significantly increase 𝜖 because the phase-space densities are very low, meaning theircontributions to 𝑇 𝑒 and 𝑛 𝑒 are negligible.The electron distribution in Figure 10a exhibits a dense low-energy electron beam par-allel to B , associated with accelerated solar wind electrons. For the direction antiparallel to B the electrons are hotter than in the parallel direction. These features are not captured by thebi-Maxwellian distribution (Figure 10b), resulting in a relatively large 𝜖 . Figure 10c showsthat the shape of the observed electron distribution differs greatly from the bi-Maxwelliandistribution, consistent with 𝜖 being unusually large. Figure 10d shows an electron distribu-tion near the ramp, where shock ripples are observed. The distribution is similar to the onein Figure 10a, except the solar wind electrons have been further accelerated parallel to B andnow occupy a smaller range of pitch angles. At 𝜃 = ◦ , and 180 ◦ we observe flat-top likedistributions, as previously observed at quasi-perpendicular shocks [ Feldman et al. , 1982;
Scudder , 1995]. Here, 𝜖 is reduced from the distribution in Figure 10a, but remains wellabove the statistical median. These distributions consisting of a beam of accelerated solarwind electrons and flat-top-like distributions have been observed previously and are due tothe cross-shock potential in the deHoffman-Teller frame. By integrating the divergence of theelectron pressure divergence over position, we estimate a maximum cross-shock potential of ∼
300 V, which is several times larger than the maximum energy of the beam of solar windelectrons in the shock and the energies where the distribution is relatively flat 𝐸 (cid:46)
100 eV.In the magnetosheath behind the bowshock the electrons are close to Maxwellian, forexample the electron distribution in Figures 10g–10i. In these panels we see little deviationfrom a Maxwellian distribution function. We conclude that the strongly enhanced 𝜖 occur ation spatial scales across the shock. Behind the shock in the magnetosheath 𝜖 is significantlyreduced, but continues to vary with position. In this subsection we investigate a region of magnetosheath turbulence behind thequasi-parallel bowshock. Figure 11 provides an overview of the region observed by MMS1on 25 October 2015. The interval is characterized by multiple current sheets, as indicated byreversal in B (Figure 11a), and narrow enhancements in J (Figure 11c). Figure 11d showsthat 𝑇 𝑒 remains relatively constant, although there are some variations in the electron fluxes.Figure 11e and 11f that 𝑇 (cid:107) / 𝑇 ⊥ varies across the interval, with 𝑇 (cid:107) / 𝑇 ⊥ ranging from 1 to 2. Wefind that the whistler and firehose thresholds for instability are not satisfied in this interval.Throughout the interval √ 𝑄 remains relatively small, although some enhancements in √ 𝑄 occur over very short intervals.In Figure 11h we plot 𝜖 , along with 𝜖 , 𝜖 , and 𝜖 . In general, 𝜖 , 𝜖 , and 𝜖 re-main relatively constant across the entire interval. We find that 𝜖 varies between 0 .
05 and0 .
17, and peaks in the region of enhanced 𝑛 𝑒 and J at 10:49:55 UT. Figure 11i shows that thepercentile of 𝜖 ranges from 1 to 99, meaning the electron distributions vary from very closeto Maxwellian to highly non-Maxwellian for magnetosheath conditions. In general, we seeno clear correlation between 𝜖 and the other local plasma parameters. For example, thereis no clear consistent correlation with J . At 10:49:55 UT there is a peak in 𝜖 and J ; how-ever, at other times there are peaks in J without enhanced 𝜖 (e.g., at 10:49:09 UT), and thereare peaks in 𝜖 where J is negligible (e.g., at 10:49:16 UT). To illustrate this further, Figure –20–onfidential manuscript submitted to JGR-Space Physics
Figure 10.
Three examples of electron distributions and the modeled bi-Maxwellian distribution based onthe electron moments from the foreshock crossing observed by MMS1 on 04 November 2015. Panels (a), (d),and (g) show a two-dimensional slice of the observed three-dimensional electron distribution in the B and E × B plane. Panels (b), (e), and (h) show the modeled bi-Maxwellian distributions in the same plane. Panels(c), (f), and (i) show the phase-space densities at pitch angles 0 ◦ (black), 90 ◦ (red), and 180 ◦ (blue) for theobserved distributions (circles) and modeled bi-Maxwellian (solid lines). The distribution in panels (a)–(c)is in the shock foot, (d)–(f) is near the shock ramp, (g)–(i) is in the magnetosheath behind the bowshock. Theelectron distribution properties and 𝜖 of the three distributions are given in panels (c), (f), and (i).–21–onfidential manuscript submitted to JGR-Space Physics
Figure 11.
Overview of the electron behavior from MMS1 in the turbulent magnetosheath observedon 2015 October 25 observed by MMS1. (a) B . (b) 𝑛 𝑒 . (c) V 𝑒 . (d) Electron omnidirectional differentialenergy flux (blue line indicates 𝑇 𝑒 ). (e) Spectrogram of the electron pitch-angle distribution for energies20 eV < 𝐸 <
500 eV. (f) 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ (black), and firehose (red) and whistler (green) thresholds (for 𝛾 / Ω 𝑐𝑒 = . 𝜖 (black), and the 10th and 90th percentiles of 𝜖 as a function of 𝑛 𝑒 (red) and median 𝜖 as a function 𝑛 𝑒 . (i) Percentile of the observed 𝜖 as a function of 𝑛 𝑒 .–22–onfidential manuscript submitted to JGR-Space Physics
12 shows the histograms of the entire dataset, with the data from the magnetosheath turbu-lence interval (from all four spacecraft) overplotted as magenta points. Figure 12a shows awide range of 𝜖 observed in this interval and no clear correlation with 𝑛 𝑒 . Similarly, thereis no correlation of 𝜖 with 𝑇 (cid:107) / 𝑇 ⊥ and √ 𝑄 (Figures 12b and 12c). Figure 12d shows thereis a slight tendency of 𝜖 to increase with J , which in this case is due to the region centeredaround 10:49:55 UT. Overall, there is little clear correlation of 𝜖 with the local plasma con-ditions. Other turbulent magnetosheath intervals, such as those investigated in Yordanovaet al. [2016] and
Voros et al. [2017] yield qualitatively similar results. We propose that theenhanced non-Maxwellian features may develop at the bowshock, and that the locally ob-served changes in 𝜖 are due to the changing magnetic connectivity to the bowshock duringthe turbulent intervals. Figure 12.
Two-dimensional histograms of 𝜖 versus different plasma conditions when 𝑛 𝑒 >
10 cm − [panels (b)–(d)]. Overplotted in magenta are scatter plots of the points in the interval shown in Figure 9 forMMS1–MMS4. (a) log 𝜖 versus log 𝑇 𝑒 . (b) 𝜖 versus log 𝑇 𝑒, (cid:107) / 𝑇 𝑒, ⊥ . (c) 𝜖 versus agyrotropy measure √ 𝑄 . (d) 𝜖 versus | J | . In Figure 13 we present three electron distributions at the times indicated by the ma-genta dashed lines in Figure 11. Each row consists of the observed electron distribution themodel bi-Maxwellian distribution and ( 𝑓 𝑒 − 𝑓 model ) 𝑣 , which indicates the regions of velocityspace that contribute most to 𝜖 . The observed distributions have 𝜖 of 0 . .
12, and 0 . 𝑇 (cid:107) / 𝑇 ⊥ = .
1. For the first distribution, Figures 13a and 13b showthat 𝑓 𝑒 is very similar to 𝑓 model , and as a result ( 𝑓 𝑒 − 𝑓 model ) 𝑣 remains small. For the sec-ond there is a beam-like feature in 𝑓 𝑒 (Figure 13d), which is not captured by 𝑓 model (Figure13e) and results in an enhanced 𝜖 . Figure 13g shows a clear flat-top distribution, which dif-fers significantly from 𝑓 model (Figure 13h). Figure 13i shows large values of ( 𝑓 𝑒 − 𝑓 model ) 𝑣 ,with ( 𝑓 𝑒 − 𝑓 model ) 𝑣 < ( 𝑓 𝑒 − 𝑓 model ) 𝑣 > –23–onfidential manuscript submitted to JGR-Space Physics ( 𝑓 𝑒 − 𝑓 model ) 𝑣 < 𝜖 . This distribution is similar tothe flat-top distributions found at the bowshock, which exhibit large 𝜖 . Figure 13.
Three examples of electron distributions and the predicted bi-Maxwellian distribution based onthe electron moments from the foreshock crossing observed by MMS1 on 25 November 2015 in the turbu-lent magnetosheath. Panels (a), (d), and (g) show a two-dimensional slice of the observed three-dimensionalelectron distribution in the B and E × B plane. Panels (b), (e), and (h) show the modeled bi-Maxwelliandistributions in the same plane. Panels (c), (f), and (i) show the modeled bi-Maxwellian distributions in thesame plane. Panels (c) and (f) show ( 𝑓 𝑒 − 𝑓 model ) 𝑣 in the same plane, which indicates the regions of velocityspace that contribute most to 𝜖 . The distribution in panels (a)–(c) is corresponds to low 𝜖 , (d)–(f) correspondto moderate 𝜖 , and (g)–(i) correspond to high 𝜖 . The values of 𝜖 of the three distributions are given in the titlesof panels (c), (f), and (i). In summary, we find that in magnetosheath turbulence 𝜖 varies significantly, but isnot directly correlated with local plasma conditions, such as density, temperature, temper-ature anisotropy, and current density. This may suggest that the local plasma turbulence doesnot significantly enhance 𝜖 . We propose that the enhanced 𝜖 seen intermittently in magne- –24–onfidential manuscript submitted to JGR-Space Physics tosheath turbulence is likely generated at the bowshock. We suggest that the changes in mag-netic connectivity to the bowshock, due to the changes in the direction of B throughout mag-netosheath turbulence affect the local values of 𝜖 . In this paper we have investigated the deviation of electron distributions from the bi-Maxwellian distribution function. We have defined a dimensionless quantity 𝜖 , which quan-tifies the deviation of the observed electron distribution from the bi-Maxwellian distributionfunction. We have calculated this quantity for the electron distributions observed by the fourMMS over a six month interval, primarly focussing on the magnetosphere, magnetopause,and magnetosheath. The key results of this study are:(1) The electron non-Maxwellianity 𝜖 scales inversely with the electron number densitynear the magnetopause. This is primarily due to the tendency of low-density magnetosphericelectron distributions having distinct cold and hot populations, which deviate significantlyfrom a single bi-Maxwellian distribution resulting in large 𝜖 , whereas in the higher-densitymagnetosheath the electron distributionsa are characterized a single temperature. By com-paring specific events with the statistical study of 𝜖 versus number density we can identifyregions of enhanced non-Maxwellianity.(2) Statistically, the electron non-Maxwellianity does not depend strongly on localplasma properties such as temperature anisotropy, agyrotropy, and current density.(3) The observed temperature anisotropies are bounded by the thresholds for the obliqueelectron firehose instability and the whistler temperature anisotropy instability in high 𝛽 plas-mas, such as at the magnetopause and in the magnetosheath. The distribitions close to thesethresholds tend to be close to bi-Maxwellian. These results suggest that these instabilitiesconstrain the electron temperature anisotropies that can develop in high- 𝛽 plasmas.(4) Enhanced 𝜖 are found in the ion and electron diffusion regions of magnetic recon-nection. While very large 𝜖 are observed in the electron diffusion region where electron dis-tributions are agyrotropic, similarly large 𝜖 can develop in the outflow regions of magneticreconnection. Thus, 𝜖 cannot uniquely identify electron diffusion regions.(5) Enhanced 𝜖 develops at the bowshock, due to the development of flat-top distribu-tions and electron beams resulting from the cross-shock potential. These 𝜖 develop over ionspatial scales and tend to decrease behind the bowshock in the magnetosheath.(6) Intermittent enhancements in 𝜖 are observed in magnetosheath turbulence. Theseincreases in 𝜖 are not well correlated with the local plasma conditions, which might suggestthat the observed 𝜖 are produced at the bowshock, and is highly variable in magnesheath tur-bulence due to the changing magnetic connectivity to the bowshock.These results show that 𝜖 can be used to identify regions where large deviations in theobserved distributions from a bi-Maxwellian distribution function, which may suggest thatlocal kinetic processes are occurring. Future work on electron non-Maxwellianity shouldinclude the following:(1) More detailed investigations of 𝜖 in magnetosheth turbulence and how it relates tolocal turbulent processes and connectivity to the bowshock.(2) Direct observation of the electron firehose instability and the resulting waves. Ourresults show that the electron temperature anisotropy is well constrained by the oblique elec-tron firehose instability; however, to our knowledge the electron firehose instability has notbeen directly observed in spacecraft data. –25–onfidential manuscript submitted to JGR-Space Physics
Acknowledgments
We thank the entire MMS team and instrument PIs for data access and support. This workwas supported by the Swedish National Space Agency (SNSA), grant 128/17. MMS data areavailable at https://lasp.colorado.edu/mms/sdc/public.
A: Non-Maxwellianity of time-averaged electron distributions in the magneto-sphere
In this Appendix we investigate the increase in 𝜖 in the magnetosphere due to the sta-tistical noise in the electron distributions measured by FPI-DES. To do this we average theelectron distributions over time to increase the overall counting statistics of the electron dis-tributions and compare the results with the unaveraged distributions. The time averaging en-sures that the observed distribution is smoother as a function of speed and angle. For phase1a of the MMS mission FPI was operating in a mode where two distinct energy tables areused, which alternate for each consecutive electron distribution [ Pollock et al. , 2016]. Thisresults in the electrons being sampled at 64 energies over two consecutive electron distribu-tions. To create time-averaged electron distributions we first combine each two consecutiveelectron distributions to construct electron distributions with 64 energy channels and sam-pled every 60 ms. We then simply average 𝑓 𝑒 of these distributions at all energies and anglesto obtain the time-averaged electron distributions. We now calculate the non-Maxwellianityfor the original distributions 𝜖 , the distributions with 64 energy channels 𝜖 , and electrondistributions averaged over 3 𝜖 𝑎𝑣, , 5 𝜖 𝑎𝑣, , and 11 𝜖 𝑎𝑣, of the 64 energy channel distribu-tions. Figures A.1 and A.2 show two examples of these calculations of 𝜖 . Figure A.1.
Calculations of 𝜖 for different time averages for the magnetopause crossing observed on 2015October 30 by MMS1. (a) Electron omnidirectional energy spectrogram with 𝑇 𝑒 (blue) and 𝑉 𝑆𝐶 (black). (b) 𝑛 𝑒 . (c) Non-Maxwellianities 𝜖 (black), 𝜖 , 𝜖 𝑎𝑣, (red), 𝜖 𝑎𝑣, (green), 𝜖 𝑎𝑣, (cyan). The magenta dashedlines indicate the times of the electron distributions shown in Figure 1.–26–onfidential manuscript submitted to JGR-Space Physics
In Figure A.1 we plot the magnetopause crossing shown in Figure 6. Figures A.1a andA.1b show the electron energy spectrogram and 𝑛 𝑒 , respectively. The low-density magne-tosphere is characterized by a single dominant electron distribution with temperature com-parable to the magnetosheath. Figure A.1c shows 𝜖 , 𝜖 , 𝜖 𝑎𝑣, , 𝜖 𝑎𝑣, , and 𝜖 𝑎𝑣, . We findthat there is little difference between 𝜖 and 𝜖 , except that the fluctuations in 𝜖 are smaller.This is not surprising because 𝜖 does not involve time averaging, so the counting statisticsare not improved for the 64 energy channel distributions. However, we find that when thedistributions are time-averaged there is a decrease in 𝜖 in the magnetosphere. In the magne-tosheath, where 𝑛 𝑒 is larger, time-averaging has only a very minor effect. In the magneto-sphere the most significant decrease occurs between 𝜖 and 𝜖 𝑎𝑣, ; taking longer time averagesdoes not result in significant further decreases in 𝜖 . In the magnetosphere 𝜖 decreases by ∼
30 % when time-averaged distributions are used. This indicates that the counting statisticsin the magnetosphere artificially increase 𝜖 in this case. Figure A.2.
Calculations of 𝜖 for different time averages for the magnetopause crossing observed on 2015December 06 by MMS1. (a) Electron omnidirectional energy spectrogram with 𝑇 𝑒 (blue) and 𝑉 𝑆𝐶 (black).(b) 𝑛 𝑒 . (c) Non-Maxwellianities 𝜖 (black), 𝜖 , 𝜖 𝑎𝑣, (red), 𝜖 𝑎𝑣, (green), 𝜖 𝑎𝑣, (cyan). The magenta dashedlines indicate the times of the electron distributions shown in Figure 3. As a second example, we plot the magnetopause crossing observed on 2015 December06 in Figure A.2. Two electron distributions from this event are shown in Figure 3. For thisevent the magnetospheric electron distributions are composed of distinct hot and cold popu-lations (Figure A.1a), in contrast to the event in Figure A.1. In this case time-averaging onlyresults in a very small decrease in 𝜖 both in the magnetosheath and in the magnetosphere. Inthe magnetosphere the very large 𝜖 results from the electrons having distinct temperatures of ∼
10 eV and (cid:38) 𝜖 is very small, and theobserved 𝜖 is physical in the magnetosphere. In this case the distinct electron temperatures ofmagnetospheric electron distributions primarily determines 𝜖 . –27–onfidential manuscript submitted to JGR-Space Physics
To illustrate the dependence of 𝜖 on 𝑛 ℎ / 𝑛 𝑐 and 𝑇 ℎ / 𝑇 𝑐 , we consider an electron distri-bution given by the sum of two stationary Maxwellian distributions with distinct densities 𝑛 𝑐 and 𝑛 ℎ , and distinct temperatures 𝑇 𝑐 and 𝑇 ℎ , where the subscripts 𝑐 and ℎ refer to the coldand hot distributions. The model Maxwellian distribition used in the calculation of 𝜖 has 𝑛 𝑒 = 𝑛 𝑐 + 𝑛 ℎ and temperature given by equation (3). The resulting 𝜖 is plotted versus 𝑛 ℎ / 𝑛 𝑐 and 𝑇 ℎ / 𝑇 𝑐 in Figure A.3. We find that a large region of parameter space has large values of 𝜖 .For 𝑛 ℎ / 𝑛 𝑐 (cid:46) 𝑇 ℎ / 𝑇 𝑐 (cid:38) 𝜖 can reach very large values. For very large 𝑇 ℎ / 𝑇 𝑐 , 𝜖 canapproach 1. Figure A.3.
Non-Maxwellianity 𝜖 as a function of 𝑛 ℎ / 𝑛 𝑐 and 𝑇 ℎ / 𝑇 𝑐 for electron distributions composed oftwo Maxwellian distributions with distinct temperatures. For Earth’s outer magnetosphere 𝑇 ℎ / 𝑇 𝑐 is often ∼ , with 𝑇 𝑐 ∼
10 eV and 𝑇 𝑐 ∼ 𝑛 𝑐 can be comparable to 𝑛 ℎ and in some cases 𝑛 𝑐 (cid:29) 𝑛 ℎ .As a result, in the outer magnetosphere 𝜖 will have very large values of 𝜖 , as seen in FigureA.2. Therefore, we expect that large values of 𝜖 to be observed in the magnetopause and atthe magnetopause when cold electrons are present. This will result in the large values of 𝜖 inthe magnetosphere, 𝑛 𝑒 (cid:46) − and can account for the statistical results in Figure 2.In Figure A.4 we statistically compare 𝜖 with 𝜖 𝑎𝑣, using the entire dataset used in sec-tion 4. For direct comparison of 𝜖 with 𝜖 𝑎𝑣, we have downsampled 𝜖 to the same samplingrate as 𝜖 𝑎𝑣, (0.3 s or ten electron distributions). In Figures A.4a and A.4b we plot the his-tograms of log ( 𝜖 ) and log ( 𝜖 𝑎𝑣, ) versus log ( 𝑛 𝑒 ) , respectivity. Both histograms arequalitatively very similar. In particular, in both plots 𝜖 and 𝜖 𝑎𝑣, statistically decrease sig-nificantly for 1 cm − (cid:46) 𝑛 𝑒 (cid:46)
10 cm − . The main difference is that there is a slight de-crease in 𝜖 𝑎𝑣, compared with 𝜖 . This can be seen clearly in Figure A.4c, which plots thehistogram of 𝜖 versus 𝜖 𝑎𝑣, . For almost all points 𝜖 𝑎𝑣, < 𝜖 , as expected from averaging theobserved distribution function with time. In Figure A.4d we plot the histogram of 𝜖 𝑎𝑣, / 𝜖 versus log ( 𝑛 𝑒 ) . For almost all points we find that 0 . < 𝜖 𝑎𝑣, / 𝜖 <
1, meaning that at most –28–onfidential manuscript submitted to
JGR-Space Physics averaging five 64 energy channel electron distributions reduces the non-Maxwellianity by afactor of 2. The black line in Figure A.4d shows the median of 𝜖 𝑎𝑣, / 𝜖 as a function of 𝑛 𝑒 .For 𝑛 𝑒 (cid:46)
20 cm − we find that the median remains large, > .
9, indicating that averagingthe electron distribution in time does not significantly change the results. For 𝑛 𝑒 (cid:38)
20 cm − we find that the median of 𝜖 𝑎𝑣, / 𝜖 decreases to 0 .
7. At 𝑛 𝑒 ∼ − there is a smaller peakin the median at 0 .
8. This 𝑛 𝑒 corresponds to the typical density in the outer magnetosphere,where hot and cold electron distributions are common. Because the decrease in 𝜖 𝑎𝑣, from 𝜖 is typically relatively small, we conclude that the increase in 𝜖 as 𝑛 𝑒 decreases is physicaland primarily due to the simultaneous presence of electron distributions with distinct temper-atures in the outer magnetosphere. Figure A.4.
Statstical comparison of 𝜖 with 𝜖 𝑎𝑣, . (a) Histogram of log ( 𝑛 𝑒 ) versus log ( 𝜖 ) . (b) His-togram of log ( 𝑛 𝑒 ) versus log ( 𝜖 𝑎𝑣, ) . The black lines in panels (a) and (b) indicate the medians (50thpercentiles) of 𝜖 and 𝜖 𝑎𝑣, as a function 𝑛 𝑒 and the lower and upper red curves indicate the 10th and 90th per-centiles as a function of 𝑛 𝑒 , respectively. (c) Histogram of log 𝜖 versus log ( 𝜖 𝑎𝑣, ) . The red line indicates 𝜖 = 𝜖 𝑎𝑣, . (d) Histogram of 𝜖 𝑎𝑣, / 𝜖 versus 𝑛 𝑒 . The black line is the median of 𝜖 𝑎𝑣, / 𝜖 as a function of 𝑛 𝑒 .For direct comparison of 𝜖 with 𝜖 𝑎𝑣, we have downsampled 𝜖 to the same sampling rate of 𝜖 𝑎𝑣, . In this Appendix we consider the effect of instrumental counting statistics on the calcu-lated 𝜖 . The key results are:(1) Improving the counting statistics by averaging the electron distributions over timebefore calculating 𝜖 results in 𝜖 decreasing. This decrease tends to be larger at lower den-sities. Nevertheless, 𝜖 increases significantly as density decreases and remains large in theouter magnetosphere and qualitatively the results do not change significantly. –29–onfidential manuscript submitted to JGR-Space Physics (2) The large values of 𝜖 in the outer magnetosphere are due to the presence of coldelectrons. The simultaneous presence of cold and hot electron distributions with distinct tem-peratures is the primary source of non-Maxwellianity in the outer magnetosphere. References
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